Biomedical Engineering Reference
In-Depth Information
⎛
⎞
c
1,1
c
2,1
...
c
m
,1
⎝
⎠
c
1,2
c
2,2
...
c
m
,2
U
occ
=
(16.10)
.
.
.
...
c
1,
n
c
2,
n
...
c
m
,
n
then the matrix 2
U
occ
(
U
occ
)
T
gives the charges and bond orders matrix
P
discussed earlier.
The next step is to express the energy in terms of the basis functions and the matrix
P
.
The one-electron contribution is
ψ
R
(
r
1
)
h
(1)
(
r
1
)
ψ
R
(
r
1
)
dτ
c
R
,
i
c
R
,
j
χ
i
(
r
1
)
h
(1)
(
r
1
)
χ
j
(
r
1
)
dτ
M
M
n
n
2
=
2
R
=
1
R
=
1
i
=
1
j
=
1
If we switch the summation signs on the right-hand side we recognize elements of the
charges and bond orders matrix
P
2
c
R
,
i
c
R
,
j
χ
i
(
r
1
)
h
(1)
(
r
1
) χ
j
(
r
1
) dτ
c
R
,
i
c
R
,
j
χ
i
(
r
1
)
h
(1)
(
r
1
) χ
j
(
r
1
) dτ
M
n
n
n
n
M
2
=
R
=
1
i
=
1
j
=
1
i
=
1
j
=
1
R
=
1
P
i
,
j
χ
i
(
r
1
)
h
(1)
(
r
1
) χ
j
(
r
1
) dτ
n
n
=
i
=
1
j
=
1
Finally, on collecting together the one-electron integrals over the basis functions into an
n
×
n
matrix
h
1
whose
i
,
j
th element is
χ
i
(
r
1
)
h
(1)
(
r
1
) χ
i
(
r
1
) dτ
1
(
h
1
)
ij
=
then the one-electron term emerges as the trace of the matrix product
Ph
1
n
n
P
ij
h
(1)
ij
=
Tr (
Ph
1
)
(16.11)
i
=
1
j
=
1
A corresponding analysis shows that the two-electron terms can be written as
2
Tr
P
(
J
2
K
)
1
1
−
where the elements of the matrices
J
and
K
depend on those of
P
in a more complicated
way:
P
kl
χ
i
(
r
1
)χ
j
(
r
1
)
n
n
J
ij
=
g
(
r
1
,
r
2
) χ
k
(
r
2
) χ
l
(
r
2
) dτ
1
dτ
2
ˆ
k
=
1
l
=
1
(16.12)
P
kl
χ
i
(
r
1
)χ
k
(
r
1
)
n
n
K
ij
=
g
(
r
1
,
r
2
) χ
j
(
r
2
) χ
l
(
r
2
) dτ
1
dτ
2
ˆ
k
=
1
l
=
1
Many authors collect together these
Coulomb and exchange matrices
into a composite
called the
electron repulsion matrix
:
1
2
K
=
−
G
J
