On the applicability of a wavefunction-free, energy-based procedure for generating first-order non-adiabatic couplings around conical intersections

The primal definition of first-order non-adiabatic couplings among electronic states relies on the knowledge of how electronic wavefunctions vary with nuclear coordinates. However, the non-adiabatic coupling between two electronic states can be obtained in the vicinity of a conical intersection from energies only, as this vector spans the branching plane along which degeneracy is lifted to first order. The gradient difference and derivative coupling are responsible of the two-dimensional cusp of a conical intersection between both potential-energy surfaces and can be identified to the non-trivial eigenvectors of the second derivative of the square energy difference, as first pointed out in Köppel and Schubert [Mol. Phys. 104(5-7), 1069 (2006)]. Such quantities can always be computed in principle for the cost of two numerical Hessians in the worst-case scenario. Analytic-derivative techniques may help in terms of accuracy and efficiency but also raise potential traps due to singularities and ill-defined derivatives at degeneracies. We compare here two approaches, one fully numerical, the other semianalytic, where analytic gradients are available but Hessians are not, and investigate their respective conditions of applicability. Benzene and 3-hydroxychromone are used as illustrative application cases. It is shown that non-adiabatic couplings can thus be estimated with decent accuracy in regions of significant size around conical intersections. This procedure is robust and could be useful in the context of on-the-fly non-adiabatic dynamics or be used for producing model representations of intersecting potential energy surfaces with complete obviation of the electronic wavefunctions.