Quantum Mechanical Insights into Biological Processes at the Electronic Level - Computational Modeling of Biological Systems - page 125

The part of the total Hamiltonian that is viewed as a perturbation in MPPT is
defined as:

J ij

2 K ij ;

X X X X

1 1

V D

(19)

j r i r j j

i i

j ¤ i j ¤ i

i.e., the difference between the true electron-electron interaction and the HF
electron-electron interaction energy. The corrected wave function is

X

n

i .i/ ;

D 0 C

lim
n !1

(20)

i D 1

where 0 is the HF wave function and .i/ are the parts of the electronic wave
function due to the perturbation. The total energy is a power-series expansion in
, too:

X

n

i E .i/ ; E D E 0 C lim

n !1

(21)

i D 1

D P i " i . Skipping the details, this particular partitioning of the total
Hamiltonian into the unperturbed part and the perturbation yields a first order
perturbation theory energy correction equal to zero, with the second order PT
correction being:

where E .0/

D 0 ˇ ˇ ˇ V ˇ ˇ ˇ n i

2

X

E .2/
0

D ; (22)

E .0/
0

E .0/

n

n ¤ 0

where 0 is the HF wave function, n are doubly excited states (terms due to the
singly and triply excited states are zero), and E .0/ D P

i

" i for either the ground
state, or the n th excited state. If the power series for the total energy is truncated
after E .2/ , the method is called MP2, and it is the most commonly used method of
the MP group. Higher order corrections can be derived fairly straightforwardly too,
to produce the MP3, MP4, and MP5 formalisms.
There is an important limitation intrinsic to the MP methods. As any PT, MP
works well only if the perturbation is small, i.e., the reference HF wave function is
already a fairly good solution. If this is not the case, MPPT results will be erroneous.
Additionally, one must keep in mind that the perturbation theory correction is
nonvariational. In other words, it is not guaranteed that the total energy obtained
with the MP methods will be the upper-bound of the true energy. What often
comes as a surprise is an oscillatory behavior of the total energies found with MP2,
MP3, MP4, and MP5, meaning that with the seeming increase in the amount of
electron correlation the energy does not consistently go down but oscillates, often
slowly converging to a particular intermediate value. This behavior is also a result
of nonvariational treatment. In such cases, the best MP solution can be found by
projecting to the limit of MP

1 .