The difference comes from the fact that the crude adiabatic basis in Ref. 32 is defined in such a way that only the CI coefficients are fixed, whereas in our definition both the CI coefficients and the Slater determinants are fixed. As a result of Brillouin's theorem, the definition in Eq. ( ) makes the nonadiabatic couplings vanish between the HF ground state and the CIS states at any geometric coordinates R. This choice is not convenient for examining the topology of potential energy surfaces at approximate crossing points; see Appendix B. When applied to compute nonadiabatic couplings between two CIS excited states, the use of Eq. ( ) versus Eq. ( ) affords couplings that differ by so-called electronic translation factors, equivalent to non-Hellmann-Feynman terms in the analytic gradient expression for the coupling. As shown in Appendix A, the nonadiabatic coupling between the Kohn-Sham DFT ground state and LR-TDDFT excited states can be defined as