Computational Chemistry

numbernof pairs of electrons in the molecule, although only thenoccupied spatial
orbitals are used to construct the Slater determinant which represents the HF
wavefunction (Section 5.2.3.1). This point, and basis sets, are discussed further in
Section 5.3.
To continue with the Roothaan–Hall approach, we substitute the expansion
(5.52) for thec’s into the Hartree–Fock equations5.47, getting (we will work
withm, notn, HF equations since there is one such equation for each MO, and ourm
basis functions will generatemMO’s):

Xm

s¼ 1

cs 1 F^fsj¼e 1

Xm

sj¼ 1

cs 1 fs

Xm

s¼ 1

cs 2 F^fs¼e 2

Xm

s¼ 1

cs 2 fs

...

Xm

s¼ 1

csmF^fs¼em

Xm

s¼ 1

csmF^fs

ð 5 : 54 Þ

(F^operates on the functionsf, not on thec’s, which have no variablesx,y,z).
Multiplying each of thesemequations byf 1 ,f 2 ,...,fm, (orf 1 * etc. if thef’s are
complex functions, as is occasionally the case) and integrating, we getmsets of
equations (one for each of the basis functionsf).
Basis functionf 1 gives

Xm

s¼ 1

cs 1 F 1 s¼e 1

Xm

s¼ 1

cs 1 S 1 s

Xm

sj¼ 1

cs 2 F 1 s¼e 2

Xm

s¼ 1

cs 2 S 1 s

...

Xm

s¼ 1

csmF 1 s¼em

Xm

s¼ 1

csmS 1 s

ð 5 : 54 ‐ 1 Þ

where

Frs¼

Z

frF^fsdv and Srs¼

Z

frfsdv ð 5 : 55 Þ

5.2 The Basic Principles of the ab initio Method 199