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

Abstract

The simulation of X-ray absorption spectra requires both scalar and spin–orbit (SO) relativistic effects to be taken into account, particularly near L- and M-edges where the SO splitting of core p and d orbitals dominates. Four-component Dirac–Coulomb Hamiltonian-based linear damped response time-dependent density functional theory (4c-DR-TDDFT) calculates spectra directly for a selected frequency region while including the relativistic effects variationally, making the method well suited for X-ray applications. In this work, we show that accurate X-ray absorption spectra near L 2,3 - and M 4,5 -edges of closed-shell transition metal and actinide compounds with different central atoms, ligands, and oxidation states can be obtained by means of 4c-DR-TDDFT. While the main absorption lines do not change noticeably with the basis set and geometry, the exchange–correlation functional has a strong influence with hybrid functionals performing the best. The energy shift compared to the experiment is shown to depend linearly on the amount of Hartee–Fock exchange with the optimal value being 60% for spectral regions above 1000 eV, providing relative errors below 0.2% and 2% for edge energies and SO splittings, respectively. Finally, the methodology calibrated in this work is used to reproduce the experimental L 2,3 -edge X-ray absorption spectra of [RuCl 2 (DMSO) 2 (Im) 2 ] and [WCl 4 (PMePh 2 ) 2 ], and resolve the broad bands into separated lines, allowing an interpretation based on ligand field theory and double point groups. These results support 4c-DR-TDDFT as a reliable method for calculating and analyzing X-ray absorption spectra of chemically interesting systems, advance the accuracy of state-of-the art relativistic DFT approaches, and provide a reference for benchmarking more approximate techniques.

Computational approaches for XANES, VtC-XES, and RIXS using linear-response time-dependent density functional theory based methods

“…LR-TDDFT/TDA is sufficiently predictive and has been applied to a broad range of systems to compute K-edge, L-edge, and M-edge XANES 29–45,84,92–103 as well as transient X-ray absorption spectroscopy. 104–109 In Fig.…”

“…Over the years many ab initio methods have been developed to compute XANES at different levels of computational complexity. 26,27 These include methods based on density functional theory (DFT), 28 linear response (LR) and real-time (RT) time-dependent density functional theory (TDDFT), [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45] Bethe-Salpeter equation (BSE), [46][47][48] algebraic-diagramatic construction (ADC), [49][50][51] linear-response density cumulant theory, 52 coupled-cluster theory using both the complex polarization propagator 53,54 and equation-of-motion [55][56][57][58] frameworks, restricted active space (RAS) multiconfigurational methods, [59][60][61][62] and multireference coupled cluster methods. 63 Over the past two decades, LR-TDDFT has become a computationally attractive approach for studying excited-state spectra, within the space of single excitations, in a wide range of molecular systems.…”

“…However, computations of the L-edge XANES are more complicated due spin-orbit (SO) splitting associated with the 2p orbitals leading to the L 2,3 XANES. The SO splitting requires at least a scalar relativistic approach with perturbative spin-orbit effects 82 or more rigorously, relativistic two-component (2c) 36,39 or four-component (4c) 40,41,44 LR-TDDFT treatments, where the scalar and SO effects are treated variationally, to be able to assign the spectral features at the L-and M-edges XANES accurately.…”

“…88 More advanced treatments of going beyond the dipole approximation have since been developed and reported by List and co-workers. [89][90][91] LR-TDDFT/TDA is sufficiently predictive and has been applied to a broad range of systems to compute K-edge, L-edge, and M-edge XANES [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45]84,[92][93][94][95][96][97][98][99][100][101][102][103] as well as transient Xray absorption spectroscopy. [104][105][106][107][108][109] In Fig.…”

Following a Silent Metal Ion: A Combined X-ray Absorption and Nuclear Magnetic Resonance Spectroscopic Study of the Zn2+ Cation Dissipative Translocation between Two Different Ligands

“…39 Finally, the experimental XAS spectrum of 1•Zn 2+ shows the presence of a shoulder at the rising edge (transition A, Figure 2c) that has been assigned to a dipole allowed 1s to 4p transition. 20 In order to better understand the structural and electronic properties of the investigated Zn 2+ species, time dependent DFT (TDDFT) theoretical spectra 50,51 were calculated by means of the ORCA code 40 starting from DFT optimized geometrical models of the 1•Zn 2+ and 2 2 •Zn 2+ complexes. The associated DFT optimized structures of 1•Zn 2+ and 2 2 •Zn 2+ are shown in Figure 2 panels a and b, respectively, while Table S1 lists the relevant structural parameters.…”