Introduction to M øller-Plesset Perturbation Theory

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 • Complete Hamiltonian • 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, 2002 .