Research

Advancement in reduced-density-matrix theory is fostering the development of a new paradigm in theoretical chemistry that promises to promote unprecedented growth in our ability to explore computationally a myriad of chemical questions, ranging from structure to reactivity. The immediate impact of my research has been the development of new electronic structure methods with improved accuracy and efficiency for small-to-medium-sized atoms and molecules - both ground and excited-state properties. These methods will assist chemists in investigating experimental properties such as molecular geometries, bond stretching, bond polarity, electron density, dissociation, and excitation energies with reliable, consistent accuracy. The new methodology is not limited to electronic structure but rather is also appropriate for other aspects of chemistry, including the prediction of vibrational and rotational molecular properties.

While both Hartree-Fock and density functional theory work within the framework of a single electron, the importance of the electron pairing in the chemical bond is well-known to every chemist. In my research, the electron pair is elevated to a more prominent role in electronic structure. The dream of rigorously describing all chemical properties through only two electrons has existed for many years. It was initially inspired by the observation that because electrons interact only two-at-a-time, the electronic energy may be expressed exactly as a simple, known functional of the coordinates of two electrons. The distribution of the two electrons, however, may not properly represent a realistic, many-electron system. The development of systematic rules for constraining two electrons to represent a collection of more-than-two electrons is called the N-representability problem (this name was first proposed by Professor John Coleman). The N signifies the number of electrons in the collection.

In 1994 Professor Carmela Valdemoro achieved an approximate solution to the problem through a mapping of the Schrödinger equation for an N-electron atom onto a contracted Schrödinger equation (CSE) for an effective two-electron atom. Through independent efforts in the late-90s, Professor Nakatsuji at Kyoto University and I at Harvard University verified and extended Valdemoro's initial success. My 1998 paper in Physical Review A introduces the term reconstruction to describe the approximation of the four-electron distribution in terms of the two-electron distribution. The paper explores the delicate relationship between the N-representability problem and reconstruction; effectively, reconstruction provides an approximate solution to the important problem of representing many-electrons by only two electrons. My research computes the reconstruction within a framework known as cumulant theory.

Motivated by the contracted Schrödinger equation, we have also recently developed variational two-electron methods with systematic, nontrivial N-representability conditions. This second class of two-electron methods directly computes the effective two-electron probability distribution of a many-electron atom or molecule without any higher-electron probability distributions. Variational optimization of the ground-energy energy, in terms of only two effective electrons, is treatable by a class of optimization techniques known as semidefinite programming. The variational two-electron method has been accurately applied to generating potential energy surfaces of molecules, including the difficult-to-predict dissociation curve for N2, where wavefunction methods fail to give physically meaningful results.

While two-electron approaches are still in their early stages, the direct determination of chemical properties by mapping any atom or molecule onto an effective two electron problem offers a new level of accuracy and efficiency for electronic structure calculations.