Pivot invariance of multiconfiguration perturbation theory via frame vectors

Multiconfiguration perturbation theory (MCPT) is a general framework for correcting the reference function of arbitrary structures. The variants of MCPT introduced so far differ in the specification of their 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 determinant expansion of the reference, rendering the PT results dependent on this choice. It is here shown that the pivot dependence of MCPT can be eliminated by using an overcomplete set of projected determinants in the space orthogonal and complementary to the reference. The projected determinants form a so-called frame, a generalization of the notion of basis, allowing for redundancy of the set. The simple structure of the frame overlap matrix facilitates overlap treatment in closed form, a feature shared by previous MCPT variants. In particular, the Moore-Penrose inverse of singular matrices appearing in frame-based MCPT can be constructed without the need for 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.