An Alternative Density Functional Theory DFT

In recent years, an alternative approach to implementing the Schrodinger equation for quantitative electronic structure calculations has appeared. Instead of calculating wave-functions of the sort we have described, these methods focus on the electron density (p) across the entire molecule. It has been shown that if p is known precisely, one can in principle determine the total energy (and all other properties of the system) precisely. In addition, p is certainly simpler than the complicated total wavefunction used in the orbital approximation (Eq. 14.13). 

Unfortunately, we do not know the mathematical function that relates p to the energy. Therefore, the function must be guessed at rather than derived. In fact, because p is a function of the coordinates of the system, the function we need to relatep to energy is actually not a function, but rather a functional. A functional is a mathematical function that has a mathematical function (in this casep) as its argument. For example, if g(x) = p, then f g(x)) is the functional that gives the energy. Thus, this theoretical approach is called density functional theory (DFT). 

The basics of DFT are embodied in Eq. 14.54. The total energy is partitioned into several terms. Each term is itself a functional of the electron density. is the electron kinetic energy term (the Bom-Oppenheimer approximation is in place, so nuclear kinetic energy is neglected). The E potential energy term includes both nuclear-electron attraction and nuclear-nuclear repulsion. The term is sometimes called the Coulomb self-interaction term, and it evaluates electron-electron repulsions. It has the form of Coulomb s law. The sum of the first three terms (E + -I- ) corresponds to the classical energy of the charge distribution. 

It is not obvious what the optimal form for the E functional is. Usually, it is broken into two parts, the exchange functional and the correlation functional (Eq. 14.55). A number of different forms have been suggested for each. A currently popular form for the exchange functional is one developed by Becke in 1988. While the exact form of this functional is not given here, it is important to realize that it contains a parameter (y) that is chosen to fit experimental data related to atomic exchange energies. As such, DFT methods have a semi-empirical flavor, in that there is a fit to experimental data (see Section 14.2.3). DFT as typically implemented is therefore not an ab initio method. 

Popular forms for the correlation functional have been developed by Perdew and Wang and by Lee, Yang, and Parr. In recent years a method known as Becke3LYP (or equivalently, B3LYP) is emerging as a favorite of DFT practitioners. This is actually a hybrid HF/DFT method, in which E is composed of both HF and DFT exchange terms (recall HF does include exchange interactions) and DFT electron correlation functionals. These various terms 

---