Physical Chemistry Third Edition

804 19 The Electronic States of Atoms. III. Higher-Order Approximations

The correct zero-order wave functions are expressed as linear combinations of the

degenerate “initial” zero-order wave functions:

Ψ

(0)

n,cor

∑g

j 1

cnjΨ

(0)

j,init (19.4-1)

wheregis the degeneracy (the number of states in the zero-order energy level). The

subscript “cor” denotes the correct zero-order functions and the subscript “init” denotes

the initial zero-order functions. In order to find thecnjcoefficients, one must solve a

set of homogeneous linear simultaneous equations that are described in Appendix G.

One solution of these equations is that all of theccoefficients vanish. This is called

the trivial solution and is not useful to us. A condition that must be satisfied for a

nontrivial solution of these equations to exist is called asecular equation.^14 Solution

of the secular equation gives the first-order corrections to the energies and allows

solution of the equations for thecnjcoefficients for each correct zero-order function.

Additional information can be found in Appendix G. The wave functions of Eq. (18.4-2)

are the correct zero-order functions for the (1s)(2s) configuration of the helium atom,

and three sets of similar functions are the correct zero-order functions for the (1s)(2p)

configuration.

Figure 19.3 shows the results of calculations to first order and to third order for

the energies of the four levels that result from the (1s)(2s) and the (1s)(2p)

250

260

268 2 68.0 eV(1s)(2s) and (1s)(2p)

2 58.4 eV(1s)(2s) singlet

2 59.2 eV(1s)(2s) triplet

2 58.1 eV(1s)(2p) triplet

2 57.8 eV(1s)(2p) singlet

2 55.7 eV(1s)(2p) triplet

2 57.8 eV(1s)(2s) triplet

2 55.4 eV(1s)(2s) singlet

2 53.9 eV(1s)(2p) singlet

E(0) E(0) 1 E(1) E(0) 1 E(1) 1 E(2) 1 E(3)

Energy of helium atom/eV

Figure 19.3 Approximate Energies of Helium Excited States.

(^14) Levine,op. cit., p. 262 (note 2).