Computational Chemistry

where
Z
f 1 H^f 1 dv¼H 11
Z
f 1 H^f 2 dv¼H 12 ¼

Z

f 2 H^f 1 dv¼H 21

Z

f 2 H^f 2 dv¼H 22

Z

f^21 dv¼S 11

Z

f 1 f 2 dv¼S 12 ¼

Z

f 2 f 1 dv¼S 21

Z

f^22 dv¼S 22

(4.44)

Note that in Eqs.4.43and4.44theHijare not operators hence are not given hats;
they are integrals involvingHˆand basis functionsf.
For any particular molecular geometry (i.e. nuclear configuration: Section 2.3,
the Born–Oppenheimer approximation) the energy of the ground electronic
state is the minimum energy possible for that particular nuclear arrangement
and the collection of electrons that goes with it. Our objective now is to minimize
the energy with respect to the basis set coefficients. We want to find thec’s
corresponding to the minimum on an energy versusc’s potential energy surface.
To do this we follow a standard calculus procedure: set∂E/∂c 1 equal to zero,
explore the consequences, then repeat for∂E/∂c 2. In theory, setting the first
derivatives equal to zero guarantees only that we will find in “MO space”
(an abstract space defined by an energy axis and two or more coefficient axes)
a stationary point (cf. Section 2.2), but examining the second derivatives shows
that the procedure gives an energy minimum if all or most of the electrons are
in bonding MOs, which is the case for most real molecules [ 35 ]. Write Eq.4.43as

Ec^21 S 11 þ 2 c 1 c 2 S 12 þc^22 S 22

'(

¼c^21 H 11 þ 2 c 1 c 2 H 12 þc^22 H 22 (4.45)

and differentiate with respect toc 1 :

@E

@c 1



c^21 S 11 þ 2 c 1 c 2 S 12 þc^22 S 22

'(

þEð 2 c 1 S 11 þ 2 c 2 S 22 Þ¼ 2 c 1 H 11 þ 2 c 2 H 12

Set@E=@c 1 ¼0:

Eð 2 c 1 S 11 þ 2 c 2 S 22 Þ¼ 2 c 1 H 11 þ 2 c 2 H 12

This can be written

ðH 11 "ES 11 Þc 1 þðH 12 "ES 12 Þc 2 ¼ 0 (4.46)

4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 121