Gradients and non-adiabatic derivative coupling terms for spin-orbit wavefunctions

Analytic gradients of electronic eigenvalues require one calculation per nuclear geometry, compared to 3n calculations for finite difference methods, where n is the number of nuclei. Analytic non-adiabatic derivative coupling terms, which are calculated in a similar fashion, are used to remove non-diagonal contributions to the kinetic energy operator, leading to more accurate nuclear dynamics calculations than those that employ the Born-Oppenheimer approximation, i.e., that assume off-diagonal contributions are zero. The current methods and underpinnings for calculating both of these quantities for MRCI-SD wavefunctions in COLUMBUS are reviewed. Before this work, these methods were not available for wavefunctions of a relativistic MRCI-SD Hamiltonian. A formalism for calculating the density matrices, analytic gradients, and analytic derivative coupling terms for those wavefunctions is presented. The results of a sample calculation using a Stuttgart basis for K He are presented. Density matrices predict the MRCI eigenvalues to approximately 10-10 hartree. Analytic gradients match finite central-difference gradients to within one percent. The non-adiabatic coupling angle calculated by integrating the radial analytic derivative coupling terms matches the same angle approximated by the Werner method to within 0.02 radians. Non-adiabatic energy surfaces for K He are presented.