Computational Chemistry

wherelijare the Lagrangian multipliers; we don’t know what they are, physically,
yet (after all, they are “undetermined”). Differentiating with respect to thec’s (each
cis a function ofx,y,z) of theS’s:

dEþd

Xn

i¼ 1

Xn

j¼ 1

lijSij¼ 0 ð 5 : 24 Þ

Substituting the expression forEfrom Eq.5.17into Eq.5.24we get

2

Xn

i¼ 1

dHiiþ

Xn

i¼ 1

Xn

j¼ 1

ð 2 dJij#dKijÞþ

Xn

i¼ 1

Xn

j¼ 1

lijdSij¼ 0 ð 5 : 25 Þ

Note that this procedure of minimizing the energy with respect to themolecular
orbitalscis somewhat analogous to the minimization of energy with respect to the
atomic orbital coefficients cin the less rigorous procedure which gave the H€uckel
secular equations in Section 4.3.4. It is also somewhat similar to finding a relative
minimum on a PES (Section 2.4), but with energy in that case being varied with
respect to geometry rather than parameters of MOs. Since the procedure starts with
Eq.5.14and varies the MO’s to find the minimum value ofE, it is called the
variation method; the variation theorem/principle (Section 5.2.3.3) assures us that
the energy we calculate from the results will be greater than or equal to the true
energy.
From the definitions ofHii,Jij,KijandSijwe get

dHii¼

Z

dc$ið 1 ÞH^

core

ð 1 Þcið 1 Þdv 1 þ

Z

c$ið 1 ÞH^

core

ð 1 Þdcið 1 Þdv 1 ð 5 : 26 Þ

dJij¼

Z

dc$ið 1 ÞJ^jð 1 Þcið 1 Þdv 1 þ

Z

dc$jð 1 ÞJ^ið 1 Þcjð 1 Þdv 1 þcomplex conjugate

ð 5 : 27 Þ

dKij¼

Z

dc$ið 1 ÞK^jð 1 Þcið 1 Þdv 1 þ

Z

dc$jð 1 ÞK^ið 1 Þcjð 1 Þdv 1 þcomplex conjugate

ð 5 : 28 Þ

where

J^ið 1 Þ¼

Z

c$ið 2 Þ

1

r 12



cið 2 Þdv 2 ð 5 : 29 Þ

and

K^ið 1 Þcjð 1 Þ¼cið 1 Þ

Z

c$ið 2 Þ

1

r 12



cjð 2 Þdv 2 ð 5 : 30 Þ

190 5 Ab initio Calculations