Quadratic Response Properties from TDDFT: Trials and Tribulations

Electronic Structures and Spectra of Donor-Acceptor Conjugated Oligomers

Narrow band gap donor-acceptor conjugated polymers present excellent paradigms in photonics and optoelectronics due to their chemical tunability, correlated electronic structures, and tunable open-shell electronic configurations. However, rational design for enhancing the properties of these molecular systems remains challenging. In this study, we employed density functional theory (DFT) calculations to investigate prototypical narrow band gap donor-acceptor conjugated oligomers, consisting of alternating cyclopentadithiophene (CPDT) donors paired with benzothiadiazole (BT), benzoselenadiazole (BSe), benzobisthiadiazole (BBT), and thiadiazoloquinoxaline (TQ) acceptors. Analyses of structures, singlet-triplet gaps, and absorption spectra of oligomers with up to ten repeat units have shown that when incorporating the BT, BSe, and TQ acceptors, the backbone curvature resulted in spiral structures that were energetically favored over their linear counterparts, causing differences in the calculated circular dichroism spectra. Oligomers with BBT-based acceptors preferred, however, a linear geometry, consistent with an open-shell electronic structure. Calculated singlet-triplet splittings demonstrated the importance of long chains and specific structures for consistency with the experiment, while effects of the solvent were also quantified. Based on the predicted low-energy conformations, one-photon absorption spectra for the considered oligomers have shown that using the Tamm-Dancoff approximation within time-dependent DFT for the large systems offers good agreement with the first absorption maxima in measured experimental spectra, thus validating the method for large donor-acceptor oligomers. Natural transition orbital analyses provided insights into the excited-state characteristics. Two-photon absorption maxima were accurately predicted, but the cross-sections were overestimated or underestimated, as dependent on the level of theory employed, to be addressed in future work.

Two-Photon Absorption Strengths of Small Molecules: Reference CC3 Values and Benchmarks

We present a large dataset of highly accurate two-photon transition strengths (δTPA) determined for standard small molecules. Our reference values have been calculated using the quadratic response implementation of the third-order coupled cluster method including iterative triples (Q-CC3). The aug-cc-pVTZ atomic basis set is used for molecules with up to five non-hydrogen atoms, while larger molecules are assessed with aug-cc-pVDZ; the differences due to the basis sets are discussed. This dataset, encompassing 82 singlet transitions of various characters (Rydberg, valence, and double excitations), enables a comprehensive benchmark of smaller basis sets and alternative wavefunction methods when Q-CC3 calculations become beyond reach as well as time-dependent density functional theory (TD-DFT) approaches. The evaluated wavefunction methods include quadratic response and equation-of-motion CCSD approximations, Q-CC2, and second-order algebraic diagrammatic construction in its intermediate state representation (I-ADC2). In the TD-DFT framework, a set of five commonly used exchange-correlation functionals are evaluted. This extensive analysis provides a quantitative assessment of these methods, revealing how different system sizes, response intensities, and types of transitions affect their performances.

Reformulation of Time-Dependent Density Functional Theory for Nonperturbative Dynamics: The Rabi Oscillation Problem Resolved

Rabi oscillations have long been thought to be out of reach in simulations using time-dependent density functional theory (TDDFT), a prominent symptom of the failure of the adiabatic approximation for nonperturbative dynamics. We present a reformulation of TDDFT which requires response quantities only, thus enabling an adiabatic approximation to predict such dynamics accurately because the functional is evaluated on a density close to the ground state, instead of on the fully nonperturbative density. Our reformulation applies to any real-time dynamics, redeeming TDDFT far from equilibrium. Examples of a resonantly-driven local excitation in a model He atom, and charge-transfer in the LiCN molecule are given.

High-Spin State of a Ferrocene Electron Donor Revealed by Optical and X-ray Transient Absorption Spectroscopy

