Biomedical Engineering Reference
In-Depth Information
ψ
P
ψ
M+1
ψ
M
ψ
3
ψ
2
ψ
1
Figure 16.2
Restricted open shell HF
ν
1
M
ψ
i
(
r
1
)
h
(1)
(
r
1
)ψ
i
(
r
1
) dτ
1
+
ψ
i
(
r
1
)
M
M
1
2
ν
1
ε
el
=
g
(
r
1
,
r
2
)ψ
j
(
r
2
) dτ
1
dτ
2
ˆ
i
=
1
i
=
1
j
=
1
ψ
i
(
r
1
)ψ
j
(
r
1
)
g
(
r
1
,
r
2
)ψ
i
(
r
2
)ψ
j
(
r
2
) dτ
1
dτ
2
+
ν
2
1
2
−
ψ
u
(
r
1
)
h
(1)
(
r
1
)ψ
u
(
r
1
) dτ
1
+
ψ
u
(
r
1
)
g
(
r
1
,
r
2
)ψ
v
(
r
2
) dτ
1
dτ
2
P
P
P
1
2
ν
2
u
=
M
+
1
u
=
M
+
1
v
=
M
+
1
ψ
u
(
r
1
)ψ
v
(
r
1
)
g
(
r
1
,
r
2
)ψ
u
(
r
2
)ψ
v
(
r
2
) dτ
1
dτ
2
−
(16.21)
+
ν
1
ν
2
M
ψ
i
(
r
1
)
g
(
r
1
,
r
2
)ψ
u
(
r
2
) dτ
1
dτ
2
P
i
=
1
u
=
M
+
1
ψ
i
(
r
1
)ψ
u
(
r
1
)
g
(
r
1
,
r
2
)ψ
i
(
r
2
)ψ
u
(
r
2
) dτ
1
dτ
2
1
2
−
In the LCAO variation, we introduce
n
basis functions. We can collect the HF-LCAO
coefficients of the
n
1
doubly occupied orbitals into columns of a matrix
U
1
(
n
×
n
1
) and the
coefficients of the
n
2
singly occupied orbitals into columns of
U
2
(
n
×
n
2
) and define the
n
×
n
density matrices
R
1
=
U
1
U
1
U
2
U
2
Repeating the analysis given for closed shell states gives
R
2
=
ν
1
Tr
R
1
h
1
+
2
G
1
ν
2
Tr
R
2
h
1
+
2
G
2
1
1
ε
el
=
+
(16.22)
