Physical Chemistry Third Edition

1290 H The Hückel Method

whereψaandψbare two of the 2pzatomic orbitals. The integralSabis anoverlap

integralifab, and is anormalization integralifab. The integralHabis a matrix

element of the effective Hamiltonian. We assume that theccoefficients are real, as are

theΨfunctions, so thatHabHbaandSabSba.

We now need to minimize this variational energy with respect to all of theccoeffi-

cients. To do this, we differentiateWiwith respect to each of theccoefficients and set

each derivative equal to zero. This gives us three simultaneous equations for the three

ccoefficients. The derivative ofWwith respect tocjis (where we temporarily omit

the superscript (i) and the limits on the sums):

∂W

∂cj



1

D

(

∑

b

cbHjb+

∑

a

caHaj

)

−

N

D^2

(

∑

b

cbSjb+

∑

a

caSaj

)



2

D

(

∑

b

cbHjb

)

−

2 N

D^2

(

∑

b

cbSjb

)



2

D

(

∑

b

cbHjb

)

−

2 W

D

(

∑

b

cbSjb

)

(H-6)

We have used the facts thatHabHbaandSabSba, and have replaced the summa-

tion indexabybin one sum (this makes no difference after summation). We set the

expression in Eq. (H-5) equal to zero to find the minimum. We multiply byD/2 and

obtain

0 

∑^3

a 1

(Haj−WSaj)ca (j1, 2, 3) (H-7)

This is a set of three simultaneous linear homogeneous equations, one for each value

ofj. The equations are satisfied by the “trivial solution” in which all thec’s vanish, but

this is not the solution that we seek.

The situation is similar to that in the degenerate perturbation theory. If we divide

each equation byc 1 , we have only two unknown variables,c 2 /c 1 andc 3 /c 1 .IfWis

arbitrary, the system of equations is therefore overdetermined, with three equations for

two variables. In order for a nontrivial solution to exist, a condition must be satisfied

that makes the equations equivalent to two independent equations. This condition is

that the determinant of the matrix of the coefficients must vanish.^13 Determinants are

discussed in Appendix B. We write the condition

det(Hab−WSab) 0 (H-8)

Equation (H-8) is called asecular equation. It is similar to but not identical to the

secular equation in degenerate perturbation theory. For each value ofWthat satisfies

this equation there is a usable set of simultaneous equations, so that for each value of

Wthere is a delocalized molecular orbital.

We assume that the 2pzorbitals are normalized, so that the normalization integrals

(Saa) equal unity. We now introduce some additional assumptions and approximations

that define the Hückel method: (1) we approximate the overlap integrals (Sabwith

ab) by zero; (2) we assume thatHaahas the same value, calledα, for every atom;

(^13) I. N. Levine,op. cit., pp. 220ff, 629ff (note 6).