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2025
L. H. Delgado-Granados, T. J. Krogmeier, L. M. Sager-Smith, I. Avdic, Z. Hu, M. Sajjan, M. Abbasi, S. E. Smart, P. Narang, S. Kais, A. W. Schlimgen, K. Head-Marsden, D. A. Mazziotti. Chem. Rev. 125 1823-1839 (2025). "Quantum Algorithms and Applications for Open Quantum Systems"
Accurate models for open quantum systemsquantum states that have nontrivial interactions with their environmentmay aid in the advancement of a diverse array of fields, including quantum computation, informatics, and the prediction of static and dynamic molecular properties. In recent years, quantum algorithms have been leveraged for the computation of open quantum systems as the predicted quantum advantage of quantum devices over classical ones may allow previously inaccessible applications. Accomplishing this goal will require input and expertise from different research perspectives, as well as the training of a diverse quantum workforce, making a compilation of current quantum methods for treating open quantum systems both useful and timely. In this Review, we first provide a succinct summary of the fundamental theory of open quantum systems and then delve into a discussion on recent quantum algorithms. We conclude with a discussion of pertinent applications, demonstrating the applicability of this field to realistic chemical, biological, and material systems.Y. Wang, C. Cianci, I. Avdic, R. Dutta, S. Warren, B. Allen, N. P. Vu, L. F. Santos, V. S. Batista, D. A. Mazziotti. J. Chem. Theory Comput. 21 1213-1221 (2025). "Characterizing Conical Intersections of Nucleobases on Quantum Computers"
Hybrid quantum-classical computing algorithms offer significant potential for accelerating the calculation of the electronic structure of strongly correlated molecules. In this work, we present the first quantum simulation of conical intersections (CIs) in a biomolecule, cytosine, using a superconducting quantum computer. We apply the contracted quantum eigensolver (CQE)with comparisons to conventional variational quantum deflation (VQD)to compute the near-degenerate ground and excited states associated with the conical intersection, a key feature governing the photostability of DNA and RNA. The CQE is based on an exact ansatz for many-electron molecules in the absence of noisea critically important property for resolving strongly correlated states at CIs. Both methods demonstrate promising accuracy when compared with exact diagonalization, even on noisy intermediate-scale quantum computers, highlighting their potential for advancing the understanding of photochemical and photobiological processes. The ability to simulate these intersections is critical for advancing our knowledge of biological processes like DNA repair and mutation, with potential implications for molecular biology and medical research.L. I. P. Torres, A. O. Schouten, L. M. Sager-Smith, D. A. Mazziotti. J. Phys. Chem. Lett. 16 1352-1366 (2025). "A Molecular Perspective of Exciton Condensation from Particle-Hole Reduced Density Matrices"
Exciton condensation, the Bose–Einstein-like condensation of quasibosonic particle-hole pairs, has been the subject of much theoretical and experimental interest and holds promise for ultraenergy-efficient technologies. Recent advances in bilayer systems, such as transition metal dichalcogenide heterostructures, have brought us closer to the experimental realization of exciton condensation without the need for high magnetic fields. In this perspective, we explore progress toward understanding and realizing exciton condensation, with a particular focus on the characteristic theoretical signature of exciton condensation: an eigenvalue greater than one in the particle-hole reduced density matrix, which signifies off-diagonal long-range order. This metric bridges the gap between theoretical predictions and experimental realizations by providing a unifying framework that connects exciton condensation to related phenomena, such as Bose–Einstein condensation and superconductivity. Furthermore, our molecular approach integrates exciton condensation with broader excitonic phenomena, including exciton-related entanglement and correlation, unlocking potential advancements in fields like quantum materials and energy transport. We discuss connections between recent experimental and theoretical work and highlight the discoveries that may arise from approaching exciton condensation from a molecular perspective.S. Warren, Y. Wang, C. L. Benavides-Riveros, D. A. Mazziotti. Quantum Sci. Technol. 10 02LT02 (2025). "Quantum algorithm for polaritonic chemistry based on an exact ansatz"
Cavity-modified chemistry uses strong light-matter interactions to modify the electronic properties of molecules in order to enable new physical phenomena such as novel reaction pathways. As cavity chemistry often involves critical regions where configurations become nearly degenerate, the ability to treat multireference problems is crucial to understanding polaritonic systems. In this Letter, we show through the use of a unitary ansatz derived from the anti-Hermitian contracted Schrödinger equation that cavity-modified systems with strong correlation, such as the deformation of rectangular H4 coupled to a cavity mode, can be solved efficiently and accurately on a quantum device. In contrast, while our quantum algorithm can be made formally exact, classical-computing methods as well as other quantum-computing algorithms often yield answers that are both quantitatively and qualitatively incorrect. Additionally, we demonstrate the current feasibility of the algorithm on near intermediate-scale quantum hardware by computing the dissociation curve of H2 strongly coupled to a bosonic bath.A. O. Schouten, D. A. Mazziotti. PRX Energy 4 013004 (2025). "Exciton-Condensate-Like Energy Transport in Light-Harvesting Complex 2"
Bose-Einstein condensation of excitons, with its potential for frictionless energy transport, has recently been observed in materials at low temperatures. Here, we show that partial exciton condensation plays a significant role in the 18-chromophore B850 ring of the light-harvesting complex 2 (LH2) in purple bacteria. Even in the single-excitation regime, we observe that excitonic entanglement across multiple sites exhibits signatures of exciton condensation in the particle-hole reduced density matrix—a partial exciton condensate. Crucially, we find that, by distributing the exciton across multiple sites of the ring, the exciton-condensate-like state sets favorable conditions for enhanced energy transfer, both before and after decoherence. Surprisingly, this discovery reveals that excitonic condensation, previously thought to require extreme conditions, can occur in a partial form in biological systems under ambient conditions, providing new insight into energy transport. These results additionally bring new insight into the long-standing debate on quantum versus classical mechanisms in photosynthetic light harvesting by showing that quantum coherence, in the form of a partial exciton condensate, indirectly initializes subsequent classical transfer. Our findings not only deepen our understanding of quantum coherence in light harvesting but also suggest design principles for materials capable of leveraging excitonic entanglement for efficient energy transport.L. H. Delgado-Granados, L. M. Sager-Smith, K. Trifonova, D. A. Mazziotti. J. Phys. Chem. Lett. 16 2231-2237 (2025). "Machine Learning of Two-Electron Reduced Density Matrices for Many-Body Problems"
We present a novel machine learning algorithm for the many-electron problem, predicting the convex combination of two-electron reduced density matrices (2-RDMs)obtained from upper- and lower-bound energy calculationsthat closely approximates the exact energy. In contrast to other recently developed approaches based on the wave function or one-electron density, our 2-RDM machine-learning approach predicts energies and properties without steep scaling or functional approximation. As conjectured by Preskill and co-workers, a small amount of data in a physics-based machine learning algorithmin this case, information about the RDMs and their violation of selected higher-order N-representability conditionsyields highly accurate electronic energies that capture both dynamic and static correlation. We demonstrate the method by predicting the potential energy curves for BH and N2 within a few millihartrees of results from exact diagonalization. This machine learning algorithm provides a general framework for improving electronic structure calculations, with the potential for wide-reaching applications to both moderately and strongly correlated molecular systems.M. Rose, S. Kais, D. A. Mazziotti. Phys. Rev. A 111 052612 (2025). "Open-quantum-system simulation through exploiting noise on quantum computers"
We present a general framework for simulating open quantum systems by exploiting the device noise. We find that pseudo-identity unitary transformations on system qubits in the presence of noise can generate nontrivial quantum channels driving open-system dynamics. Controlling parameters in these transformations allows us to simulate selective state-to-state transitions as well as the long-lived coherences found in systems such as photosynthetic light-harvesting complexes. These results suggest a general framework in which parameters in pseudo-identity unitary transformations can be chosen to generate arbitrary nonunitary system-bath channels without any additional qubits representing bath dynamics. The demonstrated controllability of the bath through these transformations establishes different possibilities for the efficient simulation of open quantum systems on quantum devices.J. Liebert, C. Schilling, D. A. Mazziotti. J. Chem. Phys. 163 044109 (2025). "Spin adaptation of the cumulant expansions of reduced density matrices"
We develop a systematic framework for the spin adaptation of the cumulants of p-particle reduced density matrices (RDMs), with explicit constructions for p = 1 to 3. These spin-adapted cumulants enable rigorous treatment of both S ̂ z and S ̂ 2 symmetries in quantum systems, providing a foundation for spin-resolved electronic structure methods. We show that complete spin adaptation—referred to as complete S-representability—can be enforced by constraining the variances of S ̂ z and S ̂ 2 , which require the 2-RDM and 4-RDM, respectively. Importantly, the cumulants of RDMs scale linearly with system size—size-extensive—making them a natural object for incorporating spin symmetries in scalable electronic structure theories. The developed formalism is applicable to density-based methods, one-particle RDM functional theories, and two-particle RDM methods. We further extend the approach to spin–orbit-coupled systems via total angular momentum adaptation. Beyond spin, the framework enables the adaptation of RDM theories to additional symmetries through the construction of suitable irreducible tensor operators.Y. Wang, D. A. Mazziotti. Phys. Rev. A 112 022403 (2025). "Quantum many-body simulations from a reinforcement-learned exponential Ansatz"
Solving for the many-body wave function represents a significant challenge on both classical and quantum devices because of the exponential scaling of the Hilbert space with system size. While the complexity of the wave function can be reduced through conventional Ansätze (e.g., the coupled-cluster Ansatz), it can still grow rapidly with system size even on quantum devices. An exact, universal two-body exponential Ansatz for the many-body wave function has been shown to be generated from the solution of the contracted Schrödinger equation (CSE), and recently this Ansatz has been implemented without classical approximation on quantum simulators and devices for the scalable simulation of many-body quantum systems. Here we combine the solution of the CSE with a form of artificial intelligence known as reinforcement learning to generate highly compact circuits that implement this Ansatz without sacrificing accuracy. As a natural extension of the CSE, we reformulate the wave-function update as a Markovian decision process and train the agent to select the optimal actions at each iteration based upon only the current CSE residual. Compact circuits with high accuracy are achieved for H3 and H4 molecules over a range of molecular geometries.Back to top
2024
S. Warren, L. M. Sager-Smith, D. A. Mazziotti. Phys. Rev. B 109 045134 (2024). "Topological phase transitions captured in the set of reduced density matrices"
Topological phase transitions fall outside of the symmetry-breaking paradigm and therefore elude many traditional analytical methods. In particular, significant geometric features found in the set of reduced density matrices (RDMs) disappear in symmetry-preserving topological systems. By returning to the fundamental properties of phase transitions, such as the divergence of correlation length and energy surface discontinuities, we demonstrate that the set of RDMs captures the critical behavior of systems with topological order. We find signatures of the gapless transition in the discontinuous movement of the ground-state RDMs. Additionally, the correlation length divergence near critical points appears in the off-diagonal character of RDMs, which is quantified through spectral and linear fitting analyses. This framework generalizes the quantum information approach to RDM critical theory, allowing for classification and visualization of all of the phases of the system—both topological and trivial—in the convex set of RDMs.S. E. Smart, D. A. Mazziotti. Phys. Rev. A 109 022802 (2024). "Verifiably exact solution of the electronic Schrödinger equation on quantum devices"
Quantum computers have the potential for a significant speedup of molecular computations. However, existing algorithms have limitations; quantum phase estimation (QPE) is intractable on current hardware while variational quantum eigensolvers (VQE) are dependent upon approximate wave functions without guaranteed convergence. In this paper we present an algorithm that yields verifiably exact solutions of the many-electron Schrödinger equation. Rather than solve the Schrödinger equation directly, we solve its contraction over all electrons except two, known as the contracted Schrödinger equation (CSE). The CSE generates a wave-function Ansatz, constructed from an iterative product of nonunitary two-body transformations, whose energy gradient with respect to the two-body operator of the current iteration vanishes if and only if the CSE is satisfied. Because the CSE implies the Schrödinger equation, the two-electron Ansatz provides a verifiably exact Ansatz for solving the many-electron Schrödinger equation. The exactness property contrasts with that of Ansätze built from the product of unitary two-body transformations where the gradient—the residual of the anti-Hermitian part of the CSE (ACSE)—can vanish without implying a solution of the Schrödinger equation. We demonstrate the algorithm on both simulators and noisy quantum computers with H2 dissociation and the rectangle-to-square transition in H4.S. W. Anferov, J. Boyn, D. A. Mazziotti, J. S. Anderson. J. Am. Chem. Soc. 146 5855-5863 (2024). "Selective Cobalt-Mediated Formation of Hydrogen Peroxide from Water under Mild Conditions via Ligand Redox Non-Innocence"
Despite the broad importance of hydrogen peroxide (H2O2) in oxidative transformations, there are comparatively few viable routes for its production. The majority of commercial H2O2 is currently produced by the stepwise reduction of dioxygen (O2) via the anthraquinone process, but direct electrochemical formation from water (H2O) would have several advantagesnamely, avoiding flammable gases or stepwise separations. However, the selective oxidation of H2O to form H2O2 over the thermodynamically favored product of O2 is a difficult synthetic challenge. Here, we present a molecular H2O oxidation system with excellent selectivity for H2O2 that functions both stoichiometrically and catalytically. We observe high efficiency for electrocatalytic H2O2 production at low overpotential with no O2 observed under any conditions. Mechanistic studies with both calculations and kinetic analyses from isolated intermediates suggest that H2O2 formation occurs in a bimolecular fashion via a dinuclear H2O2-bridged intermediate with an important role for a redox non-innocent ligand. This system showcases the ability of metal–ligand cooperativity and strategic design of the secondary coordination sphere to promote kinetically and thermodynamically challenging selectivity in oxidative catalysis.C. L. Benavides-Riveros, Y. Wang, S. Warren, D. A. Mazziotti. N. J. Phys. 26 033020 (2024). "Quantum simulation of excited states from parallel contracted quantum eigensolvers"
Computing excited-state properties of molecules and solids is considered one of the most important near-term applications of quantum computers. While many of the current excited-state quantum algorithms differ in circuit architecture, specific exploitation of quantum advantage, or result quality, one common feature is their rooting in the Schrödinger equation. However, through contracting (or projecting) the eigenvalue equation, more efficient strategies can be designed for near-term quantum devices. Here we demonstrate that when combined with the Rayleigh–Ritz variational principle for mixed quantum states, the ground-state contracted quantum eigensolver (CQE) can be generalized to compute any number of quantum eigenstates simultaneously. We introduce two excited-state (anti-Hermitian) CQEs that perform the excited-state calculation while inheriting many of the remarkable features of the original ground-state version of the algorithm, such as its scalability. To showcase our approach, we study several model and chemical Hamiltonians and investigate the performance of different implementations.Y. Wang, D. A. Mazziotti. Phys. Chem. Chem. Phys. 26 11491-11497 (2024). "Quantum simulation of conical intersections"
We explore the simulation of conical intersections (CIs) on quantum devices, setting the groundwork for potential applications in nonadiabatic quantum dynamics within molecular systems.L. E. McNamara, C. Melnychuk, J. Boyn, S. W. Anferov, D. A. Mazziotti, R. D. Schaller, J. S. Anderson. Chem 10 2266-2282 (2024). "Elucidating non-radiative decay in near-infrared lumiphores: Leveraging new design principles to develop a telecom band organic dye laser"
Organic near-infrared (NIR) lumiphores have been targeted for biological and technological applications; however, these dyes exhibit exponentially decreasing quantum yields with decreasing energy gaps. The accepted model of this phenomenon invokes C–H stretching modes as the dominant route of non-radiative energy dissipation. Reducing the number of these modes is a popular strategy for the design of bright NIR lumiphores, but quantitative experimental validation of this strategy is lacking. Here, we evaluate the role of C–H modes in non-radiative relaxation through isotopic labeling of a NIR-emitting complex. Measurements comparing relaxation between protonated and perdeuterated complexes indicate that C–H modes do not contribute significantly to non-radiative relaxation. We instead propose that skeletal modes may play a larger role and suggest that minimizing scaffold size is a promising route for bright NIR lumiphores. We demonstrate the promise of such strategies through the development of the reddest organic-based laser dye yet reported.I. Avdic, D. A. Mazziotti. Phys. Rev. Lett. 132 220802 (2024). "Fewer Measurements from Shadow Tomography with N-Representability Conditions"
Classical shadow tomography provides a randomized scheme for approximating the quantum state and its properties at reduced computational cost with applications in quantum computing. In this Letter we present an algorithm for realizing fewer measurements in the shadow tomography of many-body systems. Accelerated tomography of the two-body reduced density matrix (2-RDM) is achieved by combining classical shadows with necessary constraints for the 2-RDM to represent an N-body system, known as N-representability conditions. We compute the ground-state energies and 2-RDMs of hydrogen chains and the N2 dissociation curve. The results demonstrate a significant reduction in the number of measurements with important applications to quantum many-body simulations on near-term quantum devices.A. O. Schouten, L. M. Sager-Smith, D. A. Mazziotti. Phys. Rev. B 110 035110 (2024). "Computation of exciton binding energies in exciton condensation"
Exciton binding energies are fundamental to understanding excitonic materials, especially those with the potential for ground-state exciton condensation. However, these energies are typically defined with significant limitations in their consideration of electron correlation. Here we present a variational theory for computing exciton binding energies in ground-state exciton condensates in which we define the binding as the energy difference between fully correlated many-electron systems with M and M−1 excitons, respectively. The (M−1) system is obtained by adding a constraint to the ground-state energy minimization that removes an exciton while allowing all other electronic degrees of freedom to relax. We perform the energy minimizations with variational calculations of the two-electron reduced density matrix (2-RDM) in which the additional constraint is treated along with the N-representability conditions—necessary constraints for the 2-RDM to represent an N-electron system—by semidefinite programming. We demonstrate the theory first in the Lipkin model and then in several stacked organic and inorganic systems that exhibit the beginnings of exciton condensation. We find that in the Lipkin model the traditional exciton binding model overbinds relative to the constrained approach. This has significant implications for theoretical characterizations of exciton condensates which rely on exciton binding energy to make predictions regarding condensate stability and critical temperatures. This correlated approach to defining and computing exciton binding energies may therefore have important applications for understanding the relationship between binding and condensation, especially for the BCS-BEC crossover.L. E. McNamara, J. Boyn, S. W. Anferov, A. S. Filatov, M. W. Maloney, D. A. Mazziotti, R. D. Schaller, J. S. Anderson. J. Am. Chem. Soc. 146 17285-17295 (2024). "Variable Peripheral Ligand Donation Tunes Electronic Structure and NIR II Emission in Tetrathiafulvalene Tetrathiolate Diradicaloids"
Near-infrared (NIR) lumiphores are promising candidates for numerous imaging, communication, and sensing applications, but they typically require large, conjugated scaffolds to achieve emission in this low-energy region. Due to the extended conjugation and synthetic complexity required, it is extremely difficult to tune the photophysical properties of these systems for desired applications. Here, we report facile tuning of deep NIR-emitting diradicaloid complexes through simple modification of peripheral ligands. These new lumiphores are rare examples of air-, acid-, and water-stable emissive diradicaloids. We apply a simple Hammett parameter-based strategy to tune the electron donation of the capping ligand across a series of commercially available triarylphosphines. This minor peripheral modification significantly alters the electronic structure, and consequently, the electrochemical, photophysical, and magnetic properties of the tetrathiafulvalene tetrathiolate (TTFtt)-based lumiphores. The resultant ∼100 nm absorption and emission range spans common laser lines and the desirable telecom region (ca. 1260–1550 nm). Furthermore, these lumiphores are sensitive to local dielectrics, distinguishing them as promising candidates for ratiometric imaging and/or barcoding in the deep NIR region.R. Dutta, D. G. A. Cabral, N. Lyu, N. P. Vu, Y. Wang, B. Allen, X. Dan, R. G. Cortiñas, P. Khazaei, M. Schäfer, A. C. C. d. Albornoz, S. E. Smart, S. Nie, M. H. Devoret, D. A. Mazziotti, P. Narang, C. Wang, J. D. Whitfield, A. K. Wilson, H. P. Hendrickson, D. A. Lidar, F. Pérez-Bernal, L. F. Santos, S. Kais, E. Geva, V. S. Batista. J. Chem. Theory Comput. 20 6426-6441 (2024). "Simulating Chemistry on Bosonic Quantum Devices"
Bosonic quantum devices offer a novel approach to realize quantum computations, where the quantum two-level system (qubit) is replaced with the quantum (an)harmonic oscillator (qumode) as the fundamental building block of the quantum simulator. The simulation of chemical structure and dynamics can then be achieved by representing or mapping the system Hamiltonians in terms of bosonic operators. In this Perspective, we review recent progress and future potential of using bosonic quantum devices for addressing a wide range of challenging chemical problems, including the calculation of molecular vibronic spectra, the simulation of gas-phase and solution-phase adiabatic and nonadiabatic chemical dynamics, the efficient solution of molecular graph theory problems, and the calculations of electronic structure.L. I. P. Torres, A. O. Schouten, D. A. Mazziotti. J. Phys. Chem. A (2024). "Lifetime of Strongly Correlated States on Near-Term Quantum Computers"
Here we study the lifetime of strongly correlated stationary states on quantum computers. We find that these states develop a nontrivial time dependence due to the presence of noise on current devices. After an exciton-condensate state is prepared, its behavior is observed with respect to unitary operations that should preserve the stationarity of the state. Instead of stationarity, however, we observe nontrivial time dependence in which the large eigenvalue of the particle–hole reduced density matrixthe exciton population of the condensatedecays toward unity, reflecting the loss of entanglement and off-diagonal long-range order. The result offers insight into the challenge of simulating strongly correlated systems on near-term quantum devices and highlights the importance of developing novel strategies for error mitigation that can preserve many-body correlations.S. Warren, Y. Wang, C. L. Benavides-Riveros, D. A. Mazziotti. Phys. Rev. Lett. 133 080202 (2024). "Exact Ansatz of Fermion-Boson Systems for a Quantum Device"
We present an exact Ansatz for the eigenstate problem of mixed fermion-boson systems that can be implemented on quantum devices. Based on a generalization of the electronic contracted Schrödinger equation (CSE), our approach guides a trial wave function to the ground state of any arbitrary mixed Hamiltonian by directly measuring residuals of the mixed CSE on a quantum device. Unlike density functional and coupled cluster theories applied to electron-phonon or electron-photon systems, the accuracy of our approach is not limited by the unknown exchange-correlation functional or the uncontrolled form of the exponential Ansatz. To test the performance of the method, we study the Tavis-Cummings model, commonly used in polaritonic quantum chemistry. Our results demonstrate that the CSE is a powerful tool in the development of quantum algorithms for solving general fermion-boson many-body problems.D. Gibney, J. Boyn, D. A. Mazziotti. Phys. Rev. A 110 L040802 (2024). "Enhancing density-functional theory for static correlation in large molecules"
A critical challenge for density-functional theory (DFT) in practice is its limited ability to treat static electron correlation, leading to errors in its prediction of charges, multiradicals, and reaction barriers. Recently, we combined one- and two-electron reduced density-matrix theories with DFT to obtain a universal O(N3) generalization of DFT for static correlation. In this Letter, we enhance the theory's treatment of large molecules by renormalizing the trace of the two-electron identity matrix in the correction using Cauchy-Schwarz inequalities of the electron-electron repulsion matrix. We apply the resulting functional theory to linear hydrogen chains as well as the prediction of the singlet-triplet gap and equilibrium geometries of a series of acenes. This renormalization of the generalized DFT retains the O(N3) computational scaling of DFT while enabling the accurate treatment of static correlation for a broad range of molecules and materials.I. Avdic, D. A. Mazziotti. Phys. Rev. A 110 052407 (2024). "Enhanced shadow tomography of molecular excited states via the enforcement of N-representability conditions by semidefinite programming"
Excited-state properties of highly correlated systems are key to understanding photosynthesis, luminescence, and the development of novel optical materials, but accurately capturing their interactions is computationally costly. We present an algorithm that combines classical shadow tomography with physical constraints on the two-electron reduced density matrix (2RDM) to treat excited states. The method reduces the number of measurements of the many-electron 2RDM on quantum computers by (i) approximating the quantum state through a random sampling technique called shadow tomography and (ii) ensuring that the 2RDM represents an N-electron system through imposing N-representability constraints by semidefinite programming. This generalizes recent work on the N-representability-enhanced shadow tomography of ground-state 2RDMs. We compute excited-state energies and 2RDMs of the H4 chain and analyze the critical points along the photoexcited reaction pathway from gauche-1,3-butadiene to bicyclobutane via a conical intersection. The results show that the generalized shadow tomography retains critical multireference correlation effects while significantly reducing the number of required measurements, offering a promising avenue for the efficient treatment of electronically excited states on quantum devices.L. I. P. Torres, A. O. Schouten, D. A. Mazziotti. Chem. Sci. 15 20371-20378 (2024). "Molecular origins of exciton condensation in van der Waals heterostructure bilayers"
A “critical seed” of exciton condensation is found in molecular-scale fragments of van der Waals heterostructure bilayers via the theoretical signature for exciton condensation, a large eigenvalue in the particle-hole reduced density matrix.A. O. Schouten, L. M. Sager-Smith, D. A. Mazziotti. N. J. Phys. 26 123029 (2024). "Superconductor to exciton condensate transition in a model copper-oxide material"
Superconductivity and exciton condensation are fundamental phenomena in condensed matter physics, associated with the condensation of electron–electron and electron–hole pairs, respectively, into coherent quantum states. In this study, we present evidence of a superconductor to exciton condensate transition within the context of the three-band Hubbard model of copper-oxide-like materials. As the electron–electron repulsion increases, the superconducting phase is superseded by exciton condensation. In support of theoretical predictions—not yet realized experimentally—we observe the coexistence of the two condensates in the vicinity of the transition where the quantum states become a superposition of electron–electron and electron–hole condensates. Coexistence is rigorously computed from large eigenvalues and their eigenvectors in both the two-electron reduced density matrix (2-RDM) and the particle-hole RDM, which we obtain from a direct variational ground-state energy minimization with respect to the 2-RDM by semidefinite programming. We further discern that adjacent d orbitals and intervening p orbitals facilitate electron–electron pairing between copper orbitals, thereby supporting the superexchange mechanism for superconductivity. These observations suggest the feasibility of witnessing a superconductor to exciton condensate transition in copper-oxide analogs, bearing significant implications for identifying materials conducive to efficient transport processes.Back to top
2023
A. O. Schouten, J. E. Klevens, L. M. Sager-Smith, J. Xie, J. S. Anderson, D. A. Mazziotti. Phys. Rev. Mater. 7 045001 (2023). "Potential for exciton condensation in a highly conductive amorphous polymer"
An outstanding challenge in synthesis and theory is to develop molecular materials at ambient conditions that exhibit highly efficient energy transfer. Here we demonstrate the potential of a recently synthesized, highly conductive amorphous material—a nickel tetrathiafulvalene-tetrathiolate (NiTTFtt) polymer—to become an exciton condensate—a Bose-Einstein condensate of particle-hole pairs, known as excitons, that supports dissipationless flow of excitation energy. While exciton condensates have recently been realized in ordered materials, we show by advanced electronic structure calculations that this highly correlated phenomenon can potentially be realized in molecularly tailored, amorphous materials. In contrast to the Bechgaard salts that support superconductivity at compressed geometries requiring high pressures, we show that the recently synthesized, amorphous NiTTFtt polymer exhibits the computational signature of exciton condensation at experimentally realizable geometries, occurring at ambient pressures. Results suggest that superfluidity in this system and related systems—including van der Waals structures, molecular metals with extended-TTF dithiolate ligands, and Bechgaard salts—may occur via a nontraditional excitonic mechanism tuneable according to system composition, geometry, size, and charge. This study prompts further experimental investigation of the rational design of molecularly scaled exciton condensates with potential applications to efficient transport in technologically relevant materials.D. A. Mazziotti. Phys. Rev. Lett. 130 153001 (2023). "Quantum Many-Body Theory from a Solution of the N-Representability Problem"
Here we present a many-body theory based on a solution of the N-representability problem in which the ground-state two-particle reduced density matrix (2-RDM) is determined directly without the many-particle wave function. We derive an equation that re-expresses physical constraints on higher-order RDMs to generate direct constraints on the 2-RDM, which are required for its derivation from an N-particle density matrix, known as N-representability conditions. The approach produces a complete hierarchy of 2-RDM constraints that do not depend explicitly upon the higher RDMs or the wave function. By using the two-particle part of a unitary decomposition of higher-order constraint matrices, we can solve the energy minimization by semidefinite programming in a form where the low-rank structure of these matrices can be potentially exploited. We illustrate by computing the ground-state electronic energy and properties of the H8 ring.A. O. Schouten, L. M. Sager-Smith, D. A. Mazziotti. PRX Energy 2 023002 (2023). "Exciton-Condensate-Like Amplification of Energy Transport in Light Harvesting"
I. Avdic, L. M. Sager-Smith, D. A. Mazziotti. Commun. Phys. 6 180 (2023). "Open quantum system violates generalized Pauli constraints on quantum device"
The Pauli exclusion principle governs the fundamental structure and function of fermionic systems from molecules to materials. Nonetheless, when such a fermionic system is in a pure state, it is subject to additional restrictions known as the generalized Pauli constraints (GPCs). Here we verify experimentally the violation of the GPCs for an open quantum system using data from a superconducting-qubit quantum computer. We prepare states of systems with three-to-seven qubits directly on the quantum device and measure the one-fermion reduced density matrix (1-RDM) from which we can test the GPCs. We find that the GPCs of the 1-RDM are sufficiently sensitive to detect the openness of the 3-to-7 qubit systems in the presence of a single-qubit environment. Results confirm experimentally that the openness of a many-fermion quantum system can be decoded from only a knowledge of the 1-RDM with potential applications from quantum computing and sensing to noise-assisted energy transfer. Fermionic systems in a pure state are subject to restrictions on the natural orbital occupation, known as the generalized Pauli constraints. The authors probe the violation of such constraints in 3-to-7 qubit systems, experimentally demonstrating that the one-fermion reduced density matrix encodes the openness of a fermionic quantum system.L. M. Sager-Smith, S. E. Smart, D. A. Mazziotti. J. Phys. Chem. A 127 6032-6039 (2023). "Qubit Condensation for Assessing Efficacy of Molecular Simulation on Quantum Computers"
Quantum computers may demonstrate significant advantages over classical devices, as they are able to exploit a purely quantum-mechanical phenomenon known as entanglement in which a single quantum state simultaneously populates two-or-more classical configurations. However, due to environmental noise and device errors, elaborate quantum entanglement can be difficult to prepare on modern quantum computers. In this paper, we introduce a metric based on the condensation of qubits to assess the ability of a quantum device to simulate many-electron systems. Qubit condensation occurs when the qubits on a quantum computer condense into a single, highly correlated particle-hole state. While conventional metrics like gate errors and quantum volume do not directly target entanglement, the qubit-condensation metric measures the quantum computer’s ability to generate an entangled state that achieves nonclassical long-range order across the device. To demonstrate, we prepare qubit condensations on various quantum devices and probe the degree to which qubit condensation is realized via postmeasurement analysis. We show that the predicted ranking of the quantum devices is consistent with the errors obtained from molecular simulations of H2 using a contracted quantum eigensolver.Y. Wang, D. A. Mazziotti. Phys. Rev. A 108 022814 (2023). "Electronic excited states from a variance-based contracted quantum eigensolver"
Electronic excited states of molecules are central to many physical and chemical processes, and yet they are typically more difficult to compute than ground states. In this paper we leverage the advantages of quantum computers to develop an algorithm for the highly accurate calculation of excited states. We solve a contracted Schrödinger equation (CSE)—a contraction (projection) of the Schrödinger equation onto the space of two electrons—whose solutions correspond identically to the ground and excited states of the Schrödinger equation. While recent quantum algorithms for solving the CSE, known as contracted quantum eigensolvers (CQEs), have focused on ground states, we develop a CQE based on the variance that is designed to optimize rapidly to a ground or excited state. We apply the algorithm to compute the ground and excited states of H2, H4, and BH.Y. Wang, L. M. Sager-Smith, D. A. Mazziotti. N. J. Phys. 25 103005 (2023). "Quantum simulation of bosons with the contracted quantum eigensolver"
Quantum computers are promising tools for simulating many-body quantum systems due to their potential scaling advantage over classical computers. While significant effort has been expended on many-fermion systems, here we simulate a model entangled many-boson system with the contracted quantum eigensolver (CQE). We generalize the CQE to many-boson systems by encoding the bosonic wavefunction on qubits. The CQE provides a compact ansatz for the bosonic wave function whose gradient is proportional to the residual of a contracted Schrödinger equation. We apply the CQE to a bosonic system, where N quantum harmonic oscillators are coupled through a pairwise quadratic repulsion. The model is relevant to the study of coupled vibrations in molecular systems on quantum devices. Results demonstrate the potential efficiency of the CQE in simulating bosonic processes such as molecular vibrations with good accuracy and convergence even in the presence of noise.I. Avdic, L. M. Sager-Smith, I. Ghosh, O. C. Wedig, J. S. Higgins, G. S. Engel, D. A. Mazziotti. Phys. Rev. Res. 5 043097 (2023). "Quantum sensing using multiqubit quantum systems and the Pauli polytope"
Quantum sensing has highly practical potential applications in fields ranging from fundamental physics and quantum communication to biophysics and bioengineering. However, achieving high fidelity and control of entangled qubits that enables sensing beyond the quantum limit is still a challenging endeavor. In this paper, we present an alternative approach to quantum sensing, which we call open-system quantum sensing, where we exploit a generalization of the Pauli exclusion principle to sense the openness of a multiqubit quantum system from only measurement of the qubit occupations. Qubit occupations of a pure state obey generalized Pauli exclusion constraints that define a convex set known as the Pauli polytope, and hence violation of one of these constraints—a facet of the polytope—reveals a mixed state from the interaction of a quantum system with its environment without performing full-state tomography. We examine experimental ultrafast spectroscopic data from the photosynthetic light-harvesting complex in green sulfur bacteria and show that we can sense and decode the relaxation of the complex due to environmental noise. More generally, we can apply open-system quantum sensing with any general multiqubit quantum system, where it provides a unique, visual approach that promises enhanced sensitivity and fidelity.D. Gibney, J. Boyn, D. A. Mazziotti. Phys. Rev. Lett. 131 243003 (2023). "Universal Generalization of Density Functional Theory for Static Correlation"
A major challenge for density functional theory (DFT) is its failure to treat static correlation, yielding errors in predicted charges, band gaps, van der Waals forces, and reaction barriers. Here we combine one- and two-electron reduced density matrix (1- and 2-RDM) theories with DFT to obtain a universal O(N3) generalization of DFT for static correlation. Using the lowest unitary invariant of the cumulant 2-RDM, we generate a 1-RDM functional theory that corrects the convexity of any DFT functional to capture static correlation in its fractional orbital occupations. Importantly, the unitary invariant yields a predictive theory by revealing the dependence of the correction’s strength upon the trace of the two-electron repulsion matrix. We apply the theory to the barrier to rotation in ethylene, the relative energies of the benzynes, as well as an 11-molecule, dissociation benchmark. By inheriting the computational efficiency of DFT without sacrificing the treatment of static correlation, the theory opens new possibilities for the prediction and interpretation of significant quantum molecular effects and phenomena.C. Lien, J. Boyn, S. W. Anferov, D. A. Mazziotti, J. S. Anderson. Inorg. Chem. 62 19488-19497 (2023). "Origin of Weak Magnetic Coupling in a Dimanganese(II) Complex Bridged by the Tetrathiafulvalene-Tetrathiolate Radical"
Magnetic exchange coupling (J) between different spin centers plays a crucial role in molecule-based magnetic materials. Direct exchange coupling between an organic radical and a metal is frequently stronger than superexchange through diamagnetic ligands, and the strategy of using organic radicals to engender desirable magnetic properties has been an area of active investigation. Despite significant advances and exciting bulk properties, the magnitude of J for radical linkers bridging paramagnetic centers is still difficult to rationally predict. It is thus important to elucidate the features of organic radicals that govern this parameter. Here, we measure J for the tetrathiafulvalene-tetrathiolate radical (TTFtt3–•) in a dinuclear Mn(II) complex. Magnetometry studies show that the antiferromagnetic coupling in this complex is much weaker than that in related Mn(II)–radical compounds, in contrast to what might be expected for the S-based chelating donor atoms of TTFtt. Experimental and computational analyses suggest that this small J coupling may be attributed to poor overlap between Mn- and TTFtt-based magnetic orbitals coupled with insignificant spin density on the coordinating S-atoms. These factors override any expected increase in J from the comparatively strong S-donors. This work elucidates the magnetic coupling properties of the TTFtt3–• radical for the first time and also demonstrates how multiple competing factors must be considered in rationally designing organic radical ligands for molecular-based magnetic compounds.Back to top
2022
"Reduced‐Density‐Matrix Mechanics: With Application to Many‐Electron Atoms and Molecules" D. A. Mazziotti. (2022).
