Concise Physical Chemistry

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

THE SECULAR DETERMINANT 269

has been made in order to simplify the numerator. OperatorsHˆ for which this is true

are calledHermitianoperators. The equivalent substitution ofS 12 forS 21 has been

made in the denominator.

One can multiply through by the denominator to find

(

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

)

E=c^21 H 11 + 2 c 1 c 2 H 12 +c 2 H 22

Our objective is to differentiate the energy with respect to each of the minimization

parametersc 1 andc 2 so as to find the simultaneous minimum with respect to both

of them. This will be the minimum energy obtainable from the sum of functions

φ=c 1 u 1 +c 2 u 2. Differentiating first with respect toc 1 , we obtain

( 2 c 1 S 11 + 2 c 2 S 12 )E+

∂E

∂c 1

(

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

)

= 2 c 1 H 11 + 2 c 2 H 12

Differentiating with respect toc 2 , we get

( 2 c 1 S 12 + 2 c 2 S 22 )E+

∂E

∂c 2

(

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

)

= 2 c 1 H 12 + 2 c 2 H 22

In order to find the minimum, we set

∂E

∂c 1

=

∂E

∂c 2

= 0

This causes the two∂E/∂cterms to drop out:

∂E

∂c 1

(

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

)

=

∂E

∂c 2

(

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

)

= 0

We are left with a pair of simultaneous equations:

( 2 c 1 S 11 + 2 c 2 S 12 )E= 2 c 1 H 11 + 2 c 2 H 12

and

( 2 c 1 S 12 + 2 c 2 S 22 )E= 2 c 1 H 12 + 2 c 2 H 22

These two equations are somewhat more conformable to computer solutions if we

divide by 2 and write them in the equivalent form:

(H 11 −ES 11 )c 1 +(H 12 −ES 12 )c 2 = 0

(H 12 −ES 12 )c 1 +(H 22 −ES 22 )c 2 = 0