Beyond the electric-dipole approximation in simulations of x-ray absorption spectroscopy: Lessons from relativistic theory

Abstract

We present three schemes to go beyond the electric-dipole approximation in x-ray absorption spectroscopy calculations within a fourcomponent relativistic framework. The first is based on the full semi-classical light-matter interaction operator and the two others on a truncated interaction within the Coulomb gauge (velocity representation) and multipolar gauge (length representation). We generalize the derivation of the multipolar gauge to an arbitrary expansion point and show that the potentials corresponding to different expansion points are related by a gauge transformation, provided that the expansion is not truncated. This suggests that the observed gauge-origin dependence in the multipolar gauge is more than just a finite-basis set effect. The simplicity of the relativistic formalism enables arbitrary-order implementations of the truncated interactions, with and without rotational averaging, allowing us to test their convergence behavior numerically by comparison to the full formulation. We confirm the observation that the oscillator strength of the electric-dipole allowed ligand K-edge transition of TiCl 4 , when calculated to the second order in the wave vector, becomes negative but also show that inclusion of higher-order contributions allows convergence to the result obtained using the full light-matter interaction. However, at higher energies, the slow convergence of such expansions becomes dramatic and renders such approaches at best impractical. When going beyond the electric-dipole approximation, we therefore recommend the use of the full light-matter interaction.

Accurate X-ray Absorption Spectra near L- and M-Edges from Relativistic Four-Component Damped Response Time-Dependent Density Functional Theory

“…The right-hand side of eq 4 describes the interaction of the molecular system with the applied external electric field, which in the electric dipole approximation is mediated by the electric dipole moment operator P . While the short wavelength of X-ray radiation may require terms beyond the electric dipole, 81 − 83 we do not consider this aspect in the present study. Finally, the first term on the left-hand side of eq 4 is the generalized Hessian; for the Dirac–Coulomb Hamiltonian, it is defined by 63 where ε i and ε a are the energies of occupied and virtual reference MOs, respectively, δ pq is the Kronecker delta function (δ pq = 1 if p = q and 0 otherwise), and is the coupling matrix, with ξ denoting the admixture of HFX.…”

Probing Chirality across the Electromagnetic Spectrum with the Full Semi-classical Light--Matter Interaction

“…However, in contrast to the generalized velocity representation, we now loose origin independence of oscillator strengths. 28 In our previous work, 28 however, we have have shown that it is possible within the generalized velocity representation to distinguish electric and magnetic multipole components, while maintaining origin independence of the oscillator strengths, although its individual components still exhibit this dependency. The operator manipulations required for this separation do not involve commutators with the Hamiltonian and are therefore also valid when using finite basis sets.…”

“…The use of Fermi's golden rule for a transition between discrete states is justified by the inclusion of a (Lorentzian) lineshape function f (ω, ω f i , γ f i ) expressing the finite lifetime of excited states (see discussion in Ref. 28). Moreover, the lineshape function, having the dimension of time, has to be included in order for the absorption cross section to have the correct dimension of area.…”

“…In the case of the full light-matter interaction, the rotational average is handled numerically using a Lebedev grid. 27,28 In the case of the truncated interaction, the rotational average is expressed as…”

The DIRAC code for relativistic molecular calculations

“…138 Transition moments may be calculated with user-specified property operators. From DIRAC20 onward, three schemes 139 to go beyond the electric dipole approximation in the calculation of oscillator strengths will be available in DIRAC. The first is based on the full semi-classical lightmatter interaction operator and the two others on a truncated interaction within the Coulomb gauge (velocity representation) and multipolar gauge (length representation).…”