ISBN: 9780470106600L. M. Sager, D. A. Mazziotti. Phys. Rev. B 105 L121105 (2022). "Entangled phase of simultaneous fermion and exciton condensations realized"
Fermion-exciton condensates (FECs)—computationally and theoretically predicted states that simultaneously exhibit the character of superconducting states and exciton condensates—are novel quantum states whose properties may involve a hybridization of superconductivity and the dissipationless flow of energy. Here, we exploit prior investigations of superconducting states and exciton condensates on quantum devices to identify a tuneable quantum state preparation entangling the wave functions of the individual condensate states. Utilizing this state preparation, we prepare a variety of FEC states on quantum computers—realizing strongly correlated FEC states on current, noisy intermediate-scale quantum devices—and verify the presence of the dual condensate via postmeasurement analysis. This confirmation of the previously predicted condensate state on quantum devices as well as the form of its wave function motivates further theoretical and experimental exploration of the properties, applications, and stability of FECs.S. E. Smart, D. A. Mazziotti. Phys. Rev. A 105 062424 (2022). "Many-fermion simulation from the contracted quantum eigensolver without fermionic encoding of the wave function"
Quantum computers potentially have an exponential advantage over classical computers for the quantum simulation of many-fermion quantum systems. Nonetheless, fermions are more expensive to simulate than bosons due to the fermionic encoding—a mapping by which the qubits are encoded with fermion statistics. Here we generalize the contracted quantum eigensolver (CQE) to avoid fermionic encoding of the wave function. In contrast to the variational quantum eigensolver, the CQE solves for a many-fermion stationary state by minimizing the contraction (projection) of the Schrödinger equation onto two fermions. We avoid fermionic encoding of the wave function by contracting the Schrödinger equation onto an unencoded pair of particles. Solution of the resulting contracted equation by a series of unencoded two-body exponential transformations generates an unencoded wave function from which the energy and two-fermion reduced density matrix (2-RDM) can be computed. We apply the unencoded and the encoded CQE algorithms to the hydrogen fluoride molecule, the dissociation of oxygen O2, and a series of hydrogen chains. Both algorithms show comparable convergence towards the exact ground-state energies and 2-RDMs, but the unencoded algorithm has computational advantages in terms of state preparation and tomography.L. M. Sager, D. A. Mazziotti. Phys. Rev. Res. 4 013003 (2022). "Cooper-pair condensates with nonclassical long-range order on quantum devices"
An important problem in quantum information is the practical demonstration of nonclassical long-range order on quantum computers. One of the best known examples of a quantum system with nonclassical long-range order is a superconductor. Here we achieve Cooper-like pairing of qubits on a quantum computer, which can be interpreted as superconducting or superfluid states via a Jordan-Wigner mapping. We rigorously confirm the quantum long-range order by measuring the large O(N) eigenvalue of the two-electron reduced density matrix. The demonstration of maximal quantum long-range order is an important step toward more complex modeling of phenomena with significant quantum long-range order on quantum computers such as superconductivity and superfluidity.J. Boyn, D. A. Mazziotti. J. Chem. Phys. 156 194104 (2022). "Elucidating the molecular orbital dependence of the total electronic energy in multireference problems"
The accurate resolution of the chemical properties of strongly correlated systems, such as biradicals, requires the use of electronic structure theories that account for both multi-reference and dynamic correlation effects. A variety of methods exist that aim to resolve the dynamic correlation in multi-reference problems, commonly relying on an exponentially scaling complete-active-space self-consistent-field (CASSCF) calculation to generate reference molecular orbitals (MOs). However, while CASSCF orbitals provide the optimal solution for a selected set of correlated (active) orbitals, their suitability in the quest for the resolution of the total correlation energy has not been thoroughly investigated. Recent research has shown the ability of Kohn–Shan density functional theory to provide improved orbitals for coupled cluster (CC) and Møller–Plesset perturbation theory (MP) calculations. Here, we extend the search for optimal and more cost effective MOs to post-configuration-interaction [post-(CI)] methods, surveying the ability of the MOs obtained with various density functional theory (DFT) functionals, as well as Hartree–Fock and CC and MP calculations to accurately capture the total electronic correlation energy. Applying the anti-Hermitian contracted Schrödinger equation to the dissociation of N2, the calculation of biradical singlet–triplet gaps, and the transition states of bicylobutane isomerization, we demonstrate that DFT provides a cost-effective alternative to CASSCF in providing reference orbitals for post-CI dynamic correlation calculations.S. E. Smart, D. A. Mazziotti. J. Chem. Theory Comput. 18 5286-5296 (2022). "Accelerated Convergence of Contracted Quantum Eigensolvers through a Quasi-Second-Order, Locally Parameterized Optimization"
A contracted quantum eigensolver (CQE) finds a solution to the many-electron Schrödinger equation by solving its integration (or contraction) to the two-electron spacea contracted Schrödinger equation (CSE)on a quantum computer. When applied to the anti-Hermitian part of the CSE (ACSE), the CQE iterations optimize the wave function, with respect to a general product ansatz of two-body exponential unitary transformations that can exactly solve the Schrödinger equation. In this work, we accelerate the convergence of the CQE and its wave function ansatz via tools from classical optimization theory. By treating the CQE algorithm as an optimization in a local parameter space, we can apply quasi-second-order optimization techniques, such as quasi-Newton approaches or nonlinear conjugate gradient approaches. Practically, these algorithms result in superlinear convergence of the wave function to a solution of the ACSE. Convergence acceleration is important because it can both minimize the accumulation of noise on near-term intermediate-scale quantum (NISQ) computers and achieve highly accurate solutions on future fault-tolerant quantum devices. We demonstrate the algorithm, as well as some heuristic implementations relevant for cost-reduction considerations, comparisons with other common methods such as variational quantum eigensolvers, and a Fermionic-encoding-free form of the CQE.L. M. Sager, D. A. Mazziotti. Phys. Rev. B 105 035143 (2022). "Simultaneous fermion and exciton condensations from a model Hamiltonian"
Fermion-exciton condensation in which both fermion-pair (i.e., superconductivity) and exciton condensations occur simultaneously in a single coherent quantum state has recently been conjectured to exist. Here, we capture the fermion-exciton condensation through a model Hamiltonian that can recreate the physics of this new class of highly correlated condensation phenomena. We demonstrate that the Hamiltonian generates the large-eigenvalue signatures of fermion-pair and exciton condensations for a series of states with increasing particle numbers. The results confirm that the dual-condensate wave function arises from the entanglement of fermion-pair and exciton wave functions, which we previously predicted in the thermodynamic limit. This model Hamiltonian—generalizing well-known model Hamiltonians for either superconductivity or exciton condensation—can explore a wide variety of condensation behavior. It provides significant insights into the required forces for generating a fermion-exciton condensate, which will likely be invaluable for realizing such condensations in realistic materials with applications from superconductors to excitonic materials.D. Gibney, J. Boyn, D. A. Mazziotti. J. Phys. Chem. Lett. 13 1382-1388 (2022). "Density Functional Theory Transformed into a One-Electron Reduced-Density-Matrix Functional Theory for the Capture of Static Correlation"
Density Functional Theory (DFT), the most widely adopted method in modern computational chemistry, fails to describe accurately the electronic structure of strongly correlated systems. Here we show that DFT can be formally and practically transformed into a one-electron reduced-density-matrix (1-RDM) functional theory, which can address the limitations of DFT while retaining favorable computational scaling compared to wave function based approaches. In addition to relaxing the idempotency restriction on the 1-RDM in the kinetic energy term, we add a quadratic 1-RDM-based term to DFT’s density-based exchange-correlation functional. Our approach, which we implement by quadratic semidefinite programming at DFT’s computational scaling of O(r 3), yields substantial improvements over traditional DFT in the description of static correlation in chemical structures and processes such as singlet biradicals and bond dissociations.S. Warren, L. M. Sager-Smith, D. A. Mazziotti. Phys. Rev. B 106 165107 (2022). "Quantum phase transitions in a model Hamiltonian exhibiting entangled simultaneous fermion-pair and exciton condensations"
Quantum states of a novel Bose-Einstein condensate, in which both fermion-pair and exciton condensations are simultaneously present, have recently been realized theoretically in a model Hamiltonian system. Here we identify quantum phase transitions in that model between fermion-pair and exciton condensations based on a geometric analysis of the convex set of ground-state two-particle reduced density matrices (2-RDMs). The 2-RDM set provides a finite representation of the infinite parameter space of Hamiltonians that readily reveals a fermion-pair condensate phase and two distinct exciton condensate phases, as well as the emergence of first- and second-order phase transitions as the particle number of the system is increased. The set, furthermore, shows that the fermion-exciton condensate (FEC) lies along the second-order phase transition between the exciton and fermion-pair condensate phases. The detailed information about the exciton and fermion-pair phases, the forces behind these phases, as well as their associated transitions provides additional insight into the formation of the FEC condensate, which we anticipate will prove useful in its experimental realization.L. M. Sager-Smith, D. A. Mazziotti. J. Am. Chem. Soc. 144 18959-18966 (2022). "Reducing the Quantum Many-Electron Problem to Two Electrons with Machine Learning"
An outstanding challenge in chemical computation is the many-electron problem where computational methodologies scale prohibitively with system size. The energy of any molecule can be expressed as a weighted sum of the energies of two-electron wave functions that are computable from only a two-electron calculation. Despite the physical elegance of this extended “aufbau” principle, the determination of the distribution of weightsgeminal occupationsfor general molecular systems has remained elusive. Here we introduce a new paradigm for electronic structure where approximate geminal-occupation distributions are “learned” via a convolutional neural network. We show that the neural network learns the N-representability conditions, constraints on the distribution for it to represent an N-electron system. By training on hydrocarbon isomers with only 2–7 carbon atoms, we are able to predict the energies for isomers of octane as well as hydrocarbons with 8–15 carbons. The present work demonstrates that machine learning can be used to reduce the many-electron problem to an effective two-electron problem, opening new opportunities for accurately predicting electronic structure.D. Gibney, J. Boyn, D. A. Mazziotti. J. Chem. Theory Comput. 18 6600-6607 (2022). "Comparison of Density-Matrix Corrections to Density Functional Theory"
Density functional theory (DFT), one of the most widely utilized methods available to computational chemistry, fails to describe systems with statically correlated electrons. To address this shortcoming, in previous work, we transformed DFT into a one-electron reduced density matrix theory (1-RDMFT) via the inclusion of a quadratic one-electron reduced density matrix (1-RDM) correction. Here, we combine our 1-RDMFT approach with different DFT functionals as well as Hartree–Fock to elucidate the method’s dependence on the underlying functional selection. Furthermore, we generalize the information density matrix functional theory (iDMFT), recently developed as a correction to the Hartree–Fock method, by incorporating density functionals in place of the Hartree–Fock functional. We relate iDMFT mathematically to our approach and benchmark the two with a common set of functionals and systems.S. Warren, L. M. Sager-Smith, D. A. Mazziotti. Phys. Rev. A 106 012434 (2022). "Quantum simulation of quantum phase transitions using the convex geometry of reduced density matrices"
Transitions of many-particle quantum systems between distinct phases at absolute-zero temperature, known as quantum phase transitions, require an exacting treatment of particle correlations. In this work, we present a general quantum-computing approach to quantum phase transitions that exploits the geometric structure of reduced density matrices. While typical approaches to quantum phase transitions examine discontinuities in the order parameters, the origin of phase transitions—their order parameters and symmetry breaking—can be understood geometrically in terms of the set of two-particle reduced density matrices (2-RDMs). The convex set of 2-RDMs provides a comprehensive map of the quantum system including its distinct phases as well as the transitions connecting these phases. Because 2-RDMs can potentially be computed on quantum computers at nonexponential cost, even when the quantum system is strongly correlated, they are ideally suited for a quantum-computing approach to quantum phase transitions. We compute the convex set of 2-RDMs for a Lipkin-Meshkov-Glick spin model on IBM superconducting-qubit quantum processors. Even though computations are limited to few-particle models due to device noise, comparisons with a classically solvable 1000-particle model reveal that the finite-particle quantum solutions capture the key features of the phase transitions including the strong correlation and the symmetry breaking.L. M. Sager, A. O. Schouten, D. A. Mazziotti. J. Chem. Phys. 156 154702 (2022). "Beginnings of exciton condensation in coronene analog of graphene double layer"
Exciton condensation, a Bose–Einstein condensation of excitons into a single quantum state, has recently been achieved in low-dimensional materials including twin layers of graphene and van der Waals heterostructures. Here, we computationally examine the beginnings of exciton condensation in a double layer composed of coronene, a seven-benzene-ring patch of graphene. As a function of interlayer separation, we compute the exciton population in a single coherent quantum state, showing that the population peaks around 1.8 at distances near 2 Å. Visualization reveals interlayer excitons at the separation distance of the condensate. We determine the exciton population as a function of the twist angle between two coronene layers to reveal the magic angles at which the condensation peaks. As with previous recent calculations showing some exciton condensation in hexacene double layers and benzene stacks, the present two-electron reduced-density-matrix calculations with coronene provide computational evidence for the ability to realize exciton condensation in molecular-scale analogs of extended systems such as the graphene double layer.J. Xie, S. Ewing, J. Boyn, A. S. Filatov, B. Cheng, T. Ma, G. L. Grocke, N. Zhao, R. Itani, X. Sun, H. Cho, Z. Chen, K. W. Chapman, S. N. Patel, D. V. Talapin, J. Park, D. A. Mazziotti, J. S. Anderson. Nature 611 479-484 (2022). "Intrinsic glassy-metallic transport in an amorphous coordination polymer"
Conducting organic materials, such as doped organic polymers1, molecular conductors2,3 and emerging coordination polymers4, underpin technologies ranging from displays to flexible electronics5. Realizing high electrical conductivity in traditionally insulating organic materials necessitates tuning their electronic structure through chemical doping6. Furthermore, even organic materials that are intrinsically conductive, such as single-component molecular conductors7,8, require crystallinity for metallic behaviour. However, conducting polymers are often amorphous to aid durability and processability9. Using molecular design to produce high conductivity in undoped amorphous materials would enable tunable and robust conductivity in many applications10, but there are no intrinsically conducting organic materials that maintain high conductivity when disordered. Here we report an amorphous coordination polymer, Ni tetrathiafulvalene tetrathiolate, which displays markedly high electronic conductivity (up to 1,200 S cm−1) and intrinsic glassy-metallic behaviour. Theory shows that these properties are enabled by molecular overlap that is robust to structural perturbations. This unusual set of features results in high conductivity that is stable to humid air for weeks, pH 0–14 and temperatures up to 140 °C. These findings demonstrate that molecular design can enable metallic conductivity even in heavily disordered materials, raising fundamental questions about how metallic transport can exist without periodic structure and indicating exciting new applications for these materials. An unusual new material, NiTTFtt, is reported that is structurally amorphous, precluding a classical band structure, but detailed characterization reveals high conductivity and a metallic character.A. W. Schlimgen, K. Head-Marsden, L. M. Sager, P. Narang, D. A. Mazziotti. Phys. Rev. Res. 4 023216 (2022). "Quantum simulation of the Lindblad equation using a unitary decomposition of operators"
Accurate simulation of the time evolution of a quantum system under the influence of an environment is critical to making accurate predictions in chemistry, condensed-matter physics, and materials sciences. Whereas there has been a recent surge in interest in quantum algorithms for the prediction of nonunitary time evolution in quantum systems, few studies offer a direct quantum analog to the Lindblad equation. Here, we present a quantum algorithm—utilizing a decomposition of nonunitary operators approach—that models dynamic processes via the unraveled Lindblad equation. This algorithm is employed to probe both a two-level system in an amplitude damping channel as well as the transverse field Ising model in a variety of parameter regimes; the resulting population dynamics demonstrate excellent agreement with classical simulation, showing the promise of predicting population dynamics utilizing quantum devices for a variety of important systems in molecular energy transport, quantum optics, and other open quantum systems.L. E. McNamara, J. Boyn, C. Melnychuk, S. W. Anferov, D. A. Mazziotti, R. D. Schaller, J. S. Anderson. J. Am. Chem. Soc. 144 16447-16455 (2022). "Bright, Modular, and Switchable Near-Infrared II Emission from Compact Tetrathiafulvalene-Based Diradicaloid Complexes"
Near-infrared (NIR)-emitting molecules are promising candidates for biological sensing and imaging applications; however, many NIR dyes are large conjugated systems which frequently have issues with stability, solubility, and tunability. Here, we report a novel class of compact and tunable fluorescent diradicaloid complexes which are air-, water-, light-, and temperature-stable. These properties arise from a compressed π manifold which promotes an intense ligand-centered π–π transition in the NIR II (1000–1700 nm) region and which subsequently emits at ∼1200 nm. This emission is among the brightest known for monomolecular lumiphores with deep NIR II (>1100 nm) emission, nearly an order of magnitude brighter than the commercially available NIR II dye IR 26. Furthermore, this fluorescence is electrochemically sensitive, with efficient switching upon addition of redox agents. The brightness, stability, and modularity of this system distinguish it as a promising candidate for the development of new technologies built around NIR emission.S. E. Smart, J. Boyn, D. A. Mazziotti. Phys. Rev. A 105 022405 (2022). "Resolving correlated states of benzyne with an error-mitigated contracted quantum eigensolver"
The simulation of strongly correlated many-electron systems is one of the most promising applications for near-term quantum devices. Here we use a class of eigenvalue solvers [presented in Smart and Mazziotti, Phys. Rev. Lett. 126, 070504 (2021)10.1103/PhysRevLett.126.070504] in which a contraction of the Schrödinger equation is solved for the two-electron reduced density matrix (2-RDM) to resolve the energy splittings of the ortho-, meta-, and para-isomers of benzyne C6H4. In contrast to the traditional variational quantum eigensolver, the contracted quantum eigensolver can solve an integration (or contraction) of the many-electron Schrödinger equation onto the two-electron space. The quantum solution of the anti-Hermitian part of the contracted Schrödinger equation provides a scalable approach with few variational parameters that has its foundations in 2-RDM theory. Experimentally, a variety of error-mitigation strategies enable the calculation, including a linear shift in the 2-RDM targeting the iterative nature of the algorithm as well as a projection of the 2-RDM onto the convex set of approximately N-representable 2-RDMs defined by the 2-positive N-representability conditions. The relative energies exhibit single-digit millihartree errors, capturing a large part of the electron correlation energy, and the computed natural orbital occupations reflect the significant differences in the electron correlation of the isomers.S. E. Smart, Z. Hu, S. Kais, D. A. Mazziotti. Commun. Phys. 5 41 (2022). "Author Correction: Relaxation of stationary states on a quantum computer yields a unique spectroscopic fingerprint of the computer’s noise"
J. Boyn, L. E. McNamara, J. S. Anderson, D. A. Mazziotti. J. Phys. Chem. A 126 3329-3337 (2022). "Interplay of Electronic and Geometric Structure Tunes Organic Biradical Character in Bimetallic Tetrathiafulvalene Tetrathiolate Complexes"
The synthesis and design of organic biradicals with tunable singlet–triplet gaps have become the subject of significant research interest, owing to their possible photochemical applications and use in the development of molecular switches and conductors. Recently, tetrathiafulvalene tetrathiolate (TTFtt) has been demonstrated to exhibit such organic biradical character in doubly ionized bimetallic complexes. In this article we use high-level ab initio calculations to interrogate the electronic structure of a series of TTFtt-bridged metal complexes, resolving the factors governing their biradical character and singlet–triplet gaps. We show that the degree of biradical character correlates with a readily measured experimental predictor, the central TTFtt C–C bond length, and that it may be described by a one-parameter model, providing valuable insight for the future rational design of TTFtt based biradical compounds and materials.M. E. Czaikowski, A. J. McNeece, J. Boyn, K. A. Jesse, S. W. Anferov, A. S. Filatov, D. A. Mazziotti, J. S. Anderson. J. Am. Chem. Soc. 144 15569-15580 (2022). "Generation and Aerobic Oxidative Catalysis of a Cu(II) Superoxo Complex Supported by a Redox-Active Ligand"
Cu systems feature prominently in aerobic oxidative catalysis in both biology and synthetic chemistry. Metal ligand cooperativity is a common theme in both areas as exemplified by galactose oxidase and by aminoxyl radicals in alcohol oxidations. This has motivated investigations into the aerobic chemistry of Cu and specifically the isolation and study of Cu–superoxo species that are invoked as key catalytic intermediates. While several examples of complexes that model biologically relevant Cu(II) superoxo intermediates have been reported, they are not typically competent aerobic catalysts. Here, we report a new Cu complex of the redox-active ligand tBu,TolDHP (2,5-bis((2-t-butylhydrazono)(p-tolyl)methyl)-pyrrole) that activates O2 to generate a catalytically active Cu(II)-superoxo complex via ligand-based electron transfer. Characterization using ultraviolet (UV)–visible spectroscopy, Raman isotope labeling studies, and Cu extended X-ray absorption fine structure (EXAFS) analysis confirms the assignment of an end-on κ1 superoxo complex. This Cu–O2 complex engages in a range of aerobic catalytic oxidations with substrates including alcohols and aldehydes. These results demonstrate that bioinspired Cu systems can not only model important bioinorganic intermediates but can also mediate and provide mechanistic insight into aerobic oxidative transformations.A. O. Schouten, L. M. Sager-Smith, D. A. Mazziotti. Phys. Rev. B 105 245151 (2022). "Large cumulant eigenvalue as a signature of exciton condensation"
The Bose-Einstein condensation of excitons into a single quantum state is known as exciton condensation. Exciton condensation, which potentially supports the frictionless flow of energy, has recently been realized in graphene bilayers and van der Waals heterostructures. Here we show that exciton condensates can be predicted from a combination of reduced density matrix theory and cumulant theory. We show that exciton condensation occurs if and only if there exists a large eigenvalue in the cumulant part of the particle-hole reduced density matrix. In the thermodynamic limit we show that the large eigenvalue is bounded from above by the number of excitons. In contrast to the eigenvalues of the particle-hole matrix, the large eigenvalue of the cumulant matrix has the advantage of providing a size-extensive measure of the extent of condensation. Here we apply this signature to predict exciton condensation in both the Lipkin model and molecular stacks of benzene. The computational signature has applications to the prediction of exciton condensation in both molecules and materials.A. W. Schlimgen, K. Head-Marsden, L. M. Sager-Smith, P. Narang, D. A. Mazziotti. Phys. Rev. A 106 022414 (2022). "Quantum state preparation and nonunitary evolution with diagonal operators"
Realizing nonunitary transformations on unitary-gate-based quantum devices is critically important for simulating a variety of physical problems, including open quantum systems and subnormalized quantum states. We present a dilation-based algorithm to simulate nonunitary operations using probabilistic quantum computing with only one ancilla qubit. We utilize the singular-value decomposition (SVD) to decompose any general quantum operator into a product of two unitary operators and a diagonal nonunitary operator, which we show can be implemented by a diagonal unitary operator in a one-qubit dilated space. While dilation techniques increase the number of qubits in the calculation, and thus the gate complexity, our algorithm limits the operations required in the dilated space to a diagonal unitary operator, which has known circuit decompositions. We use this algorithm to prepare random subnormalized two-level states on a quantum device with high fidelity. Furthermore, we present the accurate nonunitary dynamics of two-level open quantum systems in a dephasing channel and an amplitude-damping channel computed on a quantum device. The algorithm presented will be most useful for implementing general nonunitary operations when the SVD can be readily computed, which is the case for most operators in the noisy intermediate-scale quantum computing era.S. E. Smart, Z. Hu, S. Kais, D. A. Mazziotti. Commun. Phys. 5 28 (2022). "Relaxation of stationary states on a quantum computer yields a unique spectroscopic fingerprint of the computer’s noise"
Quantum computing has the potential to revolutionize computing, but its significant sensitivity to noise requires sophisticated error correction and mitigation. Traditionally, noise on the quantum device is characterized directly through qubit and gate measurements, but this approach has drawbacks in that it does not adequately capture the effect of noise on realistic multi-qubit applications. In this paper, we simulate the relaxation of stationary quantum states on a quantum computer to obtain a unique spectroscopic fingerprint of the computer’s noise. In contrast to traditional approaches, we obtain the frequency profile of the noise as it is experienced by the simulated stationary quantum states. Data from multiple superconducting-qubit IBM processors show that noise generates a bath within the simulation that exhibits both colored noise and non-Markovian behavior. Our results provide a direction for noise mitigation but also suggest how to use noise for quantum simulations of open systems. Quantifying, controlling, and correcting noise related errors is one of the current challenges in quantum computing. Here, the authors study the time dependence of the relaxation of a stationary state simulated on a quantum computer, and show that such spectroscopic signature is unique and can be used to characterize the noise on individual quantum computers.Z. Hu, K. Head-Marsden, D. A. Mazziotti, P. Narang, S. Kais. Quantum 6 726 (2022). "A general quantum algorithm for open quantum dynamics demonstrated with the Fenna-Matthews-Olson complex"
Using quantum algorithms to simulate complex physical processes and correlations in quantum matter has been a major direction of quantum computing research, towards the promise of a quantum advantage over classical approaches. In this work we develop a generalized quantum algorithm to simulate any dynamical process represented by either the operator sum representation or the Lindblad master equation. We then demonstrate the quantum algorithm by simulating the dynamics of the Fenna-Matthews-Olson (FMO) complex on the IBM QASM quantum simulator. This work represents a first demonstration of a quantum algorithm for open quantum dynamics with a moderately sophisticated dynamical process involving a realistic biological structure. We discuss the complexity of the quantum algorithm relative to the classical method for the same purpose, presenting a decisive query complexity advantage of the quantum approach based on the unique property of quantum measurement.Back to top
2021
S. Ewing, D. A. Mazziotti. J. Chem. Phys. 154 214106 (2021). "Correlation-driven phenomena in periodic molecular systems from variational two-electron reduced density matrix theory"
Correlation-driven phenomena in molecular periodic systems are challenging to predict computationally not only because such systems are periodically infinite but also because they are typically strongly correlated. Here, we generalize the variational two-electron reduced density matrix (2-RDM) theory to compute the energies and properties of strongly correlated periodic systems. The 2-RDM of the unit cell is directly computed subject to necessary N-representability conditions such that the unit-cell 2-RDM represents at least one N-electron density matrix. Two canonical but non-trivial systems, periodic metallic hydrogen chains and periodic acenes, are treated to demonstrate the methodology. We show that while single-reference correlation theories do not capture the strong (static) correlation effects in either of these molecular systems, the periodic variational 2-RDM theory predicts the Mott metal-to-insulator transition in the hydrogen chains and the length-dependent polyradical formation in acenes. For both hydrogen chains and acenes, the periodic calculations are compared with previous non-periodic calculations with the results showing a significant change in energies and increase in the electron correlation from the periodic boundary conditions. The 2-RDM theory, which allows for much larger active spaces than are traditionally possible, is applicable to studying correlation-driven phenomena in general periodic molecular solids and materials.J. Boyn, D. A. Mazziotti. J. Chem. Phys. 154 134103 (2021). "Accurate singlet–triplet gaps in biradicals via the spin averaged anti-Hermitian contracted Schrödinger equation"
The accurate description of biradical systems, and in particular the resolution of their singlet–triplet gaps, has long posed a major challenge to the development of electronic structure theories. Biradicaloid singlet ground states are often marked by strong correlation and, hence, may not be accurately treated by mainstream, single-reference methods such as density functional theory or coupled cluster theory. The anti-Hermitian contracted Schrödinger equation (ACSE), whose fundamental quantity is the two-electron reduced density matrix rather than the N-electron wave function, has previously been shown to account for both dynamic and strong correlations when seeded with a strongly correlated guess from a complete active space (CAS) calculation. Here, we develop a spin-averaged implementation of the ACSE, allowing it to treat higher multiplicity states from the CAS input without additional state preparation. We apply the spin-averaged ACSE to calculate the singlet–triplet gaps in a set of small main group biradicaloids, as well as the organic four-electron biradicals trimethylenemethane and cyclobutadiene, and naphthalene, benchmarking the results against other state-of-the-art methods reported in the literature.S. E. Smart, D. A. Mazziotti. Phys. Rev. Lett. 126 070504 (2021). "Quantum Solver of Contracted Eigenvalue Equations for Scalable Molecular Simulations on Quantum Computing Devices"
The accurate computation of ground and excited states of many-fermion quantum systems is one of the most consequential, contemporary challenges in the physical and computational sciences whose solution stands to benefit significantly from the advent of quantum computing devices. Existing methodologies using phase estimation or variational algorithms have potential drawbacks such as deep circuits requiring substantial error correction or nontrivial high-dimensional classical optimization. Here, we introduce a quantum solver of contracted eigenvalue equations, the quantum analog of classical methods for the energies and reduced density matrices of ground and excited states. The solver does not require deep circuits or difficult classical optimization and achieves an exponential speed-up over its classical counterpart. We demonstrate the algorithm though computations on both a quantum simulator and two IBM quantum processing units.S. E. Smart, D. A. Mazziotti. Phys. Rev. A 103 012420 (2021). "Lowering tomography costs in quantum simulation with a symmetry projected operator basis"
Measurement in quantum simulations provides a means for extracting meaningful information from a complex quantum state, and for quantum computing, reducing the complexity of measurement will be vital for near-term applications. For most quantum simulations, the targeted state will obey several symmetries inherent to the system Hamiltonian. We obtain an alternative symmetry projected basis of measurement that reduces the number of measurements needed by a constant factor. Our scheme can be implemented at no additional cost on a quantum computer, can be implemented under various measurement or tomography schemes, and is reasonably resilient under noise.A. O. Schouten, L. M. Sager, D. A. Mazziotti. J. Phys. Chem. Lett. 12 9906-9911 (2021). "Exciton Condensation in Molecular-Scale van der Waals Stacks"
Recent experiments have realized the Bose–Einstein condensation of excitons, known as exciton condensation, in extended systems such as bilayer graphene and van der Waals heterostructures. Here we computationally demonstrate the beginnings of exciton condensation in multilayer, molecular-scale van der Waals stacks composed of benzene subunits. The populations of excitons, which are computed from the largest eigenvalue of the particle-hole reduced density matrix (RDM) through advanced variational RDM calculations, are shown to increase with the length of the stack. The large eigenvalue indicates a nonclassical long-range ordering of the excitons that can support the frictionless flow of energy. Moreover, we use chemical substitutions and geometric modifications to tune the extent of the condensation. Results suggest exciton condensation in a potentially large family of molecular systems with applications to energy-efficient transport.D. A. Mazziotti, S. E. Smart, A. R. Mazziotti. N. J. Phys. 23 113037 (2021). "Quantum simulation of molecules without fermionic encoding of the wave function"
Molecular simulations generally require fermionic encoding in which fermion statistics are encoded into the qubit representation of the wave function. Recent calculations suggest that fermionic encoding of the wave function can be bypassed, leading to more efficient quantum computations. Here we show that the two-electron reduced density matrix (2-RDM) can be expressed as a unique functional of the unencoded N-qubit-particle wave function without approximation, and hence, the energy can be expressed as a functional of the 2-RDM without fermionic encoding of the wave function. In contrast to current hardware-efficient methods, the derived functional has a unique, one-to-one (and onto) mapping between the qubit-particle wave functions and 2-RDMs, which avoids the over-parametrization that can lead to optimization difficulties such as barren plateaus. An application to computing the ground-state energy and 2-RDM of H4 is presented.D. Gibney, J. Boyn, D. A. Mazziotti. J. Phys. Chem. Lett. 12 385-391 (2021). "Toward a Resolution of the Static Correlation Problem in Density Functional Theory from Semidefinite Programming"
Kohn–Sham density functional theory (DFT) has long struggled with the accurate description of strongly correlated and open shell systems, and improvements have been minor even in the newest hybrid functionals. In this Letter we treat the static correlation in DFT when frontier orbitals are degenerate by the means of using a semidefinite programming (SDP) approach to minimize the system energy as a function of the N-representable, non-idempotent 1-electron reduced density matrix. While showing greatly improved singlet–triplet gaps for local density approximation and generalized gradient approximation (GGA) functionals, the SDP procedure reveals flaws in modern meta and hybrid GGA functionals, which show no major improvements when provided with an accurate electron density.M. Sajjan, S. Hemmatiyan, D. A. Mazziotti. J. Phys. Chem. A 125 5448-5455 (2021). "Conductance Switching in an Organometallic Single-Electron Transistor Using Current-Constrained Reduced-Density Matrix Theory"
We report switching of molecular conductance at finite bias in a binuclear organometallic complex and its cation which were previously experimentally analyzed at low voltages to see the signature of Kondo resonance. The variational reduced density matrix theory is applied to show that the system is strongly multireferenced especially in its charged form. We also study the molecular conductance of both forms using recently developed current-constrained two-electron reduced density matrix theory which is capable of handling strong electronic correlation. We compare the results against an uncorrelated 1-electron reduced density matrix version of conductance calculations using Hartree–Fock molecular orbitals. We observe that despite quantitative disagreements, the qualitative trend in the conductance is correctly predicted to be favorable for the cationic partner by both methods. We explain the results using the inherently high density of states for the low-lying excited states in the cationic partner which is also replicable from uncorrelated electronic structure methods. Our results not only indicate that the low-bias conductance trend is maintained even beyond the Kondo regime and produces quantitative agreement with that of the experiment but also identifies important physical markers that are responsible for the high conductance of the charged species.A. W. Schlimgen, K. Head-Marsden, L. M. Sager, P. Narang, D. A. Mazziotti. Phys. Rev. Lett. 127 270503 (2021). "Quantum Simulation of Open Quantum Systems Using a Unitary Decomposition of Operators"
Electron transport in realistic physical and chemical systems often involves the nontrivial exchange of energy with a large environment, requiring the definition and treatment of open quantum systems. Because the time evolution of an open quantum system employs a nonunitary operator, the simulation of open quantum systems presents a challenge for universal quantum computers constructed from only unitary operators or gates. Here, we present a general algorithm for implementing the action of any nonunitary operator on an arbitrary state on a quantum device. We show that any quantum operator can be exactly decomposed as a linear combination of at most four unitary operators. We demonstrate this method on a two-level system in both zero and finite temperature amplitude damping channels. The results are in agreement with classical calculations, showing promise in simulating nonunitary operations on intermediate-term and future quantum devices.J. Boyn, A. O. Lykhin, S. E. Smart, L. Gagliardi, D. A. Mazziotti. J. Chem. Phys. 155 244106 (2021). "Quantum-classical hybrid algorithm for the simulation of all-electron correlation"
While chemical systems containing hundreds to thousands of electrons remain beyond the reach of quantum devices, hybrid quantum-classical algorithms present a promising pathway toward a quantum advantage. Hybrid algorithms treat the exponentially scaling part of the calculation—the static correlation—on the quantum computer and the non-exponentially scaling part—the dynamic correlation—on the classical computer. While a variety of algorithms have been proposed, the dependence of many methods on the total wave function limits the development of easy-to-use classical post-processing implementations. Here, we present a novel combination of quantum and classical algorithms, which computes the all-electron energy of a strongly correlated molecular system on the classical computer from the 2-electron reduced density matrix (2-RDM) evaluated on the quantum device. Significantly, we circumvent the wave function in the all-electron calculations by using density matrix methods that only require input of the statically correlated 2-RDM. Although the algorithm is completely general, we test it with two classical density matrix methods, the anti-Hermitian contracted Schrödinger equation (ACSE) and multiconfiguration pair-density functional theories, using the recently developed quantum ACSE method for simulating the statically correlated 2-RDM. We obtain experimental accuracy for the relative energies of all three benzyne isomers and thereby demonstrate the ability of the developed algorithm to achieve chemically relevant and accurate results on noisy intermediate-scale quantum devices.K. Head-Marsden, S. Krastanov, D. A. Mazziotti, P. Narang. Phys. Rev. Res. 3 013182 (2021). "Capturing non-Markovian dynamics on near-term quantum computers"
With the rapid progress in quantum hardware, there has been an increased interest in new quantum algorithms to describe complex many-body systems searching for the still-elusive goal of “useful quantum advantage.” Surprisingly, quantum algorithms for the treatment of open quantum systems (OQSs) have remained underexplored, in part due to the inherent challenges of mapping non-unitary evolution into the framework of unitary gates. Evolving an open system unitarily necessitates dilation into a new effective system to incorporate critical environmental degrees of freedom. In this context, we present and validate a new quantum algorithm to treat non-Markovian dynamics in OQSs built on the ensemble of Lindblad's trajectories approach, invoking the Sz.-Nagy dilation theorem. Here we demonstrate our algorithm on the Jaynes-Cummings model in the strong-coupling and detuned regimes, relevant in quantum optics and driven quantum system studies. This algorithm, a key step towards generalized modeling of non-Markovian dynamics on a noisy-quantum device, captures a broad class of dynamics and opens up a new direction in OQS problems.Back to top
