After solving for the lowest three adiabatic PESs, we project this dipole operator to the truncated Hilbert space of these adiabatic states, such that μ -→ P μ P, where P = 2 a=0 |φ a (R) φ a (R)|. By defining Ûµ in this subspace, we can then find the matrix elements, V i (R) and V ij (R) by transforming Ĥel -→ Û † µ Ĥel Ûµ , where V i (R) and V ij (R) are the diagonal and off-diagonal matrix elements, respectively, in this basis.