Transition Moments Beyond the Electric-Dipole Approximation: Visualization and Basis Set Requirements

In the simulation of X-ray absorption spectroscopy, the validity of the electric dipole approximation comes into question. Three different schemes exist to go beyond thisapproximation: the first scheme is based on the full semi-classical light-matter inter-action, whereas the latter two schemes, referred to as the generalized length and velocity representation, are based on truncated multipole expansions. Even though these schemes have been successfully implemented in several quantum chemistry codes,their basis set requirements remained largely unknown. Here, we assess the basis set requirements of the full interaction operator and the generalized length and velocity representations. These three schemes were applied in the calculation of radial distributions of transition moment densities corresponding to the 1s1/2 and 7s1/2 7p1/2transition in the radium atom, representative of core and valence excitations, respectively. We have performed calculations with the dyall.aeXz (X=2,3,4) basis sets at the four-component relativistic TD-HF level of theory and compared them withequivalent finite-difference calculations. We find that the full interaction is extremelystable with respect to the choice of basis set, already being converged with dyall.ae2z.With respect to truncated interaction, we find that the length representation electric multipoles is the easiest to converge, requiring the dyall.ae2z basis for low-order multipoles and the dyall.ae4z basis at higher orders. The magnetic multipole moments follow a similar trend, although they are more diffcult to converge. The velocity representation electric multipoles are the most difficult to converge: at high orders, thedyall.ae3z and dyall.ae4z basis sets introduce artificial peaks and oscillations, which increase the overall error. These artifacts are associated with linear dependence issues in the small component space of the larger basis sets. The full interaction operator,however, does not suffer from these problems, and we therefore recommend its use in the simulation of x-ray spectroscopy.