Bivariational time-dependent wave functions with biorthogonal adaptive basis sets: General formulation and regularization of equations of motion through polar decomposition

Abstract

We derive general bivariational equations of motion (EOMs) for time-dependent wave functions with biorthogonal time-dependent basis sets. The time-dependent basis functions are linearly parametrized and their fully variational time evolution is ensured by solving a set of so-called constraint equations, which we derive for arbitrary wave function expansions. The formalism allows the division of the basis set into an active basis and a secondary basis, ensuring a flexible and compact wave function. We show how the EOMs specialize to a few common wave function forms, including coupled cluster (CC) and linearly expanded wave functions. It is demonstrated, for the first time, that the propagation of such wave functions is not unconditionally stable when a secondary basis is employed. The main signature of the instability is a strong increase in non-orthogonality that eventually causes the calculation to fail; specifically, the biorthogonal active bra and ket bases tend towards spanning different spaces. Although formally allowed, this causes severe numerical issues. We identify the source of the problem by reparametrizing the time-dependent basis set through polar decomposition. The subsequent analysis allows us to remove the instability by setting appropriate matrix elements to zero. Although this solution is not fully variational, we find essentially no deviation in terms of autocorrelation functions relative to the variational formulation. We expect that the results presented here will be useful for the formal analysis of bivariational time-dependent wave functions for electronic and nuclear dynamics in general and for the practical implementation of time-dependent CC wave functions in particular.