A Galërkin approach to electronic near-degeneracies in molecular systems

We consider superposition states of various numbers of eigenstates in order to study vibrating quantum billiards semiquantally. We discuss the relationship between Galërkin methods, inertial manifolds, and partial differential equations such as nonlinear Schrِdinger equations and Schrِdinger equations with time-dependent boundary conditions. We then use a Galërkin approach to study vibrating quantum billiards. We consider one-term, two-term, three-term, d-term, and infinite-term superposition states. The number of terms under consideration corresponds to the level of electronic near-degeneracy in the system of interest. We derive a generalized Bloch transformation that is valid for any finite-term superposition and numerically simulate three-state superpositions of the radially vibrating spherical quantum billiard with null angular-momentum eigenstates. We discuss the physical interpretation of our Galërkin approach and thereby justify its use for vibrating quantum billiards. For example, d-term superposition states of one degree-of-vibration quantum billiards may be used to study nonadiabatic behavior in polyatomic molecules with one excited nuclear mode and a d-fold electronic near-degeneracy. Finally, we apply geometric methods to analyze the symmetries of vibrating quantum billiards.