Computational Chemistry

words, we are counting each repulsion twice. The simple sum thus represents
properly the total kinetic and electron–nuclear attraction potential energy, but over-
counts the electron–electron repulsion potential energy (recall that we are working
with 2nelectrons and thusnfilled MOs):

EðoverestimatedÞ¼ 2

Xn

i¼ 1

ei ð 5 : 84 Þ

Note that we cannot just take half of this simple sum, because only the electron–
electron energy terms, not all the terms, have been doubly-counted. The solution is
to subtract from 2∑ethe superfluous repulsion energy; from our discussion
of Eq.5.50inSection 5.2.3.5we saw that the sum∑(2J#K) overnrepresents
the repulsion energy of one electron interacting with all the other electrons, so to
remove the superfluous interactions we subtract∑∑(2J#K), the sum overnof the
repulsion energy sum, to get [ 15 ]

EHF¼ 2

Xn

i¼ 1

ej#

Xn

i¼ 1

Xn

j¼ 1

ð 2 Jijð 1 Þ#Kijð 1 ÞÞð 5 : 85 Þ

EHFis the Hartree–Fock electronic energy: the sum of one-electron energies
corrected (within the average-field HF approximation) for electron–electron repul-
sion. We can get rid of the integralsJandKover MO’scand obtain an equation for
EHFin terms ofc’s andf’s. From (5.83),

Xn

i¼ 1

Xn

j¼ 1

ð 2 Jijð 1 Þ#Kijð 1 ÞÞ¼

Xn

i¼ 1

eiþ

Xn

i¼ 1

Hcoreii

and from this and (5.85) we get

EHF¼

Xn

i¼ 1

eiþ

Xn

i¼ 1

Hiicore ð 5 : 86 Þ

From the definition ofHcoreii in Eqs.5.49and5.50, i.e. from

Hcoreii ¼ cið 1 ÞjH^

core

jci

DE

ð 5 : 87 Þ

and the LCAO expansion (5.52)

ci¼

Xm

s¼ 1

csifs ð 5 : 88 ¼ 5 : 52 Þ

5.2 The Basic Principles of the ab initio Method 211