Quantum Mechanics

Density Functional Models

[Semantics: a function is a prescription or formula for producing a number from a collection of variables. A functional then is a formula for producing a number from a function.]

Density functional models are based on the Hohenberg-Kohn (note: Kohn shared the Nobel with Pople) theorem:

The minimum energy of a collection of electrons under the influence of an external field (the nuclei) is a functional of the electron density.
This energy includes the same nuclear, core, and Coulomb terms as the Hartree-Fock energy.
However, the HF exchange energy is replaced by an exchange-correlation functional, E^XC(P), leading to the Kohn-Sham equation:

That this is a unique functional, valid for all systems, can be proved, but an explicit form of the potential has been difficult to define.

Hence the large number of density functional (DFT) methods.
Functional forms often are chosen based on which ones give the best fit to a certain body of experimental data, which makes DFT a semi-empirical method.

One advantage to using electron density is that the integrals for Coulomb repulsion need to be done only over the electron density.

Electron density is a three-dimensional function
- In terms of computational complexity, it scales as N³ (N = number of basis functions)
HF calculations scale as N⁴.

The other advantage is that use of electron density automatically includes at least some of the electron correlation.

Among the simplest models are those called local density models, such as the SVWN (Slater, Vosko, Wilk, Nusair) functional.

The basis of these models is the assumption of a many-electron gas of uniform density
This model which would not be expected to be satisfactory for molecular systems, in which the electron density surely is non-uniform.
However, for band structure, they are quite satisfactory.

Substantial improvement is obtained by introducing explicit dependence on the gradient of the electron density, as well as the density itself.

Such procedures are called gradient-corrected, or non-local DFT models.

The most popular of these models are the so-called BP (Becke, Perdew) and BLYP (Becke, Lee, Yang, Parr).

A third class of DFT models combines the exact Hartree-Fock exhange with a DFT exchange term, and adds a correlation functional.

The best model of this type is termed the B3LYP, where the "3" is the number of correction terms used to balance the two exchange components.

Combined with a good, split-valence basis set such as 6-31G* to generate the orbitals and initial electron density, B3LYP is probably the best method for obtaining good structures and energies for organic molecules at reasonable computational cost.

A Few Density Functionals
Acronym	Name	Type
HFS	Hartree-Fock Slater	HF with LDA exchange
VWN	Vosko, Wilks, and Nusair	Local density
BLYP	Becke correlation functional; Lee, Yang, Parr exchange terms	Corrected gradient
B3LYP	Becke 3-term functional; Lee, Yang, Parr exchange	Hybrid
P86	Perdew 1986	Gradient corrected
B3P86	Becke 3-part functional; Perdew correlation	Hybrid

References

Hinchcliffe, A., Modelling Molecular Structures, 2nd ed., Wiley, New York, 2000; Chapter 13.

Koch, W.; Holthausen, M. C., A Chemist's Guide to Density Functional Theory, Wiley-VCH, Weinheim, 2000.

Bartolotti, L. J.; Flurchick, K., in Lipkowitz, K. B.; Boyd, D. B., eds., Reviews in Computational Chemistry, 1996; Chapter 4.

Parr, R. G.; Yang, W., Density-Functional Theory of Atoms and Molecules, Oxford University Press, 1989.

Young, D., Computational Chemistry: A Practical Guide for Applying Techniques to Real World Situations, Wiley-Interscience, 2001; Chapter 5.

Cramer, C. J., Essentials of Computational Chemistry, Wiley, 2003; Chapter 8.

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