Natural excitation framework for defining the external space: Uncontracted and internally contracted multireference nonorthogonal wavefunction theories

In this article, 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 the overlap of the external and reference spaces. The 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 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.