Time-Dependent Density Functional Theory for X-ray Absorption Spectra: Comparing the Real-Time Approach to Linear Response

Revisiting the Half-and-Half Functional

Hybrid density functionals typically provide significantly better accuracy than semilocal functionals. Conventional wisdom holds that incorporating more than 20-25% exact exchange is deleterious to thermochemical properties and should only be used as a last resort, for problems that are dominated by self-interaction error. In such cases, the Becke-Lee-Yang-Parr "half-and-half" functional (BH&H-LYP) has emerged as a go-to choice, especially in time-dependent density functional theory calculations for excitation energies. Here, we examine the assumption that 50% Hartree-Fock exchange sacrifices thermochemical accuracy. Using a sequence of functionals B(α)LYP, with different percentages of exact exchange (0 ≤ α ≤ 100), we find that BH&H-LYP (with α = 50) is nearly optimal and affords accuracy similar to B3LYP for thermochemistry, barrier heights, and excitation energies. Although BH&H-LYP is significantly less accurate than B3LYP for atomization energies, this emerges as the sole rationale for the taboo against values α > 25. Overall, BH&H-LYP is a reasonable choice for problems that are dominated by self-interaction error, including charge-transfer complexes and core-level excitation energies. While B3LYP remains more accurate for valence excitation energies, the use of 50% exact exchange appears to be an acceptable compromise, and BH&H-LYP can be used without undue concern over its diminished accuracy for ground-state properties.

Simulation of 24,000 Electron Dynamics: Real-Time Time-Dependent Density Functional Theory (TDDFT) with the Real-Space Multigrids (RMG)

We present the theory, implementation, and benchmarking of a real-time time-dependent density functional theory (RT-TDDFT) module within the RMG code, designed to simulate the electronic response of molecular systems to external perturbations. Our method offers insights into nonequilibrium dynamics and excited states across a diverse range of systems, from small organic molecules to large metallic nanoparticles. Benchmarking results demonstrate excellent agreement with established TDDFT implementations and showcase the superior stability of our time integration algorithm, enabling long-term simulations with minimal energy drift. The scalability and efficiency of RMG on massively parallel architectures allow for simulations of complex systems, such as plasmonic nanoparticles with thousands of atoms. Future extensions, including nuclear and spin dynamics, will broaden the applicability of this RT-TDDFT implementation, providing a powerful toolset for studies of photoactive materials, nanoscale devices, and other systems where real-time electronic dynamics is essential.

Simulations of Attosecond Metallization in Quartz and Diamond Probed with Inner-Shell Transient Absorption Spectroscopy

When dielectrics are hit with intense infrared (IR) laser pulses, transient metalization can occur. The initial attosecond dynamics behind this metallization are not entirely understood. Therefore, simulations are needed to understand this process and to help interpret experimental observations of it, such as with attosecond transient absorption (ATA). In this paper, we present first-principles simulations of ATA based on bulk-mimicking clusters and real-time time-dependent density functional theory (RT-TDDFT), with Koopmans-tuned range-separated hybrid functionals and Gaussian basis sets. Our method gives good agreement with the experiment for the breakdown threshold in silica and diamond. This breakdown voltage corresponds to a Keldysh parameter of approximately one and thus involves a transition to a regime where the dynamics are driven by tunneling. Pumping at an amplitude just below this value causes a mixture of multiphoton and tunneling excitations across the band gap to occur. The computed extreme ultraviolet and X-ray attosecond transient spectra also agree well with the experiment and show a decrease in optical density due to the transient population of the conduction band from the IR field. First-principles approaches such as this are valuable for interpreting the complicated modulations in a spectrum and for guiding future attosecond experiments on solids.

