Physical Chemistry , 1st ed.

determinable.) In evaluating this determinant one will get a polynomial of or-

der n, in which the highest power ofEwill be En. The polynomial will have up

to nsolutions ofE(some of which may be the same, indicating degenerate

wavefunctions). The lowest value ofEis our calculated energy of the ground

state. Although the focus has shifted abruptly from the determining the coef-

ficients to knowing the energy, we must remember that it is the energy of the

system that we are usually most interested in.

If we have the energies, we can determine the coefficients ci,j. In the exam-

ple for a two-term trial function, we will get two energies E 1 and E 2 ; the lower

of the two is the lowest-energy state. Using the simultaneous equations 12.30,

it is easy to see that the two coefficients can be expressed as ratios:



c

c

a

a

,

,

1

2



H

H

1

1

2

1





E

E

S

S

1

1

2

1



(12.33)



c

c

a

a

,

,

1

2



H

H

2

2

1

2



E

E

S

S

2

2

2

1



where Eis the energy of either state. The energy and overlap integrals are cal-

culable, and the energies have already been determined by solving the secular

determinant of equation 12.31. Equations 12.33 provide two ratiosfor ca,1and

ca,2that should yield the same ratio for each individual value of energy calcu-

lated from the secular determinant. The exact values of the coefficients are then

adjusted so that is normalized. If orthonormal basis functions are used, the

normalization condition for the approximate wavefunction is easy to express:



j

c^2 i,j 1 (12.34)

That is, the sum of the squares of the coefficients must equal 1. After deter-

mining the coefficients ca,1and ca,2,the calculation of the ground-state wave-

function is complete for this example. One also gets the approximate energy and

wavefunction for the first excited state. (In general, when one uses nideal

wavefunctions, one determines nlinear combinations for the first nenergy lev-

els of an approximated system.) These determinations—the energy and the

wavefunction of the ground state—are the goals of linear variation theory.

If the basis functions themselves are orthogonal to each other, then the trial

wavefunctions determined using variation theory are also orthogonal to each

other. Since the trial wavefunction is expressed in terms of a linear combina-

tion of other functions and it is the integrals of these other functions that must

be evaluated in solving the secular determinant, it is wise to choose such basis

functions so that their integrals can easily be evaluated and so that the deter-

minants and coefficients can be determined for any real system. This idea is the

main thrust for modern calculational quantum mechanics, which is mostly

performed by computer (which can be programmed to perform the various

linear algebraic manipulations of a preset trial wavefunction).

Example 12.12

Assume that, for a real system, a real wavefunction is a linear combination of

two orthonormal basis functions where the energy integrals are as follows:

H 11 15 (arbitrary energy units),H 22 4, and H 12 H 21 1.

Evaluate the approximate energies of the real system, and determine the co-

efficients of the expansion:

aca,1 1 ca,2 2

400 CHAPTER 12 Atoms and Molecules