2020
D. A. Mazziotti. Phys. Rev. A 102 030802 (2020). "Exact two-body expansion of the many-particle wave function"
Progress toward the solution of the strongly correlated electron problem has been stymied by the exponential complexity of the wave function. Previous work established an exact two-body exponential product expansion for the ground-state wave function. By developing a reduced density-matrix analog of Dalgarno-Lewis perturbation theory, we prove here that (i) the two-body exponential product expansion is rapidly and globally convergent with each operator representing an order of a renormalized perturbation theory, (ii) the energy of the expansion converges quadratically near the solution, and (iii) the expansion is exact for both ground and excited states. The two-body expansion offers a reduced parametrization of the many-particle wave function as well as the two-particle reduced density matrix with potential applications on both conventional and quantum computers for the study of strongly correlated quantum systems. We demonstrate the result with the exact solution of the contracted Schrödinger equation for the molecular chains H4 and H5.D. A. Mazziotti. Phys. Rev. A 102 052819 (2020). "Dual-cone variational calculation of the two-electron reduced density matrix"
The computation of strongly correlated quantum systems is challenging because of its potentially exponential scaling in the number of electron configurations. Variational calculation of the two-electron reduced density matrix (2-RDM) without the many-electron wave function exploits the pairwise nature of the electronic Coulomb interaction to compute a lower bound on the ground-state energy with polynomial computational scaling. Recently, a dual-cone formulation of the variational 2-RDM calculation was shown to generate the ground-state energy, albeit not the 2-RDM, at a substantially reduced computational cost, especially for higher N-representability conditions such as the T2 constraint. Here we generalize the dual-cone variational 2-RDM method to compute not only the ground-state energy but also the 2-RDM. The central result is that we can compute the 2-RDM from a generalization of the Hellmann-Feynman theorem. Specifically, we prove that in the Lagrangian formulation of the dual-cone optimization the 2-RDM is the Lagrange multiplier. We apply the method to computing the energies and properties of strongly correlated electrons—including atomic charges, electron densities, dipole moments, and orbital occupations—in an illustrative hydrogen chain and the nitrogen-fixation catalyst FeMoco. The dual variational computation of the 2-RDM with T2 or higher N-representability conditions provides a polynomially scaling approach to strongly correlated molecules and materials with significant applications in atomic and molecular and condensed-matter chemistry and physics.S. E. Smart, D. A. Mazziotti. Phys. Rev. Res. 2 023048 (2020). "Efficient two-electron ansatz for benchmarking quantum chemistry on a quantum computer"
Quantum chemistry provides key applications for near-term quantum computing, but these are greatly complicated by the presence of noise. In this work we present an efficient ansatz for the computation of two-electron atoms and molecules within a hybrid quantum-classical algorithm. The ansatz exploits the fundamental structure of the two-electron system, treating the nonlocal and local degrees of freedom on the quantum and classical computers, respectively. Here the nonlocal degrees of freedom scale linearly with respect to basis-set size, giving a linear ansatz with only O(1) circuit preparations required for reduced state tomography. We implement this benchmark with error mitigation on two publicly available quantum computers, calculating accurate dissociation curves for four- and six-qubit calculations of H2 and H3+.K. Head-Marsden, D. A. Mazziotti. J. Phys. Chem. A 124 4848-4854 (2020). "Active-Space Pair Two-Electron Reduced Density Matrix Theory for Strong Correlation"
An active-space variational calculation of the two-electron reduced density matrix (2-RDM) is derived and implemented where the active orbitals are correlated within the pair approximation. The pair approximation considers only doubly occupied configurations of the wave function, which enables the calculation of the 2-RDM at a computational cost of O ( r 3 ) . Calculations were performed both with the pair active-space configuration interaction (PASCI) method and the pair active-space self-consistent field (PASSCF) method. The latter includes a mixing of the active and inactive orbitals through unitary transformations. The active-space pair 2-RDM method is applied to the nitrogen molecule, the p-benzyne diradical, a newly synthesized biscobalt complex, and the nitrogenase cofactor FeMoco. The FeMoco molecule is treated in a [120,120] active space. Fractional occupations are recovered in each of these systems, indicating the presence of strong electron correlation.A. E. Raeber, D. A. Mazziotti. Phys. Chem. Chem. Phys. 22 23998-24003 (2020). "Non-equilibrium steady state conductivity in cyclo[18]carbon and its boron nitride analogue"
A ring-shaped carbon allotrope was recently synthesized for the first time, reinvigorating theoretical interest in this class of molecules. The dual π structure of these molecules allows for the possibility of novel electronic properties. In this work we use reduced density matrix theory to study the electronic structure and conductivity of cyclo[18]carbon and its boron nitride analogue, B9N9. The variational 2-RDM method replicates the experimental polyynic geometry of cyclo[18]carbon. We use a current-constrained 1-electron reduced density matrix (1-RDM) theory with Hartree–Fock molecular orbitals and energies to compute the molecular conductance in two cases: (1) conductance in the plane of the molecule and (2) conductance around the molecular ring as potentially driven by a magnetic field through the molecule's center. In-plane conductance is greater than conductance around the ring, but cyclo[18]carbon is slightly more conductive than B9N9 for both in-the-plane and in-the-ring conduction. The computed conductance per molecular orbital provides insight into how the orbitals—their energies and densities—drive the conduction.J. M. Montgomery, D. A. Mazziotti. J. Chem. Educ. 97 3658-3666 (2020). "Maple’s Quantum Chemistry Package in the Chemistry Classroom"
An introduction to the Quantum Chemistry Package (QCP), implemented in the computer algebra system Maple, is presented. The QCP combines sophisticated electronic structure methods and Maple’s easy-to-use graphical interface to enable computation and visualization of the electronic energies and properties of molecules. Here we describe how the QCP can be used in the chemistry classroom using lessons we developed that have been incorporated within the package. In this work, we present four illustrative lessons that showcase the potential implementation of the QCP: the calculation and visualization of molecular orbitals of hydrogen fluoride, the application of the particle in a box to conjugated dyes, the use of geometry optimization and normal-mode analysis for hypochlorous acid, and the thermodynamics of combustion of methane. While these topics are typically encountered at the undergraduate level, we have developed a wide range of lessons for the QCP appropriate for high school advanced placement, undergraduate, and advanced graduate curricula. A summary of implementation and student experience at our respective institutions is also provided.J. M. Montgomery, E. Alexander, D. A. Mazziotti. J. Phys. Chem. A 124 9562-9566 (2020). "Prediction of the Existence of LiCH: A Carbene-like Organometallic Molecule"
Carbenes comprise a well-known class of organometallic compounds each consisting of a neutral, divalent carbon and two unshared electrons. Carbenes can have singlet or triplet ground states, each giving rise to a distinct reactivity. Methylene (CH2), the parent hydride, is well-known to be bent in its triplet ground state. Here, we predict the existence of LiCH, a carbene-like organometallic molecule. Computationally, we treat the electronic structure with parametric and variational two-electron reduced density matrix (2-RDM) methods, which are capable of capturing multireference correlation typically associated with the singlet state of a diradical. Similar to methylene, LiCH is a triplet ground state with a predicted 15.8 kcal/mol singlet–triplet gap. However, unlike methylene, LiCH is linear in both the triplet state and the lowest excited singlet state. Furthermore, the singlet state is found to exhibit strong electron correlation as a diradical. In comparison to dissociation channels Li + CH and Li+ + CH–, the LiCH was found to be stable by approximately 77 kcal/mol.L. M. Sager, S. Safaei, D. A. Mazziotti. Phys. Rev. B 101 081107 (2020). "Potential coexistence of exciton and fermion-pair condensations"
An extensive theoretical and experimental investigation has been conducted on fermion-pair condensation and exciton condensation as distinct classes of Bose-Einstein-like condensation. In this Rapid Communication, the existence of a fermion-exciton condensate—a single quantum state in which the characters of both fermion-pair and exciton condensates coexist—is established computationally in the low-particle-number (N) limit and theoretically in the large-N thermodynamic limit. The trade-off between the fermion-pair and excitonic character of the fermion-exciton condensate is shown to be elliptic in nature. The possibility that the properties of fermion-exciton condensates could be a hybrid of the properties of fermion-pair condensates and exciton condensates is discussed, and future experimental and computational exploration of this class of condensate, which may potentially be realizable in a bilayer of superconductors, is anticipated.L. M. Sager, S. E. Smart, D. A. Mazziotti. Phys. Rev. Res. 2 043205 (2020). "Preparation of an exciton condensate of photons on a 53-qubit quantum computer"
Quantum computation promises an exponential speedup of certain classes of classical calculations through the preparation and manipulation of entangled quantum states. So far, most molecular simulations on quantum computers, however, have been limited to small numbers of particles. Here we prepare a highly entangled state on a 53-qubit IBM quantum computer, representing 53 particles, which reveals the formation of an exciton condensate of photon particles and holes. While the experimental realization of ground state exciton condensates remained elusive for more than 50 years, such condensates were recently achieved for electron-hole pairs in graphene bilayers and metal chalcogenides. Our creation of ground state photon condensates has the potential to further the exploration of exciton condensates, and this novel preparation may play a role in realizing efficient room-temperature energy transport.A. Kawamura, J. Xie, J. Boyn, K. A. Jesse, A. J. McNeece, E. A. Hill, K. A. Collins, J. A. Valdez-Moreira, A. S. Filatov, J. W. Kurutz, D. A. Mazziotti, J. S. Anderson. J. Am. Chem. Soc. 142 17670-17680 (2020). "Reversible Switching of Organic Diradical Character via Iron-Based Spin-Crossover"
Organic diradicals are uncommon species that have been intensely studied for their unique properties and potential applicability in a diverse range of innovative fields. While there is a growing class of stable and well-characterized organic diradicals, there has been recent focus on how diradical character can be controlled or modulated with external stimuli. Here we demonstrate that a diiron complex bridged by the doubly oxidized ligand tetrathiafulvalene-2,3,6,7-tetrathiolate (TTFtt2–) undergoes a thermally induced Fe-centered spin-crossover which yields significant diradical character on TTFtt2–. UV–vis–near-IR, Mössbauer, NMR, and EPR spectroscopies with magnetometry, crystallography, and advanced theoretical treatments suggest that this diradical character arises from a shrinking TTFtt2– π-manifold from the Fe(II)-centered spin-crossover. The TTFtt2–-centered diradical is predicted to have a singlet ground state by theory and variable temperature EPR. This unusual phenomenon demonstrates that inorganic spin transitions can be used to modulate organic diradical character.J. Boyn, J. Xie, J. S. Anderson, D. A. Mazziotti. J. Phys. Chem. Lett. 11 4584-4590 (2020). "Entangled Electrons Drive a Non-superexchange Mechanism in a Cobalt Quinoid Dimer Complex"
A central theme in chemistry is the understanding of the mechanisms that drive chemical transformations. A well-known, highly cited mechanism in organometallic chemistry is the superexchange mechanism in which unpaired electrons on two or more metal centers interact through an electron pair of the bridging ligand. We use a combination of novel synthesis and computation to show that such interactions may in fact occur by a more direct mechanism than superexchange that is based on direct quantum entanglement of the two metal centers. Specifically, we synthesize and experimentally characterize a novel cobalt dimer complex with benzoquinoid bridging ligands and investigate its electronic structure with the variational two-electron reduced density matrix method using large active spaces. The result draws novel connections between inorganic mechanisms and quantum entanglement, thereby opening new possibilities for the design of strongly correlated organometallic compounds whose magnetic and spin properties have applications in superconductors, energy storage, thermoelectrics, and spintronics.Back to top
2019
S. Hemmatiyan, D. A. Mazziotti. J. Phys. Chem. C 123 14619-14624 (2019). "Unraveling the Band Gap Trend in the Narrowest Graphene Nanoribbons from the Spin-Adapted Excited-Spectra Reduced Density Matrix Method"
Polybenzenes as the narrowest graphene nanoribbons with versatile electronic properties are widely studied both theoretically and technologically. Here, we examine the singlet–triplet band gap as a function of length for two members of the oligobenzene family: the acene and phenacene chains. We observe that the prediction of the band gap is highly sensitive to the accurate treatment of the electron correlation. The excited-spectra two-electron reduced density matrix (2-RDM) method, which computes the excited states from a variationally computed ground-state 2-RDM, yields finite band gaps for all finite chain lengths through 10 rings as well as in the extrapolated infinite ring limits of both acenes and phenacenes. In contrast, we find that weakly correlated methods like configuration interaction singles and time-dependent density functional theory predict a crossing of the singlet- and triplet-state energies of the acene chains at a finite ring size, with the triplet becoming the energetically lowest state at longer chain lengths. Recent experiments through decacene and 9-phenacene agree with the correlated 2-RDM calculations, showing that both acene and phenacene chains in the large polymer limit possess finite band gaps.K. Head-Marsden, D. A. Mazziotti. J. Chem. Phys. 151 034111 (2019). "Satisfying fermionic statistics in the modeling of non-Markovian dynamics with one-electron reduced density matrices"
Treatment of Markovian, many-electron dynamics from the solution of the Lindblad equation for the 1-electron reduced density matrix requires additional constraints on the bath operators to maintain fermion statistics. Recently, we generalized Lindblad’s formalism to non-Markovian dynamics through an ensemble of Lindbladian trajectories. Here we show that the fermion statistics of non-Markovian dynamics can be enforced through analogous constraints on the bath operators of each Lindbladian trajectory in the ensemble. To illustrate, we apply the non-Markovian method to three distinct systems of two fermions in three levels. While the electrons violate the fermion statistics without the constraints, correct fermion behavior is recovered with the constraints.A. E. Raeber, D. A. Mazziotti. Phys. Chem. Chem. Phys. 21 12620-12624 (2019). "Current-constrained one-electron reduced density-matrix theory for non-equilibrium steady-state molecular conductivity"
In the effort to create ever smaller electronic devices, the idea of single molecule circuit elements has sparked the imagination of scientists for nearly fifty years. While traditional theories for non-equilibrium steady-state molecular conductivity like the non-equilibrium Green's function density functional theory determine the current from an applied voltage, the recently proposed current-constrained density-matrix theory computes the voltage from a current constraint on the molecule. In the present paper we extend the current-constrained density-matrix theory from its two-electron reduced density-matrix (2-RDM) formulation to a one-electron reduced density matrix (1-RDM) formulation that is applicable to Hartree–Fock, density functional, and tight-binding theories. We demonstrate the current-constrained 1-RDM method through the computation of the theoretical, intrinsic resistance of acenes and phenacenes.K. Head-Marsden, D. A. Mazziotti. Phys. Rev. A 99 022109 (2019). "Ensemble of Lindblad's trajectories for non-Markovian dynamics"
Although Lindblad developed a general Markovian theory for open-system dynamics while maintaining the positivity of the density matrix, a practical non-Markovian analog remains a significant problem. Here, we present an extension of Lindblad's theory through an ensemble of Lindbladian trajectories originating from different times in the system's history. This approach provides an account of the system's memory while preserving the positivity of the density matrix. We apply the theory to the Jaynes-Cummings model to capture non-Markovian dynamics in the weak and strong coupling regimes.J. Boyn, D. A. Mazziotti. J. Chem. Phys. 150 144102 (2019). "Sparse non-orthogonal wave function expansions from the extension of the generalized Pauli constraints to the two-electron reduced density matrix"
Generalized Pauli constraints (GPCs) impose constraints in the form of inequalities on the natural orbital occupation numbers of the one electron reduced density matrix (1-RDM), defining the set of pure N-representable 1-RDMs, or 1-RDMs that can be derived from an N-electron wave function. Saturation of these constraints is termed “pinning” and implies a significant simplification of the N-electron wave function as the number of Slater determinants required to fully describe the system is reduced. Recent research has shown pinning to occur for the ground states of atoms and molecules with N = 3 and r = 6, where N is the number of electrons and r is the number of spin orbitals. For N = 4 and r = 8, however, pinning occurs not to the GPCs but rather to inequalities defining the pure N-representable two-electron reduced density matrices (2-RDMs). Using these more general inequalities, we derive a wave function ansatz for a system with four electrons in eight spin orbitals. We apply the ansatz to the isoelectronic series of the carbon atom and the dissociation of linear H4 where the correlation energies are recovered to fractions of a kcal/mol. These results provide a foundation for further developments in wave function and RDM theories based on “pinned” solutions, and elucidate a fundamental physical basis for the emergence of non-orthogonal bases in electronic systems of N ≥ 4.S. E. Smart, D. A. Mazziotti. Phys. Rev. A 100 022517 (2019). "Quantum-classical hybrid algorithm using an error-mitigating N-representability condition to compute the Mott metal-insulator transition"
Quantum algorithms for molecular electronic structure have been developed with lower computational scaling than their classical counterparts, but emerging quantum hardware is far from being capable of the coherence, connectivity, and gate errors required for their experimental realization. Here we propose a class of quantum-classical hybrid algorithms that computes the energy from a two-electron reduced density matrix (2-RDM). The 2-RDM is constrained by N-representability conditions, constraints for representing an N-electron wave function, which mitigate noise from the quantum circuit. We compute the strongly correlated dissociation of doublet H3 into three hydrogen atoms. The hybrid quantum-classical computer matches the energies from full configuration interaction to 0.1 kcal/mol, one-tenth of “chemical accuracy,” even in the strongly correlated limit of dissociation. Furthermore, the spatial locality of the computed one-electron RDM reveals that the quantum computer accurately predicts the Mott metal-insulator transition.S. E. Smart, D. I. Schuster, D. A. Mazziotti. Commun. Phys. 2 11 (2019). "Experimental data from a quantum computer verifies the generalized Pauli exclusion principle"
“What are the consequences… that Fermi particles cannot get into the same state…” R. P. Feynman wrote of the Pauli exclusion principle, “In fact, almost all the peculiarities of the material world hinge on this wonderful fact.” In 1972 Borland and Dennis showed that there exist powerful constraints beyond the Pauli exclusion principle on the orbital occupations of Fermi particles, providing important restrictions on quantum correlation and entanglement. Here we use computations on quantum computers to experimentally verify the existence of these additional constraints. Quantum many-fermion states are randomly prepared on the quantum computer and tested for constraint violations. Measurements show no violation and confirm the generalized Pauli exclusion principle with an error of one part in one quintillion. The Pauli exclusion principle can be formulated in a generalized form where additional constraints are imposed to the orbital degrees of freedom of electrons. In this work these constraints are experimentally verified on a five qubit quantum computer with an error of one part in one quintillion.O. Werba, A. Raeber, K. Head-Marsden, D. A. Mazziotti. Phys. Chem. Chem. Phys. 21 23900-23905 (2019). "Signature of van der Waals interactions in the cumulant density matrix"
Here we propose and implement a universal signature of the van der Waals interactions based on the cumulant part of the two-electron reduced density matrix (2-RDM). Due to the connected property of the cumulant, we can use it to detect the van der Waals interactions between two molecular moieties. In particular, we use the squared Frobenius norm of the cumulant of the 2-RDM, which has been previously shown to provide a size-extensive measure of the electron correlation. As two moieties are separated to infinity, the cumulant Frobenius norm exhibits an r−6 decay to its asymptotic limit, providing a density-based measure of the van der Waals interaction. We study this signature of van der Waals forces in a collection of small molecules of varying geometries. These computations agree with experimental trends of known literature values.J. Xie, J. Boyn, A. S. Filatov, A. J. McNeece, D. A. Mazziotti, J. S. Anderson. Chem. Sci. 11 1066-1078 (2019). "Redox, transmetalation, and stacking properties of tetrathiafulvalene-2,3,6,7-tetrathiolate bridged tin, nickel, and palladium compounds"
Here we report that capping the molecule TTFtt (TTFtt = tetrathiafulvalene-2,3,6,7-tetrathiolate) with dialkyl tin groups enables the isolation of a stable series of redox congeners and facile transmetalation to Ni and Pd. TTFtt has been proposed as an attractive building block for molecular materials for two decades as it combines the redox chemistry of TTF and dithiolene units. TTFttH4, however, is inherently unstable and the incorporation of TTFtt units into complexes or materials typically proceeds through the in situ generation of the tetraanion TTFtt4−. Capping of TTFtt4− with Bu2Sn2+ units dramatically improves the stability of the TTFtt moiety and furthermore enables the isolation of a redox series where the TTF core carries the formal charges of 0, +1, and +2. All of these redox congeners show efficient and clean transmetalation to Ni and Pd resulting in an analogous series of bimetallic complexes capped by 1,2-bis(diphenylphosphino)ethane (dppe) ligands. Furthermore, by using the same transmetalation method, we synthesized analogous palladium complexes capped by 1,1′-bis(diphenylphosphino)ferrocene (dppf) which had been previously reported. All of these species have been thoroughly characterized through a systematic survey of chemical and electronic properties by techniques including cyclic voltammetry (CV), ultraviolet-visible-near infrared spectroscopy (UV-vis-NIR), electron paramagnetic resonance spectroscopy (EPR), nuclear magnetic resonance spectroscopy (NMR) and X-ray diffraction (XRD). These detailed synthetic and spectroscopic studies highlight important differences between the transmetalation strategy presented here and previously reported synthetic methods for the installation of TTFtt. In addition, the utility of this stabilization strategy can be illustrated by the observation of unusual TTF radical–radical packing in the solid state and dimerization in the solution state. Theoretical calculations based on variational 2-electron reduced density matrix methods have been used to investigate these unusual interactions and illustrate fundamentally different levels of covalency and overlap depending on the orientations of the TTF cores. Taken together, this work demonstrates that tin-capped TTFtt units are ideal reagents for the installation of redox-tunable TTFtt ligands enabling the generation of entirely new geometric and electronic structures.Back to top
2018
S. Safaei, D. A. Mazziotti. Phys. Rev. B 98 045122 (2018). "Quantum signature of exciton condensation"
Exciton condensation, a Bose-Einstein-like condensation of excitons, was recently reported in an electronic double layer (EDL) of graphene. We show that a universal quantum signature for exciton condensation can be used to both identity and quantify exciton condensation in molecular systems from direct calculations of the two-electron reduced density matrix. Computed large eigenvalues in the particle-hole reduced-density matrices of pentacene and hexacene EDLs reveal the beginnings of condensation, suggesting the potential for exciton condensation in smaller scale molecular EDLs.M. Sajjan, D. A. Mazziotti. Commun. Chem. 1 31 (2018). "Current-constrained density-matrix theory to calculate molecular conductivity with increased accuracy"
Molecular conductivity is the quantum flow of electrons through a molecule. Since its conception by Aviram and Ratner, molecular conductivity has been realized experimentally in molecules and molecular-scale circuits. Significant challenges, however, remain for its prediction with popular theoretical methods often overpredicting conductance by as much as an order of magnitude. Here we report a current-constrained, electronic structure-based variational principle for molecular conductivity. Unlike existing theories, which set the voltage to compute the current, the current-constrained variational principle determines the voltage from an electronic structure calculation in which the current is added as a constraint. We apply the variational principle to benezenedithiol with gold and nickel leads where it matches experimental values and trends, improving upon previous theory by as much as 1–2 orders of magnitude. The current constraint produces a conducting steady state that includes all many-body effects treatable by the electronic structure calculation. Molecular conductivity, the quantum flow of electrons through a molecule, is typically overpredicted by theoretical methods to date. Here, the authors report a current-constrained, electronic structure-based method improving on existing techniques for calculating conductance by up to two orders of magnitude.R. Chakraborty, D. A. Mazziotti. J. Chem. Phys. 148 054106 (2018). "Sparsity of the wavefunction from the generalized Pauli exclusion principle"
Electron occupations that arise from pure quantum states are restricted by a stringent set of conditions that are said to generalize the Pauli exclusion principle. These generalized Pauli constraints (GPCs) define the boundary of the set of one-electron reduced density matrices (1-RDMs) that are derivable from at least one N-electron wavefunction. In this paper, we investigate the sparsity of the Slater-determinant representation of the wavefunction that is a necessary, albeit not sufficient, condition for its 1-RDM to lie on the boundary of the set of pure N-representable 1-RDMs or in other words saturate one of the GPCs. The sparse wavefunction, we show, is exact not only for 3 electrons in 6 orbitals but also for 3 electrons in 8 orbitals. For larger numbers of electrons and/or orbitals in the lowest spin state, the exact wavefunction does not generally saturate one of the GPCs, and hence, the sparse representation is typically an approximation. Because the sparsity of the wavefunction is a necessary but not sufficient condition for saturation of one of the GPCs, optimization of the sparse wavefunction Ansatz to minimize the ground-state energy does not necessarily produce a wavefunction whose 1-RDM exactly saturates one of the GPCs. While the sparse Ansatz can be employed with arbitrary orbitals or optimized orbitals, in this paper, we explore the Ansatz with the natural orbitals from full configuration interaction, which yields an upper bound to the ground-state energy that equals the exact energy for a given basis set if the full-configuration-interaction wavefunction saturates the Ansatz’s GPC. With calculations on the boron isoelectronic sequence, the dinitrogen cation N2+, hydrogen chains, and cyclic conjugated π systems, we examine the quality of the sparse wavefunction Ansatz from the amount of correlation energy recovered.A. W. Schlimgen, D. A. Mazziotti. J. Chem. Phys. 149 164111 (2018). "Analytical gradients of variational reduced-density-matrix and wavefunction-based methods from an overlap-reweighted semidefinite program"
Analytical gradients of variational two-electron reduced-density matrix (2-RDM) methods are derived by transforming the atomic-orbital reduced-density matrices to remove the dependence of the N-representability conditions on the orbital-overlap matrix. The transformation, performed through a Cholesky decomposition of the geminal-overlap matrix, generates a Hellmann-Feynman-like expression for the gradient that only depends on the derivative of the transformed reduced Hamiltonian matrix. The formulation is applicable not only to the variational 2-RDM method but also to variational wavefunction methods like the full configuration interaction and complete active-space self-consistent-field. To illustrate, we apply the analytical gradients to perform geometry optimizations on several transition metal complexes, octahedral and trigonal prismatic CrF6 as well as the (ethylene-1,2-dithiolato)nickel, or Ni(edt)2, complex.J. M. Montgomery, D. A. Mazziotti. J. Phys. Chem. A 122 4988-4996 (2018). "Strong Electron Correlation in Nitrogenase Cofactor, FeMoco"
FeMoco, MoFe7S9C, has been shown to be the active catalytic site for the reduction of nitrogen to ammonia in the nitrogenase protein. An understanding of its electronic structure including strong electron correlation is key to designing mimic catalysts capable of ambient nitrogen fixation. Active spaces ranging from [54, 54] to [65, 57] have been predicted for a quantitative description of FeMoco’s electronic structure. However, a wave function approach for a singlet state using a [54, 54] active space would require 1029 variables. In this work, we systematically explore the active-space size necessary to qualitatively capture strong correlation in FeMoco and two related moieties, MoFe3S7 and Fe4S7. Using CASSCF and 2-RDM methods, we consider active-space sizes up to [14, 14] and [30, 30], respectively, with STO-3G, 3-21G, and DZP basis sets and use fractional natural-orbital occupation numbers to assess the level of multireference electron correlation, an examination of which reveals a competition between single-reference and multireference solutions to the electronic Schrödinger equation for smaller active spaces and a consistent multireference solution for larger active spaces.M. Sajjan, K. Head-Marsden, D. A. Mazziotti. Phys. Rev. A 97 062502 (2018). "Entangling and disentangling many-electron quantum systems with an electric field"
We show that the electron correlation of a molecular system can be enhanced or diminished through the application of a homogeneous electric field antiparallel or parallel to the system's intrinsic dipole moment. More generally, we prove that any external stimulus that significantly changes the expectation value of a one-electron operator with nondegenerate minimum and maximum eigenvalues can be used to control the degree of a molecule's electron correlation. Computationally, the effect is demonstrated in HeH+, MgH+, BH, HCN, H2O, HF, formaldehyde, and a fluorescent dye. Furthermore, we show in calculations with an array of formaldehyde (CH2O) molecules that the field can control not only the electron correlation of a single formaldehyde molecule but also the entanglement among formaldehyde molecules. The quantum control of correlation and entanglement has potential applications in the design of molecules with tunable properties and the stabilization of qubits in quantum computations.S. E. Smart, P. G. Scrape, L. J. Butler, D. A. Mazziotti. J. Chem. Phys. 149 024302 (2018). "Using reduced density matrix techniques to capture static and dynamic correlation in the energy landscape for the decomposition of the CH2CH2ONO radical and support a non-IRC pathway"
The unexpected abundance of HNO in the photodecomposition of the radical 2-nitrosooxy ethyl (CH2CH2ONO) is investigated through calculations of the potential energy surface by the anti-Hermitian contracted Schrödinger equation (ACSE) method, which directly generates the 2-electron reduced density matrix. The ACSE, which is able to balance single-reference (dynamic) and multi-reference (static) correlation effects, reveals some subtle correlation effects along the intrinsic reaction coordinate (IRC) en route to NO + oxirane, an IRC which offers a potential bifurcation to the HNO + vinoxy product channel. These effects were not fully captured by either single-reference techniques, such as coupled cluster, or multi-reference techniques, such as second-order multi-reference perturbation theory. These correlation effects reveal small to moderate energy changes in key transition states, which have implications for the reaction mechanism as related to the production of HNO.Back to top
2017
R. Chakraborty, D. A. Mazziotti. J. Chem. Phys. 146 184101 (2017). "Noise-assisted energy transfer from the dilation of the set of one-electron reduced density matrices"
Noise-assisted energy transfer can be explained geometrically in terms of the set of one-electron reduced density matrices (1-RDMs) [R. Chakraborty and D. A. Mazziotti, Phys. Rev. A 91, 010101(R) (2015)]. In this paper, we examine the geometric picture of quantum noise for the seven-chromophore Fenna-Matthews-Olson (FMO) complex. Noise expands the feasible set of orbital occupation trajectories to the target state through the violation of the pure-state N-representability conditions on the 1-RDM, known as the generalized Pauli constraints. While the generalized Pauli constraints are not explicitly known for seven-electron systems, we are able to treat a seven-exciton model of the FMO complex through the use of generalized Pauli constraints for p qubits which are known for arbitrary p. In the model, we find that while dephasing noise alone produces a trajectory of ensemble states that neither expands the set of 1-RDMs nor reaches the reaction center, the inclusion of both dephasing and dissipation expands the set of 1-RDMs and exhibits an efficient energy transfer to the reaction center. The degree to which the noise expands the set of 1-RDMs, violating the generalized Pauli constraints, is quantified by the distance of the 1-RDM outside its pure set to the distance of the 1-RDM inside its ensemble set. The geometric picture of energy transfer has applications to general quantum systems in chemistry and physics.A. R. McIsaac, D. A. Mazziotti. Phys. Chem. Chem. Phys. 19 4656-4660 (2017). "Ligand non-innocence and strong correlation in manganese superoxide dismutase mimics"
We examine the 1-electron reduction of manganese porphyrin complexes Mn(III) porphyrin and Mn(III) TDE-2-ImP5+, which have attracted recent interest due to their properties as superoxide dismutase mimics. We perform a series of electronic structure calculations using the variational 2-electron reduced density matrix (2-RDM) method with a large [30,30] active space that represents a wavefunction with 1019 variables, as well as the more traditional complete active space self-consistent field (CASSCF) method with a [14,14] active space. We show that the larger 2-RDM calculation, intractable with CASSCF, is required to capture the full effects of electron correlation in the molecule and predict the non-innocence of the porphyrin ligand during the reduction. The CASSCF method predicts single-reference systems exhibiting a metal-centered reduction, but the 2-RDM method predicts a strongly correlated system exhibiting a ligand-centered reduction. Based on these results, we find that the porphyrin ligand is reduced rather than the manganese, and suggest that the electron correlation plays a role in driving the ligand non-innocence.K. Head-Marsden, D. A. Mazziotti. J. Chem. Phys. 147 084101 (2017). "Pair 2-electron reduced density matrix theory using localized orbitals"
Full configuration interaction (FCI) restricted to a pairing space yields size-extensive correlation energies but its cost scales exponentially with molecular size. Restricting the variational two-electron reduced-density-matrix (2-RDM) method to represent the same pairing space yields an accurate lower bound to the pair FCI energy at a mean-field-like computational scaling of O(r3) where r is the number of orbitals. In this paper, we show that localized molecular orbitals can be employed to generate an efficient, approximately size-extensive pair 2-RDM method. The use of localized orbitals eliminates the substantial cost of optimizing iteratively the orbitals defining the pairing space without compromising accuracy. In contrast to the localized orbitals, the use of canonical Hartree-Fock molecular orbitals is shown to be both inaccurate and non-size-extensive. The pair 2-RDM has the flexibility to describe the spectra of one-electron RDM occupation numbers from all quantum states that are invariant to time-reversal symmetry. Applications are made to hydrogen chains and their dissociation, n-acene from naphthalene through octacene, and cadmium telluride 2-, 3-, and 4-unit polymers. For the hydrogen chains, the pair 2-RDM method recovers the majority of the energy obtained from similar calculations that iteratively optimize the orbitals. The localized-orbital pair 2-RDM method with its mean-field-like computational scaling and its ability to describe multi-reference correlation has important applications to a range of strongly correlated phenomena in chemistry and physics.A. W. Schlimgen, D. A. Mazziotti. J. Phys. Chem. A 121 9377-9384 (2017). "Static and Dynamic Electron Correlation in the Ligand Noninnocent Oxidation of Nickel Dithiolates"
Metal dithiolates have a wide range of applications from catalysis to molecular conductors with the ligands being the source of electrons during electrochemical oxidation in an effect known as ligand noninnocence. Recent large-scale variational two-electron reduced-density matrix (2-RDM) calculations of the vanadium oxo complex and manganese superoxide dismutase show that quantum entanglement stabilizes the addition of an electron to the ligands, providing a quantum mechanical explanation for ligand noninnocence. In this paper, we confirm and explore the ligand noninnocence in the electron oxidation series of bis(ethylene-1,2-dithiolato)nickel or [Ni(edt2)](−2,–1,0) with variational 2-RDM calculations. While previous wave function calculations of this series have selected only the ligand π orbitals as the critical (active) orbitals to be correlated, we find that both ligand π and nickel d orbitals must be correlated to generate a realistic picture of the electron-transfer process. Using the computed 2-RDM to seed a solution of the anti-Hermitian contracted Schrödinger equation, we predict that the singlet state is lower in energy than the triplet state, which is consistent with experimental observations.E. P. Hoy, D. A. Mazziotti, T. Seideman. J. Chem. Phys. 147 184110 (2017). "Development and application of a 2-electron reduced density matrix approach to electron transport via molecular junctions"
Can an electronic device be constructed using only a single molecule? Since this question was first asked by Aviram and Ratner in the 1970s [Chem. Phys. Lett. 29, 277 (1974)], the field of molecular electronics has exploded with significant experimental advancements in the understanding of the charge transport properties of single molecule devices. Efforts to explain the results of these experiments and identify promising new candidate molecules for molecular devices have led to the development of numerous new theoretical methods including the current standard theoretical approach for studying single molecule charge transport, i.e., the non-equilibrium Green’s function formalism (NEGF). By pairing this formalism with density functional theory (DFT), a wide variety of transport problems in molecular junctions have been successfully treated. For some systems though, the conductance and current-voltage curves predicted by common DFT functionals can be several orders of magnitude above experimental results. In addition, since density functional theory relies on approximations to the exact exchange-correlation functional, the predicted transport properties can show significant variation depending on the functional chosen. As a first step to addressing this issue, the authors have replaced density functional theory in the NEGF formalism with a 2-electron reduced density matrix (2-RDM) method, creating a new approach known as the NEGF-RDM method. 2-RDM methods provide a more accurate description of electron correlation compared to density functional theory, and they have lower computational scaling compared to wavefunction based methods of similar accuracy. Additionally, 2-RDM methods are capable of capturing static electron correlation which is untreatable by existing NEGF-DFT methods. When studying dithiol alkane chains and dithiol benzene in model junctions, the authors found that the NEGF-RDM predicts conductances and currents that are 1-2 orders of magnitude below those of B3LYP and M06 DFT functionals. This suggests that the NEGF-RDM method could be a viable alternative to NEGF-DFT for molecular junction calculations.A. J. Valentine, D. A. Mazziotti. Chem. Phys. Lett. 685 300-304 (2017). "Analytical nuclear derivatives for the parametric two-electron reduced density matrix method"
Efficient and accurate nuclear gradients are essential to performing molecular optimizations. Here for the first time we present analytical nuclear gradients for the parametric two-electron reduced-density-matrix method (p2-RDM), which uses the 2-RDM as the primary variable in calculations in lieu of the many-electron wavefunction. While numerical gradients require six energy evaluations for each atom, analytical gradients require only a single calculation for each geometry sampled. We present benchmark p2-RDM geometry optimizations that show analytical gradients reduce CPU times by as much as 80%, even for small molecules. We also use p2-RDM to evaluate the bond length alternation (BLA), or the difference in length between adjacent single and double bonds, of trans-polyacetylene (PA). We find that the BLA in the extrapolated limit to be 0.080Å, in agreement with experiment and closely mirroring the prediction of the more expensive coupled-cluster with single and double excitations with perturbative triples (CCSD(T)).A. J. S. Valentine, D. V. Talapin, D. A. Mazziotti. J. Phys. Chem. A 121 3142-3147 (2017). "Orbitals, Occupation Numbers, and Band Structure of Short One-Dimensional Cadmium Telluride Polymers"
