Biomedical Engineering Reference
In-Depth Information
19
Electron Correlation
I have mentioned electron correlation at several points in the text, and I gave an operational
definition of correlation energy as the difference between the exact HF energy and the true
experimental energy. There is a great deal of small print; in general we cannot obtain exact
HF wavefunctions for molecules, only LCAO approximations. The Schrödinger treatment
takes no account of the theory of relativity, whilst we know from simple atomic spectra
that relativistic effects are non-negligible. We have to be careful to treat zero-point energy
in a consistent manner when dealing with vibrations, and so on.
19.1 Electron Density Functions
I have put a great deal of emphasis on the electron density. The wavefunction for a molecule
with
n
electrons will depend on the 3
n
spatial electron coordinates
r
1
,
r
2
,...,
r
n
together
with the
n
spin variables
s
i
(α or β). Many authors combine space and spin variables into
a composite
x
=
r
s and so we would write a wavefunction
Ψ (
x
1
,
x
2
, ...,
x
n
)
According to the Born interpretation,
Ψ
∗
(
x
1
,
x
2
, ...,
x
n
) Ψ (
x
1
,
x
2
, ...,
x
n
) dτ
1
d
s
1
dτ
2
d
s
2
... dτ
n
d
s
n
gives the probability of finding simultaneously electron 1 in spatial volume element dτ
1
with spin between
s
1
and
s
1
+
d
s
1
, electron 2 in spatial volume element dτ
2
with spin
between
s
2
and s
2
+
d
s
2
,...,electron
n
in spatial volume element dτ
n
with spin between
s
n
and
s
n
+
d
s
n
.
