Uncertainty quantification in theoretical spectroscopy: The structural sensitivity of X‐ray emission spectra

Abstract

We present a methodology for analyzing the dependence of molecular spectra calculated with quantum-chemical methods on the underlying molecular structure. This analysis is applied to investigate the structural sensitivity of calculated valence-to-core X-ray emission (VtC-XES) spectra for the test case of three iron carbonyl complexes, Fe(CO) 5 , [FeCp(CO) 2 (THF)] 1 (Cp 5 cyclopentadienyl, THF 5 tetrahydrofuran), and Fe(CO) 3 (cod) (cod 5 cyclooctadienyl). Based on this analysis, we discuss how the VtC-XES spectra depend on changes of metal-ligand bond distances and bond angles as well as on the structure of the ligands. The benefits of such an analysis of the structural sensitivity are discussed. Our methodology can serve as a first step toward quantifying and accounting for uncertainties due to the underlying model structure in theoretical spectroscopy.inverse quantum chemistry, theoretical spectroscopy, uncertainty quantification, X-ray spectroscopy 1 | I N TR ODU C TI ON Spectroscopy is an essential tool for unraveling molecular structure. However, only few spectroscopic techniques, most importantly nuclear magnetic resonance (NMR) spectroscopy or X-ray absorption fine structure spectroscopy (EXAFS), provide direct access to structural parameters such as interatomic distances or coordination numbers and thus allow for the direct determination of molecular structures and/or a direct structural refinement. In contrast, many important spectroscopic techniques such as vibrational spectroscopy, UV/Vis spectroscopy, or X-ray absorption and emission spectroscopy, only provide indirect access to structural information. In this case, spectroscopic experiments are often combined with quantum-chemical calculations [1] to connect them to specific features of the underlying molecular structure.For instance, vibrational spectroscopy in combination with quantum-chemical calculations can be used to determine the gas-phase structures of polypetides [2][3][4] or to assign the absolute configuration of chiral molecules. [5] The analysis of calculated vibrational spectra can provide detailed insights into the connection between specific spectral features and the underlying molecular structure. [6][7][8] For X-ray spectroscopy, quantumchemical calculations can provide an assignment of the peaks in X-ray emission and absorption spectra to occupied and unoccupied electronic states, and can connect those to the structure of the ligand environment in transition-metal complexes (see e.g., ).Here, we will focus on valence-to-core (VtC) X-ray emission spectroscopy (XES) [13,14] as an example. In combination with quantum-chemical calculations, VtC-XES can be used to obtain partial structural information, in particular in the vicinity of transition metal centers. VtC-XES makes it possible to identify which ligands are coordinated to a transition metal center, for instance in molecular transition metal complexes, [11,[15][16][17] for catalytic metal centers in zeolites, [18,19] and for metal clusters in enzymes. [20][21][22] The ability of...

Towards theoretical spectroscopy with error bars: systematic quantification of the structural sensitivity of calculated spectra

“…27,[29][30][31] Here, we aim at systematically quantifying a further source of uncertainty, namely the dependence of s(E; R) on the input molecular structure. 35 The structural sensitivity presents a challenging case because the calculated spectrum depends on a rather large number of independent parameters (i.e., the nuclear coordinates R). In this respect, it fundamentally differs from, e.g., uncertainties due to approximation in the quantum-chemical model, which are usually related to only a few parameters.…”

Towards Theoretical Spectroscopy with Error Bars: Systematic Quantification of the Structural Sensitivity of Calculated Spectra

“…This linear structural sensitivity can be calculated by numerical differentiation, i.e., by calculating the spectrum for displaced molecular structures [35]. Here, we employ a symmetric two-point formula [see Eq.…”

“…As an example, we consider the structural sensitivity of the calculated XES spectrum of Fe(CO) 5 . For this example, we already explored the dependence on manually selected structural distortions in our previous work [35]. An in-depth experimental and computational study of the XES spectrum of Fe(CO) 5 can be found in Ref.…”

“…All calculations of XES spectra were performed with the BP86 xc functional and a QZ4P basis set in combination with the COSMO solvation model [61] with default parameters. The calculated spectra were shifted by 180.62 eV [35,48] and a Gaussian line broadening with a full-width at half maximum of 1.5 eV was applied to each calculated transition.…”

“…As a first target, we aim at systematically quantifying this dependence of σ(E; R) on the input molecular structure defined by the Cartesian nuclear coordinates R [35]. While this is only one of many relevant sources of uncertainty, it serves as a first step towards establishing a methodological framework that can be extended to other sources of uncertainty.…”