Biomedical Engineering Reference
In-Depth Information
ψ
i
ψ
j
(
r
1
)
h
(1)
(
r
1
) ψ
j
(
r
1
) dτ
1
P
Q
(
r
1
)
h
(1)
(
r
1
) ψ
i
ε
el
=
(
r
1
) dτ
1
i
=
1
j
=
1
ψ
i
(
r
1
) ψ
i
(
r
1
)
P
P
g
(
r
1
,
r
2
)ψ
j
(
r
2
) ψ
j
(
r
2
) dτ
1
dτ
2
+
ˆ
i
=
1
j
=
1
ψ
i
(
r
1
) ψ
j
(
r
1
)
P
P
1
2
−
g
(
r
1
,
r
2
) ψ
i
(
r
2
) ψ
j
(
r
2
) dτ
1
dτ
2
ˆ
(16.23)
i
=
1
j
=
1
ψ
i
(
r
1
) ψ
i
(
r
1
)
Q
Q
g
(
r
1
,
r
2
)ψ
j
(
r
2
) ψ
j
(
r
2
) dτ
1
dτ
2
+
ˆ
i
=
1
j
=
1
ψ
i
(
r
1
) ψ
j
(
r
1
)
P
P
1
2
g
(
r
1
,
r
2
) ψ
i
(
r
2
) ψ
j
(
r
2
) dτ
1
dτ
2
−
ˆ
i
=
1
j
=
1
There are no cross terms between the two sets of orbitals because of the orthogonality of the
spin functions. We now introduce the LCAO concept; we expand each set of HF orbitals
in terms of the same basis set χ
1
, χ
2
,...,χ
n
and form two density matrices, one for the
α-spin electrons
P
α
and one for the β-spin electrons
P
β
in the obvious way. We finally
arrive at two linked HF-LCAO Hamiltonian matrices; the α-spin matrix has elements
=
χ
i
(
r
1
)
h
(1)
(
r
1
) χ
j
(
r
1
) dτ
h
F
,α
ij
k
=
1
l
=
1
P
kl
+
P
kl
χ
i
(
r
1
)χ
j
(
r
1
)
n
n
+
g
(
r
1
,
r
2
) χ
k
(
r
2
) χ
l
(
r
2
) dτ
1
dτ
2
ˆ
(16.24)
l
=
1
P
kl
χ
i
(
r
1
)χ
k
(
r
1
)
k
=
1
n
n
ˆ
−
g
(
r
1
,
r
2
) χ
j
(
r
2
) χ
l
(
r
2
) dτ
1
dτ
2
with a similar expression for the β electrons. The UHF-LCAO orbitals are found by any
of the standard techniques already discussed, for example, repeated construction of the
density matrices, the two Hamiltonians, matrix diagonalization and so on until consistency
is attained. The orbital energies can be associatedwith ionization energies using an extension
of Koopmans' theorem.
16.7.1 Three Technical Points
There are three technical points of which you should be aware.
1. I chose the example CH
3
with some care, to make sure that the electronic ground state
could be written as a single Slater determinant. There are some electronic states of very
simple molecules where this is not possible, and more advanced considerations apply;
for example, the first excited singlet state of dihydrogen cannot be represented as a single
Slater determinant.
2. The UHF method does not generally give wavefunctions that are eigenfunctions of the
spin operator
S
2
. The methyl radical UHF wavefunction will actually be a mixture of
