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 this approximation: the first scheme is based on the full semi-classical light-matter interaction, 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 basis set requirements of these three schemes. We have considered 1s1/2 and 7s1/2 → 7p1/2 transitions in the radium atom, representative of core and valence excitations, respectively, and carried out calculations with dyall.aeXz (X = 2, 3, 4) basis sets at the four-component relativistic TD-HF level of theory. Our basis set study was greatly facilitated by the generation and visualization of radial distributions of transition moment densities, allowing for a straightforward comparison with equivalent finite-difference calculations. Pertaining to the truncated interaction, we find that the length representation electric multipole 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 difficult to converge. The velocity representation electric multipoles are the most difficult to converge: at high orders, the dyall.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 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.