Recent work found that soldering CdTe quantum dots together with a molecular CdTe polymer yielded field-effect transistors with much greater electron mobility than quantum dots alone. We present a computational study of the CdTe polymer using the active-space variational two-electron reduced density matrix (2-RDM) method. While analogous complete active-space self-consistent field (CASSCF) methods scale exponentially with the number of active orbitals, the active-space variational 2-RDM method exhibits polynomial scaling. A CASSCF calculation using the (48o,64e) active space studied in this paper requires 1024 determinants and is therefore intractable, while the variational 2-RDM method in the same active space requires only 2.1 × 107 variables. Natural orbitals, natural-orbital occupations, charge gaps, and Mulliken charges are reported as a function of polymer length. The polymer, we find, is strongly correlated, despite possessing a simple sp3-hybridized bonding scheme. Calculations reveal the formation of a nearly saturated valence band as the polymer grows and a charge gap that decreases sharply with polymer length.Back to top
2016
D. A. Mazziotti. Adv. Chem. Phys. 331-342 (2016). "Reduced‐Density‐Matrix Mechanics: With Application to Many‐Electron Atoms and Molecules"
This chapter contains sections titled: Introduction Anti‐Hermitian Contracted Schrödinger Equation Reconstruction of the 3‐RDM Optimization of the 2‐RDM Applications Some Connections and a Second Look Ahead Acknowledgments References Introduction Anti‐Hermitian Contracted Schrödinger Equation Reconstruction of the 3‐RDM Optimization of the 2‐RDM Applications Some Connections and a Second Look Ahead Acknowledgments ReferencesD. A. Mazziotti. Phys. Rev. A 94 032516 (2016). "Pure-N-representability conditions of two-fermion reduced density matrices"
We derive necessary conditions for any two-fermion reduced density matrix (2-RDM) to be representable by a pure N-fermion density matrix ΨΨ* where Ψ is the wave function. These pure N-representability conditions of the 2-RDM are important because they provide stringent constraints beyond those from the Pauli and the generalized Pauli constraints on the structure of many-fermion 2-RDMs and their wave functions. The pure 2-RDM conditions are derived as generalized Pauli constraints on effective one-fermion reduced density matrices (1-RDMs) generated by the removal or addition of a fermion from the wave function. Computationally, we show for four-electron molecules that the derived pure N-representability conditions are nontrivially active for exact ground-state 2-RDMs and that they provide significant restrictions beyond the D, Q, and G ensemble N-representability conditions. Constraints on higher-order p-RDMs where p>2 are derived in a similar fashion. The constraints have potentially significant applications to computing strongly correlated many-fermion states with enhanced accuracy and decreased computational complexity.D. A. Mazziotti. Adv. Chem. Phys. 19-59 (2016). "Reduced‐Density‐Matrix Mechanics: With Application to Many‐Electron Atoms and Molecules"
This chapter contains sections titled: Introduction Theory Semidefinite programming Applications A Look Ahead Acknowledgments References Introduction Theory Semidefinite programming Applications A Look Ahead Acknowledgments ReferencesD. A. Mazziotti. Phys. Rev. Lett. 117 153001 (2016). "Enhanced Constraints for Accurate Lower Bounds on Many-Electron Quantum Energies from Variational Two-Electron Reduced Density Matrix Theory"
A central challenge of physics is the computation of strongly correlated quantum systems. The past ten years have witnessed the development and application of the variational calculation of the two-electron reduced density matrix (2-RDM) without the wave function. In this Letter we present an orders-of-magnitude improvement in the accuracy of 2-RDM calculations without an increase in their computational cost. The advance is based on a low-rank, dual formulation of an important constraint on the 2-RDM, the T2 condition. Calculations are presented for metallic chains and a cadmium-selenide dimer. The low-scaling T2 condition will have significant applications in atomic and molecular, condensed-matter, and nuclear physics.R. Chakraborty, D. A. Mazziotti. Int. J. Quantum Chem. 116 784-790 (2016). "Role of the generalized pauli constraints in the quantum chemistry of excited states"
The Pauli exclusion principle requires that spin orbitals have occupations between zero and one. For pure quantum systems, however, the occupations of the spin orbitals are constrained by additional inequalities known as the generalized Pauli constraints. If the occupation numbers saturate (or nearly saturate) the generalized Pauli constraints, then the occupation numbers are said to be pinned (or quasi‐pinned) to the constraints. Here, we assess the complexity of electron correlation of excited states from the pinning or quasi‐pinning of the occupation numbers to the generalized Pauli constraints where the degree of pinning encodes information about the structure of the wave function including correlation and entanglement. Results are presented for three‐ and four‐electron atoms and molecules, the five‐electron cyclopentadienyl radical, and the seven‐electron Fenna‐Matthews‐Olson complex in green‐sulfur bacteria. The data shows that even when the ground‐state occupation numbers are pinned, the occupation numbers of excited states manifest pinned, quasi‐pinned, and unpinned behavior, reflecting the complexity of electron correlation and entanglement in excited states. © 2016 Wiley Periodicals, Inc. More than 40 years ago, it was recognized that there are additional constraints on occupations beyond the Pauli exclusion principle that constrain the structure of the many‐fermion wave function. These conditions, known as generalized Pauli conditions, provide valuable information about correlation and entanglement in atoms and molecules. These additional constraints on the wave function highlight the structural complexity of the excited state wave functions which are typically associated with greater electron correlation.D. A. Mazziotti. Adv. Chem. Phys. 165-203 (2016). "Reduced‐Density‐Matrix Mechanics: With Application to Many‐Electron Atoms and Molecules"
This chapter contains sections titled: Introduction Contracted Schrödinger Equation Reconstruction of the 3‐ and 4‐RDMS Purification of the 2‐RDM Self‐Consistent Iteration Algorithm for Solving the CSE Applications A Look Ahead Acknowledgments Appendix: Grassmann Products References Introduction Contracted Schrödinger Equation Reconstruction of the 3‐ and 4‐RDMS Purification of the 2‐RDM Self‐Consistent Iteration Algorithm for Solving the CSE Applications A Look Ahead Acknowledgments Appendix: Grassmann Products ReferencesC. W. Heaps, D. A. Mazziotti. J. Chem. Phys. 145 064101 (2016). "Accurate non-adiabatic quantum dynamics from pseudospectral sampling of time-dependent Gaussian basis sets"
Quantum molecular dynamics requires an accurate representation of the molecular potential energy surface from a minimal number of electronic structure calculations, particularly for nonadiabatic dynamics where excited states are required. In this paper, we employ pseudospectral sampling of time-dependent Gaussian basis functions for the simulation of non-adiabatic dynamics. Unlike other methods, the pseudospectral Gaussian molecular dynamics tests the Schrödinger equation with N Dirac delta functions located at the centers of the Gaussian functions reducing the scaling of potential energy evaluations from O ( N 2 ) to O ( N ) . By projecting the Gaussian basis onto discrete points in space, the method is capable of efficiently and quantitatively describing the nonadiabatic population transfer and intra-surface quantum coherence. We investigate three model systems: the photodissociation of three coupled Morse oscillators, the bound state dynamics of two coupled Morse oscillators, and a two-dimensional model for collinear triatomic vibrational dynamics. In all cases, the pseudospectral Gaussian method is in quantitative agreement with numerically exact calculations. The results are promising for nonadiabatic molecular dynamics in molecular systems where strongly correlated ground or excited states require expensive electronic structure calculations.E. J. Sturm, D. A. Mazziotti. Mol. Phys. 114 335-343 (2016). "Highly accurate excited-state energies from direct computation of the 2-electron reduced density matrix by the anti-Hermitian contracted Schrödinger equation"
Directly solving for the 2-electron reduced density matrix (2-RDM) via the anti-Hermitian contracted Schrödinger equation (ACSE) enables computations for excited states energies without the N-electron wave function. Of particular interest are excitations and dissociation curves that exhibit strong multi-reference correlation effects. The ground and excited states of the molecules HF, H2O, and N2 are examined at both equilibrium and non-equilibrium geometries to compare the ability of the ACSE and widely used ab initio techniques to treat strong multi-reference electron correlation. Calculations are performed with double-ζ basis sets for calibration with full configuration interaction (FCI). Multi-reference second-order perturbation theory (MRPT2) and the ACSE both provide qualitative precision with respect to FCI data, although the ACSE's capability to include higher order correlation effects permits near-FCI accuracy for ground and excited states and excitation energies.C. W. Heaps, D. A. Mazziotti. J. Chem. Phys. 144 164108 (2016). "Pseudospectral Gaussian quantum dynamics: Efficient sampling of potential energy surfaces"
Trajectory-based Gaussian basis sets have been tremendously successful in describing high-dimensional quantum molecular dynamics. In this paper, we introduce a pseudospectral Gaussian-based method that achieves accurate quantum dynamics using efficient, real-space sampling of the time-dependent basis set. As in other Gaussian basis methods, we begin with a basis set expansion using time-dependent Gaussian basis functions guided by classical mechanics. Unlike other Gaussian methods but characteristic of the pseudospectral and collocation methods, the basis set is tested with N Dirac delta functions, where N is the number of basis functions, rather than using the basis function as test functions. As a result, the integration for matrix elements is reduced to function evaluation. Pseudospectral Gaussian dynamics only requires O ( N ) potential energy calculations, in contrast to O ( N 2 ) evaluations in a variational calculation. The classical trajectories allow small basis sets to sample high-dimensional potentials. Applications are made to diatomic oscillations in a Morse potential and a generalized version of the Henon-Heiles potential in two, four, and six dimensions. Comparisons are drawn to full analytical evaluation of potential energy integrals (variational) and the bra-ket averaged Taylor (BAT) expansion, an O ( N ) approximation used in Gaussian-based dynamics. In all cases, the pseudospectral Gaussian method is competitive with full variational calculations that require a global, analytical, and integrable potential energy surface. Additionally, the BAT breaks down when quantum mechanical coherence is particularly strong (i.e., barrier reflection in the Morse oscillator). The ability to obtain variational accuracy using only the potential energy at discrete points makes the pseudospectral Gaussian method a promising avenue for on-the-fly dynamics, where electronic structure calculations become computationally significant.A. W. Schlimgen, C. W. Heaps, D. A. Mazziotti. J. Phys. Chem. Lett. 7 627-631 (2016). "Entangled Electrons Foil Synthesis of Elusive Low-Valent Vanadium Oxo Complex"
We examine the recently reported first synthesis of the elusive low-valent vanadium(III) in a vanadium oxo complex with a computation representing 1021 quantum degrees of freedom. While this computation is intractable with a conventionally constructed wave function, it is performed here by a direct calculation of the system’s two-electron reduced density matrix (2-RDM), where the 2-RDM is constrained by nontrivial conditions, known as N-representability conditions, that restrict the 2-RDM to represent an N electron quantum system. We show that the added (reducing) electron becomes entangled among the five pyridine ligands. While smaller calculations predict a metal-centered addition, large-scale 2-RDM calculations show that quantum entanglement redirects the electron transfer to the pyridine ligands, resulting in a ligand-centered addition. Beyond its implications for the synthesis of low-valent vanadium oxo complexes, the result suggests new possibilities for using quantum entanglement to predict and control electron transfer in chemical and biological materials.Back to top
2015
R. Chakraborty, D. A. Mazziotti. Int. J. Quantum Chem. 115 1305-1310 (2015). "Structure of the one‐electron reduced density matrix from the generalized Pauli exclusion principle"
The Pauli exclusion principle requires that the occupations of the orbitals lie between zero and one. These Pauli conditions hold for one‐electron reduced density matrices (1‐RDMs) from both open and closed quantum systems. More than 40 years ago, it was recognized that there are additional conditions on the 1‐RDM for closed quantum systems. In this review, we discuss the structure of the 1‐RDM from the generalized Pauli exclusion principle in many‐electron atoms and molecules and the violation of the generalized Pauli principle as a sufficient condition for the openness of a many‐electron quantum system. © 2015 Wiley Periodicals, Inc. The Pauli exclusion principle requires that the occupations of the orbitals lie between zero and one. These Pauli conditions hold for one‐electron reduced density matrices (1‐RDMs) from both open and closed quantum systems. More than 40 years ago, it was recognized that there are additional conditions on the 1‐RDM for closed quantum systems. The violation of the generalized Pauli principle is discussed as a sufficient condition for the openness of a many‐electron quantum system.R. Chakraborty, D. A. Mazziotti. Phys. Rev. A 91 010101 (2015). "Sufficient condition for the openness of a many-electron quantum system from the violation of a generalized Pauli exclusion principle"
Information about the interaction of a many-electron quantum system with its environment, we show, is encoded within the one-electron density matrix (1-RDM). While the 1-RDM from an ensemble many-electron quantum system must obey the Pauli exclusion principle, the 1-RDM must obey additional constraints known as generalized Pauli conditions when it corresponds to a closed system describable by a single wave function. By examining the 1-RDM's violation of these generalized Pauli conditions, we obtain a sufficient condition at the level of a single electron for a many-electron quantum system's openness. In an application to exciton dynamics in photosynthetic light harvesting we show that the interaction of the system with the environment (quantum noise) relaxes significant constraints imposed on the exciton dynamics by the generalized Pauli conditions. This relaxation provides a geometric (kinematic) interpretation for the role of noise in enhancing exciton transport in quantum systems.S. Veeraraghavan, D. A. Mazziotti. Phys. Rev. A 92 022512 (2015). "Semidefinite programming formulation of linear-scaling electronic structure theories"
We present a linear-scaling approach based on semidefinite programs (SDPs) to compute the density matrix for effective one-electron theories. Traditional methods constrain the density matrix to represent a Slater determinant and hence rely on parameterization or purification. We eliminate the need for such a constraint by performing an energy minimization over all the convex combinations of density matrices representing Slater determinants. By not relying on purification, the SDP approach not only eliminates accumulation error present in some methods but also reduces the amount of truncation error. Sparsity in the Hamiltonian can be exploited to make the SDP approach scale linearly with system size. Crossovers in computational time with a cubically scaling algorithm are demonstrated for one-dimensional hydrogen chains ranging from H50 to H1500.A. Raeber, D. A. Mazziotti. Phys. Rev. A 92 052502 (2015). "Large eigenvalue of the cumulant part of the two-electron reduced density matrix as a measure of off-diagonal long-range order"
Off-diagonal long-range order (ODLRO) in the two-electron reduced density matrix (2-RDM) has long been recognized as a mathematical characteristic of conventional superconductors. The large eigenvalue of the 2-RDM has been shown to be a useful measure of this long-range order. The 2-RDM can be represented as the sum of a connected (cumulant) piece and an unconnected piece. In this work, we show that the cumulant 2-RDM also has a large eigenvalue in the limit of ODLRO. The largest eigenvalue of the cumulant 2-RDM, we prove, is bounded from above by N. In the limit of extreme pairing, such as Cooper pairing, the largest eigenvalue and the trace of the cumulant 2-RDM approach their extreme values of N and −N, respectively. While the trace of the cumulant 2-RDM, which is computable from only a knowledge of the 1-RDM, can reflect ODLRO, it alone does not appear to be a sufficient criterion. The large eigenvalue of the cumulant 2-RDM, we show, implies the large eigenvalue of the 2-RDM and, hence, is a natural measure of ODLRO that vanishes in the mean-field limit.A. M. Sand, D. A. Mazziotti. J. Chem. Phys. 143 134110 (2015). "Enhanced computational efficiency in the direct determination of the two-electron reduced density matrix from the anti-Hermitian contracted Schrödinger equation with application to ground and excited states of conjugated π-systems"
Determination of the two-electron reduced density matrix (2-RDM) from the solution of the anti-Hermitian contracted Schrödinger equation (ACSE) yields accurate energies and properties for both ground and excited states. Here, we develop a more efficient method to solving the ACSE that uses second-order information to select a more optimal step towards the solution. Calculations on the ground and excited states of water, hydrogen fluoride, and conjugated π systems show that the improved ACSE algorithm is 10-20 times faster than the previous ACSE algorithm. The ACSE can treat both single- and multi-reference electron correlation with the initial 2-RDM from a complete-active-space self-consistent-field (CASSCF) calculation. Using the improved algorithm, we explore the relationship between truncation of the active space in the CASSCF calculation and the accuracy of the energy and 2-RDM from the ACSE calculation. The accuracy of the ACSE, we find, is less sensitive to the size of the active space than the accuracy of other wavefunction methods, which is useful when large active space calculations are computationally infeasible.N. C. Rubin, D. A. Mazziotti. J. Phys. Chem. C 119 14706-14713 (2015). "Strong Electron Correlation in Materials from Pair-Interacting Model Hamiltonians"
Strong electron correlation in materials is explored within a class of model Hamiltonians that treat only pair interactions between electrons. The model is unique among typical spin Hamiltonians in that it does not have an effective mean-field reference wave function. The ground-state wave functions from all Hamiltonians in the model have the same one-electron reduced density matrix (1-RDM); consequently, one-electron theories such as the Hartree–Fock and density functional theories are inapplicable. In contrast, the ground-state two-electron reduced density matrix (2-RDM) has a one-to-one mapping to the ground-state wave function. For a range of lattices including linear, ladder, and square topologies, we variationally compute the 2-RDM subject to constraints, known as N-representability conditions, that are necessary for the 2-RDM to represent an N-electron ensemble density matrix. We find that for all model Hamiltonians the 2-RDM is accurately computed as long as the D, Q, and G N-representability conditions are supplemented with the T 2 condition. Energies, orbital and geminal occupation numbers, and correlation functions are computed. The model has applications to superconductivity as well as more general pairing phenomena in electronic systems. Effective methods are needed to treat strong electron correlation arising in the study of materials including the study of high-temperature superconductivity and phase transitions.K. Head-Marsden, D. A. Mazziotti. J. Chem. Phys. 142 051102 (2015). "Communication: Satisfying fermionic statistics in the modeling of open time-dependent quantum systems with one-electron reduced density matrices"
For an open, time-dependent quantum system, Lindblad derived the most general modification of the quantum Liouville equation in the Markovian approximation that models environmental effects while preserving the non-negativity of the system’s density matrix. While Lindblad’s modification is correct for N-electron density matrices, solution of the Liouville equation with a Lindblad operator causes the one-electron reduced density matrix (1-RDM) to violate the Pauli exclusion principle. Consequently, after a short time, the 1-RDM is not representable by an ensemble N-electron density matrix (not ensemble N-representable). In this communication, we derive the necessary and sufficient constraints on the Lindbladian matrix within the Lindblad operator to ensure that the 1-RDM remains N-representable for all time. The theory is illustrated by considering the relaxation of an excitation in several molecules F2, N2, CO, and BeH2 subject to environmental noise.E. P. Hoy, D. A. Mazziotti. J. Chem. Phys. 143 064103 (2015). "Positive semidefinite tensor factorizations of the two-electron integral matrix for low-scaling ab initio electronic structure"
Tensor factorization of the 2-electron integral matrix is a well-known technique for reducing the computational scaling of ab initio electronic structure methods toward that of Hartree-Fock and density functional theories. The simplest factorization that maintains the positive semidefinite character of the 2-electron integral matrix is the Cholesky factorization. In this paper, we introduce a family of positive semidefinite factorizations that generalize the Cholesky factorization. Using an implementation of the factorization within the parametric 2-RDM method [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)], we study several inorganic molecules, alkane chains, and potential energy curves and find that this generalized factorization retains the accuracy and size extensivity of the Cholesky factorization, even in the presence of multi-reference correlation. The generalized family of positive semidefinite factorizations has potential applications to low-scaling ab initio electronic structure methods that treat electron correlation with a computational cost approaching that of the Hartree-Fock method or density functional theory.A. L. McManus, E. P. Hoy, D. A. Mazziotti. Phys. Chem. Chem. Phys. 17 12521-12529 (2015). "Energies and structures in biradical chemistry from the parametric two-electron reduced-density matrix method: applications to the benzene and cyclobutadiene biradicals"
The treatment of biradical chemistry presents a challenge for electronic structure theory, especially single-reference methods, as it requires the description of varying degrees and kinds of electron correlation. In this work we assess the ability of the parametric two-electron reduced-density matrix (p2-RDM) method to describe biradical chemistry through application to the benzene and cyclobutadiene biradicals. The relative energy of o- and m-benzynes predicted by the p2-RDM method is consistent with Wenthold et al.'s experimental determinations, while the more difficult relative energy prediction of the more multi-referenced p-benzyne is within 1.4 kcal mol−1 of the experimental value [P. G. Wenthold et al., J. Am. Chem. Soc., 1998, 120, 5279], which is significantly better than traditional single-reference methods. We observe that the degree of multireference correlation in the biradicals depends upon the distance between their radical centers, with the largest radical separation displaying the largest degree of multireference correlation. In addition to relative and absolute electronic energies, we report molecular geometries, natural orbitals, and natural-orbital occupations for the benzene and cyclobutadiene biradicals.Back to top
2014
D. A. Mazziotti. arXiv (2014). "Parallel Large-scale Semidefinite Programming for Strong Electron Correlation: Using Correlation and Entanglement in the Design of Efficient Energy-Transfer Mechanisms"
Challenges addressed under the grant include: (i) improving our understanding of the many-electron quantum mechanisms by which nature uses strong electron correlation for efficient energy transfer, particularly in photosynthesis and bioluminescence, (ii) providing an innovative paradigm for energy transfer in photovoltaic materials by which new levels of solar efficiency are achieved through the use of strong electron correlation and entanglement, (iii) enhancing two-electron reduced-density-matrix (2-RDM)-based electronic-structure methods that significantly expand the range of strongly correlated molecular systems that can be studied with applications throughout science and engineering, and (iv) developing a new generation of large-scale, parallel algorithms for performing semidefinite programming with applications to problems in engineering, computer science, statistics, finance and economics. Research led to important technology transitions including the formation of RDMCHEM LLC, a software company that is developing the next generation of computational software for chemistry with applications to engineering, molecular biology, and physics.D. A. Mazziotti, N. Skochdopole. Adv. Chem. Phys. 355-370 (2014). "Quantum Information and Computation for Chemistry"
This chapter discusses two key features of the photosynthetic light harvesting: (1) the effect of strong correlation of the electrons within the chromophores on the transfer efficiency of energy to the reaction center and (2) the role of functional subsystems‐subsets of chromophores with the ability to transfer energy efficiently to the reaction center without the other chromophores‐in providing a built‐in quantum redundancy to keep photosynthesis “humming” along even with more than half of the chromophores temporarily or permanently shut down. The chapter examines the efficiency of light harvesting where each chromophore is represented by a correlated N‐electron model to treat strong electron correlation. The chapter shows that photosynthetic light harvesting exhibits quantum redundancy. The computations presented here reveal that the functional subsystems achieve their efficiencies by a quantum mechanism similar to that of the full system including the roles of entanglement and environmental noise.R. Chakraborty, D. A. Mazziotti. Phys. Rev. A 89 042505 (2014). "Generalized Pauli conditions on the spectra of one-electron reduced density matrices of atoms and molecules"
The Pauli exclusion principle requires the spectrum of the occupation numbers of the one-electron reduced density matrix (1-RDM) to be bounded by one and zero. However, for a 1-RDM from a wave function, there exist additional conditions on the spectrum of occupation numbers, known as pure N-representability conditions or generalized Pauli conditions. For atoms and molecules, we measure through a Euclidean-distance metric the proximity of the 1-RDM spectrum to the facets of the convex set (polytope) generated by the generalized Pauli conditions. For the ground state of any spin symmetry, as long as time-reversal symmetry is considered in the definition of the polytope, we find that the 1-RDM's spectrum is pinned to the boundary of the polytope. In contrast, for excited states, we find that the 1-RDM spectrum is not pinned. Proximity of the 1-RDM to the boundary of the polytope provides a measurement and classification of electron correlation and entanglement within the quantum system. For comparison, this distance to the boundary of the generalized Pauli conditions is also compared to the distance to the polytope of the traditional Pauli conditions, and the distance to the nearest 1-RDM spectrum from a Slater determinant. We explain the difference in pinning in the ground- and excited-state 1-RDMs through a connection to the N-representability conditions of the two-electron reduced density matrix.S. Veeraraghavan, D. A. Mazziotti. J. Chem. Phys. 140 124106 (2014). "Global solutions of restricted open-shell Hartree-Fock theory from semidefinite programming with applications to strongly correlated quantum systems"
We present a density matrix approach for computing global solutions of restricted open-shell Hartree-Fock theory, based on semidefinite programming (SDP), that gives upper and lower bounds on the Hartree-Fock energy of quantum systems. While wave function approaches to Hartree-Fock theory yield an upper bound to the Hartree-Fock energy, we derive a semidefinite relaxation of Hartree-Fock theory that yields a rigorous lower bound on the Hartree-Fock energy. We also develop an upper-bound algorithm in which Hartree-Fock theory is cast as a SDP with a nonconvex constraint on the rank of the matrix variable. Equality of the upper- and lower-bound energies guarantees that the computed solution is the globally optimal solution of Hartree-Fock theory. The work extends a previously presented method for closed-shell systems [S. Veeraraghavan and D. A. Mazziotti, Phys. Rev. A 89, 010502–R (2014)]. For strongly correlated systems the SDP approach provides an alternative to the locally optimized Hartree-Fock energies and densities with a certificate of global optimality. Applications are made to the potential energy curves of C2, CN, Cr 2, and NO 2.S. Veeraraghavan, D. A. Mazziotti. Phys. Rev. A 89 010502 (2014). "Global solutions of Hartree-Fock theory and their consequences for strongly correlated quantum systems"
We present a density matrix approach for computing global solutions of Hartree-Fock theory, based on semidefinite programming (SDP), that gives upper and lower bounds on the Hartree-Fock energy of quantum systems. Equality of the upper- and lower-bound energies guarantees that the computed solution is the globally optimal solution of Hartree-Fock theory. For strongly correlated systems the SDP approach provides an alternative to the locally optimized Hartree-Fock energies and densities from the standard solution of the Euler-Lagrange equations. Applications are made to the potential energy curves of the H4 dimer and the N2 molecule.G. Gidofalvi, D. A. Mazziotti. J. Phys. Chem. A 118 495-502 (2014). "Molecule-Optimized Basis Sets and Hamiltonians for Accelerated Electronic Structure Calculations of Atoms and Molecules"
Molecule-optimized basis sets, based on approximate natural orbitals, are developed for accelerating the convergence of quantum calculations with strongly correlated (multireferenced) electrons. We use a low-cost approximate solution of the anti-Hermitian contracted Schrödinger equation (ACSE) for the one- and two-electron reduced density matrices (RDMs) to generate an approximate set of natural orbitals for strongly correlated quantum systems. The natural-orbital basis set is truncated to generate a molecule-optimized basis set whose rank matches that of a standard correlation-consistent basis set optimized for the atoms. We show that basis-set truncation by approximate natural orbitals can be viewed as a one-electron unitary transformation of the Hamiltonian operator and suggest an extension of approximate natural-orbital truncations through two-electron unitary transformations of the Hamiltonian operator, such as those employed in the solution of the ACSE. The molecule-optimized basis set from the ACSE improves the accuracy of the equivalent standard atom-optimized basis set at little additional computational cost. We illustrate the method with the potential energy curves of hydrogen fluoride and diatomic nitrogen. Relative to the hydrogen fluoride potential energy curve from the ACSE in a polarized triple-ζ basis set, the ACSE curve in a molecule-optimized basis set, equivalent in size to a polarized double-ζ basis, has a nonparallelity error of 0.0154 au, which is significantly better than the nonparallelity error of 0.0252 au from the polarized double-ζ basis set.N. C. Rubin, D. A. Mazziotti. Highlights Theor. Chem. 167-175 (2014). "Isaiah Shavitt, A Memorial Festschrift from Theoretical Chemistry Accounts"
Minimizing the energy of an N-electron system as a functional of a two-electron reduced density matrix (2-RDM), constrained by necessary N-representability conditions (conditions for the 2-RDM to represent an ensemble N-electron quantum system), yields a rigorous lower bound to the ground-state energy in contrast to variational wave function methods. We characterize the performance of two sets of approximate constraints, (2,2)-positivity (DQG) and approximate (2,3)-positivity (DQGT) conditions, at capturing correlation in one-dimensional and quasi-onedimensional (ladder) Hubbard models. We find that, while both the DQG and DQGT conditions capture both the weak and strong correlation limits, the more stringent DQGT conditions improve the ground-state energies, the natural occupation numbers, the pair correlation function, the effective hopping, and the connected (cumulant) part of the 2-RDM. We observe that the DQGT conditions are effective at capturing strong electron correlation effects in both one- and quasi-one-dimensional lattices for both half filling and less-than-half filling.L. W. Bertels, D. A. Mazziotti. J. Chem. Phys. 141 044305 (2014). "Accurate prediction of diradical chemistry from a single-reference density-matrix method: Model application to the bicyclobutane to gauche-1,3-butadiene isomerization"
Multireference correlation in diradical molecules can be captured by a single-reference 2-electron reduced-density-matrix (2-RDM) calculation with only single and double excitations in the 2-RDM parametrization. The 2-RDM parametrization is determined by N-representability conditions that are non-perturbative in their treatment of the electron correlation. Conventional single-reference wave function methods cannot describe the entanglement within diradical molecules without employing triple- and potentially even higher-order excitations of the mean-field determinant. In the isomerization of bicyclobutane to gauche-1,3-butadiene the parametric 2-RDM (p2-RDM) method predicts that the diradical disrotatory transition state is 58.9 kcal/mol above bicyclobutane. This barrier is in agreement with previous multireference calculations as well as recent Monte Carlo and higher-order coupled cluster calculations. The p2-RDM method predicts the Nth natural-orbital occupation number of the transition state to be 0.635, revealing its diradical character. The optimized geometry from the p2-RDM method differs in important details from the complete-active-space self-consistent-field geometry used in many previous studies including the Monte Carlo calculation.N. C. Rubin, D. A. Mazziotti. Theor. Chem. Acc. 133 1492 (2014). "Comparison of one-dimensional and quasi-one-dimensional Hubbard models from the variational two-electron reduced-density-matrix method"
Minimizing the energy of an N-electron system as a functional of a two-electron reduced density matrix (2-RDM), constrained by necessary N-representability conditions (conditions for the 2-RDM to represent an ensemble N-electron quantum system), yields a rigorous lower bound to the ground-state energy in contrast to variational wave function methods. We characterize the performance of two sets of approximate constraints, (2,2)-positivity (DQG) and approximate (2,3)-positivity (DQGT) conditions, at capturing correlation in one-dimensional and quasi-one-dimensional (ladder) Hubbard models. We find that, while both the DQG and DQGT conditions capture both the weak and strong correlation limits, the more stringent DQGT conditions improve the ground-state energies, the natural occupation numbers, the pair correlation function, the effective hopping, and the connected (cumulant) part of the 2-RDM. We observe that the DQGT conditions are effective at capturing strong electron correlation effects in both one- and quasi-one-dimensional lattices for both half filling and less-than-half filling.C. C. Forgy, D. A. Mazziotti. J. Chem. Phys. 141 224111 (2014). "Relations between environmental noise and electronic coupling for optimal exciton transfer in one- and two-dimensional homogeneous and inhomogeneous quantum systems"
Recent studies have indicated that environmental noise may increase energy-transfer efficiency in quantum systems. For homogeneous networks of chromophores previous studies have primarily considered excitonic transport in one-dimensional (linear) networks. In our study, we expand previous research to a two-dimensional fully coupled topology of chromophore molecules. We demonstrate that not only does an optimal dephasing rate exist in both one- and two-dimensional networks but also that it increases in magnitude with increasing coupling strength between chromophores. Optimal transport occurs when the noise quenches the entanglement between local modes that prevent the exciton from moving efficiently to the target site. We find that these results are insensitive to minor site defects such as those found in realistic systems. We contrast these findings to systems with a high degree of inhomogeneity, in which the optimal dephasing rate is largely set by the system topology and does not vary significantly with respect to coupling strength. Our findings have potential applications to systems such as quantum dot arrays and carbon nanotube structures.A. M. Sand, C. Liu, A. J. S. Valentine, D. A. Mazziotti. J. Phys. Chem. A 118 6085-6091 (2014). "Modulating the Electronic Structure of Chromophores by Chemical Substituents for Efficient Energy Transfer: Application to Fluorone"
Strong electron correlation within a quasi-spin model of chromophores was recently shown to enhance exciton energy transfer significantly. Here we investigate how the modulation of the electronic structure of the chromophores by chemical substitution can enhance energy-transfer efficiency. Unlike previous work that does not consider the direct effect of the electronic structure on exciton dynamics, we add chemical substituents to the fluorone dimer to study the effect of electron-donating and electron-withdrawing substituents on exciton energy transfer. The exciton dynamics are studied from the solution of a quantum Liouville equation for an open system whose model Hamiltonian is derived from excited-state electronic structure calculations. Both van der Waals energies and coupling energies, arising from the Hellmann–Feynman force generated upon transferring the dimers from infinity to a finite separation, are built into the model Hamiltonian. Though these two effects are implicitly treated in dipole-based models, their explicit and separate treatment as discussed here is critical to forging the correct connection with the electronic structure calculations. We find that the addition of electron-donating substituents to the fluorone system results in an increase in exciton-transfer rates by factors ranging from 1.3–1.9. The computed oscillator strength is consistent with the recent experimental results on a larger heterodimer system containing fluorone. The oscillator strength increases with the addition of electron-donating substituents. Our results indicate that the study of chromophore networks via electronic structure will help in the future design of efficient synthetic light-harvesting systems.Back to top
2013
A. M. Sand, D. A. Mazziotti. Comput. Theor. Chem. 1003 44-49 (2013). "Parametric two-electron reduced-density-matrix method with application to diradical rectangular H4"
Parameterization of the two-electron reduced density matrix (2-RDM) has made possible the determination of electronic energies with greater accuracy and reduced computational cost compared to traditional electron-pair theories, including coupled cluster with single and double excitations [D.A. Mazziotti, Phys. Rev. Lett. 101 (2008) 253002]. We apply the method to an H4 model system, a rectangular arrangement of two H2 monomers (P4), which is often used for benchmark calculations of multireference methods. At the square geometry, H4 becomes a diradical. We find that the parametric 2-RDM method obtains occupation numbers of 0.5471 and 0.4489 for the Nth and (N+1)th natural orbitals, respectively, which indicate diradical character. Energies and orbital occupation numbers obtained from the parametric 2-RDM method are found to be more accurate than single-reference wavefunction methods of comparable computational cost. We report energies and natural orbital occupation numbers for several geometries in the rectangular H4 system.A. M. Sand, D. A. Mazziotti. J. Chem. Phys. 138 244102 (2013). "Effect of molecular-orbital rotations on ground-state energies in the parametric two-electron reduced density matrix method"
Different sets of molecular orbitals and the rotations connecting them are of great significance in molecular electronic structure. Most electron correlation methods depend on a reference wave function that separates the orbitals into occupied and unoccupied spaces. Energies and properties from these methods depend upon rotations between the spaces. Some electronic structure methods, such as modified coupled electron pair approximations and the recently developed parametric two-electron reduced density matrix (2-RDM) methods [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)]10.1103/PhysRevLett.101.253002, also depend upon rotations between occupied orbitals and rotations between unoccupied orbitals. In this paper, we explore the sensitivity of the ground-state energies from the parametric 2-RDM method to rotations within the occupied space and within the unoccupied space. We discuss the theoretical origin of the rotational dependence and provide computational examples at both equilibrium and non-equilibrium geometries. We also study the effect of these rotations on the size extensivity of the parametric 2-RDM method. Computations show that the orbital rotations have a small effect upon the parametric 2-RDM energies in comparison to the energy differences observed between methodologies such as coupled cluster and parametric 2-RDM. Furthermore, while the 2-RDM method is rigorously size extensive in a local molecular orbital basis set, calculations reveal negligible deviations in nonlocal molecular orbital basis sets such as those from canonical Hartree-Fock calculations.J. T. Skolnik, D. A. Mazziotti. Phys. Rev. A 88 032517 (2013). "Cumulant reduced density matrices as measures of statistical dependence and entanglement between electronic quantum domains with application to photosynthetic light harvesting"
Recent ultrafast spectroscopy experiments have demonstrated the entanglement of chromophores in photosynthetic light harvesting. Here we apply the cumulant parts of reduced density matrices (RDMs) to measure the statistical dependence and entanglement between electronic quantum domains in light harvesting. Because the cumulant RDMs are invariant to one-electron unitary transformations, they provide a measure of electron correlation and entanglement that is independent of the orbital basis set. Specifically, we apply the cumulant to a three-chromophore subsystem of the Fenna-Matthews-Olson complex, which has been shown to exhibit similar entanglement as the full system. Time-dependent Frobenius norms of the cumulant p-RDMs for 2≤p≤6 reveal correlation and entanglement in groupings of p chromophores. The results show that the entanglement of pairs of chromophores is significantly more important than the entanglement in higher groupings of the chromophores. Data from the model are generally consistent with recent findings from ultrafast spectroscopy. Beyond their application to light harvesting, the Frobenius norms of the cumulant RDMs provide a useful measure of statistical dependence, correlation, and entanglement between electronic quantum domains with applications to molecules and materials.J. J. Foley, D. A. Mazziotti. J. Phys. Chem. A 117 6712-6716 (2013). "Cage versus Prism: Electronic Energies of the Water Hexamer"