Distinguishing Organomagnesium Species in the Grignard Addition to Ketones with X-Ray Spectroscopy

The addition of Grignard reagents to ketones is a well-established textbook reaction. However, a comprehensive understanding of its mechanism has only recently begun to emerge. X-ray spectroscopy, because of its high selectivity and sensitivity, is the ideal tool for distinguishing between an ensemble of competing pathways. With this aim in mind, we investigated the concerted mechanism of the addition of methylmagnesium chloride (CH3MgCl) to acetone in tetrahydrofuran by simulating the X-ray spectra of different molecules in solution. We used electronic structure methods to calculate the X-ray absorption spectra at the Mg K- and L1-edges and the X-ray photoelectron spectra at the Mg K-edge for different organomagnesium species, which coexist in solution due to the Schlenk equilibrium. The simulated spectra show that individual species can be distinguished throughout the different stages of the reaction. Each species has a distinct spectral feature which can be used as a fingerprint in solution. The absorption and photoelectron spectra consistently show a blue shift as the reaction progressed from reagents to products.

Attosecond formation of charge-transfer-to-solvent states of aqueous ions probed using the core-hole-clock technique

Charge transfer between molecules lies at the heart of many chemical processes. Here, we focus on the ultrafast electron dynamics associated with the formation of charge-transfer-to-solvent (CTTS) states following X-ray absorption in aqueous solutions of Na+, Mg2+, and Al3+ ions. To explore the formation of such states in the aqueous phase, liquid-jet photoemission spectroscopy is employed. Using the core-hole-clock method, based on Auger-Meitner (AM) decay upon 1s excitation or ionization of the respective ions, upper limits are estimated for the metal-atom electron delocalization times to the neighboring water molecules. These delocalization processes represent the first steps in the formation of hydrated electrons, which are determined to take place on a timescale ranging from several hundred attoseconds (as) below the 1s ionization threshold to only 20 as far above the 1s ionization threshold. The decrease in the delocalization times as a function of the photon energy is continuous. This indicates that the excited electrons remain in the vicinity of the studied ions even above the ionization threshold, i.e., metal-ion electronic resonances associated with the CTTS state manifolds are formed. The three studied isoelectronic ions exhibit quantitative differences in their electron energetics and delocalization times, which are linked to the character of the respective excited states.

A new parameterization of the DFT/CIS method with applications to core-level spectroscopy

Time-dependent density functional theory (TD-DFT) within a restricted excitation space is an efficient means to compute core-level excitation energies using only a small subset of the occupied orbitals. However, core-to-valence excitation energies are significantly underestimated when standard exchange-correlation functionals are used, which is partly traceable to systemic issues with TD-DFT's description of Rydberg and charge-transfer excited states. To mitigate this, we have implemented an empirically modified combination of configuration interaction with single substitutions (CIS) based on Kohn-Sham orbitals, which is known as "DFT/CIS." This semi-empirical approach is well-suited for simulating x-ray near-edge spectra, as it contains sufficient exact exchange to model charge-transfer excitations yet retains DFT's low-cost description of dynamical electron correlation. Empirical corrections to the matrix elements enable semi-quantitative simulation of near-edge x-ray spectra without the need for significant a posteriori shifts; this should be useful in complex molecules and materials with multiple overlapping x-ray edges. Parameter optimization for use with a specific range-separated hybrid functional makes this a black-box method intended for both core and valence spectroscopy. Results herein demonstrate that realistic K-edge absorption and emission spectra can be obtained for second- and third-row elements and 3d transition metals, with promising results for L-edge spectra as well. DFT/CIS calculations require absolute shifts that are considerably smaller than what is typical in TD-DFT.

TDDFT and the x-ray absorption spectrum of liquid water: Finding the "best" functional

We investigate the performance of time-dependent density functional theory (TDDFT) for reproducing high-level reference x-ray absorption spectra of liquid water and water clusters. For this, we apply the integrated absolute difference (IAD) metric, previously used for x-ray emission spectra of liquid water [T. Fransson and L. G. M. Pettersson, J. Chem. Theory Comput. 19, 7333-7342 (2023)], in order to investigate which exchange-correlation (xc) functionals yield TDDFT spectra in best agreement to reference, as well as to investigate the suitability of IAD for x-ray absorption spectroscopy spectrum calculations. Considering highly asymmetric and symmetric six-molecule clusters, it is seen that long-range corrected xc-functionals are required to yield good agreement with the reference coupled cluster (CC) and algebraic-diagrammatic construction spectra, with 100% asymptotic Hartree-Fock exchange resulting in the lowest IADs. The xc-functionals with best agreement to reference have been adopted for larger water clusters, yielding results in line with recently published CC theory, but which still show some discrepancies in the relative intensity of the features compared to experiment.

