On nonadiabatic coupling vectors in time-dependent density functional theory

Two approaches for the calculation of nonadiabatic couplings (NACs) within linear-response time-dependent density functional theory (TDDFT) were independently developed by Tavernelli and co-workers and Sugino and co-workers. Both methods are based on the matrix formulation of the TDDFT equations that are also known under the name of Casida's equations. Despite the different formulations, the two approaches are identical, in the sense that the introduction of the auxiliary wave functions does not imply any additional approximation. Both of these approaches make use of the expansion of a suitable TDDFT response function in a sum-over-states formulation, which is then linked to the equivalent equation obtained from many-body theory (MBT) in the so-called Lehmann representation.