Biomedical Engineering Reference
In-Depth Information
16.2 The
J
and
K
Operators
It is sometimes useful to recast the equation as the expectation value of a sumof one-electron
and certain
pseudo one-electron operators
2
ψ
i
(
r
1
)
2
K
(
r
1
)
ψ
i
(
r
1
) dτ
1
M
1
h
(1)
(
r
1
)
+
J
(
r
1
)
ε
e
=
−
(16.5)
i
=
1
The operator
h
(1)
represents the kinetic energy of an electron and the nuclear attraction. The
operators
J
and
K
are called the
Coulomb
and the
exchange operators
. They can be defined
through their expectation values as follows:
ψ
R
(
r
1
)
J
(
r
1
)
ψ
R
(
r
1
)
dτ
1
=
ψ
R
(
r
1
)
M
g
(
r
1
,
r
2
)
ψ
i
(
r
2
)
dτ
1
dτ
2
ˆ
(16.6)
i
=
1
and
ψ
R
(
r
1
)
K
(
r
1
) ψ
R
(
r
1
) dτ
1
=
ψ
R
(
r
1
) ψ
i
(
r
1
)
M
ˆ
g
(
r
1
,
r
2
) ψ
R
(
r
2
) ψ
i
(
r
2
) dτ
1
dτ
2
i
=
1
(16.7)
The HF Hamiltonian is a one-electron operator, defined by
1
2
K
(
r
)
h
F
(
r
)
=
h
(1)
(
r
)
+
J
(
r
)
−
(16.8)
where the coordinates
r
refer to an arbitrary electron. HF orbitals are solutions of the
eigenvalue equation
h
F
(
r
) ψ (
r
)
=
εψ (
r
)
16.3 HF-LCAO Equations
I am going to make use of matrices and matrix algebra for many of the derived equations,
for the simple reason that they look neater than they would otherwise do.
The
n
basis functions are usually real and usually overlap each other and so they are
not necessarily orthogonal. Following the arguments of Chapters 13 and 14, we collect
together the basis function overlap integrals into an
n
×
n
real symmetric matrix
S
that has
typically an
i
,
j
element
χ
i
(
r
1
) χ
j
(
r
1
) dτ
1
S
i
,
j
=
×
It is convenient to store the LCAO coefficients in an
n
n
matrix
⎛
⎝
⎞
⎠
c
1,1
c
2,1
...
c
n
,1
c
1,2
c
2,2
...
c
n
,2
U
=
(16.9)
.
.
.
...
c
1,
n
c
2,
n
...
c
n
,
n
so that the first column collects the coefficient of the first occupied HF-LCAO orbital and
so on. If we collect together the
m
occupied LCAO orbitals into an
n
×
m
matrix
U
occ
