Physical Chemistry Third Edition

808 19 The Electronic States of Atoms. III. Higher-Order Approximations

EXAMPLE19.4

Estimate the effective number of protons in the nucleus for the 2selectron in a lithium atom

from the ionization potential, 124 kcal mol−^1. Compare this value with the value of 1.776

obtained with the variation method.

Solution

The ionization potential in electron volts is

(IP)

(124000 cal mol−^1 )(4.184 J cal−^1 )

96485 J mol−^1 eV−^1

 5 .38 eV

The energy of the 2shydrogen-like orbital is given by Eq. (17.3-19) as

E 2 −

(13.6eV)Z′^2

22

whereZ′is the effective number of protons. Setting this energy equal to 5.38 eV gives

Z′ 1 .26, which compares with the value of 1.776 obtained by the variation calculation.

When the perturbation method is applied to the lithium atom, the first-order cor-

rection to the ground-state energy is equal to 83.5 eV, resulting in an energy through

first order equal to− 192 .0eV.^20 This value is considerably less accurate than the value

obtained by the variation calculation. The SCF method is the most successful of the

three common approximation methods. A careful Hartree–Fock–Roothaan calculation

gave a ground-state energy of− 202 .3 eV, differing from the correct value by 1.2 eV.^21

Assuming that the value of− 202 .3 eV is a good approximation to the optimum Hartree–

Fock energy, 1.2 eV is presumably a good approximation to the correlation error.

One way to include dynamical electron correlation in an orbital wave function is to

construct a wave function that is a linear combination of several Slater determinants

corresponding to different electron configurations, a method that is known asconfig-

uration interaction(abbreviated by CI). For example, the ground state of the lithium

atom could be represented by

Ψc 1 Ψ 1 s 1 s 2 s+c 2 Ψ 1 s 2 s 2 s+c 3 Ψ 1 s 1 s 3 s+... (19.6-1)

wherec 1 ,c 2 , andc 3 , and so on, are variable parameters and theΨsrepresent wave

functions corresponding to the configurations specified in the subscripts. Although it is

not obvious from inspection of Eq. (19.6-1) that this wave function includes dynamical

correlation, it does depend on electron–electron distances, a fact that we discuss briefly

in the next chapter. The configuration interaction process converges slowly, so that many

configurations must be used to get good accuracy. Atomic and molecular calculations

have been carried out with as many as a million configurations.

Atoms with More Than Three Electrons

Figure 19.4 shows orbital energies in neutral atoms as a function of atomic number,

obtained by an approximation scheme called the Thomas–Fermi method.^22 We will not

(^20) I. N. Levine,op. cit., p. 297ff (note 2).
(^21) F. L. Pilar,Elementary Quantum Chemistry, McGraw-Hill, New York, 1968, p. 336.
(^22) H. A. Bethe and R. W. Jackiw,Intermediate Quantum Chemistry, 3rd ed., Benjamin-Cummings, 1985,
Chapter 5.