Ferrocene is one of the most common electron donors, and mapping its ligand-field excited states is critical to designing donor-acceptor (D-A) molecules with long-lived charge transfer states. Although 3(d-d) states are commonly invoked in the photophysics of ferrocene complexes, mention of the high-spin 5(d-d) state is scarce. Here, we provide clear evidence of 5(d-d) formation in a bimetallic D-A molecule, ferrocenyl cobaltocenium hexafluorophosphate ([FcCc]PF6). Femtosecond optical transient absorption (OTA) spectroscopy reveals two distinct electronic excited states with 30 and 500 ps lifetimes. Using a combination of ultraviolet, visible, near-infrared, and short-wave infrared probe pulses, we capture the spectral features of these states over an ultrabroadband range spanning 320 to 2200 nm. Time-dependent density functional theory (DFT) calculations of the lowest triplet and quintet states, both primarily Fe(II) (d-d) in character, qualitatively agree with the experimental OTA spectra, allowing us to assign the 30 ps state as the 3(d-d) state and the 500 ps state as the high-spin 5(d-d) state. To confirm the ferrocene-centered high-spin character of the 500 ps state, we performed X-ray transient absorption (XTA) spectroscopy at the Fe and Co K edges. The Fe K-edge XTA spectrum at 150 ps shows a red shift of the absorption edge that is consistent with an Fe(II) high-spin state, as supported by ab initio calculations. The transient signal detected at the Co K-edge is 50× weaker, confirming the ferrocene-centered character of the excited state. Fitting of the transient extended X-ray absorption fine structure region yields an Fe-C bond length increase of 0.25 ± 0.1 Å in the excited state, as expected for the high-spin state based on DFT. Altogether, these results demonstrate that the high-spin state of ferrocene should be considered when designing donor-acceptor assemblies for photocatalysis and photovoltaics.

Reduced Density Matrix Formulation of Quantum Linear Response

The prediction of spectral properties via linear response (LR) theory is an important tool in quantum chemistry for understanding photoinduced processes in molecular systems. With the advances of quantum computing, we recently adapted this method for near-term quantum hardware using a truncated active space approximation with orbital rotation, named quantum linear response (qLR). In an effort to reduce the classic cost of this hybrid approach, we here derive and implement a reduced density matrix (RDM) driven approach of qLR. This allows for the calculation of spectral properties of moderately sized molecules with much larger basis sets than so far possible. We report qLR results for benzene and R-methyloxirane with a cc-pVTZ basis set and study the effect of shot noise on the valence and oxygen K-edge absorption spectra of H2O in the cc-pVTZ basis.

The eXact integral simplified time-dependent density functional theory (XsTD-DFT)

In the framework of simplified quantum chemistry methods, we introduce the eXact integral simplified time-dependent density functional theory (XsTD-DFT). This method is based on the simplified time-dependent density functional theory (sTD-DFT), where all semi-empirical two-electron integrals are replaced by exact one- and two-center two-electron integrals, while other approximations from sTD-DFT are kept. The performance of this new parameter-free XsTD-DFT method was benchmarked on excited state and (non)linear response properties, including ultra-violet/visible absorption, first hyperpolarizability, and two-photon absorption (2PA). For a set of 77 molecules, the results from the XsTDA approach were compared to the TDA data. XsTDA/B3LYP excitation energies only deviate on average by 0.14 eV from TDA while drastically cutting computational costs by a factor of 20 or more depending on the energy threshold chosen. The absolute deviations of excitation energies with respect to the full scheme are decreasing with increasing system size, showing the suitability of XsTDA/XsTD-DFT to treat large systems. Comparing XsTDA and its predecessor sTDA, the new scheme generally improves excitation energies and oscillator strengths, in particular, for charge transfer states. TD-DFT first hyperpolarizability frequency dispersions for a set of push-pull π-conjugated molecules are faithfully reproduced by XsTD-DFT, while the previous sTD-DFT method provides redshifted resonance energy positions. Excellent performance with respect to the experiment is observed for the 2PA spectrum of the enhanced green fluorescent protein. The obtained robust accuracy similar to TD-DFT at a fraction of the computational cost opens the way for a plethora of applications for large systems and in high throughput screening studies.

Sparse-Stochastic Fragmented Exchange for Large-Scale Hybrid Time-Dependent Density Functional Theory Calculations

We extend our recently developed sparse-stochastic fragmented exchange formalism for ground-state near-gap hybrid DFT to calculate absorption spectra within linear-response time-dependent generalized Kohn-Sham DFT (LR-GKS-TDDFT) for systems consisting of thousands of valence electrons within a grid-based/plane-wave representation. A mixed deterministic/fragmented-stochastic compression of the exchange kernel, here using long-range explicit exchange functionals, provides an efficient method for accurate optical spectra. Both real-time propagation as well as frequency-resolved Casida-equation-type approaches for spectra are presented, and the method is applied to large molecular dyes.

