NoteTaking_C3: Computational Organic Chemistry

CHAPTER 1: QUANTUM MECHANICS FOR ORGANIC CHEMISTRY (3)

Density Functional Theory

1) Hohenberg-Kohn existence theorem: The exact electronic energy is a unique functional of electron density.

The electron density obeys the variational theorem.

2) Functional relates a function to a scalar quantity.

3) Kohn-Sham equation

A key point in Kohn-Sham equation is that the kinetic energy is defined as the kinetic energy of noninteracting electrons whose density is the same as the density of the real interacting electrons. The exchange-correlation functional takes care of all the other aspects of a true system.
The Kohn-Sham procedure is then to solve for the orbitals that minimize the energy.

The Kohn–Sham orbitals are separable by definition (the electrons they describe are noninteracting), analogous to the HF MOs. Therefore, the Kohn-Sham equation can be solved in a self-consistent way using similar steps as was used in the Hatree-Fork-Roothann method. 

Therefore, for a similar cost as the HF methods, DFT reproduces the energy of a molecule that includes the electron correlation.  But, there is no guidance as to the form of the density functional. 

Paraphrasing Cramer’s description of the contrast between HF and DFT, HF and the various post-HF electron correlation methods provide an exact solution to an approximate theory, but DFT provides an exact theory with an approximate solution.

4)  Exchange-Correlation Functionals

                                            E_{xc}=E_x + E_c

Local density Approximation (LDA) and Generalized Gradient Approximation (GGA)

Exchange functional:

"B" Becke exchange 

Correlational functional:

"LYP" Lee, Yang and Parr, second derivative of density

"PW" Perdew and Wang , first derivative of density

Hybrid functional: add some HF exchange