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

The emergence of state-of-the-art X-ray light sources has paved the way for novel spectroscopies that take advantage of their atomic specificity to shed light on fundamental physical, chemical, and biological processes both in the static and time domains. The success of these experiments hinges on the ability to interpret and predict core-level spectra, which has opened avenues for theory to play a key role. Over the last two decades, linear-response time-dependent density functional theory (LR-TDDFT), despite various theoretical challenges, has become a computationally attractive and versatile framework to study excited-state spectra including X-ray spectroscopies. In this context, we focus our discussion on LR-TDDFT approaches for the computation of X-ray Near-Edge Structure (XANES), Valence-to-Core X-ray Emission (VtC-XES), and Resonant Inelastic X-ray Scattering (RIXS) spectroscopies in molecular systems with an emphasis on Gaussian basis set implementations. We illustrate these approaches with applications and provide a brief outlook of possible new directions.

Intruder Peak-Free Transient Inner-Shell Spectra Using Real-Time Simulations

Real-time methods are convenient for simulating core-level absorption spectra but suffer from nonphysical intruder peaks when using atom-centered basis sets. In transient absorption spectra, these peaks exhibit highly nonphysical time-dependent modulations in their energies and oscillator strengths. In this paper, we address the origins of these intruder peaks and propose a straightforward and effective solution based on a filtered dipole operator. In combination with real-time time-dependent density functional theory (RT-TDDFT), we demonstrate how to compute intruder-free attosecond transient X-ray absorption spectra for the aminophenol (C₆H₇NO) oxygen and nitrogen K-edges and the α-quartz (SiO₂) silicon L-edge. Without filtering, the computed spectra are qualitatively wrong. This procedure is suitable for both static and transient inner-shell spectroscopy studies and can easily be implemented in a range of real-time methodologies.

Quantum Chemical Calculations of X-ray Emission Spectroscopy

The calculation of X-ray emission spectroscopy with equation of motion coupled cluster theory (EOM-CCSD), time-dependent density functional theory (TDDFT), and resolution of the identity single excitation configuration interaction with second-order perturbation theory (RI-CIS(D)) is studied. These methods can be applied to calculate X-ray emission transitions by using a reference determinant with a core-hole, and they provide a convenient approach to compute the X-ray emission spectroscopy of large systems since all of the required states can be obtained within a single calculation, removing the need to perform a separate calculation for each state. For all of the methods, basis sets with the inclusion of additional basis functions to describe core orbitals are necessary, particularly when studying transitions involving the 1s orbitals of heavier nuclei. EOM-CCSD predicts accurate transition energies when compared with experiment; however, its application to larger systems is restricted by its computational cost and difficulty in converging the CCSD equations for a core-hole reference determinant, which become increasing problematic as the size of the system studied increases. While RI-CIS(D) gives accurate transition energies for small molecules containing first row nuclei, its application to larger systems is limited by the CIS states providing a poor zeroth-order reference for perturbation theory which leads to very large errors in the computed transition energies for some states. TDDFT with standard exchange-correlation functionals predicts transition energies that are much larger than experiment. Optimization of a hybrid and short-range corrected functional to predict the X-ray emission transitions results in much closer agreement with EOM-CCSD. The most accurate exchange-correlation functional identified is a modified B3LYP hybrid functional with 66% Hartree-Fock exchange, denoted B(66)LYP, which predicts X-ray emission spectra for a range of molecules including fluorobenzene, nitrobenzene, acetone, dimethyl sulfoxide, and CF3Cl in good agreement with experiment.

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

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 so-called 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 origin-independent 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 and electric-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 time-dependent density-functional theory, and demonstrate its usefulness for the calculation of metal and ligand K-edge XAS spectra of transition metal complexes.

Time-dependent density functional theory calculations of the spectroscopy of core electrons

Recent advances in X-ray sources have led to a renaissance in spectroscopic techniques in the X-ray region. These techniques that involve the excitation of core electrons can provide an atom specific probe of electronic structure and provide powerful analytical tools that are used in many fields of research. Theoretical calculations can often play an important role in the analysis and interpretation of experimental spectra. In this perspective, we review recent developments in quantum chemical calculations of X-ray absorption spectra, focusing on the use of time-dependent density functional theory to study core excitations. The practical application of these calculations is illustrated with examples drawn from surface science and bioinorganic chemistry, and the application of these methods to study X-ray emission spectroscopy is explored.