Non-linear light-matter interactions from the Bethe-Salpeter equation

A route to assess non-linear light-matter interactions from the increasingly popular GW-Bethe-Salpeter equation (GW-BSE) method is outlined. In the present work, the necessary analytic expressions within the static-screened exchange approximation of the BSE are derived. This enables a straightforward implementation of the computation of the first hyperpolarizability as well as two-photon absorption processes for molecular systems. Benchmark calculations on small molecular systems reveal that the GW-BSE method is intriguingly accurate for predicting both first hyperpolarizabilities and two-photon absorption strengths. Using state-of-the-art Kohn-Sham references as a starting point, the accuracy of the GW-BSE method rivals that of the coupled-cluster singles-and-doubles method, outperforming both second-order coupled-cluster and time-dependent density-functional theory.

Toward Accurate Two-Photon Absorption Spectrum Simulations: Exploring the Landscape beyond the Generalized Gradient Approximation

In this Letter, we present a pioneering analysis of the density functional approximations (DFAs) beyond the generalized gradient approximation (GGA) for predicting two-photon absorption (2PA) strengths of a set of push-pull π-conjugated molecules. In more detail, we have employed a variety of meta-generalized gradient approximation (meta-GGA) functionals, including SCAN, MN15, and M06-2X, to assess their accuracy in describing the 2PA properties of a chosen set of 48 organic molecules. Analytic quadratic response theory is employed for these functionals, and their performance is compared against the previously studied DFAs and reference data obtained at the coupled-cluster CC2 level combined with the resolution-of-identity approximation (RI-CC2). A detailed analysis of the meta-GGA functional performance is provided, demonstrating that they improve upon their predecessors in capturing the key electronic features of the π-conjugated two-photon absorbers. In particular, the Minnesota functional MN15 shows very promising results as it delivers pleasingly accurate chemical rankings for two-photon transition strengths and excited-state dipole moments.

Pitfall in simulations of vibronic TD-DFT spectra: diagnosis and assessment

In this Communication, we study the effect of spurious oscillations in the profiles of energy derivatives with respect to nuclear coordinates calculated with density functional approximations (DFAs) for formaldehyde, pyridine, and furan in their ground and electronic excited states. These spurious oscillations, which can only be removed using extensive integration grids that increase enormously the CPU cost of DFA calculations, are significant in the case of third- and fourth-order energy derivatives of the ground and excited states computed by M06-2X and ωB97X functionals. The errors in question propagate to anharmonic vibronic spectra computed under the Franck-Condon approximation, i.e., positions and intensities of vibronic transitions are affected to a large extent (shifts as significant as hundreds of cm-1 were observed). On the other hand, the LC-BLYP and CAM-B3LYP functionals show a much less pronounced effect due to spurious oscillations. Based on the results presented herein, we recommend either LC-BLYP or CAM-B3LYP with integration grids (250, 974) (or larger) for numerically stable simulations of vibronic spectra including anharmonic effects.

NWChem: Recent and Ongoing Developments

This paper summarizes developments in the NWChem computational chemistry suite since the last major release (NWChem 7.0.0). Specifically, we focus on functionality, along with input blocks, that is accessible in the current stable release (NWChem 7.2.0) and in the "master" development branch, interfaces to quantum computing simulators, interfaces to external libraries, the NWChem github repository, and containerization of NWChem executable images. Some ongoing developments that will be available in the near future are also discussed.

Nonlinear Plasmonics in Nanostructured Phosphorene

Phosphorene has emerged as an atomically thin platform for optoelectronics and nanophotonics due to its excellent optical properties and the possibility of actively tuning light-matter interactions through electrical doping. While phosphorene is a two-dimensional semiconductor, plasmon resonances characterized by pronounced anisotropy and strong optical confinement are anticipated to emerge in highly doped samples. Here we show that the localized plasmons supported by phosphorene nanoribbons (PNRs) exhibit high tunability in relation to both edge termination and doping charge polarity and can trigger an intense nonlinear optical response at moderate doping levels. Our explorations are based on a second-principles theoretical framework, employing maximally localized Wannier functions constructed from ab initio electronic structure calculations, which we introduce here to describe the linear and nonlinear optical response of PNRs on mesoscopic length scales. Atomistic simulations reveal the high tunability of plasmons in doped PNRs at near-infrared frequencies, which can facilitate the synergy between the electronic band structure and plasmonic field confinement to drive efficient high-harmonic generation. Our findings establish nanostructured phosphorene as a versatile atomically thin material candidate for nonlinear plasmonics.

