Computational Chemistry

3. The third term in Eq.7.16, the classical electrostatic repulsion energy term, is
   readily calculated ifr 0 is known.


4. This leaves us with the exchange-correlation energyEXC[r 0 ] (Eq.7.15) as the
   only term for which some new method of calculation must be devised. Devising
   good exchange-correlation functionals for calculating this energy term from the
   electron density function is the main problem in DFT research. This is discussed
   in Section7.2.3.4.
   Written out more fully, then, Eq.7.16is

E 0 ¼"

X

nuclei A

ZA

Z

r 0 ðr 1 Þ

r1A

dr 1 "

1

2

X^2 n

i¼ 1

cKS 1 ð 1 Þjr^21 jcKS 1 ð 1 Þ

þ

1

2

ZZ

r 0 ðr 1 Þr 0 ðr 2 Þ

r 12

dr 1 dr 2 þEXC½r 0 Š

(7.21)

The term most subject to error is the relatively smallExc[r 0 ] term, which
contains the “unknown” (not precisely known) functional. Into this term the exact
electron correlation and exchange energies have been swept, and for it we must find
at least an approximate functional.

The Kohn–Sham Equations

The KS equations are obtained by differentiating the energy with respect to the
KS molecular orbitals, analogously to the derivation of the Hartree–Fock equa-
tions, where differentiation is with respect to wavefunction molecular orbitals
(Section 5.2.3.4). We use the fact that the electron density distribution of the
reference system, which is by decree exactly the same as that of the ground state
of our real system (see the definition at the beginning of the discussion of the
Kohn–Sham energy), is given by (reference [ 9 ])

r 0 ¼rr¼

X^2 n

i¼ 1

jcKSi ð 1 Þj^2 *(7.22)

where thecKSi are the Kohn–Sham spatial orbitals. Substituting the above expres-
sion for the electron density in terms of orbitals into the energy expression of
Eq.7.21and differentiating to varyE 0 with respect to thecKSi subject to the
constraint that these remain orthonormal (the spin orbitals of a Slater determinant
are orthonormal) leads to the Kohn–Sham equations (the derivation is discussed in
considerable detail by Parr and Yang [ 35 ]):

"

1

2

r^2 i"

X

nuclei A

ZA

r1A

þ

Z

rðr 2 Þ

r 12

dr 2 þvXCð 1 Þ

"

cKSi ð 1 Þ¼eKSi cKSi ð 1 Þ (7.23)

7.2 The Basic Principles of Density Functional Theory 455