Biomedical Engineering Reference
In-Depth Information
where the sum runs over the occupied spinorbitals. The first-order energy is
H
(1)
Ψ
(0)
dτ
ε
(1)
Ψ
(0)
=
(19.17)
and adding ε
(1)
to ε
(0)
gives the full HF energy ε
HF
. We therefore need to progress to second-
order theory in order to give any treatment of electron correlation. The levels of theory are
denoted MP2, MP3, ...,MP
n
, where
n
is the order of perturbation theory.
The second-order energy is given by
H
(1)
Ψ
j
dτ
Ψ
(0)
2
ε
(2)
=
(19.18)
ε
(0)
−
ε
j
j
and the first-order correction to the wavefunction is
Ψ
(0)
H
(1)
Ψ
j
dτ
ε
(0)
Ψ
(1)
=
Ψ
j
(19.19)
−
ε
j
j
where the sum runs over all excited states, written Ψ
j
, energies ε
j
. The ground state Ψ
(0)
is
a HF wavefunction and so the integral vanishes for all singly excited states because of the
Brillouin theorem. It is also zero when the excited state differs from Ψ
(0)
by more than two
spin orbitals, by the Slater-Condon-Shortley rules. Hence we only need consider doubly
excited states in order to find the second order energy. The ε
(2)
numerator is nonnegative
since it is the square of a modulus. The denominator is negative because Ψ
(0)
refers to the
ground state and Ψ
j
to excited states. The second-order energy correction is therefore always
negative or zero. Higher orders of perturbation theory may give corrections of either sign.
The MP1 energy is therefore identical to the HF energy and MP2 is the simplest prac-
ticable perturbation treatment for electron correlation. It includes only the effect of double
excitations.
By similar considerations, the third, fourth and higher orders of perturbation theory can
be determined. The terms rapidly become algebraically complicated and the higher orders
are increasingly costly to apply. For
n
basis functions, HF-LCAO theory scales as
n
4
, MP2
as
n
5
, MP3 as
n
6
and MP4 as
n
7
. These should be seen as theoretical upper bounds, since
sophisticated use of symmetry and integral cut-offs mean that practical calculations need
considerably less resource.
A great simplifying feature of MP2 is that a full four-index transformation is not neces-
sary, all we have to do is to semitransform the two-electron integrals and this leads to an
immense saving in time compared to conventional CI treatments. MP3 also includes only
double excitations, whilst MP4 includes a description of single excitations together with
some triples and quadruples. The triple contributions in MP4 are the most expensive. In the
MP4(SDQ) variation we just include the least expensive singles, doubles and quadruple
excitations
Analytical energy gradients have been determined for MP2, MP3 and MP4 making for
very effective geometry searching. I can illustrate some points of interest by considering
two examples. First is the geometry optimization of ethane, shown in Table 19.2. I ran all
calculations from the same starting point on the molecular potential energy surface. The
HF/STO-3G calculation would these days be regarded as woefully inadequate.
