1. Introduction to Møller-Plesset Perturbation Theory
Kelsie Betsch
Chem 381
Spring 2004

2. Møller-Plesset: Subset of Perturbation Theory
Rayleigh-Schrödinger Perturbation Theory
H = H<0> + V
Møller-Plesset
Assumption that H<0> is Hartree-Fock hamiltonian

3. 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

4. Complete Hamiltonian and Energy Expression
Hartree-Fock energy is sum of zeroth- and first-order corrections
Expression for correlation energy
EHF = E0<0> + E0<1>

5. Calculating Correlation Energies
Promote electrons from occupied to unoccupied (virtual) orbitals
Electrons have more room
Decreased interelectronic repulsion lowers energy
MP with 2nd order correction (MP2)
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))

6. 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

7. 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

8. Basis Set Selection Ideally, complete basis set
Yields an infinite number of virtual orbitals
More accurate correlation energy
Complete basis sets not available
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)

9. 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

10. 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

11. Summary
Møller-Plesset perturbation theory assumes Hartree-Fock hamiltonian as the zero-order perturbation
Hartree-Fock energy is sum of zeroth- and first-order energies
Correlation energy begins with second-order perturbation
How an MP calculation is carried out
Strengths and weaknesses of MP vs. CI

12. Acknowledgements
Dr. Brian Moore
Dr. Arlen Viste

13. 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,