Computational Chemistry

EtotalHF ¼EHFþVNN¼

Xn

i¼ 1

eiþ

1

2

Xm

r¼ 1

Xm

s¼ 1

PrsHcorers þVNN ð 5 : 149 ¼ 5 : 93 Þ

EtotalHF, which is what is normally meant by the Hartree–Fock energy, is printed by
the program at the end of a single-point calculation or a geometry optimization, or
by some programs at the end of each step of a geometry optimization.
Using the energy levels and density matrix elements from the first cycle
(Table5.1), with the Hcoreelements from Eq. 5.120, Eq.5.147gives for the
electronic energy

EHF¼e 1 þ

1

2

X^2

r¼ 1

X^2

s¼ 1

PrsHcorers

¼e 1 þ

1

2

X^2

r¼ 1

Pr 1 Hcorer 1 þPr 2 Hcorer 2

+,

¼e 1 þ

1

2

P 11 Hcore 11 þP 12 Hcore 12 þP 21 H 21 coreþP 22 H 22 core

+,

¼# 1 :4027 hþ

1

2

½ 0 : 1885 ð# 1 : 6606 Þþ 0 : 4973 ð# 1 : 3160 Þ

þ 0 : 4973 ð# 1 : 3160 Þþ 1 : 3122 ð# 2 : 3030 ފh

¼# 3 :7222 h ð 5 : 150 Þ

From Eq.5.148¼5.92the internuclear repulsion energy is

VNN¼

ZHZHe

rHHe

¼

1 ð 2 Þ

1 : 5117

h¼ 1 :3230 h

ð 5 : 151 Þ

and from Eq.5.149¼5.93the total Hartree–Fock energy is

EtotalHF ¼EHFþVNN¼# 3 :7222 hþ 1 :3230 h¼# 2 :3992 h ð 5 : 152 Þ

The Hartree–Fock energies for the five SCF cycles are given in Table5.1.
Instead of starting with eigenvectors from a non-SCF method like the extended
H€uckel method, as was done in this illustrative procedure, an SCF calculation is
occasionally initiated by takingHcoreas the Fock matrix, that is, by initially
ignoring electron–electron repulsion, setting equal to zero the second term in
Eq.5.82, orGin Eq.5.100, whereuponFrsbecomesHcorers. This is usually a poor
initial guess, but is occasionally useful. You are urged to work your way through
several SCF cycles starting with this Fock matrix; this tedious calculation will help
you to appreciate the power and utility of modern electronic computers and may
enhance your respect for those who pioneered complex numerical calculations
when the only arithmetical aids were mathematical tables and mechanical calcula-
tors (mechanical calculators were machines with rotating wheels, operated by

5.2 The Basic Principles of the ab initio Method 229