The most general results of the nuclear gradient [∇ α V ] ij is thus presented in Eq. 28, with the matrix elements of V ij expressed in Eq. 30. To express the nuclear gradients in Eq. 28, we need the gradients ∇ α V ij and derivative coupling matrix [d α ] which often can be directly obtained from ab initio electronic structure calculations or numerical differential techniques. To construct the derivative coupling matrix [d α ], one can simply use