Recent experiments show that the cage isomer of the water hexamer is lower in energy than the prism isomer near 0 K, and yet state-of-the-art electronic structure calculations predict the prism to be lower in energy than the cage at 0 K. Here, we study the relative energies of the water hexamers from the parametric two-electron reduced density matrix (2-RDM) method in which the 2-RDM rather than the wave function is the basic variable of the calculations. In agreement with experiment and in contrast with traditional wave function methods, the 2-RDM calculations predict the cage to be more stable than the prism after vibrational zero-point correction. Multiple configurations from the hydrogen bonding are captured by the method. More generally, the results are consistent with our previous 2-RDM applications in that they reveal how multireference correlation in molecular systems is important for resolving small energy differences from hydrogen bonding as well as other types of intermolecular forces, even in systems that are not conventionally considered strongly correlated.E. P. Hoy, N. Shenvi, D. A. Mazziotti. J. Chem. Phys. 139 034105 (2013). "Comparison of low-rank tensor expansions for the acceleration of quantum chemistry computations"
Low-rank spectral expansion and tensor hypercontraction are two promising techniques for reducing the size of the two-electron excitation tensor by factorizing it into products of smaller tensors. Both methods can potentially realize an O(r4) quantum chemistry method where r is the number of one-electron orbitals. We compare the two factorizations in this paper by applying them to the parametric 2-electron reduced density matrix method with the M functional [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)]10.1103/PhysRevLett.101.253002. We study several inorganic molecules, alkane chains, and potential curves as well as reaction and dissociation energies. The low-rank spectral expansion, we find, is typically more efficient than tensor hypercontraction due to a faster convergence of the energy and a smaller constant prefactor in the energy optimization. Both factorizations are applicable to the acceleration of a wide range of wavefunction and reduced-density-matrix methods.A. J. S. Valentine, D. A. Mazziotti. J. Phys. Chem. A 117 9746-9752 (2013). "Theoretical Prediction of the Structures and Energies of Olympicene and its Isomers"
Pentacene, a linear five-ringed polyaromatic hydrocarbon, has recently been used as an organic semiconductor in field-effect transistors. The recently synthesized olympicene molecule, so named because of its resemblance to the olympic rings, is a more compact five-ringed structure. This paper offers the first theoretical study of the kinetic stability of olympicene and its isomers. We use the parametric two-electron reduced density matrix (2-RDM) method, which takes the 2-RDM as the basic variable in lieu of the traditional wave function in calculations [ Mazziotti D. A. Phys. Rev. Lett. 2008, 101, 253002 ]. Our calculations demonstrate that olympicene’s isomers may be separated into aromatic and diradical isomers, the latter of which require accurate treatment of strong electron correlation to detect multireference character. Albeit formally a single-reference method, the parametric 2-RDM captures the multireference correlation of the diradical isomers; relative to olympicene, the 2-RDM predicts five diradical isomers that are 16–22 kcal/mol lower in energy than those from coupled cluster with single and double excitationsa significant change that causes these isomers to be stable to dissociation by 2–20 kcal/mol. We characterize the transition states between olympicene’s isomers, observe differences in aromaticity among the different isomers, and compare the electronic properties of olympicene to those of pentacene. The olympicene molecule has the potential to complement pentacene as an organic semiconductor.E. P. Hoy, C. A. Schwerdtfeger, D. A. Mazziotti. J. Phys. Chem. A 117 1817-1825 (2013). "Relative Energies and Geometries of the cis- and trans-HO3 Radicals from the Parametric 2‑Electron Density Matrix Method"
The parametric 2-electron reduced density matrix (2-RDM) method employing the M functional [ Mazziotti D. A. Phys. Rev. Lett. 2008, 101, 253002 ], also known as the 2-RDM(M) method, improves on the accuracy of coupled electron-pair theories including coupled cluster with single–double excitations at the computational cost of configuration interaction with single–double excitations. The cis- and trans-HO3 isomers along with their isomerization transition state were examined using the recent extension of 2-RDM(M) to nonsinglet open-shell states [ Schwerdtfeger C. A. ; Mazziotti J. Chem. Phys. 2012, 137, 034107 ] and several coupled cluster methods. We report the calculated energies, geometries, natural-orbital occupation numbers, and reaction barriers for the HO3 isomers. We find that the 2-RDM(M) method predicts that the trans isomer of HO3 is lower in energy than the cis isomer by 1.71 kcal/mol in the correlation-consistent polarized valence quadruple-ζ (cc-pVQZ) basis set and 1.84 kcal/mol in the augmented correlation-consistent polarized valence quadruple-ζ (aug-cc-pVQZ) basis set. Results include the harmonic zero-point vibrational energies calculated in the correlation-consistent polarized valence double-ζ basis set. On the basis of the results of a geometry optimization in the augmented correlation consistent polarized valence triple-ζ basis set, the parametric 2-RDM(M) method predicts a central oxygen–oxygen bond of 1.6187 Å. We compare these energies and geometries to those predicted by three single-reference coupled cluster methods and experimental results and find that the inclusion of multireference correlation is important to describe properly the relative energies of the cis- and trans-HO3 isomers and improve agreement with experimental geometries.Back to top
2012
D. A. Mazziotti. Chem. Rev. 112 244-262 (2012). "Two-Electron Reduced Density Matrix as the Basic Variable in Many-Electron Quantum Chemistry and Physics"
D. A. Mazziotti. J. Chem. Phys. 137 074117 (2012). "Effect of strong electron correlation on the efficiency of photosynthetic light harvesting"
Research into the efficiency of photosynthetic light harvesting has focused on two factors: (1) entanglement of chromophores, and (2) environmental noise. While chromophores are conjugated π-bonding molecules with strongly correlated electrons, previous models have treated this correlation implicitly without a mathematical variable to gauge correlation-enhanced efficiency. Here we generalize the single-electron/exciton models to a multi-electron/exciton model that explicitly shows the effects of enhanced electron correlation within chromophores on the efficiency of energy transfer. The model provides more detailed insight into the interplay of electron correlation within chromophores and electron entanglement between chromophores. Exploiting this interplay is assisting in the design of new energy-efficient materials, which are just beginning to emerge.D. A. Mazziotti. Phys. Rev. A 85 062507 (2012). "Significant conditions for the two-electron reduced density matrix from the constructive solution of N representability"
We recently presented a constructive solution to the N-representability problem of the two-electron reduced density matrix (2-RDM)—a systematic approach to constructing complete conditions to ensure that the 2-RDM represents a realistic N-electron quantum system [D. A. Mazziotti, Phys. Rev. Lett. (to be published)]. In this paper we provide additional details and derive further N-representability conditions on the 2-RDM that follow from the constructive solution. The resulting conditions can be classified into a hierarchy of constraints, known as the (2,q)-positivity conditions, where the q indicates their derivation from the non-negativity of q-body operators. In addition to the known T1 and T2 conditions, we derive another class of (2,3)-positivity conditions. We also derive 3 classes of (2,4)-positivity conditions, 6 classes of (2,5)-positivity conditions, and 24 classes of (2,6)-positivity conditions. The constraints obtained can be divided into two general types: (i) lifting conditions, that is, conditions which arise from lifting lower (2,q)-positivity conditions to higher (2,q+1)-positivity conditions, and (ii) pure conditions, that is, conditions which cannot be derived from a simple lifting of the lower conditions. All of the lifting conditions and the pure (2,q)-positivity conditions for q>3 require tensor decompositions of the coefficients in the model Hamiltonians. Subsets of the derived N-representability conditions can be employed with the previously known conditions to achieve polynomially scaling calculations of ground-state energies and 2-RDMs of many-electron quantum systems even in the presence of strong electron correlation.C. A. Schwerdtfeger, D. A. Mazziotti. J. Chem. Phys. 137 034107 (2012). "Treating molecules in arbitrary spin states using the parametric two-electron reduced-density-matrix method"
Minimizing the electronic energy with respect to a parameterized two-electron reduced density matrix (2-RDM) is known as a parametric variational 2-RDM method. The parametric 2-RDM method with the M 2-RDM parametrization [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)]10.1103/PhysRevLett.101.253002 is extended to treat molecules in arbitrary spin states. Like its singlet counterpart, the M parametric 2-RDM method for arbitrary spin states is derived using approximate N-representability conditions, which allow it to capture more correlation energy than coupled cluster with single and double excitations at a lower computational cost. We present energies, optimized bond lengths, potential energy curves, and occupation numbers for a set of molecules in a variety of spin states using the M and K parametric 2-RDM methods as well as several wavefunction methods. We show that the M parametric 2-RDM method can describe bond breaking of open-shell molecules like triplet B2 and singlet and triplet OH + even in the presence of strong correlation. Finally, the computed 2-RDMs are shown to be nearly N-representable at both equilibrium and non-equilibrium geometries.J. J. Foley, D. A. Mazziotti. Phys. Rev. A 86 012512 (2012). "Measurement-driven reconstruction of many-particle quantum processes by semidefinite programming with application to photosynthetic light harvesting"
Quantum measurements provide a trove of information about a quantum system or process without solution of the Schrödnger equation, and in principle, the associated density matrix is a function of these measurements. Inversion of the measurements can produce an estimate of the density matrix, but this estimate may be unphysical, especially when the measurements are noisy or incomplete. We develop a general approach based on semidefinite programming [D. A. Mazziotti, Phys. Rev. Lett. 106, 083001 (2011)PRLTAO0031-900710.1103/PhysRevLett.106.083001] for reconstructing the density matrix from quantum measurements which leads naturally to nonnegative solutions, a critical attribute of physically realistic solutions. We discuss the use of this methodology for reconstructing p-particle reduced density matrices (p-RDMs) of N-particle systems where additional semidefinite constraints, known as N-representability conditions, are essential because they ensure that the p-RDM represents an N-particle system. Special attention is given to the N-representability conditions for the experimentally important cases where p=1 or 2. We apply the methodology to reconstructing the time-dependent quantum process of exciton transfer in a photosynthetic light-harvesting complex.C. A. Schwerdtfeger, D. A. Mazziotti. J. Chem. Phys. 137 244103 (2012). "Low-rank spectral expansions of two electron excitations for the acceleration of quantum chemistry calculations"
Treatment of two-electron excitations is a fundamental but computationally expensive part of ab initio calculations of many-electron correlation. In this paper we develop a low-rank spectral expansion of two-electron excitations for accelerated electronic-structure calculations. The spectral expansion differs from previous approaches by relying upon both (i) a sum of three expansions to increase the rank reduction of the tensor and (ii) a factorization of the tensor into geminal (rank-two) tensors rather than orbital (rank-one) tensors. We combine three spectral expansions from the three distinct forms of the two-electron reduced density matrix (2-RDM), (i) the two-particle 2D, (ii) the two-hole 2Q, and the (iii) particle-hole 2G matrices, to produce a single spectral expansion with significantly accelerated convergence. While the resulting expansion is applicable to any quantum-chemistry calculation with two-particle excitation amplitudes, it is employed here in the parametric 2-RDM method [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)]10.1103/PhysRevLett.101.253002. The low-rank parametric 2-RDM method scales quartically with the basis-set size, but like its full-rank version it can capture multi-reference correlation effects that are difficult to treat efficiently by traditional single-reference wavefunction methods. Applications are made to computing potential energy curves of HF and triplet OH+, equilibrium bond distances and frequencies, the HCN-HNC isomerization, and the energies of hydrocarbon chains. Computed 2-RDMs nearly satisfy necessary N-representability conditions. The low-rank spectral expansion has the potential to expand the applicability of the parametric 2-RDM method as well as other ab initio methods to large-scale molecular systems that are often only treatable by mean-field or density functional theories.A. E. DePrince, D. A. Mazziotti. Mol. Phys. 110 1917-1925 (2012). "Connection of an elementary class of parametric two-electron reduced-density-matrix methods to the coupled electron-pair approximations"
Parametric two-electron reduced-density-matrix (p-2RDM) methods have recently been shown to exhibit accuracies greater than coupled cluster with single and double substitutions (CCSD) at a lower computational cost. In this paper we derive an elementary class of parametric 2-RDM methods with connections to the coupled electron pair approximations (CEPA). Three parametric 2-RDM methods p-2RDM/n are presented that correspond to the CEPA/n family where n = 1, 2, 3. We isolate the function distinguishing the stationary condition of the parametric 2-RDM methods from the nonlinear equations of CEPA. Calculations of energies, geometries, and harmonic frequencies show that p-2RDM/n and CEPA/n are very similar for a variety of closed-shell systems. Finally, each of the p-2RDM/n methods is extended to satisfy particle–hole symmetry by the removal of exclusion principle violating terms in the virtual space. These extensions denoted p-2RDM′/n are shown to be essential for the proper dissociation of CH radical. Both p-2RDM/n and p-2RDM′/n form an elementary class of parametric 2-RDM methods with accuracy like CCSD; more general parametric 2-RDM methods [D.A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)] have an accuracy approaching CCSD with full triple excitations.S. Pabst, L. Greenman, D. A. Mazziotti, R. Santra. Phys. Rev. A 85 023411 (2012). "Impact of multichannel and multipole effects on the Cooper minimum in the high-order-harmonic spectrum of argon"
We investigate the relevance of multiple-orbital and multipole effects during high-harmonic generation (HHG). The time-dependent configuration interaction singles (TDCIS) approach is used to study the impact of the detailed description of the residual electron-ion interaction on the HHG spectrum. We find that the shape and position of the Cooper minimum in the HHG spectrum of argon changes significantly whether or not interchannel interactions are taken into account. The HHG yield can be underestimated by up to 2 orders of magnitude in the energy region of 30–50 eV. We show that the argument of low ionization probability is not sufficient to justify ignoring multiple-orbital contributions. Additionally, we find the HHG yield is sensitive to the nonspherical multipole character of the electron-ion interaction.A. M. Sand, C. A. Schwerdtfeger, D. A. Mazziotti. J. Chem. Phys. 136 034112 (2012). "Strongly correlated barriers to rotation from parametric two-electron reduced-density-matrix methods in application to the isomerization of diazene"
Recently, parameterization of the two-electron reduced density matrix (2-RDM) has made possible the determination of electronic energies with greater accuracy and lower cost than traditional electron-pair theories including coupled cluster with single and double excitations [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)]. We examine the method's performance for strongly correlated barriers to rotation; in particular, we study two distinct pathways in the isomerization of diazene (N2H2) from cis to trans: (i) a strongly correlated rotational pathway and (ii) a moderately correlated inversion pathway. While single reference wavefunction methods predict that the rotational barrier is higher than the inversional barrier, the parametric 2-RDM method predicts that the rotational barrier is lower than the inversional barrier by 3.1 kcal/mol in the extrapolated basis set limit. The parametric 2-RDM results are in agreement with those from multireference methods including multireference perturbation theory and the solution to the anti-Hermitian contracted Schrödinger equation. We report energies, optimized structures, and natural orbital occupation numbers for three diazene minima and two transition states.D. Roca‐Sanjuán, M. Lundberg, D. A. Mazziotti, R. Lindh. J. Comput. Chem. 33 2124-2126 (2012). "Comment on “Density functional theory study of 1,2‐dioxetanone decomposition in condensed phase”"
In the preceding paper results are presented, which are in serious conflict with state‐of‐the‐art ab initio method. Based on these new results the authors propose a new explanation of the reason for the preferential production of a phosphorescent state. Here we show that these controversial results are flawed, since the model use exclude biradical electron structures. © 2012 Wiley Periodicals, Inc. This comment demonstrates that the recently reported DFT study (see the previous article) of the thermal dissociation of 1,2‐dioxetanone is flawed since open‐shell solutions to the DFT equations where not included in the simulations. This renders the suggested reaction mechanism proposed in the previous paper to be an unsupported hypothesis.E. P. Hoy, C. A. Schwerdtfeger, D. A. Mazziotti. Mol. Phys. 110 765-773 (2012). "Isoelectronic analogue of oxywater: a parametric two-electron reduced-density-matrix study of ammonia oxide"
A parametrization of the two-electron reduced density matrix (2-RDM) provides energies that improve on the accuracy of coupled electron-pair theories including coupled cluster with single-double excitations at the computational cost of configuration interaction with single-double excitations [Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)]. This parametric 2-RDM method was recently employed to study the isomerization of oxywater to hydrogen peroxide where it predicted a lower energy barrier from oxywater (2.1 kcal mol−1) than coupled cluster methods (4.2 kcal mol−1). In this paper we study an isoelectronic analogue, the isomerization of ammonia oxide to hydroxylamine. In the extrapolated basis-set limit, using the augmented correlation-consistent polarized valance quadruple-zeta (aug-cc-pVQZ) basis set, the parametric 2-RDM method predicts a 27.5 kcal mol−1 barrier from ammonia oxide to hydroxylamine. We report reaction energies, barriers, geometries, and natural-orbital occupation numbers for the ammonia-oxide reaction and compare them to those from the oxywater reaction. We find that the parametric 2-RDM method agrees with dynamic correlation wavefunction methods when the multi-reference character of the system is small as in the ammonia-oxide isomerization computed here but that it captures additional multi-reference correlation, usually requiring a multi-reference method, when such correlation increases as in the oxywater isomerization.B. Friedrich, S. Kais, D. Mazziotti. Mol. Phys. 110 1537-1537 (2012). "Scaling Mount Impossible: A Festschrift for Dudley Herschbach"
D. A. Mazziotti. Phys. Rev. Lett. 108 263002 (2012). "Structure of Fermionic Density Matrices: Complete N-Representability Conditions"
We present a constructive solution to the N-representability problem: a full characterization of the conditions for constraining the two-electron reduced density matrix to represent an N-electron density matrix. Previously known conditions, while rigorous, were incomplete. Here, we derive a hierarchy of constraints built upon (i) the bipolar theorem and (ii) tensor decompositions of model Hamiltonians. Existing conditions D, Q, G, T1, and T2, known classical conditions, and new conditions appear naturally. Subsets of the conditions are amenable to polynomial-time computations of strongly correlated systems.Back to top
2011
L. Greenman, D. A. Mazziotti. J. Chem. Phys. 134 174110 (2011). "Balancing single- and multi-reference correlation in the chemiluminescent reaction of dioxetanone using the anti-Hermitian contracted Schrödinger equation"
Direct computation of energies and two-electron reduced density matrices (2-RDMs) from the anti-Hermitian contracted Schrödinger equation (ACSE) [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)], it is shown, recovers both single- and multi-reference electron correlation in the chemiluminescent reaction of dioxetanone especially in the vicinity of the conical intersection where strong correlation is important. Dioxetanone, the light-producing moiety of firefly luciferin, efficiently converts chemical energy into light by accessing its excited-state surface via a conical intersection. Our previous active-space 2-RDM study of dioxetanone [L. Greenman and D. A. Mazziotti, J. Chem. Phys. 133, 164110 (2010)] concluded that correlating 16 electrons in 13 (active) orbitals is required for realistic surfaces without correlating the remaining (inactive) orbitals. In this paper we pursue two complementary goals: (i) to correlate the inactive orbitals in 2-RDMs along dioxetanone's reaction coordinate and compare these results with those from multireference second-order perturbation theory (MRPT2) and (ii) to assess the size of the active space—the number of correlated electrons and orbitals—required by both MRPT2 and ACSE for accurate energies and surfaces. While MRPT2 recovers very different amounts of correlation with (4,4) and (16,13) active spaces, the ACSE obtains a similar amount of correlation energy with either active space. Nevertheless, subtle differences in excitation energies near the conical intersection suggest that the (16,13) active space is necessary to determine both energetic details and properties. Strong electron correlation is further assessed through several RDM-based metrics including (i) total and relative energies, (ii) the von Neumann entropy based on the 1-electron RDM, as well as the (iii) infinity and (iv) squared Frobenius norms based on the cumulant 2-RDM.N. Skochdopole, D. A. Mazziotti. J. Phys. Chem. Lett. 2 2989-2993 (2011). "Functional Subsystems and Quantum Redundancy in Photosynthetic Light Harvesting"
The Fenna–Matthews–Olson (FMO) antennae complex, responsible for light harvesting in green sulfur bacteria, consists of three monomers, each with seven chromophores. Here we show that multiple subsystems of the seven chromophores can transfer energy from either chromophore 1 or 6 to the reaction center with an efficiency matching or in many cases exceeding that of the full seven chromophore system. In the FMO complex, these functional subsystems support multiple quantum pathways for efficient energy transfer that provide a built-in quantum redundancy. There are many instances of redundancy in nature, providing reliability and protection, and in photosynthetic light harvesting this quantum redundancy provides protection against the temporary or permanent loss of one or more chromophores. The complete characterization of functional subsystems within the FMO complex offers a detailed map of the energy flow within the FMO complex, which has potential applications to the design of more efficient photovoltaic devices.D. A. Mazziotti. Phys. Rev. Lett. 106 083001 (2011). "Large-Scale Semidefinite Programming for Many-Electron Quantum Mechanics"
The energy of a many-electron quantum system can be approximated by a constrained optimization of the two-electron reduced density matrix (2-RDM) that is solvable in polynomial time by semidefinite programming (SDP). Here we develop a SDP method for computing strongly correlated 2-RDMs that is 10–20 times faster than previous methods [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)PRLTAO0031-900710.1103/PhysRevLett.93.213001]. We illustrate with (i) the dissociation of N2 and (ii) the metal-to-insulator transition of H50. For H50 the SDP problem has 9.4×106 variables. This advance also expands the feasibility of large-scale applications in quantum information, control, statistics, and economics.P. Popelier, D. A. Mazziotti. arXiv 61-90 (2011). "Solving the Schrödinger Equation"
C. A. Schwerdtfeger, D. A. Mazziotti. J. Phys. Chem. A 115 12011-12016 (2011). "Populations of Carbonic Acid Isomers at 210 K from a Fast Two-Electron Reduced-Density Matrix Theory"
Parametrization of the 2-electron reduced density matrix (2-RDM) rather than the many-electron wave function yields a new family of electronic-structure methods that are faster and more accurate than traditional coupled electron-pair methods including coupled cluster with single and double excitations. Deriving the parametrization from N-representability conditions generates a 2-RDM that captures significant correlation from triple and higher-order excitations at the cost of double excitations. We apply the parametric 2-RDM method to confirm recent experiments determining the relative thermodynamic populations of the cis–cis and cis–trans isomers of carbonic acid. In 2010 Bernard et al. showed by infrared spectroscopy that the populations of cis–cis and cis–trans isomers have a 10:1 ratio at 210 K. By use of the parametric 2-RDM method, we predict a 8:1 ratio at 210 K. Comparable ab initio methods overestimate the stability of the cis–cis isomer with 24:1 and 21:1 ratios. These 2-RDM-based methods promise to have significant applications throughout chemistry.J. W. S. Jr, D. A. Mazziotti. Phys. Chem. Chem. Phys. 14 1660-1667 (2011). "Photoexcited tautomerization of vinyl alcohol to acetylaldehyde via a conical intersection from contracted Schrödinger theory"
The photoexcited tautomerization of vinyl alcohol to acetylaldehydevia a conical intersection is analyzed through the direct calculation of two-electron reduced density matrices (2-RDMs) from solutions to the anti-Hermitian contracted Schrödinger equation (ACSE). The study utilizes the recent generalization of the ACSE method for the treatment of excited states [G. Gidofalvi and D. A. Mazziotti, Phys. Rev. A, 2009, 80, 022507]. We computed absolute energies of the critical points as well as various intermediate points along the ground- and excited-state potential energy surface of vinyl alcohol and acetylaldehyde. The ACSE, MCSCF, second-order multireference many-body perturbation theory (MRMP2), and various coupled cluster methods all demonstrate the existence of a family of pathways from vinyl alcohol to acetylaldehydevia a conical intersection that are monotonically decreasing in energy. The conical intersection, proposed for the first time in this paper, is both structurally and energetically similar to the ground-state transition state. We observe a relationship between conical intersections and transition states both in this paper and in our previous work on bicyclobutane's ring conical intersection [J. W. Snyder, Jr. and D. A. Mazziotti, J. Chem. Phys., 2011, 135, 024107]. To treat multireference correlation, we seeded the ACSE with an initial 2-RDM guess from a multiconfiguration self-consistent field (MCSCF) calculation. The ACSE recovers more single-reference correlation energy than MRMP2 and more multireference correlation energy than comparable single-reference wave function methods. The 2-RDMs from the ACSE nearly satisfy all necessary N-representability conditions.J. W. Snyder, D. A. Mazziotti. J. Phys. Chem. A 115 14120-14126 (2011). "Conical Intersection of the Ground and First Excited States of Water: Energies and Reduced Density Matrices from the Anti-Hermitian Contracted Schrödinger Equation"
A conical intersection between the ground and first-excited states of water is computed through the direct calculation of two-electron reduced density matrices (2-RDMs) from solutions of the anti-Hermitian contracted Schrödinger equation (ACSE). This study is an extension of a previous study in which the ACSE was used to compute the energies around a conical intersection in the triplet excited states of methylene [Snyder, J. W., Jr.; Rothman, A. E.; Foley, J. J.; Mazziotti, D. A. J. Chem. Phys. 2010, 132, 154109]. We compute absolute energies of the 11 A′ and 21 A′ states of water (H2O) and the location of the conical intersection. The ACSE energies are compared to those from ab initio wave function methods. To treat multireference correlation, we seed the ACSE with an initial 2-RDM from a multiconfiguration self-consistent field (MCSCF) calculation. Unlike the situation for methylene, the two states in the vicinity of the conical intersection of water both have the same spatial symmetry. Hence, the study demonstrates the ability of the ACSE to resolve states of the same spatial symmetry that are nearly degenerate in energy. The 2-RDMs from the ACSE nearly satisfy necessary N-representability conditions. Comparison of the results from double-ζ and augmented double-ζ basis sets demonstrates the importance of augmented (or diffuse) functions for determining the location of the conical intersection.J. W. Snyder, D. A. Mazziotti. J. Chem. Phys. 135 024107 (2011). "Photoexcited conversion of gauche-1,3-butadiene to bicyclobutane via a conical intersection: Energies and reduced density matrices from the anti-Hermitian contracted Schrödinger equation"
The photoexcited reaction pathway from gauche-1,3-butadiene to bicyclobutane via a conical intersection is analyzed through the direct calculation of two-electron reduced density matrices (2-RDMs) from solutions to the anti-Hermitian contracted Schrödinger equation (ACSE). The study utilizes the recent generalization of the ACSE method for the treatment of excited states [G. Gidofalvi and D. A. Mazziotti, Phys. Rev. A 80, 022507 (2009)10.1103/PhysRevA.80.022507]. We computed absolute energies of the critical points as well as various intermediate points along the ground-and excited-state potential energy surface of gauche-1,3-butadiene and bicyclobutane. To treat multi-reference correlation, we seeded the ACSE with an initial 2-RDM from a multi-configuration self-consistent field (MCSCF) calculation. The ACSE, MCSCF, and second-order multi-reference many-body perturbation theory (MRPT2) all demonstrate that there exists a family of pathways from gauche-1,3-butadiene to bicyclobutane via a conical intersection that are monotonically decreasing in energy, confirming a conjecture by Sicilia et al. [J. Phys. Chem. A 111, 2182 (2007)10.1021/jp067614w]. The ACSE recovers more single-reference correlation energy than MRPT2 and more multi-reference correlation energy than comparable single-reference wave function methods. The 2-RDMs from the ACSE nearly satisfy all necessary N-representability conditions.J. J. Foley, A. E. Rothman, D. A. Mazziotti. J. Chem. Phys. 134 034111 (2011). "Strongly correlated mechanisms of a photoexcited radical reaction from the anti-Hermitian contracted Schrödinger equation"
Photoexcited radical reactions are critical to processes in both nature and materials, and yet they can be challenging for electronic structure methods due to the presence of strong electron correlation. Reduced-density-matrix (RDM) methods, based on solving the anti-Hermitian contracted Schrödinger equation (ACSE) for the two-electron RDM (2-RDM), are examined for studying the strongly correlated mechanisms of these reactions with application to the electrocyclic interconversion of allyl and cyclopropyl radicals. We combine recent extensions of the ACSE to excited states [G. Gidofalvi and D. A. Mazziotti, Phys. Rev. A 80, 022507 (2009)] and arbitrary spin states [A. E. Rothman, J. J. Foley IV, and D. A. Mazziotti, Phys. Rev. A 80, 052508 (2009)]. The ACSE predicts that the ground-state ring closure of the allyl radical has a high 52.5 kcal/mol activation energy that is consistent with experimental data, while the closure of an excited allyl radical can occur by disrotatory and conrotatory pathways whose transition states are essentially barrierless. Comparisons are made with multireference second- and third-order perturbation theories and multireference configuration interaction. While predicted energy differences do not vary greatly between methods, the ACSE appears to improve these differences when they involve a strongly and a weakly correlated radical by capturing a greater share of single-reference correlation that increases the stability of the weakly correlated radicals. For example, the ACSE predicts a −39.6 kcal/mol conversion of the excited allyl radical to the ground-state cyclopropyl radical in comparison to the −32.6 to −37.3 kcal/mol conversions predicted by multireference methods. In addition, the ACSE reduces the computational scaling with the number of strongly correlated orbitals from exponential (traditional multireference methods) to quadratic. Computed ground- and excited-state 2-RDMs are nearly N-representable.C. A. Schwerdtfeger, A. E. DePrince, D. A. Mazziotti. J. Chem. Phys. 134 174102 (2011). "Testing the parametric two-electron reduced-density-matrix method with improved functionals: Application to the conversion of hydrogen peroxide to oxywater"
Parametrization of the two-electron reduced density matrix (2-RDM) has recently enabled the direct calculation of electronic energies and 2-RDMs at the computational cost of configuration interaction with single and double excitations. While the original Kollmar energy functional yields energies slightly better than those from coupled cluster with single-double excitations, a general family of energy functionals has recently been developed whose energies approach those from coupled cluster with triple excitations [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)]. In this paper we test the parametric 2-RDM method with one of these improved functionals through its application to the conversion of hydrogen peroxide to oxywater. Previous work has predicted the barrier from oxywater to hydrogen peroxide with zero-point energy correction to be 3.3-to-3.9 kcal/mol from coupled cluster with perturbative triple excitations [CCSD(T)] and -2.3 kcal/mol from complete active-space second-order perturbation theory (CASPT2) in augmented polarized triple-zeta basis sets. Using a larger basis set than previously employed for this reaction—an augmented polarized quadruple-zeta basis set (aug-cc-pVQZ)—with extrapolation to the complete basis-set limit, we examined the barrier with two parametric 2-RDM methods and three coupled cluster methods. In the basis-set limit the M parametric 2-RDM method predicts an activation energy of 2.1 kcal/mol while the CCSD(T) barrier becomes 4.2 kcal/mol. The dissociation energy of hydrogen peroxide to hydroxyl radicals is also compared to the activation energy for oxywater formation. We report energies, optimal geometries, dipole moments, and natural occupation numbers. Computed 2-RDMs nearly satisfy necessary N-representability conditions.K. Naftchi-Ardebili, N. W. Hau, D. A. Mazziotti. Phys. Rev. A 84 052506 (2011). "Rank restriction for the variational calculation of two-electron reduced density matrices of many-electron atoms and molecules"
Variational minimization of the ground-state energy as a function of the two-electron reduced density matrix (2-RDM), constrained by necessary N-representability conditions, provides a polynomial-scaling approach to studying strongly correlated molecules without computing the many-electron wave function. Here we introduce a route to enhancing necessary conditions for N representability through rank restriction of the 2-RDM. Rather than adding computationally more expensive N-representability conditions, we directly enhance the accuracy of two-particle (2-positivity) conditions through rank restriction, which removes degrees of freedom in the 2-RDM that are not sufficiently constrained. We select the rank of the particle-hole 2-RDM by deriving the ranks associated with model wave functions, including both mean-field and antisymmetrized geminal power (AGP) wave functions. Because the 2-positivity conditions are exact for quantum systems with AGP ground states, the rank of the particle-hole 2-RDM from the AGP ansatz provides a minimum for its value in variational 2-RDM calculations of general quantum systems. To implement the rank-restricted conditions, we extend a first-order algorithm for large-scale semidefinite programming. The rank-restricted conditions significantly improve the accuracy of the energies; for example, the percentages of correlation energies recovered for HF, CO, and N2 improve from 115.2%, 121.7%, and 121.5% without rank restriction to 97.8%, 101.1%, and 100.0% with rank restriction. Similar results are found at both equilibrium and nonequilibrium geometries. While more accurate, the rank-restricted N-representability conditions are less expensive computationally than the full-rank conditions.K. Pelzer, L. Greenman, G. Gidofalvi, D. A. Mazziotti. J. Phys. Chem. A 115 5632-5640 (2011). "Strong Correlation in Acene Sheets from the Active-Space Variational Two-Electron Reduced Density Matrix Method: Effects of Symmetry and Size"
Polyaromatic hydrocarbons (PAHs) are a class of organic molecules with importance in several branches of science, including medicine, combustion chemistry, and materials science. The delocalized π-orbital systems in PAHs require highly accurate electronic structure methods to capture strong electron correlation. Treating correlation in PAHs has been challenging because (i) traditional wave function methods for strong correlation have not been applicable since they scale exponentially in the number of strongly correlated orbitals, and (ii) alternative methods such as the density-matrix renormalization group and variational two-electron reduced density matrix (2-RDM) methods have not been applied beyond linear acene chains. In this paper we extend the earlier results from active-space variational 2-RDM theory [Gidofalvi, G.; Mazziotti, D. A. J. Chem. Phys. 2008, 129, 134108] to the more general two-dimensional arrangement of ringsacene sheetsto study the relationship between geometry and electron correlation in PAHs. The acene-sheet calculations, if performed with conventional wave function methods, would require wave function expansions with as many as 1.5 × 1017 configuration state functions. To measure electron correlation, we employ several RDM-based metrics: (i) natural-orbital occupation numbers, (ii) the 1-RDM von Neumann entropy, (iii) the correlation energy per carbon atom, and (iv) the squared Frobenius norm of the cumulant 2-RDM. The results confirm a trend of increasing polyradical character with increasing molecular size previously observed in linear PAHs and reveal a corresponding trend in two-dimensional (arch-shaped) PAHs. Furthermore, in PAHs of similar size they show significant variations in correlation with geometry. PAHs with the strictly linear geometry (chains) exhibit more electron correlation than PAHs with nonlinear geometries (sheets).S. Pabst, L. Greenman, P. J. Ho, D. A. Mazziotti, R. Santra. Phys. Rev. Lett. 106 053003 (2011). "Decoherence in Attosecond Photoionization"
The creation of superpositions of hole states via single-photon ionization using attosecond extreme-ultraviolet pulses is studied with the time-dependent configuration-interaction singles (TDCIS) method. Specifically, the degree of coherence between hole states in atomic xenon is investigated. We find that interchannel coupling not only affects the hole populations, but it also enhances the entanglement between the photoelectron and the remaining ion, thereby reducing the coherence within the ion. As a consequence, even if the spectral bandwidth of the ionizing pulse exceeds the energy splittings among the hole states involved, perfectly coherent hole wave packets cannot be formed. For sufficiently large spectral bandwidth, the coherence can only be increased by increasing the mean photon energy.Back to top
2010
D. A. Mazziotti. Phys. Rev. A 81 062515 (2010). "Parametrization of the two-electron reduced density matrix for its direct calculation without the many-electron wave function: Generalizations and applications"
An improved parametrization of the two-electron reduced density matrix (2-RDM) [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)PRLTAO0031-900710.1103/PhysRevLett.101.253002] was recently shown to yield energies and properties that are markedly better than those calculated by traditional ab initio methods of similar computational scaling. In this paper a family of such energy functionals, generalizing the ones obtained previously, is derived through the use of (i) p-particle contraction relations based on the contraction of the cumulant expansions of p-particle RDMs and (ii) Cauchy-Schwarz relations that arise from an important set of N-representability constraints known as the two-positivity conditions. The 2-RDMs are explicitly parameterized in terms of the first-order part of the cumulant 2-RDM and, for the inclusion of single excitations, a second-order part of the 1-RDM. In contrast to earlier formulations based on the coefficients from configuration interaction with single and double excitations (CISD), the cumulant-based parametric 2-RDM methods, from the properties of cumulants, are rigorously size extensive. We also show that writing the energy functionals in terms of correlated 1-RDMs and cumulant 2-RDMs reduces the computational cost of the parametric 2-RDM methods to that of CISD. Applications are made to ground-state energies of several molecules, equilibrium bond distances, and frequencies of HF, F2, and CO, the relative energy of the cis and trans isomers of HO3-, and the HCN-HNC isomerization reaction. For bond breaking in hydrogen fluoride the improved and more efficient parametric 2-RDM methods yield energies with similar accuracies at both equilibrium and nonequilibrium geometries in 6-31G** and polarized valence quadruple-ζ basis sets. Computed 2-RDMs very nearly satisfy well-known N-representability conditions.L. Greenman, D. A. Mazziotti. J. Phys. Chem. A 114 583-588 (2010). "Energy Barriers of Vinylidene Carbene Reactions from the Anti-Hermitian Contracted Schrödinger Equation"
Computational studies of carbenes must take into account the possibility of multireference correlation because the highest occupied and lowest unoccupied molecular orbitals can be nearly energetically degenerate. We apply the anti-Hermitian contracted Schrödinger equation (ACSE) [Mazziotti, D. A. Phys. Rev. Lett. 2006, 97, 143002] to compute two-electron reduced density matrices (2-RDMs) and their energies for two carbene reactions: (i) the acetylene-vinylidene rearrangement and (ii) the rearrangement of pent-1-en-4-yn-3-one to acryloylvinylidene, which then cyclizes to cyclopenta-2,4-dienone. The ACSE has some unique advantages in the treatment of carbene reactions and more general families of reactions in which the importance of multireference correlation is not known a priori: (i) the ACSE is more reliable than single-reference methods for confirming the presence or absence of multireference correlation and (ii) in the absence of multireference correlation, unlike multireference second-order perturbation theory (MRPT2), the ACSE recovers more single-reference correlation energy than similarly scaling coupled-cluster methods. Because MRPT2 does not recover as much single-reference correlation as the coupled-cluster or ACSE methods, it tends to underestimate reaction barriers within the carbene reactions. For example, in the rearrangement of pent-1-en-4-yn-3-one, the ACSE and CCSD(T) methods produce cyclization barriers of 18.9 and 18.7 kcal/mol with the 6-31G(d) basis set, whereas MRPT2 predicts this barrier to be 12.1 kcal/mol; furthermore, both the ACSE and CCSD(T) determine the energy of the transition state for acryloylvinylidene formation to be 6.6−6.7 kcal/mol above that of the carbene, and yet, MRPT2 does not predict a transition state.L. Greenman, D. A. Mazziotti. J. Chem. Phys. 133 164110 (2010). "Strong electron correlation in the decomposition reaction of dioxetanone with implications for firefly bioluminescence"
