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

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 length scale of the transition. For the short wavelengths used in hard X-ray spectroscopy, the dipole approximation may not be adequate. In particular, for metal K-edge X-ray absorption spectroscopy (XAS), it becomes necessary to include higher-order contributions. In quantum-chemical approaches to X-ray spectroscopy, these socalled 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 originindependent 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 andelectric-octupole and of the electric-dipole and magnetic-quadrupole transition moments, respectively. We have implemented such an origin-independent calculation of quadrupole intensities in XAS within timedependent density-functional theory, and demonstrate its usefulness for the calculation of metal and ligand K-edge XAS spectra of transition metal complexes.2

Quantum Chemical Calculations of X-ray Emission Spectroscopy

“…25 Although, in the context of X-ray absorption spectroscopy, it has been shown that for heavier the quadrupole contributions to the intensity are significant. 26 Within this approach, several different approaches to describing the orbitals of the core-excited state have been used. These include using the frozen orbital approximation where the ground state orbitals are used, 25,[27][28][29] the Z+1 approximation where an increased nuclear charge is used for the appropriate atom, 3 a transition potential approach where a half filled core orbital is used providing a balance between final and initial states 27 and a fully relaxed approach where the orbitals of the core-hole state are optimized in a separate SCF procedure.…”

Applications to metal K pre-edges of transition metal dimers illustrate the approximate origin independence for the intensities in the length representation

“…However, an important detail is that the expansion of the origin-independent formulation in reference [22] is only strictly valid in the velocity representation. Transformation to the length representation is typically done through commutator relations, which, however, requires that these hold exactly.…”

“…Bernadotte et al [22] showed that full origin independence of the oscillator strengths can be achieved by including all terms of a given order in the expansion of the observable oscillator strength and not in the transition moments, and implemented this approach for time-dependent density-functional theory (TD-DFT) calculations of K edges. The electric dipole is the only contribution to the zerothorder expansion.…”

Revisiting the Dependence of Cu K-Edge X-Ray Absorption Spectra on Oxidation State and Coordination Environment

“…[37]. For complexes with more than one copper center, no localization of the core orbitals was performed (see also Ref.…”

“…Therefore, an additional pre-edge peak at lower energies arising from the 1s → 3d transition is expected. However, this transition is usually dipole-forbidden and only a very weak quadrupole transition can be observed [34,37].…”