Evaluating the Impact of the Tamm-Dancoff Approximation on X-ray Spectrum Calculations

The impact of the Tamm-Dancoff approximation (TDA) for time-dependent density functional theory (TDDFT) calculations of X-ray absorption and X-ray emission spectra (XAS and XES) is investigated, showing small discrepancies in the excitation energies and intensities. Through explicit diagonalization of the TDDFT Hessian, XES was considered by using full TDDFT with a core-hole reference state. This has previously not been possible with most TDDFT implementations as a result of the presence of negative eigenvalues. Furthermore, a core-valence separation (CVS) scheme for XES is presented, in which only elements including the core-hole are considered, resulting in a small Hessian with the dimension of the number of remaining occupied orbitals of the same spin as the core-hole (CH). The resulting spectra are in surprisingly good agreement with the full-space counterpart, illustrating the weak coupling between the valence-valence and valence-CH transitions. Complications resulting from contributions from the discretized continuum are discussed, which can occur for TDDFT calculations of XAS and XES and for TDA calculations of XAS. In conclusion, we recommend that TDA be used when calculating X-ray emission spectra, and either CVS-TDA or CVS-TDDFT can be used for X-ray absorption spectra.

Visualizing and characterizing excited states from time-dependent density functional theory

Time-dependent density functional theory (TD-DFT) is the most widely-used electronic structure method for excited states, due to a favorable combination of low cost and semi-quantitative accuracy in many contexts, even if there are well recognized limitations. This Perspective describes various ways in which excited states from TD-DFT calculations can be visualized and analyzed, both qualitatively and quantitatively. This includes not just orbitals and densities but also well-defined statistical measures of electron-hole separation and of Frenkel-type exciton delocalization. Emphasis is placed on mathematical connections between methods that have often been discussed separately. Particular attention is paid to charge-transfer diagnostics, which provide indicators of when TD-DFT may not be trustworthy due to its categorical failure to describe long-range electron transfer. Measures of exciton size and charge separation that are directly connected to the underlying transition density are recommended over more ad hoc metrics for quantifying charge-transfer character.

Generalization of One-Center Nonorthogonal Configuration Interaction Singles to Open-Shell Singlet Reference States: Theory and Application to Valence-Core Pump-Probe States in Acetylacetone

We formulate a one-center nonorthogonal configuration interaction singles (1C-NOCIS) theory for the computation of core excited states of an initial singlet state with two unpaired electrons. This model, which we refer to as 1C-NOCIS two-electron open-shell (2eOS), is appropriate for computing the K-edge near-edge X-ray absorption spectra (NEXAS) of the valence excited states of closed-shell molecules relevant to pump-probe time-resolved (TR) NEXAS experiments. With the inclusion of core-hole relaxation effects and explicit spin adaptation, 1C-NOCIS 2eOS requires mild shifts to match experiment, is free of artifacts due to spin contamination, and can capture the high-energy region of the spectrum beyond the transitions into the singly occupied molecular orbitals (SOMOs). Calculations on water and thymine illustrate the different key features of excited-state NEXAS, namely, the core-to-SOMO transitions as well as shifts and spin-splittings in the transitions analogous to those of the ground state. Simulations of the TR-NEXAS of acetylacetone after excitation to its π → π* singlet excited state at the carbon K-edge, an experiment carried out recently, showcase the ability of 1C-NOCIS 2eOS to efficiently simulate NEXAS based on nonadiabatic molecular dynamics simulations.