N-electron valence state perturbation theory

In quantum chemistry, N-Electron Valence state Perturbation Theory (NEVPT) is a perturbative treatment applicable to multireference CASCI-type wavefunctions. It can be considered as a generalization of the well-known Møller-Plesset perturbation theory to multireference Complete Active Space cases.

Theory

Defining $\Psi _{m}^{(0)}$ as the zero-order CASCI wavefunction obtained diagonalizing the true Hamiltonian ${\hat {H}}$ inside the CAS space

${\hat {\mathcal {P}}}_{CAS}{\hat {H}}{\hat {\mathcal {P}}}_{CAS}\left|\Psi _{m}^{(0)}\right\rangle =E_{m}^{(0)}\left|\Psi _{m}^{(0)}\right\rangle$

where ${\hat {\mathcal {P}}}_{CAS}$ is the projector inside the CASCI space.

Zero-order perturber wavefunctions in NEVPT are defined in the outer space (external to CAS) where $k$ electrons are removed from the inactive part (core and virtual orbitals) and added to the valence part (active orbitals). At second order of perturbation $-2\leq k\leq 2$. The pattern of inactive orbitals involved in the procedure can be grouped as a collective index $l$, so to represent the various perturber wavefunctions as $\Psi _{l,\mu }^{k}$, with $\mu$ an enumerator index for the different wavefunctions. Supposing indexes $i$ and $j$ referring to core orbitals, $a$ and $b$ referring to active orbitals and $r$ and $s$ 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 $k=0$)
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 $k=+1$)
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 $k=-1$)
4. two electrons from core orbitals to active orbitals (active space enriched with two electrons, $k=+2$)
5. two electrons from active orbitals to virtual orbitals (active space depleted with two electrons, $k=-2$)

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 ($k=0$)
2. one electron from a core orbital to an active orbital, and an internal active-active excitation ($k=+1$)
3. one electron from an active orbital to a virtual orbital, and an internal active-active excitation ($k=-1$)

All these classes are characterized by Hilbert spaces $S_{l}^{k}$ made of those determinants with given k and l labels. It is interesting to note that the determinants characterizing these spaces can be written as a partition comprising the same inactive (core + virtual) part $\Phi _{l}^{-k}$ and all possible valence (active) parts $\Psi _{I}^{k}$

$S_{l}^{k}\equiv \{\Phi _{l}^{-k}\Psi _{I}^{k}\}$

Properties

NEVPT is blessed with many important properties, making the approach very solid and reliable. These properties arise both from the theoretical approach used and on the Dyall's Hamiltonian particular structure:

• Size consistency: NEVPT is size consistent (strict separable). Briefly, if A and B are two non-interacting systems, the energy of the supersystem A-B is equal to the sum of the energy of A plus the energy of B taken by themselves ($E(A-B)=E(A)+E(B)$). This property is of particular importance to obtain correctly behaving dissociation curves.

• Absence of intruder states: in perturbation theory, divergencies can occur if the energy of some perturber happens to be nearly equal to the energy of the zero-order wavefunction. This situation, which is due to the presence of an energy difference at the denominator, can be avoided if the energies associated to the perturbers are guaranteed to be never nearly equal to the zero-order energy. Møller-Plesset perturbation theory satisfies this requisite, the energy difference being $\epsilon _{r}+\epsilon _{s}-\epsilon _{i}-\epsilon _{j}$, with $\epsilon$ orbital energies of the virtual ($r,s$ indexes) and core ($i,j$ indexes). NEVPT also satisfies this requirement, with a resulting expression for the energy differences conceptually similar to the MP2.

• Invariance under active orbital rotation: The NEVPT results are stable if an intraclass active-active orbital mixing occurs. This arises both from the structure of the Dyall Hamiltonian and the properties of a CASSCF wavefunction. This property has been also extended to the intraclass core-core and virtual-virtual mixing, thanks to the Non Canonical NEVPT approach, allowing to apply a NEVPT evaluation without performing an orbital canonization (which is required, as we saw previously)

• Spin purity is guaranteed: The resulting wavefunctions are guaranteed to be spin pure, due to the spin-free formalism.

• Efficiency: although not a formal theoretical property, computational efficiency is highly important for the evaluation on medium-size molecular systems. The current limit of the NEVPT application is largely dependent on the feasibility of the previous CASSCF evaluation, which scales factorially with respect to the active space size. The NEVPT implementation using the Dyall's Hamiltonian involves the evaluation of Koopmans' matrixes and density matrixes up to the four-particle density matrix spanning only active orbitals. This is particularly convenient, given the small size of currently used active spaces.

See also

• Electron correlation
• Perturbation theory (quantum mechanics)
• Post-Hartree-Fock

References

• Angeli C., Cimiraglia R., Evangelisti S., Leininger T., Malrieu J.-P., Introduction of n-electron valence states for multireference perturbation theory J. Chem. Phys., 114 (23) 10252 (2001)
• Angeli C., Cimiraglia R., Malrieu J.-P., n-electron valence state perturbation theory: a fast implementation of the strongly contracted variant Chem. Phys. Lett., 350 (3-4) 297 (2001)
• Angeli C., Cimiraglia R., Malrieu J.-P., n-Electron Valence State Perturbation Theory. A spinless formulation and an efficient implementation of the strongly contracted and of the partially contracted variants, J. Chem. Phys., 117 (20) 9138 (2002)