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

Abstract

An efficient method is proposed for evaluating first- and second-order nonadiabatic matrix elements of the form 〈${\ensuremath{\psi}}_{i}$(q;Q)\ensuremath{\Vert}\ensuremath{\delta}${\ensuremath{\psi}}_{j}$(q;Q)/\ensuremath{\delta}Q〉 and 〈${\ensuremath{\psi}}_{i}$(q;Q)\ensuremath{\Vert}${\ensuremath{\delta}}^{2}$${\ensuremath{\psi}}_{j}$(q;Q)/\ensuremath{\delta}${Q}^{2}$ 〉, where ${\ensuremath{\psi}}_{i}$(q;Q) and ${\ensuremath{\psi}}_{j}$(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 〈${\ensuremath{\psi}}_{i}$(q,${Q}_{0}$-x)\ensuremath{\Vert}${\ensuremath{\psi}}_{j}$(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 ${\ensuremath{\psi}}_{i}$(q;Q) and ${\ensuremath{\psi}}_{j}$(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 $^{2}\mathrm{\ensuremath{\Sigma}}^{+}$ states of ${\mathrm{NaLi}}^{+}$, relevant for analyzing the asymmetric charge exchange in the ion-atom collision Na+${\mathrm{Li}}^{+}$\ensuremath{\rightarrow}${\mathrm{Na}}^{+}$+Li. .AE