Biomedical Engineering Reference
In-Depth Information
and the averaged ones from HF theory as the perturbation. This gives us Møller-Plesset
perturbation theory. The method had been in the literature since 1934, but its potential
for molecular calculations was not appreciated until the 1970s. It is of immense historical
interest because it gave us the first realistic route for a treatment of small-molecule correl-
ation energy, and you ought to read the original paper. The synopsis is reproduced below
(Møller and Plesset 1934).
A Perturbation Theory is developed for treating a system of n electrons in which the Hartree
Fock solution appears as the zero-order approximation. It is shown by this development that
the first order correction for the energy and the charge density of the system is zero. The
expression for the second order correction for the energy greatly simplifies because of the
special property of the zero order solution. It is pointed out that the development of the higher
order approximation involves only calculations based on a definite one-body problem.
The HF model averages over electron repulsions, and the HF pseudo-one-electron
operator for each electron has the form (from Chapters 14 and 16)
1
2
K
(
r
i
)
The unperturbed Hamiltonian is taken as the sum of the HF operators for each of the
n
electrons
h
F
(
r
i
)
=
h
(1)
(
r
i
)
+
J
(
r
i
)
−
n
H
(0)
h
F
(
r
i
)
=
i
=
1
2
K
(
r
i
)
n
1
h
(1)
(
r
i
)
+
J
(
r
i
)
=
−
(19.14)
i
=
1
whilst the true Hamiltonian makes no reference to averaging:
n
n
−
1
n
e
2
4πε
0
1
r
ij
H
h
(1)
(
r
i
)
=
+
i
=
1
i
=
1
j
=
i
+
1
The first few key equations of the perturbation expansion, taken from Chapter 14 but
simplified for our present discussion, are shown below. The electronic state of interest is the
ground state, denoted Ψ
(0)
, with energy ε
(0)
. I have therefore dropped the (
i
) subscripts, since
we are dealing with just the one electronic state. Promoting electrons from the occupied to
the virtual HF spinorbitals gives the excited states, and we consider singe, double, triple,
etc. excitations in the usual fashion of CI:
H
(0)
Ψ
(0)
=
ε
(0)
Ψ
(0)
ε
(0)
Ψ
(1)
ε
(1)
−
H
(1)
Ψ
(0)
H
(0)
−
=
(19.15)
ε
(
0
)
Ψ
(
2
)
H
(
0
)
−
H
(
1
)
Ψ
(
1
)
−
=
ε
(
2
)
Ψ
(
0
)
+
ε
(
1
)
Ψ
(
1
)
Since the zero-order Hamiltonian is a sum of HF operators, the zero-order energy is a
sum of orbital energies
ε
(0)
=
ε
i
(19.16)
i
