Computational Chemistry

1. The first term (the integral of the density times the external potential) is

Z

r 0 ðrÞvðrÞdr¼

Z

r 0 ðr 1 Þ

X

nuclei A

"

ZA

r1A

"

dr 1 ¼"

X

nuclei A

ZA

Z

r 0 ðr 1 Þ

r1A

dr 1 (7.17)

We integrate the potential energy of attraction of each nucleus for an infinitesi-

mal portion of the charge cloud and sum for all the nuclei. If we knowr 0 the

integrals to be summed are readily calculated.

2. The second term (the electronic kinetic energy of the noninteracting-electrons
   reference system) is the expectation value of the sum of the one-electron kinetic
   energy operators over the ground state multielectron wavefunction of the refer-
   ence system (Parr and Yang explain this in detail [ 33 ]). Using the compact Dirac
   notation for integrals:

hiT½r 0 Šref¼ crj

X^2 n

i¼ 1

"

1

2

r^2 ijcr

*+

(7.18)

Since these hypothetical electrons are noninteractingcrcan be writtenexactly
(for a closed-shell system) as a single Slater determinant of occupied spin molecular
orbitals (Section 5.2.3.1). For arealsystem, the electrons interact and using a single
determinant causes errors due to neglect of electron correlation (Section 5.4), the
root of most of our troubles in wavefunction methods. Thus for a four-electron
system

cr¼

1

ffiffiffiffi

4!

p

cKS 1 ð 1 Það 1 Þ cKS 1 ð 1 Þbð 1 Þ cKS 2 ð 1 Það 1 Þ cKS 2 ð 1 Þbð 1 Þ

cKS 1 ð 2 Það 2 Þ cKS 1 ð 2 Þbð 2 Þ cKS 2 ð 2 Það 2 Þ cKS 2 ð 2 Þbð 2 Þ

cKS 1 ð 3 Það 3 Þ cKS 1 ð 3 Þbð 3 Þ cKS 2 ð 3 Það 3 Þ cKS 2 ð 3 Þbð 3 Þ

cKS 1 ð 4 Það 4 Þ cKS 1 ð 4 Þbð 4 Þ cKS 2 ð 4 Það 4 Þ cKS 2 ð 4 Þbð 4 Þ

(^)
(^)
(7.19)
The 16 spin orbitals in this determinant are theKohn–Sham spin orbitalsof the
reference system; each is the product of a Kohn–Sham spatial orbitalcKSi and a spin
functionaorb. Equation7.18can be written in terms of thespatialKS orbitals by
invoking a set of rules (the Slater–Condon rules [ 34 ]) for simplifying integrals
involving Slater determinants:
hiT½r 0 Šref¼"

1

2

X^2 n

i¼ 1

cKS 1 ð 1 Þjr^21 jcKS 1 ð 1 Þ

(7.20)

The integrals to be summed are readily calculated. Note that DFTper sedoes not
involve wavefunctions, and the Kohn–Sham approach to DFT uses orbitals only as
a kind of subterfuge to calculate the noninteracting-system kinetic energy and the
electron density function; see below.

454 7 Density Functional Calculations