Origin-independent calculation of quadrupole intensities in X-ray spectroscopy

TL;DR

This paper develops an origin-independent method for calculating quadrupole intensities in X-ray spectroscopy, addressing the limitations of previous approaches that depended on the coordinate system origin.

Contribution

It introduces a consistent inclusion of all relevant contributions for origin independence in quadrupole intensity calculations within time-dependent density-functional theory.

Findings

Implemented origin-independent quadrupole intensity calculations in XAS.

Demonstrated the method on transition metal complexes.

Showed improved accuracy in metal and ligand K-edge spectra.

Abstract

For electronic excitations in the ultraviolet and visible range of the electromagnetic spectrum, the intensities are usually calculated within the dipole approximation, which assumes that the oscillating electric field is constant over the whole molecule. For the short wavelengths used in X-ray spectroscopy, this dipole approximation breaks down and it becomes necessary to include higher-order contributions. In quantum-chemical approaches to X-ray spectroscopy, these so-called quadrupole intensities have so far been calculated by including contributions depending on the square of the electric-quadrupole and magnetic-dipole transition moments. However, the resulting quadrupole intensities depend on the choice of the origin of the coordinate system. Here, we show that for obtaining an origin-independent theory, one has to include all contributions that are of the same order in the wave vector consistently. This leads to two additional contributions depending on products of the electric-dipole and electric-octupole and of the electric-dipole and magnetic-quadrupole transition moments, respectively. We have implemented such an origin-independent calculation of quadrupole intensities in X-ray absorption spectroscopy (XAS) within time-dependent density-functional theory, and demonstrate its usefulness for the calculation of metal and ligand K-edge XAS spectra of transition metal complexes.