790 19 The Electronic States of Atoms. III. Higher-Order Approximations
19.1 The Variation Method and Its Application to
the Helium Atom
The zero-order orbital approximation employed in Chapter 18 completely neglects the
electron–electron repulsions in atoms and therefore gives poor energy values. There
are several commonly used methods that go beyond this approximation. Thevariation
methodis based on the variation theorem.The Variation Theorem
The expectation value of the energy corresponding to a time-independent wave function
ψis given by Eq. (16.4-4):〈E〉
∫
ψ∗Hψdq̂
∫
ψ∗ψdq(19.1-1)
whereĤis the Hamiltonian operator of the system and where the coordinates of all of
the particles of the system are abbreviated byq. The integration is to be done over all
values of all coordinates.
Thevariationtheoremstates:Theexpectationvalueoftheenergycalculatedwithany
functionφobeying the same boundary conditions as the correct system wave functions
cannot be lower thanEgs,the correct ground-state energy eigenvalue of the system:∫
φ∗Hφdq̂
∫
φ∗φdq≥Egs (variation theorem) (19.1-2)The correct Hamiltonian operator must be used. This expectation value is equal toEgs
if and only if the functionφis the same function as the correct ground-state energy
eigenfunction. The proof of the theorem is assigned in Problem 19.1.The Variation Method
The variation theorem suggests the variation method for finding an approximate
ground-state energy and wave function. The first step of the method is to choose a
family of possible approximate wave functions. The second step is to calculate the
expectation value of the energy using the different members of the family of functions.
This expectation value is called thevariation energy, and is usually denoted byW. The
final step is to find the member of the family that gives a lower (more negative) value
ofWthan any other member of the family. This value ofWis a better approximation to
the ground-state energy than is obtained from any other member of the family of func-
tions. The theorem does not guarantee that this function is a better approximation to
the correct wave function than any other member of the family, but it is likely to be so.
A typical family of functions is represented by a formula containing one or more
variable parameters. Such a formula is called avariation functionor avariation trial