Dioxetanone, a key component of the bioluminescence of firefly luciferin, is itself a chemiluminescent molecule due to two conical intersections on its decomposition reaction surface. While recent calculations of firefly luciferin have employed four electrons in four active orbitals [(4,4)] for the dioxetanone moiety, a study of dioxetanone [F. Liu et al., J. Am. Chem. Soc. 131, 6181 (2009)] indicates that a much larger active space is required. Using a variational calculation of the two-electron reduced-density-matrix (2-RDM) [D. A. Mazziotti, Acc. Chem. Res. 39, 207 (2006)], we present the ground-state potential energy surface as a function of active spaces from (4,4) to (20,17) to determine the number of molecular orbitals required for a correct treatment of the strong electron correlation near the conical intersections. Because the 2-RDM method replaces exponentially scaling diagonalizations with polynomially scaling semidefinite optimizations, we readily computed large (18,15) and (20,17) active spaces that are inaccessible to traditional wave function methods. Convergence of the electron correlation with active-space size was measured with complementary RDM-based metrics, the von Neumann entropy of the one-electron RDM as well as the Frobenius and infinity norms of the cumulant 2-RDM. Results show that the electron correlation is not correctly described until the (14,12) active space with small variations present through the (20,17) space. Specifically, for active spaces smaller than (14,12), we demonstrate that at the first conical intersection, the electron in the σ∗ orbital of the oxygen-oxygen bond is substantially undercorrelated with the electron of the σ orbital and overcorrelated with the electron of the carbonyl oxygen's p orbital. Based on these results, we estimate that in contrast to previous treatments, an accurate calculation of the strong electron correlation in firefly luciferin requires an active space of 28 electrons in 25 orbitals, beyond the capacity of traditional multireference wave function methods.A. E. Rothman, D. A. Mazziotti. J. Chem. Phys. 132 104112 (2010). "Nonequilibrium, steady-state electron transport with N-representable density matrices from the anti-Hermitian contracted Schrödinger equation"
We study molecular conductivity for a one-electron, bath-molecule-bath model Hamiltonian. The primary quantum-mechanical variable is the one-electron reduced density matrix (1-RDM). By identifying similarities between the steady-state Liouville equation and the anti-Hermitian contracted Schrödinger equation (ACSE) [D. A. Mazziotti, Phys. Rev. A 75, 022505 (2007)], we develop a way of enforcing nonequilibrium, steady-state behavior in a time-independent theory. Our results illustrate the relationship between current and voltage in molecular junctions assuming that the total number of electrons under consideration can be fixed across all driving potentials. The impetus for this work is a recent study by Subotnik et al. that also uses the 1-RDM to study molecular conductivity under different assumptions regarding the total number of electrons [J. E. Subotnik et al., J. Chem. Phys. 130, 144105 (2009)]. Unlike calculations in the previous study, our calculations result in 1-RDMs that are fully N-representable. The present work maintains N-representability through a bath-bath mixing that is related to a time-independent relaxation of the baths in the absence of the molecule, as governed by the ACSE. A lack of N-representability can be important since it corresponds to occupying energy states in the molecule or baths with more than one electron or hole (the absence of an electron) in violation of the Pauli principle. For this reason the present work may serve as an important, albeit preliminary, step in designing a 2-RDM/ACSE method for studying steady-state molecular conductivity with an explicit treatment of electron correlation.A. E. DePrince, D. A. Mazziotti. J. Chem. Phys. 133 034112 (2010). "Isomerization of nitrosomethane to formaldoxime: Energies, geometries, and frequencies from the parametric variational two-electron reduced-density-matrix method"
The isomerization of nitrosomethane to trans-formaldoxime is treated with the parametric variational two-electron reduced-density-matrix (2-RDM) method. In the parametric 2-RDM method, the ground-state energy is minimized with respect to a 2-RDM that is parameterized to be both size extensive and nearly N-representable. The calculations were performed with an efficient version of the 2-RDM method that we developed as an extension of the PSI3 ab initio package. Details of the implementation, which scales like configuration interaction with single and double excitations, are provided as well as a comparison of two optimization algorithms for minimizing the energy functional. The conversion of nitrosomethane to trans-formaldoxime can occur by one of two pathways: (i) a 1,3-sigmatropic hydrogen shift or (ii) two successive 1,2-sigmatropic hydrogen shifts. The parametric 2-RDM method predicts that the reaction channel involving two sequential 1,2-shifts is about 10 kcal/mol more favorable than the channel with a single 1,3-shift, which is consistent with calculations from other ab initio methods. We computed geometric parameters and harmonic frequencies for each stationary point on the reaction surfaces. Transition-state energies, geometries, and frequencies from the 2-RDM method are often more accurate than those from traditional wave function methods of a similar computational cost. Although electronic-structure methods generally agree that the 1,2-shift is more efficient, the energy ordering of the reactant nitrosomethane and the 1,2-shift intermediate formaldonitrone is unresolved in the literature. With an extrapolation to the complete-basis-set limit the parametric 2-RDM method predicts formaldonitrone to be very slightly more stable than nitrosomethane.J. J. Foley, D. A. Mazziotti. Mol. Phys. 108 2543-2550 (2010). "Efficient geometry optimization by Hellmann–Feynman forces with the anti-Hermitian contracted Schrödinger equation"
An efficient method for geometry optimization based on solving the anti-Hermitian contracted Schrödinger equation (ACSE) is presented. We formulate a reduced version of the Hellmann–Feynman theorem (HFT) in terms of the two-electron reduced Hamiltonian operator and the two-electron reduced density matrix (2-RDM). The HFT offers a considerable reduction in computational cost over methods which rely on numerical derivatives. While previous geometry optimizations with numerical gradients required 2M evaluations of the ACSE where M is the number of nuclear degrees of freedom, the HFT requires only a single ACSE calculation of the 2-RDM per gradient. Synthesizing geometry optimization techniques with recent extensions of the ACSE theory to arbitrary electronic and spin states provides an important suite of tools for accurately determining equilibrium and transition-state structures of ground- and excited-state molecules in closed- and open-shell configurations. The ability of the ACSE to balance single- and multi-reference correlation is particularly advantageous in the determination of excited-state geometries where the electronic configurations differ greatly from the ground-state reference. Applications are made to closed-shell molecules N2, CO, H2O, the open-shell molecules B2 and CH, and the excited state molecules N2, B2, and BH. We also study the HCN ↔ HNC isomerization and the geometry optimization of hydroxyurea, a molecule which has a significant role in the treatment of sickle-cell anaemia.A. E. DePrince, D. A. Mazziotti. J. Chem. Phys. 132 034110 (2010). "Exploiting the spatial locality of electron correlation within the parametric two-electron reduced-density-matrix method"
The parametric variational two-electron reduced-density-matrix (2-RDM) method is applied to computing electronic correlation energies of medium-to-large molecular systems by exploiting the spatial locality of electron correlation within the framework of the cluster-in-molecule (CIM) approximation [S. Li et al., J. Comput. Chem. 23, 238 (2002); J. Chem. Phys. 125, 074109 (2006)]. The 2-RDMs of individual molecular fragments within a molecule are determined, and selected portions of these 2-RDMs are recombined to yield an accurate approximation to the correlation energy of the entire molecule. In addition to extending CIM to the parametric 2-RDM method, we (i) suggest a more systematic selection of atomic-orbital domains than that presented in previous CIM studies and (ii) generalize the CIM method for open-shell quantum systems. The resulting method is tested with a series of polyacetylene molecules, water clusters, and diazobenzene derivatives in minimal and nonminimal basis sets. Calculations show that the computational cost of the method scales linearly with system size. We also compute hydrogen-abstraction energies for a series of hydroxyurea derivatives. Abstraction of hydrogen from hydroxyurea is thought to be a key step in its treatment of sickle cell anemia; the design of hydroxyurea derivatives that oxidize more rapidly is one approach to devising more effective treatments.A. V. Sinitskiy, L. Greenman, D. A. Mazziotti. J. Chem. Phys. 133 014104 (2010). "Strong correlation in hydrogen chains and lattices using the variational two-electron reduced density matrix method"
The variational two-electron reduced-density-matrix (2-RDM) method, scaling polynomially with the size of the system, was applied to linear chains and three-dimensional clusters of atomic hydrogen as large as H64. In the case of the 4×4×4 hydrogen lattice of 64 hydrogen atoms, a correct description of the dissociation requires about 1018 equally weighted determinants in the wave function, which is too large for traditional multireference methods. The correct energy in the dissociation limit was obtained from the variational 2-RDM method in contrast to Hartree–Fock and single-reference methods. Analysis of the occupation numbers demonstrates that even for 1.0 Å bond distances the presence of strong electron correlation requires a multireference method. Three-dimensional systems exhibit a marked increase in electron correlation from one-dimensional systems regardless of size. The metal-to-insulator transition upon expansion of the clusters was studied using the decay of the 1-RDM off-diagonal elements. The variational 2-RDM method was shown to capture the metal-to-insulator transition and dissociation behavior accurately for all systems.L. Greenman, P. J. Ho, S. Pabst, E. Kamarchik, D. A. Mazziotti, R. Santra. Phys. Rev. A 82 023406 (2010). "Implementation of the time-dependent configuration-interaction singles method for atomic strong-field processes"
We present an implementation of the time-dependent configuration-interaction singles (TDCIS) method for treating atomic strong-field processes. In order to absorb the photoelectron wave packet when it reaches the end of the spatial grid, we add to the exact nonrelativistic many-electron Hamiltonian a radial complex absorbing potential (CAP). We determine the orbitals for the TDCIS calculation by diagonalizing the sum of the Fock operator and the CAP using a flexible pseudospectral grid for the radial degree of freedom and spherical harmonics for the angular degrees of freedom. The CAP is chosen such that the occupied orbitals in the Hartree-Fock ground state remain unaffected. Within TDCIS, the many-electron wave packet is expanded in terms of the Hartree-Fock ground state and its single excitations. The virtual orbitals satisfy nonstandard orthogonality relations, which must be taken into consideration in the calculation of the dipole and Coulomb matrix elements required for the TDCIS equations of motion. We employ a stable propagation scheme derived by second-order finite differencing of the TDCIS equations of motion in the interaction picture and subsequent transformation to the Schrödinger picture. Using the TDCIS wave packet, we calculate the expectation value of the dipole acceleration and the reduced density matrix of the residual ion. The technique implemented will allow one to study electronic channel-coupling effects in strong-field processes.J. W. Snyder, A. E. Rothman, J. J. Foley, D. A. Mazziotti. J. Chem. Phys. 132 154109 (2010). "Conical intersections in triplet excited states of methylene from the anti-Hermitian contracted Schrödinger equation"
A conical intersection in triplet excited states of methylene is computed through the direct calculation of two-electron reduced density matrices (2-RDMs) from solutions of the anti-Hermitian contracted Schrödinger equation (ACSE). The study synthesizes recent extensions of the ACSE method for the treatment of excited states [G. Gidofalvi and D. A. Mazziotti, Phys. Rev. A 80, 022507 (2009)] and arbitrary-spin states [A. E. Rothman, J. J. Foley, and D. A. Mazziotti, Phys. Rev. A 80, 052508 (2009)]. We compute absolute energies of the 1 B31, 1 A32, and 2 B31 states of methylene (CH2) and the location of the conical intersection along the 1 A32−2 B31 potential-energy surfaces. To treat multireference correlation, we seed the ACSE with an initial 2-RDM from a multiconfiguration self-consistent field (MCSCF) calculation. The ACSE produces energies that significantly improve upon those from MCSCF and second-order multireference many-body perturbation theory, and the 2-RDMs from the ACSE nearly satisfy necessary N-representability conditions. Comparison of the results from augmented double-zeta and triple-zeta basis sets demonstrates the importance of augmented (or diffuse) functions for determining the location of the conical intersection.Back to top
2009
E. Kamarchik, D. A. Mazziotti. Phys. Rev. A 79 012502 (2009). "Coupled nuclear and electronic ground-state motion from variational reduced-density-matrix theory with applications to molecules with floppy or resonant hydrogens"
The variational two-electron reduced-density-matrix (2RDM) method for electronic systems [Phys. Rev. Lett. 93, 213001 (2004)] is extended to compute ground-state distributions of electrons and hydrogen nuclei in molecules beyond the Born-Oppenheimer approximation. While traditional methods for nuclei rely on the construction of expensive potential energy surfaces or other approximations, the variational 2RDM method has the advantage of treating both electrons and hydrogen nuclei as quantum-mechanical particles simultaneously. Because these particles interact by pairwise Coulombic potentials, the ground-state energy is expressible as a linear functional of three 2RDMs corresponding to two electrons, two hydrogens, and one electron and one hydrogen. Nuclei other than hydrogen are treated in the Born-Oppenheimer approximation. Variational optimization of the ground-state energy requires that the 2RDMs be restricted by N-representability conditions to represent a realistic N-particle system where N is the total number of electrons and hydrogens. Recent progress in electronic systems with (i) developing necessary N-representability conditions and (ii) optimizing the ground-state energy subject to these conditions is extended to systems with two types of particles, electrons and nuclei. The nuclear-electronic 2RDM method can be applied to studying macroscopic quantum phenomena in molecules with “floppy” or resonant hydrogens. Illustrative applications are made to (i) large-scale hydrogen motion in hydrogen-bonded molecules and protonated acetylene C2H3+ and (ii) hydrogen resonance in malonaldehyde C3H4O2 and ammonia NH3.G. Gidofalvi, D. A. Mazziotti. Phys. Rev. A 80 022507 (2009). "Direct calculation of excited-state electronic energies and two-electron reduced density matrices from the anti-Hermitian contracted Schrödinger equation"
Direct calculation of the ground-state two-electron reduced density matrix (2-RDM) and its energy has recently been achieved for many-electron atoms and molecules by solving the anti-Hermitian part of the contracted Schrödinger equation (ACSE) [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. In this paper the ACSE method is extended to computing the 2-RDMs and energies of excited states without the many-electron wave function. The contracted Schrödinger equation (CSE) is an important ingredient for excited-state 2-RDM methods because it is a stationary-state condition for both ground and excited states. We develop the theoretical framework for the ACSE as a stationary-state condition through its connections to the CSE and the Schrödinger equation. As in previous ground-state calculations, the indeterminacy of the ACSE is removed by reconstructing its 3-RDM as a functional of its 2-RDM through a cumulant theory for RDMs [D. A. Mazziotti, Chem. Phys. Lett. 289, 419 (1998)]. We calculate the initial 2-RDM from a multiconfiguration self-consistent-field calculation that includes multireference electron correlation, which can be especially important for excited states. The excited-state ACSE method is applied to computing absolute excited-state energies and vertical excitation energies of the molecules HF, H2O, and N2 as well as ground and excited potential-energy curves of HF. Comparisons are made to traditional multireference methods as well as full configuration interaction. Computed excited-state 2-RDMs nearly satisfy necessary N-representability conditions.L. Greenman, D. A. Mazziotti. J. Chem. Phys. 130 184101 (2009). "Highly multireferenced arynes studied with large active spaces using two-electron reduced density matrices"
Using the active-space two-electron reduced density matrix (2-RDM) method, which scales polynomially with the size of the active space [G. Gidofalvi and D. A. Mazziotti, J. Chem. Phys. 129, 134108 (2008)], we were able to use active spaces as large as 24 electrons in 24 orbitals in computing the ground-state energies and properties of highly multireferenced arynes. Because the conventional complete-active-space self-consistent-field (CASSCF) method scales exponentially with the size of the active space, its application to arynes was mainly limited to active spaces of 12 electrons in 12 orbitals. For these smaller active spaces the active-space 2-RDM method accurately reproduces the results of CASSCF. However, we show that the larger active spaces are necessary for describing changes in energies and properties with aryne chain length such as the emergence of polyradical character. Furthermore, the addition of further electron correlation by multireference perturbation theory is demonstrated to be inadequate for removing the limitations of the smaller active spaces.C. A. Schwerdtfeger, D. A. Mazziotti. J. Chem. Phys. 130 224102 (2009). "Convex-set description of quantum phase transitions in the transverse Ising model using reduced-density-matrix theory"
Quantum phase transitions in N-particle systems can be identified and characterized by the movement of the two-particle reduced density matrix (2-RDM) along the boundary of its N-representable convex set as a function of the Hamiltonian parameter controlling the phase transition [G. Gidofalvi and D. A. Mazziotti, Phys. Rev. A 74, 012501 (2006)]. For the one-dimensional transverse Ising model quantum phase transitions as well as their finite-lattice analogs are computed and characterized by the 2-RDM movement with respect to the transverse magnetic field strength g. The definition of a 2-RDM “speed” quantifies the movement of the 2-RDM per unit of g, which reaches its maximum at the critical point of the phase transition. For the infinite lattice the convex set of 2-RDMs and the 2-RDM speed are computed from the exact solution of the 2-RDM in the thermodynamic limit of infinite N [P. Pfeuty, Ann. Phys. 57, 79 (1970)]. For the finite lattices we compute the 2-RDM convex set and its speed by the variational 2-RDM method [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] in which approximate ground-state 2-RDMs are calculated without N-particle wave functions by using constraints, known as N-representability conditions, to restrict the 2-RDMs to represent quantum system of N fermions. Advantages of the method include: (i) rigorous lower bounds on the ground-state energies, (ii) polynomial scaling of the calculation with N, and (iii) independence of the N-representability conditions from a reference wave function, which enables the modeling of multiple quantum phases. Comparing the 2-RDM convex sets for the finite- and infinite-site lattices reveals that the variational 2-RDM method accurately captures the shape of the convex set and the signature of the phase transition in the 2-RDM movement. From the 2-RDM all one- and two-particle expectation values (or order parameters) of the quantum Ising model can also be computed including the pair correlation function, which decays rapidly around the critical field strength g.A. E. DePrince, D. A. Mazziotti. J. Chem. Phys. 130 164109 (2009). "Open-shell molecular electronic states from the parametric two-electron reduced-density-matrix method"
The parametric variational two-electron reduced-density-matrix (2-RDM) method, developed from an analysis of positivity (N-representability) constraints on the 2-RDM, is extended to treat both closed- and open-shell molecules in singlet, doublet, and triplet spin states. The parametric 2-RDM method can be viewed as using N-representability conditions to modify the 2-RDM from a configuration interaction singles-doubles wave function to make the energy size extensive while keeping the 2-RDM approximately N-representable [J. Kollmar, Chem. Phys. 125, 084108 (2006); A. E. DePrince and D. A. Mazziotti, Phys. Rev. A 76, 049903 (2007)]. Vertical excitation energies between triplet and singlet states are computed in a polarized valence triple-zeta basis set. In comparison to traditional single-reference wave function methods, the parametric 2-RDM method recovers a larger percentage of the multireference correlation in the singlet excited states, which improves the accuracy of the vertical excitation energies. Furthermore, we show that molecular geometry optimization within the parametric 2-RDM method can be efficiently performed through a Hellmann–Feynman-like relation for the energy gradient with respect to nuclear coordinates. Both the open-shell extension and the energy-gradient relation are applied to computing relative energies and barrier heights for the isomerization reaction HCN+↔HNC+. The computed 2-RDMs very nearly satisfy well known, necessary N-representability conditions.A. E. Rothman, J. J. Foley, D. A. Mazziotti. Phys. Rev. A 80 052508 (2009). "Open-shell energies and two-electron reduced density matrices from the anti-Hermitian contracted Schrödinger equation: A spin-coupled approach"
Two-electron reduced density matrices (2-RDMs) and their energies have recently been computed accurately for singlet molecular states through the solution of the anti-Hermitian part of the contracted Schrödinger equation (ACSE) [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. In this work we directly solve the ACSE for the 2-RDMs of open-shell molecular systems in arbitrary spin states. This generalization is achieved by spin coupling the open-shell molecule to one or more hydrogen atoms to form a singlet composite system, which can be solved by existing ACSE algorithms. Advantages of the spin-coupled approach include: (i) applicability of the singlet ACSE algorithm to high-spin open-shell molecules, (ii) consistent treatment of arbitrary spin states that avoids spin-free or other approximations to the cumulant reconstruction of higher RDMs, and (iii) computational reduction in floating-point operations and storage from exploiting the singlet spin symmetry of the composite system. The ACSE energy of the composite system we showed is the sum of the open-shell energy and the energies of the hydrogens, and the open-shell 2-RDM can be readily extracted from the composite 2-RDM. With the open-shell ACSE method we calculate energies and natural-orbital occupation numbers for a variety of doublet and triplet open-shell molecules including potential-energy curves for CH, B2, N2+, and H2O+. The ACSE produces energies that are consistently more accurate than those from either multireference second-order perturbation theory or coupled cluster singles-doubles, and as with singlet states, it provides a balanced description of single-reference and multireference correlation, which is exemplified in its dissociation of open-shell N2+. Computed 2-RDMs nearly satisfy necessary N-representability constraints.J. J. Foley, A. E. Rothman, D. A. Mazziotti. J. Chem. Phys. 130 184112 (2009). "Activation energies of sigmatropic shifts in propene and acetone enolate from the anti-Hermitian contracted Schrödinger equation"
The hydrogen [1,3]-sigmatropic shift in propene is predicted by the Woodward–Hoffman rules to occur by an antarafacial pathway, yet the lack of experimental evidence suggests that this pathway is not favorable. Two natural questions arise: (i) can the [1,3]-shift be made more favorable by a symmetry-forbidden multistep pathway, and (ii) can the energetics be influenced by a substituent on propene? As in many chemical reactions, describing the energetics of these reactions requires a balanced treatment of both single-reference and multireference electron correlations, and yet traditional wave function methods often excel in treating only one kind of correlation. An equitable description of correlation effects, however, can be achieved, at a cost similar to efficient single-reference methods, by computing the two-electron reduced density matrix (2-RDM) from the anti-Hermitian part of the contracted Schrödinger equation (ACSE) [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. As with the contracted Schrödinger equation, the indeterminacy of the ACSE is removed without the many-electron wave function by reconstructing the 3-RDM from the 2-RDM via cumulant theory [D. A. Mazziotti, Chem. Phys. Lett. 289, 419 (1998)]. In this paper we apply the ACSE to study sigmatropic shifts in both propene and acetone enolate while extending its formalism to treat doublet spin states. In the 6-311G∗∗ basis set the ACSE predicts the activation energy of the trimethylene-to-propene rearrangement to be 8.8 kcal/mol while multireference perturbation theory yields a smaller barrier of 2.2 kcal/mol and coupled cluster singles-doubles predicts a negative barrier. We further find that the [1,3]-shift in acetone enolate is more favorable by ≈30 kcal/mol than the [1,3]-shift in propene, which is consistent with a prior theoretical investigation as well as experimental observations of these shifts in 2-butanone enolate.Back to top
2008
D. A. Mazziotti. Phys. Rev. Lett. 101 253002 (2008). "Parametrization of the Two-Electron Reduced Density Matrix for its Direct Calculation without the Many-Electron Wave Function"
Parametrization of a molecular two-electron reduced density matrix (2-RDM) enables energies and properties to be directly computed at a highly efficient computational cost. In this Letter an improved 2-RDM parametrization yields energies and properties that are markedly better than those calculated with a similar computational scaling by traditional ab initio methods. Computed 2-RDMs very nearly satisfy well-known N-representability conditions. Applications are made to ground-state energies of several molecules and equilibrium bond distances and harmonic frequencies of HF, F2, and CO. The method for single-bond breaking in HF and CH4 yields similar accuracy at equilibrium and nonequilibrium geometries.D. A. Mazziotti. J. Phys. Chem. A 112 13684-13690 (2008). "Energy Barriers in the Conversion of Bicyclobutane to gauche-1,3-Butadiene from the Anti-Hermitian Contracted Schrödinger Equation"
In 1959 Charles Coulson popularized the idea of computing a molecule’s ground-state energy as a functional of the two-electron reduced density matrix (2-RDM) without the many-electron wave function. For 50 years, however, a practical, direct calculation of the 2-RDM was not achieved because the 2-RDM must be constrained by N-representability conditions to represent an N-electron system. Recently, two general approaches to the direct calculation of the 2-RDM have emerged including the solution of the anti-Hermitian contracted Schrödinger equation (ACSE) [Mazziotti, Phys. Rev. Lett., 2006, 97, 143002]. In this article, after further extending the theoretical development of the ACSE method for multireference correlation, we apply the ACSE to studying an unresolved question regarding the opening of bicyclobutane to gauche-1,3-butadiene by conrotatory and disrotatory pathways. Previous theoretical values for the disrotatory energy barrier reveal a disagreement between correlation methods on the order of 10 kcal/mol. By capturing significantly more correlation energy than traditional multireference methods, the ACSE provides new insight into this discrepancy. The ACSE energy for the conrotatory energy barrier agrees with the 40.6 ± 2.5 kcal/mol experimental value.L. Greenman, D. A. Mazziotti. J. Chem. Phys. 128 114109 (2008). "Electronic excited-state energies from a linear response theory based on the ground-state two-electron reduced density matrix"
Ground-state two-particle reduced density matrices (2-RDMs) are used to calculate excited-state energy spectra. Solving the Schrödinger equation for excited states dominated by single excitations from the ground-state wavefunction requires the ground-state 2- and 3-RDMs. The excited states, however, can be obtained without a knowledge of the ground-state 3-RDM by two methods: (i) cumulant expansion methods which build the 3-RDM from the 2-RDM, and (ii) double commutator methods which eliminate the 3-RDM. Previous work [Mazziotti, Phys. Rev. A 68, 052501 (2003)] examined the accuracy of excited states extracted from ground-state 2-RDMs, which were calculated by full configuration interaction or the variational 2-RDM method. In this work we employ (i) advances in semidefinite programming to treat the excited states of water and hydrogen fluoride and chains of hydrogen atoms, and (ii) the addition of partial three-particle N-representability conditions to compute more accurate ground-state 2-RDMs. With the hydrogen chains we examine the metal-to-insulator transition as measured by the band gap (the difference between the ground-state and the first excited-state energies), which is difficult for excited-state methods to capture.G. Gidofalvi, D. A. Mazziotti. J. Chem. Phys. 129 134108 (2008). "Active-space two-electron reduced-density-matrix method: Complete active-space calculations without diagonalization of the N-electron Hamiltonian"
Molecular systems in chemistry often have wave functions with substantial contributions from two-or-more electronic configurations. Because traditional complete-active-space self-consistent-field (CASSCF) methods scale exponentially with the number N of active electrons, their applicability is limited to small active spaces. In this paper we develop an active-space variational two-electron reduced-density-matrix (2-RDM) method in which the expensive diagonalization is replaced by a variational 2-RDM calculation where the 2-RDM is constrained by approximate N-representability conditions. Optimization of the constrained 2-RDM is accomplished by large-scale semidefinite programming [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Because the computational cost of the active-space 2-RDM method scales polynomially as ra6 where ra is the number of active orbitals, the method can be applied to treat active spaces that are too large for conventional CASSCF. The active-space 2-RDM method performs two steps: (i) variational calculation of the 2-RDM in the active space and (ii) optimization of the active orbitals by Jacobi rotations. For large basis sets this two-step 2-RDM method is more efficient than the one-step, low-rank variational 2-RDM method [Gidofalvi and Mazziotti, J. Chem. Phys. 127, 244105 (2007)]. Applications are made to HF, H2O, and N2 as well as n-acene chains for n=2–8. When n>4, the acenes cannot be treated by conventional CASSCF methods; for example, when n=8, CASSCF requires optimization over approximately 1.47×1017 configuration state functions. The natural occupation numbers of the n-acenes show the emergence of bi- and polyradical character with increasing chain length.A. E. Rothman, D. A. Mazziotti. Phys. Rev. A 78 032510 (2008). "Variational reduced-density-matrix theory applied to the electronic structure of few-electron quantum dots"
Variational two-electron reduced-density-matrix (2RDM) theory is applied to computing energy spectra and properties of few-electron quantum dots. The model system is a two-dimensional electron gas with a central confinement potential. For each orbital angular momentum J, the energy and 2RDM are computed by the variational 2RDM method in which the energy is minimized as a functional of the 2RDM. In the minimization, which is performed by semidefinite programming, the 2RDM is constrained to represent a N-electron wave function with angular momentum J by N- and J-representability conditions [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Advantages of the variational 2RDM method include (i) lower bounds on the energies for all J values, (ii) calculation of approximate 2RDMs in polynomial time without many-electron wave functions, (iii) exploitation of angular symmetry in the sparse block-diagonal structure of the 2RDM, (iv) accurate description of multireference correlation (entanglement) effects, and (v) direct calculation of one- and two-electron properties from the 2RDM. With the 2RDM we directly compute pair-correlation functions, radial charge densities, and average radial electron displacements in the quantum dot. Energies and properties are compared to those from solving the N-electron Schrödinger equation by large-scale, exact diagonalization. It is found that the accuracy of the variational 2RDM approach is sensitive to the total orbital angular momentum and the symmetry of the final wave function. For quantum dots of high symmetry, the variational algorithm isolates a highly accurate solution that recovers the correlation energy within a few percent.A. E. Rothman, D. A. Mazziotti. Phys. Rev. A 77 012507 (2008). "Geminal-based statistics for the energies of many-electron molecular systems"
In 1959, Bopp developed a lower bound to atomic and molecular ground-state energies by summing the lowest eigenvalues of the two-particle reduced Hamiltonian, K̂ 2. His approximation is accurate only for very small systems (fewer than about four electrons), with the results degenerating rapidly for larger problems. In this paper, we extend and improve Bopp’s work by introducing a flexible distribution function, guided by familiar Fermi-Dirac statistics, in order to generate occupation numbers for the energy levels of K̂ 2. The distribution function and the resulting energy are parametrized by a correlation temperature T. For a given system, characteristic temperatures may be identified that yield the true energy or any other benchmark energy of the system. Using a geometric argument and the empirical properties of the energy vs temperature curve, the two-electron statistics are investigated as a predictive tool for a variety of small atoms and molecules.A. E. D. III, D. A. Mazziotti. J. Phys. Chem. B 112 16158-16162 (2008). "Molecular Geometries and Harmonic Frequencies from the Parametric Two-Electron Reduced Density Matrix Method with Application to the HCN ↔ HNC Isomerization"
Energies, geometries, and harmonic frequencies of the chemical species in the HCN ↔ HNC isomerization including the transition state are computed with the parametric variational two-electron reduced density matrix (2-RDM) method. The parametric 2-RDM method parametrizes the 2-RDM with single- and double-excitation coefficients to be both size-consistent and nearly N-representable [ DePrince A. E. III ; Mazziotti D. A. Phys. Rev. A 2007, 73, 042501. ]. With the inclusion of the zero-point energies of both species, the energy of HNC relative to HCN in a polarized valence triple-ζ basis set is 14.2 kcal/mol, which agrees with the experimental value of 14.8 ± 2 kcal/mol. The present calculations provide the first assessment of the method for computing harmonic frequencies from a molecular geometry optimization. Bond lengths, angles, and harmonic frequencies are also computed for HF, CO, and H2O.A. E. DePrince, E. Kamarchik, D. A. Mazziotti. J. Chem. Phys. 128 234103 (2008). "Parametric two-electron reduced-density-matrix method applied to computing molecular energies and properties at nonequilibrium geometries"
A parametric approach to the variational calculation of the two-electron reduced density matrix (2-RDM) for many-electron atoms and molecules has recently been developed in which the 2-RDM is parametrized to be both size consistent and nearly N-representable [C. Kollmar, J. Chem. Phys. 125, 084108 (2006); A. E. DePrince and D. A. Mazziotti, Phys. Rev. A 76, 049903 (2007)]. The parametric variational 2-RDM method is applied to computing ground-state molecular energies and properties at nonequilibrium geometries in significantly larger basis sets than previously employed. We study hydrogen abstraction from the hydroxide groups of H2O, NH3OH, and CH3OH. The 2-RDM method, parametrized by single and double excitations, shows significant improvement over coupled-cluster methods with similar excitations in predicting the shape of potential energy curves and bond-dissociation energies. Previous work completes the parametrization of the energy and 2-RDM by a system of n2h2 normalization constraints, where n and h are the number of occupied and unoccupied orbitals, respectively. In the present paper, however, we show that the constraints can be eliminated by incorporating them into the energy and 2-RDM functions and, hence, the constrained optimization of the ground-state energy can be reformulated as an unconstrained optimization. The 2-RDMs from the parametric method are very nearly N-representable, and as measured by an l2 norm, they are more accurate than the 2-RDMs from configuration interaction truncated at single and double excitations by an order of magnitude.Back to top
2007
D. A. Mazziotti. Phys. Rev. A 75 022505 (2007). "Anti-Hermitian part of the contracted Schrödinger equation for the direct calculation of two-electron reduced density matrices"
A recent advance in the theory of the contracted Schrödinger equation (CSE), in which only the anti-Hermitian part of the equation is solved, permits the direct determination of ground-state two-electron reduced density matrices (2-RDM’s) that yield 95%–100% of the correlation energy of atoms and molecules [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. Here we discuss in detail the anti-Hermitian contracted Schrödinger equation (ACSE) and its comparison to the CSE with regard to cumulant reconstruction of RDM’s, the role of Nakatsuji’s theorem, and the structure of the wave function. The ACSE is also formulated in the Heisenberg representation and related to canonical diagonalization. The solution of the ACSE is illustrated with a variety of molecules including H2O, CH2, NH4+, HF, and N2, and potential energy and dipole-moment surfaces are computed for boron hydride in a polarized double-ζ basis set. The computed 2-RDM’s very closely satisfy known N-representability conditions.D. A. Mazziotti. J. Phys. Chem. A 111 12635-12640 (2007). "Determining the Energy Gap between the Cis and Trans Isomers of HO3 - Using Geometry Optimization within the Anti-Hermitian Contracted Schrödinger and Coupled Cluster Methods †"
The cis and trans isomers of the HO3 - anion, which are important in proposed mechanisms for ozonization, are studied computationally. Relative energies, geometries, and normal-mode frequencies are calculated with anti-Hermitian contracted Schrödinger equation (ACSE) and coupled cluster methods. Both the ACSE method and the coupled cluster method with single and double excitations (CCSD) are applied in a correlation-consistent polarized double-ζ basis set (cc-pVDZ). Using coupled cluster with singles, doubles, and perturbative triples (CCSD(T)), we treat the problem with larger basis sets than those in previous work, including correlation-consistent polarized quadruple-ζ basis sets with (aug-cc-pVQZ) and without (cc-pVQZ) diffuse functions, which permit extrapolation of the cis and trans energies to the complete-basis-set limit. The cis isomer is found to be lower in energy than the trans isomer by −3.5 kcal/mol, which is 50% larger in magnitude than the best previous result of −2.2 kcal/mol. The bond lengths between the O2 and OH fragments of the cis- and trans-HO3 are calculated to be 1.713 and 1.857 Å, respectively, where both bond lengths are significantly longer than the 1.464 Å O−O bond in hydrogen peroxide. In this paper, we extend the ACSE method [Mazziotti, D. A. J. Chem. Phys. 2007, 126, 184101], which computes the two-electron reduced density matrix directly, to include geometry optimization by a Newton's method with numerical derivatives. Calculation of the cis- and trans-HO3 - isomers by the ACSE yields energies, geometries, and frequencies that are closer to those from CCSD(T) than those from CCSD.D. A. Mazziotti. Phys. Rev. A 76 052502 (2007). "Multireference many-electron correlation energies from two-electron reduced density matrices computed by solving the anti-Hermitian contracted Schrödinger equation"
Two-electron reduced density matrices (2-RDMs) have recently been directly calculated by solving the anti-Hermitian contracted Schrödinger equation (ACSE) to obtain 95–100% of the ground-state correlation energy of atoms and molecules with the accuracy increasing with the size of the one-electron basis set [Mazziotti, Phys. Rev. Lett. 97, 143002 (2006).] In this paper, the ACSE method is extended to treat strong multireference correlation effects that are often important at nonequilibrium molecular geometries. While previous ACSE calculations have employed an initial 2-RDM from the Hartree-Fock method, we initialize the solution of the ACSE with a 2-RDM guess from a multiconfiguration self-consistent field calculation. Applications are made to multireference correlation in the potential energy surfaces of the molecules HF, H2O, and C2 in polarized valence double-zeta basis sets while N2 is treated in polarized valence double- and triple-zeta basis sets. Accurate ground-state energies and 1-RDM occupation numbers are obtained at both equilibrium and nonequilibrium geometries where the energies are within a few millihartrees of those from full configuration interaction.D. A. Mazziotti. ESAIM: Math. Model. Numer. Anal. 41 249-259 (2007). "First-order semidefinite programming for the two-electron treatment of many-electron atoms and molecules"
The ground-state energy and properties of any many-electron atom or molecule may be rigorously computed by variationally computing the two-electron reduced density matrix rather than the many-electron wavefunction. While early attempts fifty years ago to compute the ground-state 2-RDM directly were stymied because the 2-RDM must be constrained to represent an N-electron wavefunction, recent advances in theory and optimization have made direct computation of the 2-RDM possible. The constraints in the variational calculation of the 2-RDM require a special optimization known as a semidefinite programming. Development of first-order semidefinite programming for the 2-RDM method has reduced the computational costs of the calculation by orders of magnitude [Mazziotti, Phys. Rev. Lett. 93 (2004) 213001]. The variational 2-RDM approach is effective at capturing multi-reference correlation effects that are especially important at non-equilibrium molecular geometries. Recent work on 2-RDM methods will be reviewed and illustrated with particular emphasis on the importance of advances in large-scale semidefinite programming.
