Physical Chemistry Third Edition

19.3 The Perturbation Method and Its Application to the Ground State of the Helium Atom 799

The self-consistent field method can converge to the best orbital wave function, but

it does not include any dynamical electron correlation. The difference between the best

energy calculated with an orbital wave function and the energy corresponding to the

exact solution of the Schrödinger equation is called thecorrelation energyor thecorre-

lation error. The 1.1 eV error of Clementi and Roetti is presumably an approximation

to the correlation error. Theconfiguration interactionmethod eliminates part of the

correlation error by constructing a wave function that is a sum of terms, each of which

corresponds to a different electron configuration. We discuss this method briefly later

in this chapter and in the next chapter.

PROBLEMS

Section 19.2: The Self-Consistent Field Method

19.16For an electron configuration of a helium atom in which
the two electrons occupy different space orbitals, there
must be two simultaneous integrodifferential equations in

the SCF method. Write the equations for the (1s)(2s)

configuration.

19.17Write the simultaneous SCF integrodifferential equations

for the (1s)^2 (2s) configuration of the lithium atom. Are

three equations needed?

19.3 The Perturbation Method and Its Application

to the Ground State of the Helium Atom

We consider a Hamiltonian operator that can be separated into two terms,

ĤĤ(0)+̂H′ (19.3-1)

such that̂H(0)gives a Schrödinger equation that can be solved:

Ĥ(0)Ψ(0)E(0)Ψ(0) (19.3-2)

The wave functionΨ(0)is called thezero-order wave functionorunperturbed wave

function. The energy eigenvalueE(0)is called thezero-order energy eigenvalueor

unperturbed energy eigenvalue, and̂H(0)is called thezero-order Hamiltonian. The

term̂H′in the Hamiltonian operator is called theperturbation. To apply the method to

the helium atom, we takêH(0)from Chapter 18 to be the zero-order Hamiltonian and

the electron–electron repulsion energy to bêH′.

We construct a new Hamiltonian operator by multiplying the perturbation term by

a fictitious parameter,λ:

̂H(λ)̂H(0)+λ̂H′ (19.3-3)

At first, this seems to make an insoluble problem even more complicated, but it will

turn out to be useful when we express energies and wave function as power series in

λ. The new Schrödinger equation for a particular energy eigenfunctionΨnis

Ĥ(λ)Ψn(q,λ)En(λ)Ψn(q,λ) (19.3-4)

where the wave functionΨnnow depends onλas well as on the coordinates of the

system, which we abbreviate byqin our usual way. The correct wave function would

be obtained by lettingλ1.