Natural Excitation Framework for Defining the External Space: Uncontracted and Internally Contracted Multireference Nonorthogonal Wavefunction Theories

TL;DR

The paper addresses the challenge of defining the external space in nonorthogonal reference expansions used for post-NOCI multireference methods, where internal contamination and linear dependencies can cause correlation double counting. It introduces a natural excitation framework based on natural orbitals (nos) derived from the one-particle density matrix (1PDM) to provide a common MO basis, partitioning orbitals into occupied, virtual, and active sets and employing a projector that removes internal contamination, enabling excitations up to order $l$. The approach yields both uncontracted and internally contracted post-NOCI variants within this framework, enabling reduced scaling, a cleaner separation of excitation types, and easier translation from orthogonal to nonorthogonal methodologies, by imposing a block-diagonal structure in the excitation space. A VH dissociation benchmark demonstrates that the method captures mixed $3d^3$/$3d^4$ configurations with a modest active space, where the active-space dimension is 39 out of 96 natural orbitals, and reduces the external-space construction cost while preserving accuracy.

Abstract

In this communication, we examine new formalisms for the construction of the external space when correlating reference wavefunctions built from nonorthogonal determinant expansions. Defining the external space in nonorthogonal approaches is challenging, as every substitution from the reference wavefunction can potentially mix both internal and external configurations. As a result, post-nonorthogonal methods are plagued by internal contamination and linear dependencies in the external space, which may lead to correlation double counting that results from overlap of the external and reference spaces. Removal of these internal configurations and orthonormalization of the excited space basis can be computationally expensive. In particular, as the excitation operators cannot be subdivided by their action on orbital subspaces, the external space cannot be partitioned into non-overlapping subsets as is possible in orthogonal methods. To resolve these issues, we propose both uncontracted and internally-contracted approaches based on a natural excitation framework that allows for reduced scaling, more straightforward separation of excitation types that lead to external and internal spaces, and allows for a facile translation of orthogonal methods to a nonorthogonal framework. Several proofs and a numerical demonstration using vanadium monohydride (VH) are provided to illustrate the viability of the proposed approach, using a method-agnostic presentation to highlight the generality of the approach.

Figures (2)

• Figure 1: Depiction of a small nonorthogonal expansion of configurations in their original molecular orbitals basis and the set of natural orbital with the proposed orbital partitioning. Closed (inactive) natural orbitals map entirely on to orbitals that are occupied in all nonorthogonal determinants, while virtual natural orbitals map entirely on to orbitals that are unoccupied. The fractionally occupied active natural orbitals map on to orbitals that are occupied in some configurations but virtual in other configurations.
• Figure 2: (a) Quintet and septet vanadium monohydride (VH) dissociation potential energy curves using a valence double zeta-type basis set. Hollow marks indicates the reference occupation-specific (OS) states and filled marks are the states formed by the nonorthogonal coupled OS reference states. (b) Number of orbitals with natural occupation ($n_o$) between zero and one given a color and shaped coded threshold ($\nu<$$n_o$$<1-\nu$). Dashed lines and hollow markers represents the occupation of the full 1PDM, while full lines and markers represents the the state-specific (ground state 1PDM) occupations. As a consequence, for a given occupation threshold, the dashed lines are a ceiling for the state-specific spaces.