Biomedical Engineering Reference
In-Depth Information
It can be shown (e.g. Eyring
et al.
1944) that if Ψ
i
and Ψ
j
are eigenfunctions of an operator
A
that commutes with the Hamiltonian
H
, and that the eigenvalues are different
A
Ψ
i
=
a
i
Ψ
i
A
Ψ
j
=
a
j
Ψ
j
a
i
=
a
j
then
Ψ
i
H
Ψ
j
dτ
=
0
This means that the Hamiltonian matrix will have a simple form
⎛
⎞
H
11
H
12
000
H
16
H
21
H
22
000
H
26
00
H
33
000
000
H
44
00
0000
H
55
0
H
61
H
62
000
H
66
⎝
⎠
so instead of a 6
×
6 matrix eigenvalue problem, we have a 3
×
3and3@1
×
1. There is
nothing particularly hard about solving a 6
6 matrix eigenvalue problem, but this simple
example demonstrates howangularmomentumcan be used to helpwith atomic calculations.
In the case of polyatomic molecules, things are not so easy. Angular momentum operators
do not generally commute with molecular Hamiltonians and so molecular problems are
much harder than atomic ones. Spin operators commute with molecular Hamiltonians, as
do symmetry operators.
×
14.7 Slater Determinants
Solution of the appropriate electronic Schrödinger equation is only one aspect of a calcu-
lation; we have to also take account of electron spin and because electrons are fermions
the electronic wavefunction has to satisfy the Pauli principle. Neither electron spin nor the
Pauli principle appears from the Schrödinger treatment. As I mentioned in Chapters 12 and
13, Pauli's principle can be stated in a number of different ways; I am going to restate it as
Electronic wavefunctions must be antisymmetric to exchange of electron names.
I produced the helium orbital wavefunctions in Table 14.1 by a somewhat
ad hoc
method;
I constructed suitable spatial parts and spin parts, which I combined together in such a
way that the Pauli principle was satisfied. A more systematic method for constructing
antisymmetric orbital wavefunctions is needed.
Electron spin can be conveniently treated by combining spatial orbitals with the spin
functions α and β; for a given spatial orbital ψ(
r
) we work with two space and spin
wavefunctions that we write ψ(
r
) α(
s
) and ψ(
r
)β(
s
); these are usually called
spinorbitals