TURBOMOLE: Today and Tomorrow

TURBOMOLE is a highly optimized software suite for large-scale quantum-chemical and materials science simulations of molecules, clusters, extended systems, and periodic solids. TURBOMOLE uses Gaussian basis sets and has been designed with robust and fast quantum-chemical applications in mind, ranging from homogeneous and heterogeneous catalysis to inorganic and organic chemistry and various types of spectroscopy, light-matter interactions, and biochemistry. This Perspective briefly surveys TURBOMOLE's functionality and highlights recent developments that have taken place between 2020 and 2023, comprising new electronic structure methods for molecules and solids, previously unavailable molecular properties, embedding, and molecular dynamics approaches. Select features under development are reviewed to illustrate the continuous growth of the program suite, including nuclear electronic orbital methods, Hartree-Fock-based adiabatic connection models, simplified time-dependent density functional theory, relativistic effects and magnetic properties, and multiscale modeling of optical properties.

Gauge-Invariant Excited-State Linear and Quadratic Response Properties within the Meta-Generalized Gradient Approximation

Gauge invariance is a fundamental symmetry connected to charge conservation and is widely accepted as indispensable for any electronic structure method. Hence, the gauge variance of the time-dependent kinetic energy density τ used in many meta-generalized gradient approximations (MGGAs) to the exchange-correlation (XC) functional presents a major obstacle for applying MGGAs within time-dependent density functional theory (TDDFT). Replacing τ by the gauge-invariant generalized kinetic energy density τ̂ significantly improves the accuracy of various functionals for vertical excitation energies [R. Grotjahn, F. Furche, and M. Kaupp. J. Chem. Phys. 2022, 157, 111102]. However, the dependence of the resulting current-MGGAs (cMGGAs) on the paramagnetic current density gives rise to new exchange-correlation kernels and hyper-kernels ignored in previous implementations of quadratic and higher-order response properties. Here we report the first implementation of cMGGAs and hybrid cMGGAs for excited-state gradients and dipole moments, as well as an extension to quadratic response properties including dynamic hyperpolarizabilities and two-photon absorption cross sections. In the first comprehensive benchmark study of MGGAs and cMGGAs for two-photon absorption cross sections, the M06-2X functional is found to be superior to the GGA hybrid PBE0. Additionally, two case studies from the literature for the practical prediction of nonlinear optical properties are revisited and potential advantages of hybrid (c)MGGAs compared to hybrid GGAs are discussed. The effect of restoring gauge invariance varies depending on the employed MGGA functional, the type of excitation, and the property under investigation: While some individual excited-state equilibrium structures are significantly affected, on average, these changes result in marginal improvements when compared against high-level reference data. Although the gauge-variant MGGA quadratic response properties are generally close to their gauge-invariant counterparts, the resulting errors are not bounded and significantly exceed typical method errors in some of the cases studied. Despite the limited effects seen in benchmark studies, gauge-invariant implementations of cMGGAs for excited-state properties are desirable from a fundamental perspective, entail little additional computational cost, and are necessary for response properties consistent with cMGGA linear response calculations such as excitation energies.

TDDFT calculations of the PETN's ultraviolet absorption spectrum under the electric field loading

CONTEXT

The UV(ultraviolet) absorption spectrum of PETN under different electric field loading directions(X, Y, and Z) with the value of strength range from 0.001 a.u. to 0.006 a.u. was calculated with the TDDFT(Time-dependent density functional) in this work. With the increase of electric field strength, the absorbance of PETN in the ultraviolet band decreases. To explain the action mechanism of the electric field on PETN UV(ultraviolet) absorption spectrum, we analyzed and counted the contribution rate, oscillator strength, and vertical excitation energy of the main excitation process whose contribution rate to the UV absorption spectrum is greater than 10%. The contribution of PETN to the UV spectrum in all directions without an electric field was also listed to investigate the anisotropy of PETN in the excitation process under an electric field. The hole-electron analysis showed that the electric field will enhance the charge transfer characteristics in the excitation process of PETN. To investigate the anisotropy of the response under different electric field application directions, the contribution of the UV absorption spectrum in different directions was studied.

METHODS

Optimization and TDDFT calculation were performed at the level of M06-2X/def2-TZVP and PBE0/def2-TZVP respectively, with Gaussian09 program. The hole-electron analysis and UV absorption spectrum plotting were performed with Multiwfn3.8.

