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 〈${\psi }_i$(q;Q)‖δ${\psi }_j$(q;Q)/δQ〉 and 〈${\psi }_i$(q;Q)‖${\delta }^2$ ${\psi }_j$(q;Q)/δ$Q^2$ 〉, where ${\psi }_i$(q;Q) and ${\psi }_j$(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 〈${\psi }_i$(q,$Q_0$-x)‖${\psi }_j$(q,$Q_0$ +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 ${\psi }_i$(q;Q) and ${\psi }_j$(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 ${}_{}{}^2{}_{}{}^{}\mathrm{\Sigma }_{}^{+}{}_{}{}^{}$ states of ${\mathrm{NaLi}}^{+}$, relevant for analyzing the asymmetric charge exchange in the ion-atom collision Na+${\mathrm{Li}}^{+}$→${\mathrm{Na}}^{+}$+Li. .AE