Electronic Diabatic Framework: Restrictions Due to Quantization of the Nonadiabatic Coupling Matrix

Abstract

In this article, two issues related to the size of the electronic diabatic potential energy matrix are treated. (a) We frequently mention the fact that the dimension of a diabatic matrix obtained by a unitary transformation from the adiabatic framework is determined by the way the nonadiabatic coupling matrix τ breaks up into blocks. In this article, we prove for the first time that the size of the diabatic matrix as obtained in a direct way is determined in the same way. In other words, if the dimension of the above-mentioned decoupled block is N, then the dimension of any diabatic potential energy matrix with physical relevance has to be N as well, regardless of how it was derived. This number, N, is also equal to the number of coupled diabatic−Schrödinger equations to be solved. (b) The second issue is, consequently, related to the actual required number of coupled Schrödinger equations to be solved to obtain a well-converged solution. Starting with the earlier introduced number N, we show that this number can be reduced, and in fact, it is most likely equal to the number of energetically open adiabatic states (for a given energy). While doing that, we rigorously derived the relevant diabatic potential matrix for this reduced case. We also worked out in detail an example related to a three-state case and derived the relevant 2 × 2 diabatic potential matrix.