We have now considered the improvement of ab initio calculations by extensions of the basis set. Improvement here, of course, is defined in the spirit of the variation principle; that energy is best that is lowest.

However, the semi-empirical and ab initio calculations we have discussed to this point are self-consistent field (SCF) calculations.

• That is, they treat interactions between electrons in the form of the interaction of one electron with the average field of all the rest.

• Electron interactions are much more specific than this, and include Pauli repulsions as well as electrostatic ones.

• Consequently, the best calculation that could possibly be made, using the largest basis set with polarization and diffuse functions, will still reach its limit in an energy that is higher than the true energy. We might call this energy, as Hinchcliffe does, the SCF limit.

The gap between the "real" energy and the SCF limit is the correlation energy, so named to reflect its origin in the correlated movement of electrons seeking to remain as far from each other as possible. The final step we can take in improving MO calculations is to recover some of this correlation energy. Two general approaches, of several that exist, will be described here: configuration interaction and Moeller-Plesset perturbation theory.

Configuration Interaction

Suppose we wish to treat electron correlation in a lithium atom, which has the ground state configuration 1s², 2s¹.

• Excited electronic configurations also are available to the atom; for example, 1s², 3s¹, a singly excited state.

• We might improve our description of the lithium atom by arguing that electron correlation could lead to an instantaneous population of this excited state
  • Taking a linear combination of the wavefunctions for the ground state and this excited state should lead to a better (lower) energy for the atom.

  • Indeed, the lower energy solution of the resulting linear equations is a better description of the ground state, and the higher energy one is a better description of the excited state.

• This process is called configuration interaction.

• Within an SCF model, one chooses a basis set and carries out a calculation of the ground state.

• Excited state wave functions then can be generated by exciting electrons from the filled orbitals to virtual ones.
  • Commonly, both single excitations and double excitations are considered, leading to what is referred to as a CISD calculation (configuration interaction, single and double).

  • Although one might think it would be better to examine the likely effect of a given excitation, for reasons of ease of calculation usually all single and double excitations are included.

  • However, we usually exclude all core electrons and a corresponding number of the highest energy virtual orbitals from the calculation.

Configuration interaction calculations are possible within SPARTAN's semi-empirical module, by the use of keywords entered in the OPTIONS dialog box. SPARTAN '04 also offers this option directly from the calculation setup module.

• The default for closed-shell molecules is the so-called 3 x 3 CI, in which the ground state and the two singlet states obtained by promoting one and two electrons from the HOMO to the LUMO are used.

• Both GAUSSIAN and GAMESS allow CI calculations for ab initio methods as well.

• Consult the appropriate user manuals for the keywords and methods of specifying the excitations to be considered.

Moeller-Plesset Perturbation Theory

In Moeller-Plesset theory, the mixing in of excited states is treated as a perturbation:

If the original wavefunction is an RHF type, we label the calculation as MP2, MP3, and so on depending upon where the series is truncated. For a UHF wavefunction, we refer to UMP2, UMP3, and so on.

Computationally, MP correlation is less laborious than CI, and thus has displaced the CI type in many ab initio calculations, where the computational labor is already high.

• SPARTAN provides MP2 and UMP2 options as built-ins in the ab initio module.

• GAMESS and GAUSSIAN provide MP correlation up to MP5; however, only MP2 may be done in conjunction with geometry optimization.

• Again, the appropriate manuals should be consulted for instructions on carrying out these calculations.

1. Hehre, W. J.; Radom, L.; Schleyer, P. v.R.; Pople, J. A. Ab Initio Molecular Orbital Theory, John Wiley and Sons, New York, 1986.

2. Hinchcliffe, A. Computational Quantum Chemistry, John Wiley and Sons, Chichester, 1988.

3. Salem, L.; Rowland, C. Angew. Chem. Int. Ed. Engl., 1972, 11, 92.

4. Moeller, C.; Plesset, M. S. Phys. Rev., 1934, 46, 618.

Cramer, C. J., Essentials of Computational Chemistry, Wiley, 2003; Chapter 7.