D. A. Mazziotti. J. Chem. Phys. 126 184101 (2007). "Two-electron reduced density matrices from the anti-Hermitian contracted Schrödinger equation: Enhanced energies and properties with larger basis sets"
Two-electron reduced density matrices (2-RDMs) have recently been directly determined from the solution of the anti-Hermitian contracted Schrödinger equation (ACSE) to obtain 95%–100% of the ground-state correlation energy of atoms and molecules, which significantly improves upon the accuracy of the contracted Schrödinger equation (CSE) [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. Two subsets of the CSE, the ACSE and the contraction of the CSE onto the one-particle space, known as the 1,3-CSE, have two important properties: (i) dependence upon only the 3-RDM and (ii) inclusion of all second-order terms when the 3-RDM is reconstructed as only a first-order functional of the 2-RDM. The error in the 1,3-CSE has an important role as a stopping criterion in solving the ACSE for the 2-RDM. Using a computationally more efficient implementation of the ACSE, the author treats a variety of molecules, including H2O, NH3, HCN, and HO3−, in larger basis sets such as correlation-consistent polarized double- and triple-zeta. The ground-state energy of neon is also calculated in a polarized quadruple-zeta basis set with extrapolation to the complete basis-set limit, and the equilibrium bond length and harmonic frequency of N2 are computed with comparison to experimental values. The author observes that increasing the basis set enhances the ability of the ACSE to capture correlation effects in ground-state energies and properties. In the triple-zeta basis set, for example, the ACSE yields energies and properties that are closer in accuracy to coupled cluster with single, double, and triple excitations than to coupled cluster with single and double excitations. In all basis sets, the computed 2-RDMs very closely satisfy known N-representability conditions.E. Kamarchik, D. A. Mazziotti. Phys. Rev. A 75 013203 (2007). "Variational reduced-density-matrix method for ground-state nuclear motion"
Recent advances have realized the direct variational calculation of the two-particle reduced density matrix (RDM) for electronic systems [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)], but such an approach has yet to be explored for studying the ground-state motion of nuclei. In this paper we develop a few-particle variational RDM theory for treating ground-state nuclear motion in atomic and molecular clusters and potentially molecules. Two features of the RDM method for nuclear motion that differ from the electronic theory are the derivation and application of generalized N-representability conditions for (i) multiple types of particles and (ii) three- or higher-body interactions. Illustrative applications are made to helium nuclei in one and two dimensions and a simple organic molecule where the effects of particle statistics and classical and quantum limits are examined. Calculations are performed in computationally expensive and yet flexible numerical grid basis sets. As interparticle interactions are increased, the emergence of molecular structure from a Bose condensate is observed. Limitations and possibilities for the treatment of general molecular systems are discussed.G. Gidofalvi, D. A. Mazziotti. J. Chem. Phys. 126 024105 (2007). "Molecular properties from variational reduced-density-matrix theory with three-particle N-representability conditions"
Molecular ground-state energies and two-electron reduced density matrices (2-RDMs) have recently been computed without the many-electron wave function by constraining the 2-RDM to satisfy a complete set of three-positivity conditions for N representability [D. A. Mazziotti, Phys. Rev. A 74, 032501 (2006)]. Energies at both equilibrium and nonequilibrium geometries are obtained within 0.3% of the correlation energy. In this paper the authors extend this work to examine the accuracy of molecular properties, including multipole moments and components of the ground-state energy, relative to full configuration interaction (FCI). Comparisons are also made with 2-RDM methods with two-positivity conditions and two-positivity plus the generalized T1T2 conditions as well as several approximate wave function methods. Using the 2-RDM method with three-positivity conditions, the authors obtain dipole, quadrupole, and octupole moments for BeH2, BH, H2O, CO, and NH3 at equilibrium geometries that are within 0.04% of their FCI values. In addition, for the potential energy surface of N2, the 2-RDM method with three-positivity yields not only accurate total ground-state energies but also accurate expectation values of the kinetic energy operator, the electron-nuclei potential, and electron-electron repulsion.G. Gidofalvi, D. A. Mazziotti. J. Chem. Phys. 127 244105 (2007). "Multireference self-consistent-field energies without the many-electron wave function through a variational low-rank two-electron reduced-density-matrix method"
The variational two-electron reduced-density-matrix (2-RDM) method allows for the computation of accurate ground-state energies and 2-RDMs of atoms and molecules without the explicit construction of an N-electron wave function. While previous work on variational 2-RDM theory has focused on calculating full configuration-interaction energies, this work presents the first application toward approximating multiconfiguration self-consistent-field (MCSCF) energies via low-rank restrictions on the 1- and 2-RDMs. The 2-RDM method with two- or three-particle N-representability conditions reduces the exponential active-space scaling of MCSCF methods to a polynomial scaling. Because the first-order algorithm [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] represents each form of the 1- and 2-RDMs by a matrix factorization, the RDMs are readily defined to have a low rank rather than a full rank by setting the matrix factors to be rectangular rather than square. Results for the potential energy surfaces of hydrogen fluoride, water, and the nitrogen molecule show that the low-rank 2-RDM method yields accurate approximations to the MCSCF energies. We also compute the energies along the symmetric stretch of a 20-atom hydrogen chain where traditional MCSCF calculations, requiring more than 17×109 determinants in the active space, could not be performed.E. Kamarchik, D. A. Mazziotti. Phys. Rev. Lett. 99 243002 (2007). "Global Energy Minima of Molecular Clusters Computed in Polynomial Time with Semidefinite Programming"
The global energy minima of pure and binary molecular clusters with 5–12 particles interacting pairwise are computed in polynomial time as a function of only the two-particle reduced density function (2-RDF). We derive linear matrix inequalities from the classical analogue of quantum N-representability constraints to ensure that the 2-RDF represents realistic N-particle configurations. The 2-RDF reformulation relaxes a combinatorial optimization into a convex optimization that scales polynomially in computer time. Clusters are optimized with a code for large-scale semidefinite programming developed for the quantum representability problem [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)PRLTAO0031-900710.1103/PhysRevLett.93.213001].K. B. Shakman, D. A. Mazziotti. J. Phys. Chem. A 111 7223-7226 (2007). "Assessing the Efficacy of Nonsteroidal Anti-Inflammatory Drugs Through the Quantum Computation of Molecular Ionization Energies"
The clinical efficacy of nonsteroidal anti-inflammatory drugs has been related to ionization energies [Mehler and Gerhards, Int. J. Quantum Chem. 1989, 25, 205]. In this paper we employ modern quantum-chemical calculations to re-examine the statistical correlation between clinical efficacy and ionization energies. Ionization energies are computed by density functional theory, with and without Koopman's theorem, for a series of salicylic acids and phenols whose activities, or efficacy, are known. Using a regression analysis, we show that improving the treatment of electron correlation beyond previous studies enhances the statistical correlation between clinical activities and ionization energies.A. E. DePrince, D. A. Mazziotti. Phys. Rev. A 76 042501 (2007). "Parametric approach to variational two-electron reduced-density-matrix theory"
Two general variational paradigms for computing ground-state energies and properties of molecular quantum systems are (i) the parametrization of the N-particle wave function, as in truncated configuration interaction, which yields an upper bound on the energy in a given basis set and (ii) the constraint of the two-electron reduced-density matrix (2-RDM) by necessary N-representability conditions (without using the wave function) which yields a lower bound on the energy in a given basis set [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. In this paper we synthesize these two directions in a class of techniques which we call parametric variational 2-RDM methods. The 2-RDM in these methods is parametrized to be size consistent while approximately satisfying the N-representability conditions. We extend an energy functional of Kollmar [C. Kollmar, J. Chem. Phys. 125, 084108 (2006)], which modifies configuration interaction with double excitations to be size consistent, by including not only double but also single excitations explicitly. Using the 2-RDM parametrization, we calculate ground-state energies at both equilibrium and nonequilibrium geometries in correlation-consistent polarized valance double-zeta (cc-pVDZ) basis sets. Energies as well as properties from the parametric variational 2-RDM method, particularly at nonequilibrium geometries, are better in accuracy than those obtained from coupled cluster with single and double excitations. The present work shows clearly that, except in the dissociation of N2, the deviation of the 2-RDM from the well-known N-representability conditions, such as the D, Q, and G conditions, is negligible. Furthermore, calculations with helium atoms demonstrate the size consistency of the method. The computational results on N representability and size consistency are especially important because they legitimatize the selected parametrization of the 2-RDM.A. E. DePrince, D. A. Mazziotti. Phys. Rev. A 76 049903 (2007). "Publisher's Note: Parametric approach to variational two-electron reduced-density-matrix theory [Phys. Rev. A 76, 042501 (2007)]"
A. E. DePrince, D. A. Mazziotti. J. Chem. Phys. 127 104104 (2007). "Cumulant reconstruction of the three-electron reduced density matrix in the anti-Hermitian contracted Schrödinger equation"
Differing perspectives on the accuracy of three-electron reduced-density-matrix (3-RDM) reconstruction in nonminimal basis sets exist in the literature. This paper demonstrates the accuracy of cumulant-based reconstructions, developed by Valdemoro (V) [F. Colmenero et al., Phys. Rev. A 47, 971 (1993)], Nakatsuji and Yasuda (NY) [Phys. Rev. Lett. 76, 1039 (1996)], Mazziotti (M) [Phys. Rev. A 60, 3618 (1999)], and Valdemoro–Tel–Pérez–Romero (VTP) [Many-electron Densities and Density Matrices, edited by J. Cioslowski (Kluwer, Boston, 2000)]. Computationally, we extend previous investigations to study a variety of molecules, including LiH, HF, NH3, H2O, and N2, in Slater-type, double-zeta, and polarized double-zeta basis sets at both equilibrium and nonequilibrium geometries. The reconstructed 3-RDMs, compared with 3-RDMs from full configuration interaction, demonstrate in nonminimal basis sets the accuracy of the first-order expansion (V) as well as the important role of the second-order corrections (NY, M, and VTP). Calculations at nonequilibrium geometries further show that cumulant functionals can reconstruct the 3-RDM from a multireferenced 2-RDM with reasonable accuracy, which is relevant to recent multireferenced formulations of the anti-Hermitian contracted Schrödinger equation (ACSE) and canonical diagonalization. Theoretically, we perform a detailed perturbative analysis of the M functional to identify its second-order components. With these second-order components we connect the M, NY, and VTP reconstructions for the first time by deriving both the NY and VTP functionals from the M functional. Finally, these 3-RDM reconstructions are employed within the ACSE [D. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)] to compute ground-state energies which are compared with the energies from the contracted Schrödinger equation and several wave function methods.T. Juhász, N. Shenvi, D. A. Mazziotti. Chem. Phys. Lett. 445 79-83 (2007). "Recursively generated linear constraints for variational two-particle reduced-density-matrix theory"
The variational two-electron reduced-density-matrix (2-RDM) method can compute many-electron energies and properties by constraining the 2-RDM with two- or three-particle positivity conditions. While milliHartree accuracy of the energy is obtained with 3-positivity conditions at both equilibrium and highly stretched geometries, the addition of these conditions to the 2-positivity constraints significantly increases the computational work. In this Letter, we relax a subset of three-particle positivity conditions (called T2) to a recursively generated set of linear inequality constraints that produces most of T2’s accuracy with lower computational cost. Illustrative applications are reported.Back to top
2006
D. A. Mazziotti. Phys. Rev. Lett. 97 143002 (2006). "Anti-Hermitian Contracted Schrödinger Equation: Direct Determination of the Two-Electron Reduced Density Matrices of Many-Electron Molecules"
Two-electron reduced density matrices (2-RDMs) of many-electron molecules are directly determined without calculation of their wave functions by solving the anti-Hermitian contracted Schrödinger equation. Approximation of the 3-RDM in the anti-Hermitian contracted Schrödinger equation by a corrected cumulant expansion [Mazziotti, Phys. Rev. A 60, 3618 (1999)PLRAAN1050-294710.1103/PhysRevA.60.3618] permits the direct calculation of the energy and 2-RDM with many high-order correlation effects included. The method is illustrated for the molecules BeH2, H2O, NH3, CH4, and CO as well as the dissociation of BH. Correlation energies are obtained within 95%–100% of full-configuration interaction, and 2-RDMs very nearly satisfy known N-representability conditions.D. A. Mazziotti. Phys. Rev. A 74 032501 (2006). "Variational reduced-density-matrix method using three-particle N-representability conditions with application to many-electron molecules"
Molecular two-electron reduced density matrices (2-RDMs) are computed variationally without the many-electron wave function by constraining the 2-RDM with a set of three-particle N-representability conditions known as three-positivity conditions. These constraints restrict four distinct three-particle probability distributions, which can be defined for any N-particle system, to be nonnegative. The variational calculation of the 2-RDM with full three-positivity conditions is implemented with the first-order semidefinite programming algorithm [D.A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. We also derive and implement a generalization of the T2 representability condition, which is a subset of the three-positivity conditions. Energies and 2-RDMs are computed for several molecules as well as the nitrogen molecule at both equilibrium and nonequilibrium geometries. The ground-state energies for nitrogen are within 0.0015a.u. of the values from full configuration interaction at all internuclear distances.D. A. Mazziotti. Acc. Chem. Res. 39 207-215 (2006). "Quantum Chemistry without Wave Functions: Two-Electron Reduced Density Matrices"
For 50 years, progress toward the direct calculation of the ground-state two-electron reduced density matrix (2-RDM) was stymied from an inability to constrain the 2-RDM to represent an N-electron wave function. Recent advances in theory and optimization have realized the direct calculation of the 2-RDM. A variational 2-RDM procedure, using first-order semidefinite programming, has been shown to capture multireference correlation effects important at nonequilibrium geometries [Mazziotti, Phys. Rev. Lett. 2004, 93, No. 213001]. This method emerged from research on a nonvariational calculation of the 2-RDM by the contracted Schrödinger equation. Both approaches will be discussed and illustrated.T. Juhász, D. A. Mazziotti. J. Chem. Phys. 125 174105 (2006). "The cumulant two-particle reduced density matrix as a measure of electron correlation and entanglement"
Several measures of electron correlation are compared based on two criteria: (i) the presence of a unique mapping between the reduced variables in the measure and the many-electron wave function and (ii) the linear scaling of the measure and its variables with system size. We propose the squared Frobenius norm of the cumulant part of the two-particle reduced density matrix (2-RDM) as a measure of electron correlation that satisfies these criteria. An advantage of this cumulant-based norm is its ability to measure the correlation from spin entanglement, which is not contained in the correlation energy. Alternative measures based on the 2-RDM, such as the von Neumann entropy, do not scale linearly with system size. Properties of the measures are demonstrated with Be, F2, HF, N2, and a hydrogen chain.G. Gidofalvi, D. A. Mazziotti. J. Chem. Phys. 125 144102 (2006). "Computation of dipole, quadrupole, and octupole surfaces from the variational two-electron reduced density matrix method"
Recent advances in the direct determination of the two-electron reduced density matrix (2-RDM) by imposing known N-representability conditions have mostly focused on the accuracy of molecular potential energy surfaces where multireference effects are significant. While the norm of the 2-RDM’s deviation from full configuration interaction has been computed, few properties have been carefully investigated as a function of molecular geometry. Here the dipole, quadrupole, and octupole moments are computed for a range of molecular geometries. The addition of Erdahl’s T2 condition [Int. J. Quantum Chem. 13, 697 (1978)] to the D, Q, and G conditions produces dipole and multipole moments that agree with full configuration interaction in a double-zeta basis set at all internuclear distances.G. Gidofalvi, D. A. Mazziotti. J. Phys. Chem. A 110 5481-5486 (2006). "Variational Reduced-Density-Matrix Theory Applied to the Potential Energy Surfaces of Carbon Monoxide in the Presence of Electric Fields †"
The variational optimization of the energy with respect to the two-electron reduced-density matrix (2-RDM), constrained by N-representability conditions, can determine the shape of molecular potential energy surfaces with useful accuracy. In this paper, we apply the 2-RDM method with a first-order optimization algorithm [Mazziotti, D. A. Phys. Rev. Lett. 2004, 93, 213001] to investigating the potential energy surfaces of carbon monoxide in the presence and absence of an electric field. Two beneficial characteristics of the 2-RDM method for computing potential energy surfaces include the following: (i) its ability to capture multireference effects without specifying any reference wave function or density matrix and (ii) its guarantee of a global energy minimum in the variational optimization. The 2-RDM method produces electronic ground-state energies with similar accuracy at equilibrium and nonequilibrium geometries in both the presence and the absence of the electric field. Computed dipole moments are similar in accuracy to the values from the computationally expensive configuration interaction with single, double, triple, and quadruple excitations. These surfaces have important applications in quantum molecular control theory.J. R. Hammond, D. A. Mazziotti. Phys. Rev. A 73 012509 (2006). "Variational reduced-density-matrix calculations on radicals: An alternative approach to open-shell ab initio quantum chemistry"
An alternative approach to open-shell molecular calculations using the variational two-electron reduced-density-matrix (2-RDM) theory [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] is presented. The energy and 2-RDM of the open-shell molecule (or radical) are computed from the limit of dissociating one or more hydrogen atoms from a molecule in a singlet state. Because the ground-state energy of an “infinitely” separated hydrogen atom in a given finite basis is known, we can determine the energy of the radical by subtracting the energy of one or more hydrogen atoms from the energy of the total dissociated system. The 2-RDM is constrained to have singlet symmetry in all calculations. Two sets of N-representability conditions are employed: (i) two-positivity conditions, and (ii) two-positivity conditions plus the T2 condition, which is a subset of the three-positivity conditions. Optimization of the energy with respect to the 2-RDM is performed with a first-order algorithm for solving the semidefinite program within the variational 2-RDM method. We present calculations of several radicals near equilibrium as well as the dissociation curves of the diatomic radicals CH and OH.G. Gidofalvi, D. A. Mazziotti. Phys. Rev. A 74 012501 (2006). "Computation of quantum phase transitions by reduced-density-matrix mechanics"
Quantum phase transitions are explored with reduced-density-matrix (RDM) mechanics. While in wave mechanics the quantum phase transition is identified by a crossing or avoided crossing between ground- and excited-state energies, in RDM mechanics the transition is characterized by movement of the ground-state two-electron RDM (2-RDM) along the boundary of the convex set of 2-RDMs between regions with dramatically different expectation values (order parameters) of one or more operators. With recent advances the ground-state 2-RDM can be directly computed without the many-particle wave function by variational optimization of the energy with the 2-RDM [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Because the variational calculation of the 2-RDM does not depend on a reference wave function, it can accurately predict the energies and properties of a system both near and far from the quantum phase transition.J. R. Hammond, D. A. Mazziotti. Phys. Rev. A 73 062505 (2006). "Variational reduced-density-matrix calculation of the one-dimensional Hubbard model"
Variational reduced-density-matrix theory is applied to calculating the ground-state energy and two-electron reduced density matrices (2-RDMs) of the one-dimensional Hubbard model for a range of interaction strengths. The 2-RDM is constrained to represent an N-particle wave function by two sets of N-representability conditions, known as the 2- and partial 3-positivity conditions. Variational optimization of the energy with the 2-RDM constrained by N-representability conditions is performed using a first-order semidefinite-programming algorithm that was developed for treating atoms and molecules [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Accurate energies for a broad range of interaction strengths indicate that the variational 2-RDM method is a valuable tool for studying strongly correlated electrons.J. D. Farnum, G. Gidofalvi, D. A. Mazziotti. J. Chem. Phys. 124 234103 (2006). "Modeling the influence of a laser pulse on the potential energy surface in optimal molecular control theory"
Understanding and modeling the interaction between light and matter is essential to the theory of optical molecular control. While the effect of the electric field on a molecule’s electronic structure is often not included in control theory, it can be modeled in an optimal control algorithm by a set or toolkit of potential energy surfaces indexed by discrete values of the electric field strength where the surfaces are generated by Born-Oppenheimer electronic structure calculations that directly include the electric field. Using a new optimal control algorithm with a trigonometric mapping to limit the maximum field strength explicitly, we apply the surface-toolkit method to control the hydrogen fluoride molecule. Potential energy surfaces in the presence and absence of the electric field are created with two-electron reduced-density-matrix techniques. The population dynamics show that adjusting for changes in the electronic structure of the molecule beyond the static dipole approximation can be significant for designing a field that drives a realistic quantum system to its target observable.Back to top
2005
D. A. Mazziotti. Phys. Rev. A 72 032510 (2005). "Variational two-electron reduced density matrix theory for many-electron atoms and molecules: Implementation of the spin- and symmetry-adapted T2 condition through first-order semidefinite programming"
The energy and properties of a many-electron atom or molecule may be directly computed from a variational optimization of a two-electron reduced density matrix (2RDM) that is constrained to represent many-electron quantum systems. In this paper we implement a variational 2RDM method with a representability constraint, known as the T2 condition. The optimization of the 2RDM is performed with a first-order algorithm for semidefinite programming [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] which, because of its lower computational cost in comparison to second-order methods, allows the treatment of larger basis sets. We also derive and implement a spin- and symmetry-adapted formulation of the T2 condition that significantly decreases the size of the largest block in the T2 matrix. The T2 condition, originally derived by Erdahl [Int. J. Quantum Chem. 13, 697 (1978)], was recently applied via a second-order algorithm to atoms and molecules [Z. Zhao , J. Chem. Phys. 120, 2095 (2004)]. While these calculations were restricted to molecules at equilibrium geometries in minimal basis sets, we apply the 2RDM method with the T2 condition to compute the electronic energies of molecules in both minimal and nonminimal basis sets at equilibrium as well as nonequilibrium geometries. Accurate potential energies curves are produced for BH, HF, and N2. Results are compared with the 2RDM method without the T2 condition as well as several wave-function methods.G. Gidofalvi, D. A. Mazziotti. J. Chem. Phys. 122 194104 (2005). "Application of variational reduced-density-matrix theory to the potential energy surfaces of the nitrogen and carbon dimers"
The acceleration of the variational two-electron reduced-density-matrix (2-RDM) method, using a new first-order algorithm [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)], has shown its usefulness in the accurate description of potential energy surfaces in nontrivial basis sets. Here we apply the first-order 2-RDM method to the potential energy surfaces of the nitrogen and carbon dimers in polarized valence double-ζ basis sets for which benchmark full-configuration-interaction calculations exist. In a wave function formalism accurately stretching the triple bond of the nitrogen dimer requires at least six-particle excitations from the Hartree–Fock reference. Furthermore, cleaving the double bond of C2 should produce a “non-Morse”-like potential curve because the ground state near equilibrium (XΣg+1) has an avoided crossing with the second excited state (B′Σg+1) and a level crossing with the first excited state (BΔg1). Because the 2-RDM method variationally optimizes the energy over correlated 2-RDMs on the two-electron space without parametrization of the many-electron wave function, it captures multireference correlations that are difficult to describe with approximate wave functions. The 2-RDM method yields for N2 a potential energy surface with features and spectroscopic constants that are more accurate than those from single-reference methods and similar in accuracy to multireference techniques, and it describes the non-Morse-like behavior of C2 which is not captured by single-reference methods.G. Gidofalvi, D. A. Mazziotti. J. Chem. Phys. 122 094107 (2005). "Application of variational reduced-density-matrix theory to organic molecules"
Variational calculation of the two-electron reduced-density matrix (2-RDM), using a new first-order algorithm [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)], is applied to medium-sized organic molecules. The calculations reveal systematic trends in the accuracy of the energy with well-known chemical concepts such as hybridization, electronegativity, and atomic size. Furthermore, correlation energies from hydrocarbon chains indicate that the calculation of the 2-RDM subject to two-positivity conditions is size extensive, that is, the energy grows linearly with the number of electrons. Because organic molecules have a well-defined set of functional groups, we employ the trends in energy accuracy of the functional groups to design a correction to the 2-RDM energy for an arbitrary organic molecule. We apply the 2-RDM calculations with the functional-group correction to a large set of organic molecules with different functional groups. Energies with millihartree accuracy are obtained both at equilibrium and nonequilibrium geometries.T. Juhász, D. A. Mazziotti. J. Chem. Phys. 122 124101 (2005). "Improved perturbative treatment of electronic energies from a minimal-norm approach to many-body perturbation theory"
We propose a zeroth-order Hamiltonian for many-body perturbation theory based on the unitary decomposition of the two-particle reduced Hamiltonian. For the zeroth-order Hamiltonian constrained to be diagonal in the Hartree–Fock basis set, the two-particle reduced perturbation matrix is chosen to have a minimal Frobenius norm. When compared with the Møller–Plesset partitioning, the method yields more accurate second-order energies.G. Gidofalvi, D. A. Mazziotti. Phys. Rev. A 72 052505 (2005). "Spin and symmetry adaptation of the variational two-electron reduced-density-matrix method"
The variational two-electron reduced-density-matrix (2-RDM) method computes the ground-state energy and 2-RDM of an atom or molecule without calculation of the many-electron wave function. Recently, the computational efficiency of the 2-RDM method has been significantly enhanced through the use of a first-order algorithm for semidefinite programming [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. In this paper we develop a spin- and symmetry-adapted formulation of the method that further improves its efficiency by incorporating both the spin and spatial symmetries of many-electron atoms and molecules. While previous work on density-matrix symmetry focused on only one form of the 2-RDM, the variational method employs three different forms of the 2-RDM, known as the D, Q, and G matrices, to restrict the 2-RDM to be approximately N-representable, that is representable by an N-electron wave function. We apply spin symmetries to the three forms of the 2-RDM, each of which breaks into four diagonal spin-blocks, namely one singlet and three triplet blocks. If the molecules have point-group symmetry, each of the 2-RDMs may be further subdivided into smaller diagonal blocks according to the spatial symmetry of the basis functions. The subdivision of the 2-RDMs into diagonal blocks generates significant computational savings in both floating-point operations and memory storage. Calculations illustrate the computational savings. Spin adaptation also enforces the correct expectation value of the Ŝ2 operator, which in earlier work is applied as a separate constraint.D. K. Jordan, D. A. Mazziotti. J. Chem. Phys. 122 084114 (2005). "Comparison of two genres for linear scaling in density functional theory: Purification and density matrix minimization methods"
Two classes of linear-scaling methods to replace diagonalization of the one-particle Hamiltonian matrix in density functional theory are compared to each other. Purification takes a density matrix with the correct eigenfunctions and corrects the occupation numbers; density matrix minimization takes a density matrix with correct occupation numbers and corrects the eigenfunctions by rotating the orbitals. Computational comparisons are performed through modification of the MondoSCF program on water clusters and the protein endothelin. A purification scheme and a density matrix minimization scheme, based on the 1,2-contracted Schrödinger equation [D. A. Mazziotti, J. Chem. Phys. 115, 8305 (2001)] are implemented in large systems.J. R. Hammond, D. A. Mazziotti. Phys. Rev. A 71 062503 (2005). "Variational two-electron reduced-density-matrix theory: Partial 3-positivity conditions for N-representability"
Variationally calculating the ground state of a many-electron quantum system using only the two-electron reduced-density-matrix (2-RDM) requires N-representability conditions that constrain the 2-RDM to correspond to an N-electron wave function. A systematic hierarchy of N-representability conditions, known as p-positivity conditions, has been developed [D. A. Mazziotti and R. M. Erdahl, Phys. Rev. A 63, 042113 (2001)], and many-electron atoms and molecules in nonminimal basis sets have been solved with useful accuracy by a variational 2-RDM method with 2-positivity conditions [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. This paper considers two forms of partial 3-positivity conditions, the lifting conditions and the T1∕T2 conditions, to further enhance the accuracy of the 2-RDM methods without the computational cost of full 3-positivity conditions. Variational 2-RDM methods with different N-representability constraints including 2-positivity conditions, the two types of partial 3-positivity conditions, as well as the complete 3-positivity conditions are applied to compute the ground state of the Lipkin spin model. The energies and 2-RDMs are compared to the results from full and truncated configuration interaction, many-body perturbation theory, and couple cluster theory with single and double excitations. Implications of using partial 3-positivity for variational 2-RDM calculations of many-electron atoms and molecules will be discussed.J. D. Farnum, D. A. Mazziotti. Chem. Phys. Lett. 416 142-146 (2005). "Trigonometric mapping for the electric field strength in molecular optimal control theory"
In molecular optimal control theory a laser pulse is designed to drive a molecular system to a target value of an observable. To be feasible in the laboratory, the optimal electric field must have a finite amplitude. Current algorithms limit the maximum amplitude indirectly by constraining the field’s energy. We introduce a trigonometric mapping for the field amplitude to limit the maximum amplitude explicitly. In a calculation with hydrogen fluoride we show that an explicit amplitude constraint is particularly important when we account for the electric field’s effect on the electronic structure.B. A. Rohrman, D. A. Mazziotti. J. Phys. Chem. B 109 13392-13396 (2005). "Quantum Chemical Design of Hydroxyurea Derivatives for the Treatment of Sickle-Cell Anemia"
Treatment of sickle-cell anemia by hydroxyurea has been shown to decrease patient mortality by 40%. In a rate-limiting step, hydroxyurea reacts with hemoglobin to form the nitroxide radical, which then decomposes to yield nitric oxide (NO). In this paper, we examine derivatives of hydroxyurea and their radicals by quantum chemical methods to identify derivatives that generate NO-producing radicals at a faster rate than hydroxyurea. The molecules are treated with Hartree−Fock theory, correlated wave function methods such as perturbation theory and coupled-cluster methods, and density functional theory. We observe that the inclusion of the correlation energy is important for an accurate comparison of the energy changes associated with modifications of the hydroxyurea molecule and its radical. The computational results are compared with available experimental data. All 19 derivatives of hydroxyurea, including a new medication for asthma Zileuton, manifest changes in their electronic energies that mark them as candidates for a faster formation of NO-producing radicals.Back to top
2004
D. A. Mazziotti. Phys. Rev. A 69 012507 (2004). "Exactness of wave functions from two-body exponential transformations in many-body quantum theory"
Recent studies have considered the possibility that the exact ground-state wavefunction from any Hamiltonian with two-particle interactions may be generated from a single finite two-body exponential transformation acting on an arbitrary Slater determinant [Piecuch et al., Phys. Rev. Lett. 90, 113001 (2003)]. Using the Campbell-Baker-Hausdorff relation, we show that it is difficult for the variational minimum of this trial wave function to satisfy the contracted Schrödinger equation which is a necessary and sufficient condition for the wave function to satisfy the Schrödinger equation. A counterexample is presented through the Lipkin quasispin model with 4–50 fermions. When the number of fermions exceeds four, the wave function from a finite two-body exponential transformation is shown to be inexact. If the trial wave function ansatz is extended to include products of finite two-body exponential transformations acting on an arbitrary Slater-determinant reference, then we show that the ansatz includes the exact ground-state wave function from any Hamiltonian with only two-particle interactions. Connections between the two-body exponential transformation of the wave function and recent research on two-body exponential similarity transformations of the Hamiltonian [S.R. White, J. Chem. Phys. 117, 7472 (2002)] are discussed.D. A. Mazziotti. Phys. Rev. Lett. 93 213001 (2004). "Realization of Quantum Chemistry without Wave Functions through First-Order Semidefinite Programming"
Determining the energy and properties of an N-electron molecule through a two-electron variational optimization has been a dream for more than half a century. While optimizations, using two-electron reduced density matrices constrained to represent N electrons, have recently been achieved, the computational costs are prohibitive. In this report an efficient algorithm with an order-of-magnitude reduction in floating-point operations and memory usage is presented. Because the optimization occurs on the space of two electrons, this method automatically treats strong, multireference correlation. Application is made to N2 and H6 where the method yields consistent accuracy at all geometries.G. Gidofalvi, D. A. Mazziotti. Chem. Phys. Lett. 398 434-439 (2004). "Variational reduced-density-matrix theory: strength of Hamiltonian-dependent positivity conditions"
Variational reduced-density-matrix (RDM) theory, employing the 2-RDM as the primary variable, has the potential to overcome the scaling limitations of configuration interaction to provide accurate electronic ground-state energies. A significant aspect of variational RDM theory is the inclusion of N-representability conditions which ensure that the 2-RDM corresponds to the N-particle wavefunction. Recent implementations of the method have mainly considered Hamiltonian-independent positivity conditions. In this Letter, we evaluate the strength of two Hamiltonian-dependent conditions. While one of the conditions is proven inactive, the positivity of the matrix SHji=〈ψ|Cˆi†[Hˆ,Cˆj]|ψ〉 provides additional N-representability conditions that may be beneficial in future RDM calculations.D. A. Mazziotti. J. Chem. Phys. 121 10957-10966 (2004). "First-order semidefinite programming for the direct determination of two-electron reduced density matrices with application to many-electron atoms and molecules"
Direct variational calculation of two-electron reduced density matrices (2-RDMs) for many-electron atoms and molecules in nonminimal basis sets has recently been achieved through the use of first-order semidefinite programming [D. A. Mazziotti, Phys. Rev. Lett. (in press)]. With semidefinite programming, the electronic ground-state energy of a molecule is minimized with respect to the 2-RDM subject to N-representability constraints known as positivity conditions. Here we present a detailed account of the first-order algorithm for semidefinite programming and its comparison with the primal-dual interior-point algorithms employed in earlier variational 2-RDM calculations. The first-order semidefinite-programming algorithm, computations show, offers an orders-of-magnitude reduction in floating-point operations and storage in comparison with previous implementations. We also examine the ability of the positivity conditions to treat strong correlation and multireference effects through an analysis of the Hamiltonians for which the conditions are exact. Calculations are performed in nonminimal basis sets for a variety of atoms and molecules and the potential-energy curves for CO and H2O.J. D. Farnum, D. A. Mazziotti. Chem. Phys. Lett. 400 90-93 (2004). "Extraction of ionization energies from the ground-state two-particle reduced density matrix"
