N-electron Valence State Perturbation Theory - Theory

Theory

Let be a zero-order CASCI wavefunction, defined as a linear combination of Slater determinants

obtained diagonalizing the true Hamiltonian inside the CASCI space

where is the projector inside the CASCI space. It is possible to define perturber wavefunctions in NEVPT as zero-order wavefunctions of the outer space (external to CAS) where electrons are removed from the inactive part (core and virtual orbitals) and added to the valence part (active orbitals). At second order of perturbation . Decomposing the zero-order CASCI wavefunction as an antisymmetrized product of the inactive part and a valence part

then the perturber wavefunctions can be written as

The pattern of inactive orbitals involved in the procedure can be grouped as a collective index, so to represent the various perturber wavefunctions as, with an enumerator index for the different wavefunctions. The number of these functions is relative to the degree of contraction of the resulting perturbative space.

Supposing indexes and referring to core orbitals, and referring to active orbitals and and referring to virtual orbitals, the possible excitation schemes are:

1. two electrons from core orbitals to virtual orbitals (the active space is not enriched nor depleted of electrons, therefore )
2. one electron from a core orbital to a virtual orbital, and one electron from a core orbital to an active orbital (the active space is enriched with one electron, therefore )
3. one electron from a core orbital to a virtual orbital, and one electron from an active orbital to a virtual orbital (the active space is depleted with one electron, therefore )
4. two electrons from core orbitals to active orbitals (active space enriched with two electrons, )
5. two electrons from active orbitals to virtual orbitals (active space depleted with two electrons, )

These cases always represent situations where interclass electronic excitations happen. Other three excitation schemes involve a single interclass excitation plus an intraclass excitation internal to the active space:

1. one electron from a core orbital to a virtual orbital, and an internal active-active excitation
2. one electron from a core orbital to an active orbital, and an internal active-active excitation
3. one electron from an active orbital to a virtual orbital, and an internal active-active excitation