Evaluation of first- and second-order nonadiabatic coupling elements from large multiconfigurational self-consistent-field wave functions

An efficient method is proposed for evaluating first- and second-order nonadiabatic matrix elements of the form 〈ψᵢ(q;Q){}{δ}ψⱼ(q;Q)/{δ}Q〉 and 〈ψᵢ(q;Q){}δ²{{ψ}}ⱼ(q;Q)/δ{Q}²$ 〉, where ${{ψ}}ᵢ$(q;Q) and ${{ψ}}ⱼ$(q;Q) denote multiconfigurational self-consistent-field electron wave functions. The method is based on a finite-difference procedure and requires the numerical computation of symmetric overlaps of the type 〈ψᵢ(q,Q₀-x){}ψⱼ(q,Q₀ +x)〉. It gives an accuracy which is quadratic in the nuclear displacement x for both the first- and second-order nonadiabatic coupling constants. The wave functions are separately optimized for each state and obtained through the direct second-order MCSCF method. The biorthogonal scheme of Malmquist is implemented that expresses ψᵢ(q;Q) and ψⱼ(q:Q) in an orthogonal common basis. The method is applied for the calculation of the nonadiabatic coupling elements and the Born-Oppenheimer corrections to the two lowest ²Σ⁺ states of NaLi⁺, relevant for analyzing the asymmetric charge exchange in the ion-atom collision Na+Li⁺{→}Na⁺+Li. .AE