Concise Physical Chemistry

c17 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

268 THE VARIATIONAL METHOD: ATOMS

In our treatment we shall unencumber this expression from some of its notation

by writing the expectation value of the energy simply as

E=

∫

φHˆφdτ

∫

φφdτ

In order to evaluate this energy, we need a trial functionφ, which we are free to

choose, and the HamiltonianHˆ, which is determined by the system.

17.2 THE SECULAR DETERMINANT

In very many cases, the wave function will not be expressed as a single function at

all but as a sum of functions. To illustrate, let us take the sum of two additive terms

(strictly, basis vectors in a vector space.)

φ=c 1 u 1 +c 2 u 2

For a variational treatment of the sum as an approximate wave function, we need the

integrals

E=

∫

φHˆφdτ

∫

φφdτ

=

∫

c 1 u 1 +c 2 u 2 Hcˆ 1 u 1 +c 2 u 2 dτ

∫

(c 1 u 1 +c 2 u 2 )(c 1 u 1 +c 2 u 2 )dτ

=

c 12

∫

u 1 Huˆ 1 dτ+c 1 c 2

∫

u 1 Huˆ 2 dτ+c 1 c 2

∫

u 2 Huˆ 1 dτ+c 22

∫

u 2 Huˆ 2 dτ

c 12

∫

u^21 + 2 c 1 c 2

∫

u 2 u 1 +c^22

∫

u^22 dτ

E=

c 12 H 11 + 2 c 1 c 2 H 12 +c 2 H 22

c^21 S 11 + 2 c 1 c 2 S 12 +c^22 S 22

The integration problem has been broken into sums of smaller integrals. Each of

the smaller integrals above has been given a separate symbol. The integrals in the

numerator are denotedHijand the integrals in the denominator are denotedSij.The

assumption that

∫

u 1 Huˆ 2 dτ=

∫

u 2 Huˆ 1 dτ