Introduction to M

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 Hlt0gt V
• Møller-Plesset
• Assumption that Hlt0gt is Hartree-Fock hamiltonian

3
Parts of the hamiltonian

• Hlt0gt 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

• Complete Hamiltonian
• Hartree-Fock energy is sum of zeroth- and
  first-order corrections
• Expression for correlation energy

• EHF E0lt0gt E0lt1gt

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

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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., 46618
  (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 .