3.2 Excited states of helium 47Another wavefunction has the same energy, namely
ψspace=u1s(2)unl(1). (3.9)These two states are related by a permutation of the labels on the elec-
trons, 1↔2; the energy cannot depend on the labeling of identical
particles so there isexchange degeneracy. To consider the effect of the
repulsive term on this pair of wavefunctions with the same energy (de-
generate states) we need degenerate perturbation theory. There are two
approaches. The look-before-you-leap approach is first to form eigen-
states of the perturbation from linear combinations of the initial states.^77 This is guided by looking for eigen-
states of symmetry operators that com-
mute with the Hamiltonian for the in-
teraction, as in Section 4.5.
In this new basis the determination of the eigenenergies of the states is
simple. It is instructive, however, simply to press ahead and go through
the algebra once.^88
In the light of this experience one can
We rewrite the Schr ̈odinger equation (eqn 3.1) as take the short cut in future.
(H 0 +H′)ψ=Eψ , (3.10)whereH 0 =H 1 +H 2 , and we consider the mutual repulsion of the
electronsH′=e^2 / 4 π 0 r 12 as a perturbation. We also rewrite eqn 3.2 as
H 0 ψ=E(0)ψ, (3.11)whereE(0)=E 1 +E 2 is the unperturbed energy. Subtraction of eqn
3.11 from eqn 3.10 gives the energy change produced by the perturbation,
∆E=E−E(0),as
H′ψ=∆Eψ. (3.12)
A general expression for the wavefunction with energyE(0)is a linear
combination of expressions 3.8 and 3.9, with arbitrary constantsaand
b,
ψ=au1s(1)unl(2) +bu1s(2)unl(1). (3.13)
Substitution into eqn 3.12, multiplication by eitheru∗1s(1)u∗nl(2) or
u∗1s(2)u∗nl(1), and then integration over the spatial coordinates for each
electron (r 1 ,θ 1 ,φ 1 andr 2 ,θ 2 ,φ 2 ) gives two coupled equations that we
write as (
JK
KJ
)(
a
b)
=∆E
(
a
b)
. (3.14)
This is eqn 3.12 in matrix form. Thedirect integralis
J=
1
4 π 0∫∫
|u1s(1)|^2e^2
r 12|unl(2)|^2 dr^31 dr^32=
1
4 π 0∫∫
ρ1s(r 1 )ρnl(r 2 )
r 12
dr^31 dr^32 , (3.15)whereρ1s(1) =−e|u1s(1)|^2 is the charge density distribution for elec-
tron 1, and similarly forρnl(2). This direct integral represents the
Coulomb repulsion of these charge clouds (Fig. 3.1). Theexchange inte-
gralis
K=
1
4 π 0∫∫
u∗1s(1)u∗nl(2)e^2
r 12u1s(2)unl(1) dr^31 dr^32. (3.16)