Solving the assignment problem in model-based diabatization with Target Overlap Assignment Diabatization (TOAD)

The quantum dynamics in excited states are often strongly influenced by non-adiabatic interactions and are treated best in a quasi-diabatic representation. Diagonalization of the diabatic model Hamiltonian yields the energy and wave function for each state in an adiabatic order. It is generally assumed that these model values can be compared directly to the reference data in an adiabatic order. We show that comparison of these adiabatic energies generates spurious local minima in the loss function due to an incorrect assignment of the corresponding eigenstates. Therefore, a new approach is presented called Target Overlap Assignment Diabatization (TOAD) using wave function information to remove the ambiguity in energy ordering of the ansatz or even the hybrid diabatization approach. The method is based on a simple permutation algorithm utilizing the overlap matrix between reference and model states to correctly assign the model-target energy pairs of the loss function of the non-linear optimization. A simple E × e Jahn-Teller model is used to analyze the problem, and it is shown that the multi-minima structure vanishes completely using this new approach. The power of the TOAD method compared to standard diabatization by ansatz is demonstrated for the complex example of a nine dimensional 5 × 5 diabatic Jahn-Teller Hamiltonian of the methyl fluoride cation. We show that the TOAD method enforces the correct physical behavior onto the resulting potential energy surfaces (PESs) consistently, resulting in an overall better description of the PESs in functional form, position of intersections, avoided crossings, and energy splittings between states.