Mechanical nanolattices printed using nanocluster-based photoresists

Natural materials exhibit emergent mechanical properties as a result of their nanoarchitected, nanocomposite structures with optimized hierarchy, anisotropy, and nanoporosity. Fabrication of such complex systems is currently challenging because high-quality three-dimensional (3D) nanoprinting is mostly limited to simple, homogeneous materials. We report a strategy for the rapid nanoprinting of complex structural nanocomposites using metal nanoclusters. These ultrasmall, quantum-confined nanoclusters function as highly sensitive two-photon activators and simultaneously serve as precursors for mechanical reinforcements and nanoscale porogens. Nanocomposites with complex 3D architectures are printed, as well as structures with tunable, hierarchical, and anisotropic nanoporosity. Nanocluster-polymer nanolattices exhibit high specific strength, energy absorption, deformability, and recoverability. This framework provides a generalizable, versatile approach for the use of photoactive nanomaterials in additive manufacturing of complex systems with emergent mechanical properties.

Analytic high-order energy derivatives for metal nanoparticle-mediated infrared and Raman scattering spectra within the framework of quantum mechanics/molecular mechanics model with induced charges and dipoles

This work is devoted to deriving and implementing analytic second- and third-order energy derivatives with respect to the nuclear coordinates and external electric field within the framework of the hybrid quantum mechanics/molecular mechanics method with induced charges and dipoles (QM/DIM). Using these analytic energy derivatives, one can efficiently compute the harmonic vibrational frequencies, infrared (IR) and Raman scattering (RS) spectra of the molecule in the proximity of noble metal clusters/nanoparticles. The validity and accuracy of these analytic implementations are demonstrated by the comparison of results obtained by the finite-difference method and the analytic approaches and by the full QM and QM/DIM calculations. The complexes formed by pyridine and two sizes of gold clusters (Au18 and Au32) at varying intersystem distances of 3, 4, and 5 Å are used as the test systems, and Raman spectra of 4,4'-bipyridine in the proximity of Au2057 and Ag2057 metal nanoparticles (MNP) are calculated by the QM/DIM method and compared with experimental results as well. We find that the QM/DIM model can well reproduce the IR spectra obtained from full QM calculations for all the configurations, while although it properly enhances some of the vibrational modes, it artificially overestimates RS spectral intensities of several modes for the systems with very short intersystem distance. We show that this could be improved, however, by incorporating the hyperpolarizability of the gold metal cluster in the evaluation of RS intensities. Additionally, we address the potential impact of charge migration between the adsorbate and MNPs.

Ultrafast Evaluation of Two-Photon Absorption with Simplified Time-Dependent Density Functional Theory

This work presents the theoretical background to evaluate two-photon absorption (2PA) cross-sections in the framework of simplified time-dependent density functional theory (sTD-DFT). Our new implementation allows the ultrafast evaluation of 2PA cross-sections for large molecules based on a regular DFT ground-state determinant as well as a variant employing our tight-binding sTD-DFT-xTX flavor for very large systems. The method is benchmarked against higher-level calculations for trans-stilbene and typical fluorescent protein chromophores. For eGFP, a quadrupolar chromophore and its branched version, the flavine mono-nucleotide, and the iLOV protein, we compare sTD-DFT 2PA spectra to experimental ones. This includes extension and testing of our all-atom quantum chemistry methodology for the evaluation of 2PA for a system of ∼2000 atoms, providing striking agreement with the experimental spectrum.

Controlling Aggregation-Induced Two-Photon Absorption Enhancement via Intermolecular Interactions

Historically, two-photon absorption (2PA) cross sections reported in the literature have been derived from solution-phase measurements. However, such techniques fail to grasp the implications of how these cross sections can be impacted by varying degrees of aggregation or in the condensed phase as bulk solids or thin films. For a precise determination of how aggregation impacts 2PA at a molecular level, computational methods present themselves as ideal. Herein, a series of quadrupolar π-conjugated dyes were simulated by molecular dynamics (MD) in the gas phase and condensed phase. In the condensed phase, their intermolecular interactions and electronic coupling behavior were fully characterized, both quantitatively and qualitatively. Using quadratic-response time-dependent density functional theory, 2PA cross sections of structures derived from MD trajectories were calculated. Comparisons are made between gas-phase and condensed-phase results, and enhancement factors are defined to show how certain dyes may experience changes in their respective 2PA cross sections as a function of aggregation. It was found that these cross sections depend heavily on conformational locking in the condensed phase and relative stacking arrangements. J-aggregates were associated with enhanced 2PA and H-aggregates with quenched 2PA activity. However, in a highly disordered aggregate, the effects of these stacking arrangements are averaged out of the bulk result, and the effects of conformational locking dominate.

