Physical Chemistry Third Edition

806 19 The Electronic States of Atoms. III. Higher-Order Approximations

of an independent variable. For example, a definite integral is a functional that depends

on the integrand function. If we have the definite integral fromxatoxb,

I

∫

f(x)dx (19.5-2)

The definite integralIhas a value that depends on the choice of the integrand

functionfand is a functional off. The definite integral with different limits would be

a different functional, since it would give a different value for the same function. We

write the functional relationship symbolically as

II[f] (19.5-3)

The symbolI[···] stands for the functional, which in this case means taking the

definite integral fromxatoxbof the independent function.

According to the Hohenberg–Kohn theorem, the ground-state energyEgsis a

functional of the ground-state electron probability density, which we denote byρgs:

EgsE

[

ρgs

]

(19.5-4)

Two things are needed to provide a useful method: We need a way to determine the

electron probability density without first finding the entire wave function, and we

need a way to obtain the functionalE[ρgs]. Hohenberg and Kohn proved a variation

theorem that provides some information. If the correct functional is available, using

it on an incorrect electron probability distribution leads to an energy that cannot be

lower than the correct ground-state energy. However, this theorem does not lead to a

calculational method as did the quantum-mechanical variation theorem, because it does

not specify the functional. Practical approximations must be found for the functional

in Eq. (19.5-4). Much work has been done in this area, but discussion of it is beyond

the scope of this book.^17 However, some software packages for molecular quantum

mechanics can perform density functional calculations, even if the user does not have a

detailed understanding of the method. It has been found that the approximation schemes

that have been developed work at least as well as Hartree–Fock–Roothaan methods

with configuration interaction for most molecular properties, such as bond lengths and

energies of molecular ground states.

19.6 Atoms with More Than Two Electrons

The treatment of other atoms is similar to that of helium. In zero order we neglect

electron–electron repulsions, and in higher-order calculations these repulsions are

treated with the same approximation methods as used with the helium atom. From

this point on in the chapter we will discuss only the results of such calculations.

The Lithium Atom

An application of the variation method to the lithium atom ground state uses an orbital

wave function containing hydrogen-like orbitals with variable orbital exponents (effec-

tive nuclear charges) similar to that used with helium, except that different effective

(^17) I. N. Levine,op. cit., pp. 576–592, and references cited therein.