Physical Chemistry , 1st ed.

12.8 Linear Variation Theory

Computer-assisted variation theory is especially powerful when there are a

large number of variables in the trial function. One of the common ways for

this to occur is to assume that the trial function iis a linear combination of

a set of known functions {j} called a basis set:

i

j

ci,jj (12.25)

where jis an individual basis function(for example, a wavefunction of a

model system or a function that can easily be integrated) and the ci,jvalues are

the expansion coefficients that must be determined as part of the solution. So

not only is the minimum energy not known yet, but neither are the values of

the expansion coefficients. As stated earlier, in order to find the lowest energy,

the energy must be minimized with respect to all variables simultaneously:







c

E

i,1







c

E

i,2







c

E

i,3

... 0 (12.26)

There turns out to be a way to determine not only the energy but also the co-

efficients. This powerful use of variation theory is called linear variation theory.

This form of variation theory is also best illustrated by example. Although

the same idea can be applied to a trial wavefunction having any number of

terms, a simple example involves the use of a two-term linear combination for

the trial wavefunction:

aca,1 1 ca,2 2

In this example, the basis set {j} is composed of the two basis functions

 1 and  2. This form of the trial function can be substituted into equation

12.24 and expanded into several terms, keeping in mind that the ordering of

the basis functions is important because of the complex conjugate operation:

E 1 (12.27)

By making the following simplifying definitions:

H 11 * 1 Hˆ 1 d

H 22  2 *Hˆ 2 d

H 12 H 21  2 *Hˆ 1 d

(12.28)

S 11  1 * 1 d

S 22  2 * 2 d

S 12 S 21  2 * 1 d

substitutions can be made into equation 12.27 to yield:

EtrialEE 1 (12.29)

The Hijintegrals are average energy integrals. The Sijintegrals are called over-

lap integrals.For orthonormal wavefunctions, the Sijvalues are either 0 or 1,

but in many instances non-orthonormal wavefunctions are used. For simplifi-

cation, the subscript on the energy is omitted.

c^2 a,1H 11  2 ca,1ca,2H 12 c^2 a,2H 22



c^2 a,1S 11  2 ca,1ca,2S 12 ca,2^2 S 22

(ca,1 1 ca,2 2 )*Hˆ(ca,1 1 ca,2 2 ) d



(ca,1 1 ca,2 2 )*(ca,1 1 ca,2 2 ) d

398 CHAPTER 12 Atoms and Molecules