Biomedical Engineering Reference
In-Depth Information
19.1.1 Fermi Correlation
The two-electron functions tell us how the two electrons are correlated. In the one-
determinant HF model, we find that the
P
's are related as follows:
P
αα
2
(
r
1
,
r
2
)
=
P
1
(
r
1
)
P
1
(
r
2
)
−
X
(
r
1
,
r
2
)
(19.7)
P
α
2
(
r
1
,
r
2
)
P
1
(
r
1
)
P
1
(
r
2
)
=
where
X
is a function whose form need not concern us, except to note that it exactly
cancels the first term when
r
1
=
r
2
. From these results we get a clear picture of the electron
correlation shown by standard closed-shell HF theory. The form of
P
αβ
2
shows that there
is no correlation between electrons of opposite spin, since the simultaneous probability is
just the product of the individual ones. This is a defect of HF theory. Electrons of like spin
are clearly correlated and they are never found at the same point in space, and HF theory
is satisfactory here. This type of correlation arises from antisymmetry and applies to all
fermions. It is usually called
Fermi correlation
.
19.2 Configuration Interaction
I have mentioned configuration interaction (CI) at various points in the text, in the dihydro-
gen discussion of Chapter 15 and in the CIS treatment of excited states in Chapter 17. The
idea of modern CI is to start with a reference wavefunction that could be a HF-LCAO closed
shell wavefunction, and systematically replace the occupied spinorbitals with virtual ones.
So if A, B, ... represent occupied orbitals and X, Y, ... represent virtual ones we would
seek to write
Ψ
CI
Ψ
HF
c
A
Ψ
A
c
XY
AB
Ψ
XY
=
+
+
+···
(19.8)
AB
A,X
A,B,X,Y
The expansion coefficients can be found by solving the variational problem, in the usual
way. This involves finding the eigenvalues and eigenvectors of the Hamiltonian matrix;
usually one is only interested in a few of the electronic states, and methods are available
for finding just a small number of eigenvalues and eigenvectors of very large matrices.
In a complete CI calculation we would include every possible excited configuration
formed by promoting electrons from the occupied spinorbitals to the virtual ones; for a
closed shell singlet state molecule with
m
electron pairs and
n
basis functions there are
n
!
(
n
+
1)
!
m
!
(
m
+
1)
!
(
n
−
m
)
!
(
n
−
m
+
1)
!
possible Slater determinants. If we consider a HF/6-311G** wavefunction for benzene, we
have
m
=
21 and
n
=
144, giving approximately 5
×
10
50
possibilities. We therefore must
truncate the CI expansion.
In the CIS model we just take singly excited states. If the reference wavefunction is a
closed-shall HF one, the single excitations do not mix with the ground state and CIS tends
to be used as a low-level method for studying excited states.
The next logical step is to include the double excitations. If this is done together with
the single excitations we have the CISD (
CI singles and doubles
) model. If the double
