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# Theoretical X‐ray spectroscopy of transition metal compounds

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Abstract X‐ray spectroscopy is one of the most powerful tools to access structure and properties of matter in different states of aggregation as it allows to trace atomic and molecular energy levels in course of various physical and chemical processes. X‐ray spectroscopic techniques probe the local electronic structure of a particular atom in its environment, in contrast to ultraviolet/visible (UV/Vis) spectroscopy, where transitions generally occur between delocalized molecular orbitals. Complementary information is provided by using a combination of different absorption, emission, scattering as well as photo‐ and autoionization X‐ray methods. However, interpretation of the complex experimental spectra and verification of experimental hypotheses is a nontrivial task and powerful first principles theoretical approaches that allow for a systematic investigation of a broad class of systems are needed. Focusing on transition metal compounds, L‐edge spectra are of particular relevance as they probe the frontier d‐orbitals involved in metal–ligand bonding. Here, near‐degeneracy effects in combination with spin‐orbit coupling lead to a complicated multiplet energy level structure, which poses a serious challenge to quantum chemical methods. Multiconfigurational self‐consistent field (MCSCF) theory has been shown to be capable of providing a rather detailed understanding of experimental X‐ray spectroscopy. However, it cannot be considered as a “blackbox” tool and its application requires not only a command of formal theoretical aspects, but also a broad knowledge of already existing applications. Both aspects are covered in this overview. This article is categorized under: Theoretical and Physical Chemistry > Spectroscopy Electronic Structure Theory > Ab Initio Electronic Structure Methods
(a) Left: calculated normalized X‐ray absorption spectrum (XAS) for [heme B‐Cl]0 (black) and [heme B‐H2O]+ (red filled curves). Right: normalized resonant inelastic X‐ray scattering (RIXS) spectra for selected excitation energies. (b) Calculated RIXS spectra for the hemin dimer (red filled curves) with the COOH groups pointing in the same direction. The monomer spectrum is also shown (×2, black lines). Note that monomer and dimer XAS are indistinguishable on this scale. (c) Different choices of excitonic bases and the corresponding transitions included in the dimer coupling. For the one‐particle basis, only transitions from or to the ground state have been included. For the two‐particle basis, de‐excitations from an arbitrary core‐excited state to any other state are allowed. (Adapted from Reference )
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(a) Experimental and calculated (restricted active space self‐consistent field [RASSCF]/restricted active space perturbation theory (second order) [RASPT2]) X‐ray absorption spectra (XAS) for Mn species in different oxidation states. (b) Contributions of states with ΔS = 0, ± 1 to the spin‐orbit coupled wave functions for the different species of panel a. The labels (val, a–e) correspond to the bands marked in panel a where the widths of the shaded areas comprise the state numbers of panel b (Adapted from Reference )
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(a) Experimental and theoretical resonant inelastic X‐ray scattering (RIXS) spectra of the Fe(CO)5 complex at iron L‐edge and oxygen K‐edge. The colored areas show the range of metal‐to‐ligand (MLCT) and ligand‐to‐metal (LMCT) charge‐transfer transitions which carry information about the strength of covalent bonding as explained in the respective insets. (b) Transition density difference plots which show the localization of the excited electron (red) and the electron which is refilling the core hole (blue) within the complex. (Adapted from Reference )
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Experimental and calculated (restricted active space self‐consistent field [RASSCF]/restricted active space perturbation theory (second order) [RASPT2]) spectra of the aqueous Fe2+ ion (an [Fe(H2O)6]2+ cluster is used in simulations, see inset). (a) Absorption spectra obtained in different modes: True X‐ray absorption spectrum (XAS) (calculated—black), valence partial fluorescence yield (PFY) (Equation (3)) with ω ∈ [660, 720] eV denoted as PVFY (experiment—blue, calculation—magenta) and core PFY ω ∈ [605, 640] eV denoted as PCFY (experiment—red; calculation—green). (b) Respective resonant inelastic X‐ray scattering (RIXS) spectra in energy loss (ω − Ω in Equation (2)) representation. The green and gray rectangles denote the energy range of the spin‐allowed and formally spin‐forbidden transitions, respectively. (c) Orbital scheme showing different radiative relaxation channels. (Adapted from References )
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(a) Illustration of the projection of the full Hamiltonian to the subspace of limited valence‐ and core‐excited configurations (blue blocks), neglecting the respective off‐diagonal coupling blocks (red crosses). (b) Principal molecular orbitals (MO) subspaces used in the restricted active space self‐consistent field (RASSCF) method. The typical active space (AS) is exemplified for the Fe(CO)5 complex. The table on the right demonstrates the level of orbital optimization (HF) and CI for each subspace. (Valence electronic excitations are depicted by red and core excitations by blue arrows, respectively)
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Overview on different methods discussed in this review, grouped according to three criteria: (i) frequency (energy) versus time domain, (ii) explicit orbital optimization, that is, self‐consistent field (SCF) versus non‐SCF, and (iii) reference for construction of excited state basis is based on a single or multiple electronic configurations. (MR‐CI, MR‐CC, and MR‐PT are multi‐reference configuration interaction (CI), coupled clusters (CC), and perturbation theory (PT), respectively. Other abbreviations are defined in the text)
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(a) Processes relevant for X‐ray spectroscopy viewed from the molecular orbital (MO) picture standpoint. For absorption, the respective process in ultraviolet/visible (UV/Vis) range is also shown. (b) Processes relevant for X‐ray spectroscopy in the many‐body state picture. (Color code for arrows: Photon absorption—black, radiative decay of a core hole—red, nonradiative decay—blue). Abbreviations: photoelectron spectrum (PES), nonresonant X‐ray emission spectrum (NXES), resonant inelastic X‐ray scattering (RIXS), resonant X‐ray emission spectrum (RXES), Auger Electron Spectrum (AES), Coster–Kronig (CK)
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