Computational Chemistry

Addressing next electronicpotentialenergy, let us define a termDhVeeias the
deviation of the real electron–electron repulsion energy from a classical charge-
cloud coulomb repulsion energy. The classical electrostatic repulsion energy is the
summation of the repulsion energies for pairs of infinitesimal volume elements
r(r 1 )dr 1 andr(r 2 )dr 2 (in a classical, nonquantum cloud of negative charge) sepa-
rated by a distancer 12 , multiplied by one-half (so that we do not count ther 1 /r 2
repulsion energy and again ther 2 /r 1 energy). The sum of infinitesimals is an
integral and so

DhiVee½r 0 Š ¼hiVee½r 0 Šrea"

1

2

ZZ

r 0 ðr 1 Þr 0 ðr 2 Þ

r 12

dr 1 dr 2 (7.13)

Actually, the classical charge-cloud repulsion is somewhat inappropriate for
electrons in that smearing an electron (a particle) out into a cloud forces it to
repel itself, as any two regions of the cloud interact repulsively. One way to
compensate for this physically incorrectelectron self-interactionis with a good
exchange-correlation functional (below).
Using (7.12) and (7.13), Eq.7.11can be written

E 0 ¼

Z

r 0 ðrÞvðrÞdrþhiT½r 0 Šrefþ

1

2

ZZ

r 0 ðr 1 Þr 0 ðr 2 Þ

r 12

dr 1 dr 2

þDhiT½r 0 ŠþDhiVee½r 0 Š

(7.14)

The two “delta terms” which have been placed side by side encapsulate the
main problem with DFT: the sum of the kinetic energy deviation from the reference
system and the electron–electron repulsion energy deviation from the classi-
cal system, called theexchange-correlation energy. In each term an unknown
functional transforms electron density into an energy, kinetic and potential respec-
tively. This exchange-correlation energy is a functional of the electron density
function:

EXC½r 0 ŠDhiT½r 0 Š þDhiVee½r 0 Š (7.15)

TheDhTiterm represents the kinetic correlation energy of the electrons and the
hDVeeiterm the potential correlation and exchange energy (although exchange and
correlation energy in DFT do not have exactly the same significance as in HF theory
[ 32 ]). Using Eq.7.15, Eq.7.14becomes

E 0 ¼

Z

r 0 ðrÞvðrÞdrþhiT½r 0 Šrefþ

1

2

ZZ

r 0 ðr 1 Þr 0 ðr 2 Þ

r 12

dr 1 dr 2 þEXC½r 0 Š (7.16)

Let’s look at the four terms in the expression for the molecular electronic energy
E 0 of Eq.7.16.

7.2 The Basic Principles of Density Functional Theory 453