Quantum Mechanical Insights into Biological Processes at the Electronic Level - Computational Modeling of Biological Systems - page 129

based on the density functional theory (DFT) [ 14 ]. DFT methods are often a good
alternative to electron-correlated methods, because they are relatively inexpensive,
and yet fairly accurate in many cases. DFT states that all properties of the system
are uniquely determined by its ground state electron probability density, 0 .x;y; z /,
which is a function of only three variables. In particular, the ground state energy is a
functional of 0 W E 0 D E 0 Π0 . The theory in principle should completely avoid the
use of the wave function. However, the problem is that in practice 0 is usually found
from the wave function, which then anyhow needs to be found in the first place.
What makes DFT practical is the use of the so-called Kohn-Sham orbitals. The
system of interest can be represented as a system of noninteracting electrons all
experiencing the same “external potential,” s . r i /, so as to reproduce the exact
electron probability density:

X
i D 1 j ' i j

N

2 ;

0 D (27)

where i are the spatial Kohn-Sham orbitals. The ground state wave function is
then the Slater determinant, in i . The exact ground state electronic energy can be
written in terms of one-electron Kohn-Sham orbitals and one-electron density as

Z ' i . r / r

X

1
2

2

EΠD

1 ' i . r /d r

i

Z . r 1 /. r 2 /
j

X

Z a 1

2

. r /d r C d r 1 d r 2 C E XC Œ: (28)

j r R a j r 1 r 2 j

a

The last term is the exchange-correlation energy, which is always empirical, and
makes DFT not exact. It is also the term that is the most difficult to get right, because
it is very problem dependent.
The Kohn-Sham orbitals are found variationally, through iteratively solving the
Kohn-Sham equations (SCF procedure). i are eigenfunctions of the one-electron
operators,

H KS
i

, with the eigenenergies being the orbital energies, " i :

H K i i D " i i ;

(29)

or

!

Z . r 2 /
j

X

1
2 r

Z a

2

1

C d r 2 C V XC . r 1 / ' i . r 1 / D " i ' i . r 1 /; (30)

j r 1 R a j r 1 r 2 j

a

ıE XC Œ
ı , was introduced. Since
the Kohn-Sham orbitals by definition represent noninteracting electrons, the total
Hamiltonian is the sum of the one-electron operators:

where the exchange-correlation potential, V XC ΠD

X

N

H s D H K i :

(31)

i D 1