Biomedical Engineering Reference
In-Depth Information
and electrons are allocated to spinorbitals.Aspinorbital can hold amaximumof one electron.
Suppose then that we have four electrons that we wish to allocate to the four spinorbitals
ψ
A
(
r
) α(
s
), ψ
B
(
r
)α(
s
), ψ
A
(
r
)β(
s
) and ψ
B
(
r
)β(
s
). One possible allocation is
ψ
A
(
r
1
) α (
s
1
) ψ
A
(
r
2
) β (
s
2
) ψ
B
(
r
3
) α (
s
3
) ψ
B
(
r
4
) β (
s
4
)
but we must now allow for the indistinguishability of the electrons and take account of all
the remaining 4! - 1 permutations of electrons through the spinorbitals, for example
ψ
A
(
r
2
) α (
s
2
) ψ
A
(
r
1
) β (
s
1
) ψ
B
(
r
3
) α (
s
3
) ψ
B
(
r
4
) β (
s
4
)
which has been obtained from the original allocation by permuting the names of electrons
1 and 2. All the 4! permutations have to appear with equal weight in the total wavefunction,
and in order to satisfy the Pauli principle we multiply each of them by the sign of the
permutation. This is
1 if we permute
an odd number.Atotal orbital wavefunction that satisfies the Pauli principlewill therefore be
−
1 if we permute an even number of electrons, and
+
Ψ
=
ψ
A
(
r
1
) α (
s
1
) ψ
A
(
r
2
) β (
s
2
) ψ
B
(
r
3
) α (
s
3
) ψ
B
(
r
4
) β (
s
4
)
−
ψ
A
(
r
2
) α (
s
2
) ψ
A
(
r
1
) β (
s
1
) ψ
B
(
r
3
) α (
s
3
) ψ
B
(
r
4
) β (
s
4
)
+···
John C. Slater is credited with having noticed that these terms could be written as a
determinant (of order 4, in this case), which we construct as
ψ
A
(
r
1
)
α
(
s
1
)
ψ
A
(
r
1
)
β
(
s
1
)
ψ
B
(
r
1
)
α
(
s
1
)
ψ
B
(
r
1
)
β
(
s
1
)
ψ
A
(
r
2
)
α
(
s
2
)
ψ
A
(
r
2
)
β
(
s
2
)
ψ
B
(
r
2
)
α
(
s
2
)
ψ
B
(
r
2
)
β
(
s
2
)
=
Ψ
(14.24)
ψ
A
(
r
3
) α (
s
3
)
ψ
A
(
r
3
) β (
s
3
)
ψ
B
(
r
3
) α (
s
3
)
ψ
B
(
r
3
) β (
s
3
)
ψ
A
(
r
4
) α (
s
4
)
ψ
A
(
r
4
) β (
s
4
)
ψ
B
(
r
4
) α (
s
4
)
ψ
B
(
r
4
) β (
s
4
)
Some authors write the determinants with rows and columns interchanged, which of course
leaves the value of the determinant unchanged:
ψ
A
(
r
1
) α (
s
1
)
ψ
A
(
r
2
) α (
s
2
)
ψ
A
(
r
3
) α (
s
3
)
ψ
A
(
r
4
) α (
s
4
)
ψ
A
(
r
1
) β (
s
1
)
ψ
A
(
r
2
) β (
s
2
)
ψ
A
(
r
3
) β (
s
2
)
ψ
A
(
r
4
) β (
s
2
)
Ψ
=
ψ
B
(
r
1
) α (
s
1
)
ψ
B
(
r
2
) α (
s
2
)
ψ
B
(
r
3
) α (
s
3
)
ψ
B
(
r
3
) α (
s
3
)
ψ
B
(
r
1
) β (
s
1
)
ψ
B
(
r
2
) β (
s
2
)
ψ
B
(
r
3
) β (
s
3
)
ψ
B
(
r
4
) β (
s
4
)
It is an attractive property of determinants that they change sign if we interchange two
rows (or columns), and this is formally equivalent to interchanging the name of two of the
electrons. Also, if two columns are the same then the determinant is zero, which is formally
equivalent to letting two electrons occupy the same spinorbital.
Not every electronic state of every atom or molecule can be written as a single Slater
determinant and linear combinations are then needed. For example, of the wavefunctions
shown in Table 14.1, we see by inspection that Ψ
1
, Ψ
3
, Ψ
5
and Ψ
6
can be written as single
Slater determinants, for example
1
2
1
2
1
s
(
r
1
) α (
s
1
)
1
s
(
r
1
) β (
s
1
)
1
s
(
r
1
) α (
s
1
)
2
s
(
r
1
) α (
s
1
)
Ψ
1
=
Ψ
3
=
1
s
(
r
2
) α (
s
2
)
1
s
(
r
2
) β (
s
2
)
1
s
(
r
2
) α (
s
2
)
2
s
(
r
2
) α (
s
2
)
