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, PySCF 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.
be a zero-order CASCI wavefunction, defined as a linear combination of Slater determinants obtained diagonalizing the true Hamiltonian
Decomposing the zero-order CASCI wavefunction as an antisymmetrized product of the inactive 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
The number of these functions is relative to the degree of contraction of the resulting perturbative space.
The determinants characterizing these spaces can be written as a partition comprising the same inactive (core + virtual) part
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
Computationally, is preferable to improve the theoretical development making use of the modified Dyall's Hamiltonian
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
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.
A different choice in the development of the NEVPT approach is to choose a single function for each space
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.
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
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
, 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
The term that still needs to be evaluated is the bracket involving the commutator.
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.
An alternative approach, named Partially Contracted (PC) is to define the perturber wavefunctions in a subspace
with dimensionality higher than one (like in case of the Strongly Contracted approach).
operator The Partially Contracted approach makes use of functions
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.
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: