What is? 

History 

B3LYP was first described in the literature in 1994 by Stevens and colleagues who used it to perform calculations using the GAUSSIAN 92/DFT software. They note that B3LYP appeared to be inspired by a functional published by Axel Becke in the previous year (1993), sometimes called B3, which was the first to combine density functional and Hartree-Fock exchange, i.e. the first hybrid GGA. B3 was inspired by the adiabatic-connection of Harris (1984), which shows that the true exchange-correlation functional can be found analytically by interpolating between interacting and non-interacting systems of electrons.  B3 functional captured this idea by including a mixture of LSDA and HF exchange energies (governed by a single mixing parameter) along with an additional parameter-weighted exchange energy from Becke’s own "B88" GGA functional.  Becke found a value for the mixing parameter by fitting calculations with his hybrid functional to a database of atomization energies. B3LYP is identical to B3 the exchange terms, and differs only in the correlation terms, where it uses the LYP functional in place of the PW98 functional for the GGA correlation and uses the VWN functional for LSDA correlation.

In the intervening years, B3LYP has become one of the most popular exchange-correlation functionals used in DFT, sometimes called the industry standard. This is likely because it was the first functional to give high enough accuracy in DFT for chemist to carry out calculations that were comparable enough to experiment to be useful. In recent years, this default status has been challenged by some researchers who argue that since its invention in 1993/1994, numerous improvements have been made to hybrid-GGA functionals. This includes the development new families of hybrid GGAs, new classes of functionals (e.g., hybrid meta GGAs), and improved B3LYP variants such as one designed to explicitly correct for B3LYP's neglect of dispersion interactions, B3LYP-D3(BJ). 

Details 

The name "B3LYP" is an acrynym for "Becke 3-paramter Lee-Yang-Parr", as it uses three parameters to determine the mixing of exchange and correlation approximations including Becke's B88 GGA exchange functional and Lee, Yang and Parr's GGA correlation functional. 

The full functional form of B3LYP is 

where subscripts and denote exchange and correlation, the three parameters and are scalar coefficients, and VWN is the Vosko-Wilk-Nusair LSDA correlation functional. The parameter values are determined empirically by minimizing the error of some DFT calculations against a reference (usually experimental data).  

The mixture of density-functional with Hartree-Fock exchange was inspired by the adiabatic connection of Harris, which begins by considering perturbed KS Hamiltonian  

for . When there is no electron-electron interaction ( disappears). When  is called the fully interacting limit, or a real system to which we have no solution, where we have both and the electron-nuclear interaction potential. When is adjusted such that the density is constant over . Harris showed that the that would give an exact solution to the Schrodinger equation is  

Assuming that is linear in Becke approximated this integral as 

in the case, there is no correlation and only exchange, so the HF equation can be used. Meanwhile, we can approximate the fully correlated system with a density functional. This lead to Becke's "half and half" functional given by 

Becke's 93 functional (and B3LYP) is an extension of this idea to arbitrary mixtures of HF and LSDA exchange with the addition of GGAs for their accuracy improbements over LSDAs. 

Example 

We will now demonstrate how to use B3LYP for DFT calculations in the Maple Quantum Chemistry Toolbox. 

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We can use diatomic Nitrogen as an example molecule by importing its geometry 

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(4.1)

 

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We can then compute electronic properties of benzene with DFT, the B3LYP functional, and the cc-pVDZ basis set.  

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The result is a Maple table of different values obtained from the DFT calculation. We can see the names of these values: 

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(4.2)

 

The total energy is  

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(4.3)

 

The orbital energies are 

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(4.4)

 

And their occupations are 

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(4.5)

 

To find the highest occupied molecular orbital (HOMO), we just find the last orbital that is occupied 

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(4.6)

 

We can visualize this orbital: 

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We know from the MO diagram of N2 that this is a bonding orbital, and what we see appears like two orbitals with a bonding σ overlap. 

Of course, the LUMO is just the next orbital above the HOMO: 

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(4.7)

 

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Which is a anti-bonding orbital. 

With the HOMO and LUMO energies, we can calculate the transition energy (a.k.a. HOMO-LUMO gap, band gap) here reported in Hartrees. 

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	(4.8)

 

Because DFT is relatively efficient compared to most wavefunction methods that model electron correlation, we can quickly visualize the orbitals of a much larger molecule, say caffiene 

>	caffeine := MolecularGeometry("caffeine"): data := DensityFunctional(caffeine, xc = "B3LYP", basis = "cc-pVDZ"):

 

We can find and plot the HOMO:  

>	HOMO := ListTools:-SelectLast(o -> 0 < o, data[mo_occ], output = indices);

 

(4.9)

 

>	DensityPlot3D(caffeine, data, basis = "cc-pVDZ", orbitalindex = HOMO, gridspacing = 0.00100000, maximumpoints = 400000)

 

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References