Pivot invariance of multiconfiguration perturbation theory via frame vectors

Abstract

Multiconfiguration perturbation theory (MCPT) is a general framework for correcting a reference function of arbitrary structure. Variants of MCPT introduced so far differ in the specification of the zero-order Hamiltonian, i.e. the partitioning. A common characteristic of MCPT variants is that no numerical procedure is invoked when handling the overlap of the reference function and determinants spanning the configuration space. This comes at the price of pinpointing a principal term in the determinantal expansion of reference, rendering the PT results dependent on this choice.It is here shown, that pivot dependence of MCPT can be eliminated by using the overcomplete set of projected determinants in the space orthogonal and complementary to the reference. Projected determinants form a so-called frame, a generalization of the notion of basis, allowing for redundancy of the set. Simple structure of the frame overlap matrix facilitates overlap treatment in closed form, a feature shared by previous MCPT variants. In particular, Moore-Penrose inverse of singular matrices appearing in frame-based MCPT can be constructed without the need of any pivoting algorithm or numerical zero threshold.Pilot numerical studies are performed for the singlet-triplet gap of biradicaloid systems, relying on geminal based, incomplete model space reference function. Comparison with previous MCPT variants as well as illustration of pivot invariance is provided.

Solvability of a system of linear equations—an approach based on the generalized inverses determined by the Penrose equations ★

“…In what follows we demonstrate validity of the representations of the sets A{1} provided in (21) and (22), A{1, 4} provided in (27) and (28), and A{1, 3} provided in (31) and (32).…”

“…In this case, by taking V = 0 and W = 0, we arrive at the fixed inner inverse A − . Formal derivations of the representations of the set A{1} provided in (21) and (22) are available in Appendix.…”

“…Despite of not so common knowledge of the Moore-Penrose inverse within physics community, relevance of the notion for the research carried out in physics is reflected by a steady stream of physics papers published annually, which refer to the inverse. In order to update the list of references provided in [3] with the most recent articles taking advantage of the Moore-Penrose inverse, let us mention that: in [16] the inverse of a stiffness matrix was used to solve the characteristic finite-element equations in considerations dealing with modeling of elastic deformation and non-local brittle fracture of solids, in [17] the inverse of a correlation matrix was used in a study of correlated pulsar-pairs, in [18] the inverse of operator matrices in a Hilbert space framework was used to derive solutions of constrained linear equations using variational principles in considerations of a class of problems (called Z-problems) that arise in effective media theory, in [19] the inverse was used to derive numerical solutions to particular nonlinear ordinary differential equations and differential algebraic equations involved in neural networks analysis, in [20] the inverse of a key matrix was used to derive a novel image encryption algorithm, in [21] the inverse was used to construct an image fusion algorithm based on a matrix factorization, which aims at optimizing computational workload, in [22] the inverse was used to solve a set of linear equations resulting from a novel variant of the multiconfiguration perturbation theory proposed in order to ensure its pivot invariance, and in [23] the inverse was used to specify new mode time coefficient (equivalent to the best-fit solution in the least squares sense) for dynamic mode decomposition utilized to extract dominant modes of unsteady flow fields. Additionally, in [24] the Moore-Penrose inverse was used in comparison of alternative models of velocity and acceleration level inverse kinematic for redundant manipulators, with the inverse of a Jacobian matrix involved in each model; orthogonal projectors onto the fundamental subspaces associated with the Jacobian matrix were utilized in the analysis as well.…”