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 second-order Møller–Plesset perturbation theory to multireference Complete Active Space cases. The theory is directly integrated into many quantum chemistry packages such as MOLCAS, Molpro, DALTON and ORCA.
The research performed into the development of this theory led to various implementations. The theory here presented refers to the deployment for the Single-State NEVPT, where the perturbative correction is applied to a single electronic state.
Research implementations has been also developed for Quasi-Degenerate cases, where a set of electronic states undergo the perturbative correction at the same time, allowing interaction among themselves. The theory development makes use of the quasi-degenerate formalism by Lindgren and the Hamiltonian multipartitioning technique from Zaitsevskii and Malrieu.
Theory.
Let formula_1 be a zero-order CASCI wavefunction, defined as a linear combination of Slater determinants
obtained diagonalizing the true Hamiltonian formula_3 inside the CASCI space
where formula_5 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 formula_6 electrons are removed from the inactive part (core and virtual orbitals) and added to the valence part (active orbitals). At second order of perturbation formula_7. Decomposing the zero-order CASCI wavefunction as an antisymmetrized product of the inactive part formula_8 and a valence part formula_9
then the perturber wavefunctions can be written as
The pattern of inactive orbitals involved in the procedure can be grouped as a collective index formula_12, so to represent the various perturber wavefunctions as formula_13, with formula_14 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 formula_15 and formula_16 referring to core orbitals, formula_17 and formula_18 referring to active orbitals and formula_19 and formula_20 referring to virtual orbitals, the possible excitation schemes are:
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:
Totally Uncontracted Approach.
A possible approach is to define the perturber wavefunctions into Hilbert spaces formula_29 defined by those determinants with given k and l labels. The determinants characterizing these spaces can be written as a partition comprising the same inactive (core + virtual) part formula_30 and all possible valence (active) parts formula_31
The full dimensionality of these spaces can be exploited to obtain the definition of the perturbers, by diagonalizing the Hamiltonian inside them
This procedure is impractical given its high computational cost: for each formula_29 space, a diagonalization of the true Hamiltonian must be performed. Computationally, is preferable to improve the theoretical development making use of the modified Dyall's Hamiltonian formula_35. This Hamiltonian behaves like the true Hamiltonian inside the CAS space, having the same eigenvalues and eigenvectors of the true Hamiltonian projected onto the CAS space. Also, given the decomposition for the wavefunction defined before, the action of the Dyall's Hamiltonian can be partitioned into
stripping out the constant contribution of the inactive part and leaving a
subsystem to be solved for the valence part
The total energy formula_38 is the sum of formula_39 and the energies of the orbitals involved in the definition of the inactive part formula_30. This introduces the possibility to perform a single diagonalization of the valence Dyall's Hamiltonian on the CASCI zero-order wavefunction and evaluate the perturber energies using the property depicted above.
Strongly Contracted Approach.
A different choice in the development of the NEVPT approach is to choose a single function for each space formula_29, leading to the Strongly Contracted (SC) scheme. A set of perturbative operators are used to produce a single function for each space, defined as the projection inside each space formula_42 of the application of the Hamiltonian to the contracted zero order wavefunction. In other words,
where formula_42 is the projector onto the subspace. This can be equivalently written as the application of a specific part of the Hamiltonian to the zero-order wavefunction
For each space, appropriate operators can be devised. We will not present their definition, as it could result overkilling. Suffice to say that the resulting perturbers are not normalized, and their norm
plays an important role in the Strongly Contracted development. To evaluate these norms, the spinless density matrix of rank not higher than three between the formula_1 functions are needed.
An important property of the formula_48 is that any other function of the space formula_29 which is orthogonal to formula_48 do not interact with the zero-order wavefunction through the true Hamiltonian. It is possible to use the formula_48 functions as a basis set for the expansion of the first-order correction to the wavefunction, and also for the expression of the zero-order Hamiltonian by means of a spectral decomposition
where formula_53 are the normalized formula_54.
The expression for the first-order correction to the wavefunction is therefore
and for the energy is
This result still misses a definition of the perturber energies formula_57, which can be defined in a computationally advantageous approach by means of the Dyall's Hamiltonian
leading to
Developing the first term and extracting the inactive part of the Dyall's Hamiltonian it can be obtained
with formula_61 equal to the sum of the orbital energies of the newly occupied virtual orbitals minus the orbital energies of the unoccupied core orbitals.
The term that still needs to be evaluated is the bracket involving the commutator. This can be obtained developing each formula_62 operator and substituting. To obtain the final result it is necessary to evaluate Koopmans matrices and density matrices involving only active indexes. An interesting case is represented by the contribution for the formula_63 case, which is trivial and can be demonstrated identical to the Møller–Plesset second-order contribution
NEVPT2 can therefore be seen as a generalized form of MP2 to multireference wavefunctions.
Partially Contracted Approach.
An alternative approach, named Partially Contracted (PC) is to define the perturber wavefunctions in a subspace formula_65 of formula_29 with dimensionality higher than one (like in case of the Strongly Contracted approach). To define this subspace, a set of functions formula_67 is generated by means of the formula_68 operators, after decontraction of their formulation. For example, in the case of the formula_69 operator
The Partially Contracted approach makes use of functions formula_71 and formula_72. These functions must be orthonormalized and purged of linear dependencies which may arise. The resulting set spans the formula_73 space.
Once all the formula_65 spaces have been defined, we can obtain as usual a set of perturbers from the diagonalization of the Hamiltonian (true or Dyall) inside this space
As usual, the evaluation of the Partially Contracted perturbative correction by means of the Dyall Hamiltonian involves simply manageable entities for nowadays computers.
Although the Strongly Contracted approach makes use of a perturbative space with very low flexibility, in general it provides values in very good agreement with those obtained by the more decontracted space defined for the Partially Contracted approach. This can be probably explained by the fact that the Strongly Contracted perturbers are a good average of the totally decontracted perturbative space.
The Partially Contracted evaluation has a very little overhead in computational cost with respect to the Strongly Contracted one, therefore they are normally evaluated together.
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: