Introduction to Møller-Plesset
Perturbation Theory
Kelsie Betsch
Chem 381
Spring 2004
Møller-Plesset: Subset of Perturbation
Theory

Rayleigh-Schrödinger Perturbation
Theory


H = H<0> + V
Møller-Plesset

Assumption that H<0> is Hartree-Fock
hamiltonian
Parts of the hamiltonian

H<0> is Hartree-Fock operator
Counts electron-electron repulsion twice


V corrects using Coulomb and
exchange integrals
gij = fluctuation potential

H
0
N

N
N
N
 Fi   ( h i   J ij  K ij ) )   h i  2  V ee 
i 1
V HH
i 1
0
j 1
 (J
i , j 1
h

N
i
i 1
i 1
N
 V ee 
N
N
ij
 K ij )  V ee  2  V ee 

i 1, j  i

g
ij
i, j
N
g ij 
g
i , j 1
ij


Complete Hamiltonian and Energy
Expression



Complete
Hamiltonian
Hartree-Fock
energy is sum of
zeroth- and firstorder corrections
Expression for
correlation energy
N
H 
h
N
i

i 1
g
ij
i 1, j  i
EHF = E0<0> + E0<1>

E corr 

n2
n
E0
Calculating Correlation Energies

Promote electrons from occupied to
unoccupied (virtual) orbitals



MP with 2nd order correction (MP2)


Electrons have more room
Decreased interelectronic repulsion lowers
energy
Two-electron operator
 Single, triple, quadruple excitations contribute
nothing
Corrections to other orders may have
S,D,T,Q, etc. contributions

Select methods may leave some contributions
out (MP4(SDQ))
How close do the methods come?




MP2 ~ 80-90% of correlation
energy
MP3 ~ 90-95%
MP4 ~ 95-98%
Higher order corrections are not
generally employed

Time demands
How to make an MP calculation


Select basis set
Carry out Self Consistent Field
(SCF) calculation on basis set


Obtain wavefunction, Hartree-Fock
energy, and virtual orbitals
Calculate correlation energy to
desired degree

Integrate spin-orbital integrals in terms
of integrals over basis functions
Basis Set Selection

Ideally, complete basis set


Complete basis sets not available



Yields an infinite number of virtual orbitals
 More accurate correlation energy
Finite basis sets lead to finite number of virtual
orbitals
 Less accurate correlation energy
Smallest basis set used: 6-31G*
Error due to truncation of basis set is
always greater than that due to
truncation of MP perturbation energy
(MP2 vs. MP3)
Advantages and Disadvantages

PT calculations not variational



Difficult to make comparisons
No such upper bound to exact
energy in PT as in variational
calculations
PT often overestimates correlation
energies

Energies lower than experimental
values
Advantages and Disadvantages

Interest in relative energies


Variational calculations, such as CI, are
poor
MP perturbation theory is size-extensive



Gives MPPT superiority
MP calculations much faster than CI
 Most ab initio programs can do them
MP calculations good close to
equilibrium geometry, poor if far from
equilibrium
Summary





Møller-Plesset perturbation theory
assumes Hartree-Fock hamiltonian as the
zero-order perturbation
Hartree-Fock energy is sum of zerothand first-order energies
Correlation energy begins with secondorder perturbation
How an MP calculation is carried out
Strengths and weaknesses of MP vs. CI
Acknowledgements


Dr. Brian Moore
Dr. Arlen Viste
References







P. Atkins and J. de Paula, Physical Chemistry, 7th ed.
W.H. Freeman and Company, New York, 2002.
A. Szabo and N.S. Ostlund, Modern Quantum Chemistry:
Introduction to Advanced Electronic Structure Theory,
Dover Publications, Inc., Mineola, NY, 1989.
C. Møller and M.S. Plesset, Phys. Rev., 46:618 (1934).
F.L. Pilar, Elementary Quantum Chemistry, 2nd ed. Dover
Publications, Inc., Mineola, NY, 1990.
F. Jensen, Introduction to Computational Chemistry, John
Wiley & Sons, Chichester, 1999.
E. Lewars, Introduction to the Teory and Applications of
Molecular and Quantum Mechanics, Kluwer Academic
Publishers, Boston, 2003.
I.N. Levine, Quantum Chemistry, 5th ed. Prentice Hall,
Upper Saddle River, NJ, 2002 .