Two methods for extracting ionization energies from the two-particle reduced density matrix (2-RDM) are explored: (i) diagonalization of the Hamiltonian in the basis of a single annihilation operator applied to the ground-state wave function |Ψg〉 and (ii) diagonalization of the Hamiltonian in the basis of two annihilation and one creation operators applied to the ground-state |Ψg〉. While the second basis set is more accurate, its Hamiltonian matrix elements depend upon the 3- and 4-RDMs. Using cumulant theory, however, we can approximate the 3- and 4-RDMs as functionals of the 2-RDM. Both methods are illustrated with calculations on a series of atoms and molecules.T. Juhász, D. A. Mazziotti. J. Chem. Phys. 121 1201-1205 (2004). "Perturbation theory corrections to the two-particle reduced density matrix variational method"
In the variational 2-particle-reduced-density-matrix (2-RDM) method, the ground-state energy is minimized with respect to the 2-particle reduced density matrix, constrained by N-representability conditions. Consider the N-electron Hamiltonian H(λ) as a function of the parameter λ where we recover the Fock Hamiltonian at λ=0 and we recover the fully correlated Hamiltonian at λ=1. We explore using the accuracy of perturbation theory at small λ to correct the 2-RDM variational energies at λ=1 where the Hamiltonian represents correlated atoms and molecules. A key assumption in the correction is that the 2-RDM method will capture a fairly constant percentage of the correlation energy for λ∈(0,1] because the nonperturbative 2-RDM approach depends more significantly upon the nature rather than the strength of the two-body Hamiltonian interaction. For a variety of molecules we observe that this correction improves the 2-RDM energies in the equilibrium bonding region, while the 2-RDM energies at stretched or nearly dissociated geometries, already highly accurate, are not significantly changed. At equilibrium geometries the corrected 2-RDM energies are similar in accuracy to those from coupled-cluster singles and doubles (CCSD), but at nonequilibrium geometries the 2-RDM energies are often dramatically more accurate as shown in the bond stretching and dissociation data for water and nitrogen.G. Gidofalvi, D. A. Mazziotti. Phys. Rev. A 69 042511 (2004). "Boson correlation energies via variational minimization with the two-particle reduced density matrix: Exact N-representability conditions for harmonic interactions"
A many-body theory for interacting bosons is developed within the framework of minimizing the ground-state energy with respect to the two-particle reduced-density matrix (2-RDM) subject to N-representability conditions. The N-representability conditions, which ensure that the 2-RDM may be derived from an N-particle wave function, are imposed through a hierarchy of positivity conditions where the p-positivity conditions restrict the metric matrices for p∕2-body operators to be positive semidefinite. Using two-positivity, we minimize the ground-state energies of 5–10 000 harmonically interacting bosons in a harmonic external potential. The energies and 2-RDMs obtained are in agreement with the exact solution except for round-off errors, which implies that for this class of boson interactions two-positivity conditions alone yield exact results for any interaction strength. The ground-state energies obtained at strong interactions are more accurate than many-body perturbative techniques by many orders of magnitude.J. D. Farnum, D. A. Mazziotti. J. Chem. Phys. 120 5962-5967 (2004). "Spectral difference Lanczos method for efficient time propagation in quantum control theory"
Spectral difference methods represent the real-space Hamiltonian of a quantum system as a banded matrix which possesses the accuracy of the discrete variable representation (DVR) and the efficiency of finite differences. When applied to time-dependent quantum mechanics, spectral differences enhance the efficiency of propagation methods for evolving the Schrödinger equation. We develop a spectral difference Lanczos method which is computationally more economical than the sinc-DVR Lanczos method, the split-operator technique, and even the fast-Fourier-Transform Lanczos method. Application of fast propagation is made to quantum control theory where chirped laser pulses are designed to dissociate both diatomic and polyatomic molecules. The specificity of the chirped laser fields is also tested as a possible method for molecular identification and discrimination.M. D. Benayoun, A. Lu, D. A. Mazziotti. Chem. Phys. Lett. 387 485-489 (2004). "Invariance of the cumulant expansion under 1-particle unitary transformations in reduced density matrix theory"
In the iterative solution of the contracted Schrödinger equation (CSE) the 3- and 4-particle reduced density matrices (RDMs) are reconstructed from the 2-RDM via cumulant expansions. Under 1-particle unitary transformations, we establish that the connected (or cumulant) part of an RDM maps onto the connected part of the RDM in the transformed basis set. Consequently, neglecting the connected RDM in the CSE produces an error which is invariant under unitary transformations of the one-particle basis set. We illustrate this result with calculations on beryllium. The present results are applicable to unitary localization in linear-scaling RDM calculations for large molecules.Back to top
2003
D. A. Mazziotti. Phys. Rev. A 68 052501 (2003). "Extraction of electronic excited states from the ground-state two-particle reduced density matrix"
Electronic excited states are computed through a technique for extracting them from a knowledge of the ground-state two-particle reduced density matrix (2-RDM). Diagonalization of the Hamiltonian in the basis set of one-particle excitations, deexcitations, and projections from the ground-state wave function yields those excited states dominated by one-particle excitations; however, explicit evaluation of the Hamiltonian requires a knowledge of the ground-state 4-RDM. Two techniques are developed for evaluating the Hamiltonian elements with only a knowledge of the 2-RDM: (i) the cumulant method in which the higher RDMs are reconstructed from the 2-RDM via cumulant theory and (ii) the Hermitian or anti-Hermitian operator method in which the higher RDMs are eliminated by selecting the basis functions to be Hermitian or anti-Hermitian. These techniques are extendable to the diagonalization of the Hamiltonian in basis sets of m-excitations where m>1. Excited-state energies with an accuracy between 0.1% and 0.001% of their absolute values are determined in illustrative calculations on Be, LiH, H2O, and BH.D. A. Mazziotti. Phys. Rev. E 68 066701 (2003). "Towards idempotent reduced density matrices via particle-hole duality: McWeeny’s purification and beyond"
Generalizations of McWeeny’s purification formula are developed within the formalism of the particle-hole duality from the theory of reduced density matrices. Each of the generalized purification formulas is expressed as a sum of the one-particle reduced density matrix (1-RDM) and a finite series in the product of the one-particle and the one-hole RDMs, a product which vanishes in the limit that the 1-RDM is idempotent. Two categories of purification formulas are explored: (i) formulas which treat the “occupied” and the “virtual” occupation numbers equivalently and (ii) formulas which treat these occupation numbers differently. The latter category includes and extends the purification formulas derived in the context of the 1,2-contracted Schrödinger equation [D. A. Mazziotti, J. Chem. Phys. 115, 8305 (2001)]. While the McWeeny purification minimizes the absolute error in the occupation numbers quadratically, the generalized purification formulas exhibit faster than quadratic convergence of the 1-RDM towards idempotency. Application of these purification formulas in existing algorithms for linear scaling will be explored and discussed including illustrative calculations on sodium wires of length 10, 20, 30, and 40 atoms.F. L. Yip, D. A. Mazziotti, H. Rabitz. J. Phys. Chem. A 107 7264-7268 (2003). "A Local-Time Algorithm for Achieving Quantum Control †"
A local-time algorithm (LTA) is developed for designing electric fields to guide a quantum system toward a desired observable. The LTA is a noniterative forward marching procedure based on making a choice for the control field over the next immediate small time increment t i +1 − t i = Δ solely on the ability of the local field value ε i in that increment to take the system closer to the target goal. Each locally optimal field value ε i , i = 1, 2,... is chosen from a fixed toolkit of discretized members {ε j } that sample the dynamic range εmin ≤ ε ≤ εmax of the control. Despite the strictly local myopic design process, the LTA is shown to be capable of achieving good quality control results in model systems. The LTA has no fixed time to reach the target, and the time it takes to produce good quality control primarily depends on the “distance” between the initial and target states, as measured by the number of intermediate state linkages connecting them and their strength. A comparison is made between the behavior of optimal control theory (OCT) and the LTA; each has different characteristics, and it is shown that the LTA can be computationally very efficient. LTA and traditional OCT methods can be viewed as extreme cases of a larger class of time-windowed approaches to control.F. Yip, D. Mazziotti, H. Rabitz. J. Chem. Phys. 118 8168-8172 (2003). "A propagation toolkit to design quantum controls"
A toolkit of time-propagation operators, to be stored and recalled as needed, is incorporated into the algorithms for the optimal control of quantum systems. Typically, the control field ε(t) revisits the same values many times during the full time evolution. This repetition may be utilized to enhance efficiency through a convenient toolkit of propagators where the propagators are computed and stored only at a small number of discrete electric-field values in the dynamic range εmin⩽ε(t)⩽εmax. At each time step of the controlled evolution a specific member of the pre-calculated toolkit is selected as dictated by the local control field value. The toolkit can reduce the cost of control field design by a factor scaling as ∼N for quantum systems described in a basis set of N states. Optimal control with the toolkit is demonstrated for systems up to dimension N=30.Back to top
2002
D. A. Mazziotti. J. Chem. Phys. 116 1239-1249 (2002). "Variational method for solving the contracted Schrödinger equation through a projection of the N-particle power method onto the two-particle space"
The power method for solving N-particle eigenvalue equations is contracted onto the two-particle space to produce a reduced “variational” method for solving the contracted Schrödinger equation (CSE), also known as the density equation. In contrast to the methods which solve a system of approximate nonlinear equations to determine the two-particle reduced density matrix (2-RDM) nonvariationally, the contracted power method updates the 2-RDM iteratively through a “gradient” of the N-particle energy. After each power iteration we modify the 2-RDM to satisfy certain N-representability conditions through an extension of purification to correlated RDMs. The contracted power method is illustrated with a variety of molecules. Significant features of the present calculations include (i) accurate results for both first- and second-order functionals for building the 3- and the 4-RDM’s from the 2-RDM’s; (ii) the first molecular implementation of the Mazziotti correction within the CSE [Mazziotti, Phys. Rev. A 60, 3618 (1999)]; (iii) a spin–orbital formulation; (iv) the treatment of both core and valence orbitals as active; and; (v) a reduction of the CSE computational scaling through fast summation and the natural-orbital transformation.D. A. Mazziotti. Phys. Rev. A 65 062511 (2002). "Variational minimization of atomic and molecular ground-state energies via the two-particle reduced density matrix"
Atomic and molecular ground-state energies are variationally determined by constraining the two-particle reduced density matrix (2-RDM) to satisfy positivity conditions. Because each positivity condition corresponds to correcting the ground-state energies for a class of Hamiltonians with two-particle interactions, these conditions collectively provide a new approach to many-body theory that, unlike perturbation theory, can capture significantly correlated phenomena including the multireference effects of potential-energy surfaces. The D, Q, and G conditions for the 2-RDM are extended through generalized lifting operators inspired from the formal solution of N-representability. These lifted conditions agree with the hierarchy of positivity conditions presented by Mazziotti and Erdahl [Phys. Rev. A 63, 042113 (2001)]. The connection between positivity and the formal solution explains how constraining higher RDMs to be positive semidefinite improves the N representability of the 2-RDM and suggests using pieces of higher positivity conditions that computationally scale like the D condition. With the D, Q, and G conditions as well as pieces of higher positivity the electronic energies for Be, LiH, H2O, and BH are computed through a primal-dual interior-point algorithm for positive semidefinite programming. The variational method produces potential-energy surfaces that are highly accurate even far from the equilibrium geometry where single-reference perturbation-based methods often fail to produce realistic energies.D. A. Mazziotti. Phys. Rev. E 65 026704 (2002). "Purification of correlated reduced density matrices"
The notion of purification is generalized to treat correlated reduced density matrices. Traditionally, purification denotes the process by which a one-particle reduced density matrix (1-RDM) is made idempotent, that is, its eigenvalues are mapped to either 0 or 1. Purification of correlated RDMs is defined as the iterative process by which an arbitrary RDM is forced to satisfy several necessary N-representability conditions. Using the unitary decomposition of RDMs and the positivity conditions, we develop an algorithm to purify the 2-RDM. The algorithm is applied within the solution of the contracted Schrödinger equation CSE for the 2-RDM [D. A. Mazziotti, Phys. Rev. A 57, 4219 (1998)]. Previous attempts to solve the CSE by powerlike methods have frequently produced divergent energies, but we show that the purification process eliminates the divergent behavior for systematic and reliable convergence of the contracted power method to the N-particle energy.D. A. Mazziotti. Phys. Rev. A 66 062503 (2002). "Solution of the 1,3-contracted Schrödinger equation through positivity conditions on the two-particle reduced density matrix"
Correlation energies and reduced density matrices (RDMs) of atoms and molecules are directly computed by solving the 1,3-contracted Schrödinger equation (1,3-CSE). The solution of the 1,3-CSE synthesizes two optimization strategies recently employed for the direct determination of the 2-RDM: (i) variational minimization of the energy with respect to a 2-RDM constrained by positivity conditions [D. A. Mazziotti, Phys. Rev. A 65, 062511 (2002)] and (ii) the contracted power method for solving the 2,4-CSE [D. A. Mazziotti, J. Chem. Phys. 116, 1239 (2002)]. While both the 3- and the 4-RDMs in the 2,4-CSE are reconstructed from the 2-RDM by cumulant expansions, similar techniques cannot be directly applied to the 1,3-CSE because constructing the 2-RDM from the 1-RDM with cumulant theory does not improve upon the mean-field approximation. We, however, establish a unique mapping from the 1-RDM to the 2-RDM by searching for the 2-RDM, constrained by contraction and N-representability conditions, which minimizes the energy. The 2-RDM constrained search is practically implemented through recent advances in positive semidefinite programming. With the variational reconstruction of the 2-RDM and a cumulant reconstruction of the 3-RDM, the 1,3-CSE may be solved via a contracted power method for the ground-state energy and RDMs. The initial RDMs, it is shown, need not be N representable for the contracted power method to converge; this allows us to choose the original RDMs from a variational calculation with approximate N-representability conditions on the 2-RDM. Application of the 1,3-CSE algorithm to atoms and molecules yields highly accurate correlation energies both near and far from equilibrium geometries.D. A. Mazziotti. J. Chem. Phys. 117 2455-2468 (2002). "Spectral difference methods for solving the differential equations of chemical physics"
Spectral differences [D. A. Mazziotti, Chem. Phys. Lett. 299, 473 (1999)] is a family of techniques for solving differential equations in which the summation in the numerical derivative is accelerated to produce a matrix representation that is not only exponentially convergent like the discrete variable representation (DVR) and other spectral methods but also sparse like traditional finite differences and finite elements. Building upon important work by Boyd [Comput. Methods Appl. Mech. Eng. 116, 1 (1994)] and Gray and Goldfield [J. Chem. Phys. 115, 8331 (2001)], we explore a new class of spectral difference methods which yields solutions that are more accurate than high-order finite differences by several orders of magnitude. With the generating weight for Gegenbauer polynomials we design a new spectral difference method where the limits of an adjustable parameter α generate both finite differences (α=∞), emphasizing the low Fourier frequencies, and a truncated sinc-DVR (α=0), emphasizing all Fourier frequencies below the aliasing limit of the grid. A range of choices for α∈[0,∞] produces solutions which are significantly better than the equivalent order of finite differences. We compare the Gegenbauer-weighted spectral differences with methods by Boyd as well as Gray and Goldfield which employ a hyperbolic secant and a step function as frequency weights, respectively. The solutions from the Gegenbauer- and the sech-weighted differences are shown to be less sensitive to parameter selection than the step-weighted differences. We illustrate all of the spectral difference methods through vibrational and quantum control calculations with diatomic iodine and the van der Waals cluster NeCO. Spectral differences also have important applications in molecular dynamics and electronic structure as well as other areas of science and engineering.Back to top
2001
D. A. Mazziotti. J. Chem. Phys. 115 6794-6795 (2001). "Comment on “High order finite difference algorithms for solving the Schrödinger equation in molecular dynamics” [J. Chem. Phys. 111, 10827 (1999)]"
The spectral difference methods [D. A. Mazziotti, Chem. Phys. Lett. 299, 473 (1999)] for solving differential equations in chemical physics combine the useful features of matrix sparsity and rapid convergence. In their recent article [J. Chem. Phys. 111, 10827 (1999)] Guantes and Farantos incorrectly classify the Lagrange distributed approximating functional (LDAF) method in the category of finite differences. This comment clarifies the connections among higher-order finite difference, Lagrange distributed approximating functionals, and other spectral difference methods.D. A. Mazziotti. Chem. Phys. Lett. 338 323-328 (2001). "Energy functional of the one-particle reduced density matrix: a geminal approach"
A significant portion of the energy functional of the one-particle reduced density matrix (1-RDM) is elucidated through geminal functional theory (GFT) [D.A. Mazziotti, J. Chem. Phys. 112 (2000) 10125]. We optimize the functional through an iterative solution of the 1,3-contracted Schrödinger equation (1,3-CSE). The method yields energies which are (i) above the true energy, (ii) more accurate than Hartree–Fock, and (iii) exact for a general family of correlated Hamiltonians. By applying a learning algorithm to these results, we determine correlation energies within 10% for atoms and molecules.D. A. Mazziotti. J. Chem. Phys. 115 8305-8311 (2001). "Linear scaling and the 1,2-contracted Schrödinger equation"
A contracted Schrödinger equation (1,2-CSE) is derived for the class of Hamiltonians without explicit interactions including those from Hartree–Fock and density functional theories. With cumulant reconstruction of the two-particle reduced density matrix (2-RDM) from the one-particle-RDM (1-RDM), the 1,2-CSE may be expressed solely in terms of the 1-RDM. We prove that a 1-RDM satisfies the 1,2-CSE if and only if it is an eigenstate of the N-particle Schrödinger equation. The 1,2-CSE is solved through the development and implementation of a reduced, linear-scaling analog of the ordinary power method for finding matrix eigenvalues. The power formula for updating the 1-RDM requires fewer matrix operations than the gradient procedure derived by Li et al. [Phys. Rev. B 47, 10891 (1993)] and Daw [Phys. Rev. B 47, 10895 (1993)]. Convergence of the contracted power method with purification is illustrated with several molecules. While providing a new tool for semiempirical, Hartree–Fock, and density functional calculations, the 1,2-CSE also represents an initial step toward a linear-scaling algorithm for solving higher CSEs which explicitly treat electron correlation.D. A. Mazziotti, R. M. Erdahl. Phys. Rev. A 63 042113 (2001). "Uncertainty relations and reduced density matrices: Mapping many-body quantum mechanics onto four particles"
For the description of ground-state correlation phenomena an accurate mapping of many-body quantum mechanics onto four particles is developed. The energy for a quantum system with no more than two-particle interactions may be expressed in terms of a two-particle reduced density matrix (2-RDM), but variational optimization of the 2-RDM requires that it corresponds to an N-particle wave function. We derive N-representability conditions on the 2-RDM that guarantee the validity of the uncertainty relations for all operators with two-particle interactions. One of these conditions is shown to be necessary and sufficient to make the RDM solutions of the dispersion condition equivalent to those from the contracted Schrödinger equation (CSE) [Mazziotti, Phys. Rev. A 57, 4219 (1998)]. In general, the CSE is a stronger N-representability condition than the dispersion condition because the CSE implies the dispersion condition as well as additional N-representability constraints from the Hellmann-Feynman theorem. Energy minimization subject to the representability constraints is performed for a boson model with 10, 30, and 75 particles. Even when traditional wave-function methods fail at large perturbations, the present method yields correlation energies within 2%.Back to top
2000
D. A. Mazziotti. Math. Comput. Chem. 139-163 (2000). "Many-Electron Densities and Reduced Density Matrices"
The quantum-mechanical wave function of an N-electron system contains much more information than is required to compute the expectation values for most observables. Because the interactions between electrons are pairwise within the Hamiltonian, the energy may be determined exactly through a knowledge of the two-particle reduced density matrix (2-RDM) [1, 2]. Unlike the unknown dependence of the energy on the one-particle density in density functional theory (DFT) [3], the dependence of the energy on the 2-RDM is linear. The 2-RDM, however, has not replaced the wave function as the fundamental parameter for many-body calculations because not every 2-particle density matrix is derivable from an N-particle wave function. The need for a simple set of necessary and sufficient conditions for ensuring that the 2-RDM may be represented by an N-particle wave function is known as the N-representability problem [4, 5]. Recent theoretical and computational results with the contracted Schrödinger equation (CSE), also known as the density equation, indicate that the CSE offers an accurate, versatile method for generating the 2-RDM without the wave function [6–17]. In the present article we will review the foundations of the CSE method.D. A. Mazziotti. J. Chem. Phys. 112 10125-10130 (2000). "Geminal functional theory: A synthesis of density and density matrix methods"
The energy of any atom or molecule with an even number N of electrons is shown to be an exact functional of a single geminal where the functionals for both the kinetic energy and the external potential are explicitly known. We derive the foundations for geminal functional theory (GFT) through a generalized constrained search and the use of two theorems which demonstrate that all one-particle properties of atoms and molecules with even N may be parametrized by a single geminal [A. J. Coleman, Int. J. Quantum Chem. 63, 23 (1997); D. W. Smith, Phys. Rev. 147, 896 (1966)]. By generalizing constrained search to optimize the universal functionals with respect to the 2-RDM (two particle reduced density matrix) rather than the wave function, we closely connect the one-density, the 1-RDM (one-particle reduced density matrix), and the geminal functional theories with 2-RDM minimization of the energy. Constrained search with the 2-RDM emphasizes that all energy functional methods must implicitly account for the N-representability of the 2-RDM within their universal functionals. An approximate universal functional for GFT, equivalent to a variational ansatz using the antisymmetrized geminal power wave function, yields energies that are significantly better than those from Hartree–Fock and yet rigorously above the exact energy.D. A. Mazziotti. Chem. Phys. Lett. 326 212-218 (2000). "Complete reconstruction of reduced density matrices"
Different from traditional electronic structure methods, the contracted Schrödinger equation with reduced-density-matrix (RDM) reconstruction may be exact when only the 2-particle RDM is employed as the fundamental parameter. Although Rosina's theorem indicates that the 3 and the 4-RDMs are functionals of the 2-RDM, cumulant theory generates only those terms expressible as antisymmetrized products of lower RDMs. We present a formal solution for reconstruction where the approximate cumulant formulas are systematically corrected through contraction conditions. Using a part of the formal 3-RDM reconstruction, the CSE is compared with other methods through a quasi-spin model containing as many as eight-hundred fermions.D. A. Mazziotti, D. R. Herschbach. Phys. Rev. A 62 043603 (2000). "Boson correlation energies and density matrices from reduced Hamiltonian interpolation"
The ground-state energies of interacting bosons are computed beyond the mean-field approximation by a new method known as reduced Hamiltonian interpolation (RHI) [D. A. Mazziotti and D. R. Herschbach, Phys. Rev. Lett. 83, 5185 (1999)]. In the RHI, the N-particle Hamiltonian is represented through a sequence of p-particle expanded and reduced Hamiltonians that give upper and lower bounds on the true energy. Assimilating ideas from N representability and dimensional interpolation, the RHI technique interpolates over the number p of quasiparticles to determine the N-particle energy with close upper and lower bounds. With the Hellmann-Feynman theorem, we extend the RHI to compute the two-particle reduced density matrix (2RDM) as well as the energy. We examine both the computational advantages of RHI in comparison with traditional methods and the possibility of extending the RHI to treat fermion correlation. Applied to bosons with harmonic interactions as well as a two-level system, the RHI technique yields more than 99% of the correlation energy and an accurate correlation correction for the elements of the 2RDM.D. A. Mazziotti, H. A. Rabitz. J. Phys. Chem. A 104 9770-9776 (2000). "Determining Quantum Molecular Potentials from Spectroscopic Energy Levels Using Parametric Equations of Motion"
A system of differential equations is presented for evolving the quantum potential as a function of its energy levels. These inverse parametric equations of motion (i-PEM) offer a novel approach to determining quantum molecular potentials from spectroscopic energy levels. The technique uses singular-value decomposition to ensure that the chosen trajectory through energy space is representable by a smooth potential trajectory. The i-PEM are facilitated by discretizing the vibrational Schrödinger equation with a spectral element method which combines the features of Hamiltonian sparsity and exponential convergence of the wave function. Often, spectroscopic data significantly affect only a specific region of the potential, and the spectral elements offer a natural framework for identifying the appropriate portion of the potential. The i-PEM with spectral elements are applied in a simulation for determining the potential of hydrogen fluoride.Back to top
1999
D. A. Mazziotti. Chem. Phys. Lett. 299 473-480 (1999). "Spectral difference methods for solving differential equations"
A family of recently developed techniques is explored for achieving both matrix sparsity and rapid convergence when numerically solving differential and eigenvalue equations without domain decomposition. These methods, which we call spectral differences, include Boyd's sum acceleration techniques and the Lagrange distributed approximating functional (LDAF) approach. A formula is developed for estimating the unknown Gaussian parameter within LDAF. We implement these methods to calculate the Morse vibrational energies for diatomic iodine. For equivalent bandwidths the sum acceleration with finite difference weights generates energies which are between two and three orders of magnitude more accurate than those from LDAF.D. A. Mazziotti. Phys. Rev. A 60 3618-3626 (1999). "Pursuit of N-representability for the contracted Schrödinger equation through density-matrix reconstruction"
The solution of N-representability for the two-particle reduced density matrix (2-RDM) is intimately connected through the contracted Schrödinger equation with the problem of reconstructing the 3- and 4-RDM’s from the 2-RDM [D. A. Mazziotti, Phys. Rev. A 57, 4219 (1998)]. We derive an explicit formula for building the 3-RDM from the 2-RDM which significantly improves the accuracy of the 3-RDM cumulant expansion. Calculations indicate that the improvement is better than that achieved by the correction of Nakatsuji and Yasuda (NY) [Phys. Rev. Lett. 76, 1039 (1996); Phys. Rev. A 56, 2648 (1997)]. We also present a more efficient formulation of the NY correction. Both the present and NY reconstructions are shown to satisfy the particle-hole duality. Comparison of the reconstruction methods is made through several molecules LiH, BeH2, BH3, H2O, and CH4, as well as systems with more general pairwise correlation.D. A. Mazziotti. Phys. Rev. A 60 4396-4408 (1999). "Comparison of contracted Schrödinger and coupled-cluster theories"
The theory of the contracted Schrödinger equation (CSE) [D. A. Mazziotti, Phys. Rev. A 57, 4219 (1998)] is connected with traditional methods of electronic structure including configuration-interaction (CI) and coupled-cluster (CC) theory. We derive a transition contracted Schrödinger equation (TCSE) which depends on the wave function ψ as well as another N-particle function χ through the two-, three-, and four-particle reduced transition matrices (RTMs). By reconstructing the 3 and 4 RTMs approximately from the 2-RTM, the indeterminacy of the equation may be removed. The choice of the reconstruction and the function χ determines whether one obtains the CI, CC, or CSE theory. Through cumulant theory and Grassmann algebra we derive reconstruction formulas for the 3- and 4-RTMs which generalize both the reduced density matrix (RDM) cumulant expansions as well as the exponential ansatz for the CC wave function. This produces a fresh approach to CC theory through RTMs. Two theoretical differences between the CC and the CSE theories are established for energetically nondegenerate states: (i) while the CSE has a single exact solution when the 3- and 4-RDMs are N-representable, the CC equations with N-representable 3- and 4-RTMs have a family of solutions. Thus, N-representability conditions offer a medium for improving the CSE solution but not the CC solution, and (ii) while the 2-RDM for an electronic Hamiltonian reconstructs to unique N-representable 3- and 4-RDMs, the 2-RTM builds to a family of N-representable 3- and 4-RTMs. Hence, renormalized reconstructions beyond the cumulant expansion may be developed for the 2-RDM but not for the 2-RTM without explicit use of the Hamiltonian. In the applications we implement our recently developed reconstruction formula for the 3-RDM which extends beyond the cumulant approximation. Calculations compare the 3-RDM and 3-RTM reconstructions for the molecules LiH, BeH2, BH3, and H2O as well as for systems with more general two-particle interactions. The TCSE offers a unified approach to electronic structure.D. A. Mazziotti, D. R. Herschbach. Phys. Rev. Lett. 83 5185-5189 (1999). "Boson Correlation Energies from Reduced Hamiltonian Interpolation"
The ground-state energies of interacting bosons are computed beyond the mean-field approximation by a new method which we call reduced Hamiltonian interpolation (RHI). In this interpolation the N-particle Hamiltonian is represented through a sequence of p-particle expanded and reduced Hamiltonians that give upper and lower bounds on the true energy. A synthesis of ideas from N-representability and dimensional interpolation, the RHI interpolates over the number p of quasiparticles (equivalent to spatial dimension) to calculate the N-particle energy as the mean of close upper and lower bounds. Application to bosons with harmonic interactions yields more than 99% of the correlation energy.Back to top
1998
D. A. Mazziotti. Int. J. Quantum Chem. 70 557-570 (1998). "3,5‐contracted Schrödinger equation: Determining quantum energies and reduced density matrices without wave functions"
Through the 3,5‐contracted Schrödinger equation (3,5‐CSchE) quantum energies and 3‐particle reduced density matrices (3‐RDMs) are determined directly without wave functions. Since the 3,5‐CSchE involves the 5‐RDM, its solution is indeterminate without N‐representability conditions. However, the indeterminacy of the 3,5‐CSchE may be removed through a reconstruction strategy for building the 4‐ and 5‐RDMs from the 3‐RDM. We present a systematic procedure for obtaining corrections for Valdemoro's reconstruction functionals from two complementary approaches, the particle–hole duality and the theory of cumulants. With the cumulants we are able to demonstrate that we have obtained all terms in the reconstruction functionals which may be written as antisymmetric products of the lower rdms. The cumulants allow us to understand the reconstruction functionals in terms of a renormalized many‐body perturbation theory. The reconstruction functionals also lead to a natural generalization of Wick's theorem for evaluating expectation values of fermionic annihilation and creation operators with respect to correlated reference states. Previous work [Phys. Rev. A 57, 4219 (1998)] has explored the determination of correlation energy and 2‐RDMs through the 2,4‐CSchE, also known as the density equation. Because the reconstruction functionals employed with the 3,5‐CSchE depend only on the antisymmetric products of lower RDMs in constrast to those used with the 2,4‐CSchE, the 3,5‐CSchE method presented here does not require the solution of systems of linear equations during reconstruction or the storage of the reconstructed RDMs. Application of the 3,5‐CSchE technique to a quasi‐spin model generates ground‐state energies and 2‐RDMs similar in accuracy to single–double configuration interaction (SDCI). We employ a simple iterative procedure for the solution of the 3,5‐CSchE without traditional diagonalization. The CSchE techniques offer an approximate solution of the N‐representability problem and a new approach to electron correlation. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 557–570, 1998D. A. Mazziotti. Chem. Phys. Lett. 289 419-427 (1998). "Approximate solution for electron correlation through the use of Schwinger probes"
Quantum energies and two-particle reduced density matrices (2-rdms) may be determined without using the N-particle wavefunction by combining the reconstruction of higher rdms from lower rdms and the contracted Schrödinger equation. In this letter we derive a systematic procedure for obtaining reconstruction functionals through the use of Schwinger probes. Previous functionals for the 3 and 4-rdms are generated as well as new functionals for higher rdms. Through a quasi-spin model we demonstrate that the 5-rdm, normalized to unity, may be reconstructed from lower rdms with the accuracy of its elements ranging from 10−4 for five fermions to 10−8 for fifty fermions.D. A. Mazziotti. Phys. Rev. A 57 4219-4234 (1998). "Contracted Schrödinger equation: Determining quantum energies and two-particle density matrices without wave functions"
The contracted Schrödinger equation (CSE) technique through its direct determination of the two-particle reduced density matrix (2RDM) without the wave function may offer a fresh alternative to traditional many-body quantum calculations. Without additional information the CSE, also known as the density equation, cannot be solved for the 2RDM because it also requires a knowledge of the 4RDM. We provide theoretical foundations through a reconstruction theorem for recent attempts at generating higher RDMs from the 2RDM to remove the indeterminacy of the CSE. With Grassmann algebra a more concise representation for Valdemoro’s reconstruction functionals [F. Colmenero, C. Perez del Valle, and C. Valdemoro, Phys. Rev. A 47, 971 (1993)] is presented. From the perspective of the particle-hole equivalence we obtain Nakatsuji and Yasuda’s correction for the 4RDM formula [H. Nakatsuji and K. Yasuda, Phys. Rev. Lett. 76, 1039 (1996)] as well as a corrective approach for the 3RDM functional. A different reconstruction strategy, the ensemble representability method (ERM), is introduced to build the 3- and 4-RDMs by enforcing four-ensemble representability and contraction conditions. We derive the CSE in second quantization without Valdemoro’s matrix contraction mapping and offer the first proof of Nakatsuji’s theorem for the second-quantized CSE. Both the functional and ERM reconstruction strategies are employed with the CSE to solve for the energies and the 2RDMs of a quasispin model without wave functions. We elucidate the iterative solution of the CSE through an analogy with the power method for eigenvalue equations. Resulting energies of the CSE methods are comparable to single-double configuration-interaction (SDCI) energies, and the 2RDMs are more accurate by an order of magnitude than those from SDCI. While the CSE has been applied to systems with 14 electrons, we present results for as many as 40 particles. Results indicate that the 2RDM remains accurate as the number of particles increases. We also report a direct determination of excited-state 2RDMs through the CSE. By circumventing the wave function, the CSE presents new possibilities for treating electron correlation.Back to top
1996
D. A. MAZZIOTTI. Mol. Phys. 89 171-193 (1996). "Determining quantum bound-state eigenvalues and eigenvectors as functions of parameters in the Hamiltonian: an efficient evolutionary approach"
This paper addresses the problem of finding the quantum bound-state energy eigenvalues and eigenvectors as functions of a set of continuous parameters characterizing a Hamiltonian. A recent paper introduced a parametric equations of motion (PEM) method for this purpose, and the present work extends the method to allow for the analysis ofa single energy level and its wavefunction. After solving the Schrodinger equation for its nth eigenvalue and eigenvector, evaluated at a reference value of the Hamiltonian's parameters, the differential equations of the single-state PEM (ss-PEM) method are used to propagate the nth energy level and its eigenfunction through the entire parameter space of the Hamiltonian. The new ss-PEM method, which reduces the number of differential equations to be solved, appears more efficient than diagonalization when the energy is sought at a moderate number of values for the parameters in the Hamiltonian. The PEM methods are extended to treat non-orthogonal basis sets that facilitate more rapid convergence of the solutions. The energy of the ss-PEM, which is always an upper bound to the true energy, is exact in the limit of a complete basis set. Connections of the method are made to linear variational calculations, Dalgarno-Lewis perturbation theory and the original PEM methods. Sets of non-orthogonal Chebyshev polynomials are employed in illustrations of the ss-PEM method to determine (a) the ground-state energy as a function of internuclear separation in the hydrogen molecular ion, and (b) the ground-state energy of two electron ions as a function of nuclear charge. The calculation withthe two-electron ions involves two parameters, the nuclear charge and a basis set parameter that influences the distribution of the nodes of the Chebyshev basis functions. Evolution of the basis set parameter to improve the energies of the ions suggests an additional application of the ss-PEM method in which quantum energies are minimized with respect to nonlinear basis set parameters. The ss-PEM method offers an effective tool for mapping the solutions of the Schro dinger equation as a function of model parameters in the Hamiltonian.Back to top