Accurate Calculation of Excited-State Absorption for Small-to-Medium-Sized Conjugated Oligomers: Multiconfigurational Treatment vs Quadratic Response TD-DFT

Excited-state absorption (ESA) spectra of π-conjugated compounds are frequently calculated by (quadratic response) time-dependent density functional theory, (QR) TD-DFT, often giving a reasonable representation of the experimental results despite the (known) incomplete electronic description. To investigate whether this is inherent to the method, we calculate here the ESA spectra of small-to-medium-sized oligophenylenevinylenes (nPV) and oligothiophenes (nT) using QR TD-DFT as well as CASPT2 based on CASSCF geometries. CASPT2 gives indeed a reliable, theoretically correct description of the ESA features for all compounds; the computational effort can be reduced without significant loss of accuracy using TD-DFT geometries. QR TD-DFT, based on BHandHLYP and CAM-/B3LYP functionals, fails on short nTs but provides a reasonable description for spectral positions of nPVs and long nTs. The failure on short nTs is, however, only partly due to the incomplete configuration description but, in particular, related to an improper MO description, resulting in an asymmetric energy spacing of the occupied vs unoccupied MOs in the DFT scheme. Longer nTs, on the other side, adapt approximately the MO scheme for alternant hydrocarbons just like in nPVs, while contributions by two triplet excitations combined to a singlet (which inhibits an accurate treatment of polyenes with standard TD-DFT) do not play a relevant role in the current case. For such "well-behaved" systems, a reasonable representation of ESA spectra is found at the QR TD-DFT level due to the rather small energy shifts when including higher-order excitations.

Nonadiabatic Molecular Dynamics Simulations Based on Time-Dependent Density Functional Tight-Binding Method

Nonadiabatic excited-state molecular dynamics underpin many photophysical and photochemical phenomena, such as exciton dynamics, charge separation and transport. In this work, we present an efficient nonadiabatic molecular dynamic (NAMD) simulation method based on time-dependent density functional tight-binding (TDDFTB) theory. Specifically, the adiabatic electronic structure, an essential NAMD input, is described at the TDDFTB level. The nonadiabatic effects originating from the coupled motions of electrons and nuclei are treated by the trajectory surface hopping algorithm. To improve the computational efficiency, nonadiabatic couplings between excited states within the TDDFTB method are derived and implemented using an analytical approach. Further, the time-dependent nonadiabatic coupling scalars are calculated based on the overlap between molecular orbitals rather than the Slater determinants to speed up the simulations. In addition, the electronic decoherence scheme and a state reassigned unavoided crossings algorithm, which has been implemented in the NEXMD software, are used to improve the accuracy of the simulated dynamics and handle trivial unavoided crossings. Finally, the photoinduced nonadiabatic dynamics of a benzene molecule are simulated to demonstrate our implementation. The results for excited state NAMD simulations of benzene molecule based on TDDFTB method compare well that obtained with numerically expensive time-dependent density functional theory. The proposed methodology provides an attractive theoretical simulation tool for predicting the photophysical and photochemical properties of complex materials.

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.

NMR Coupling Constants Based on the Bethe-Salpeter Equation in the GW Approximation

We present the first steps to extend the Green's function GW method and the Bethe-Salpeter equation (BSE) to molecular response properties such as nuclear magnetic resonance (NMR) indirect spin-spin coupling constants. We discuss both a nonrelativistic one-component and a quasi-relativistic two-component formalism. The latter describes scalar-relativistic and spin-orbit effects and allows us to study heavy-element systems with reasonable accuracy. Efficiency is maintained by the application of the resolution of the identity approximation throughout. The performance is demonstrated using conventional central processing units (CPUs) and modern graphics processing units (GPUs) for molecules involving several thousand basis functions. Our results show that a large amount of Hartree-Fock exchange is vital to provide a sufficient Kohn-Sham starting point to compute the GW quasi-particle energies. As the GW-BSE approach is generally less accurate for triplet excitations or related properties such as the Fermi-contact interaction, the admixture of the Kohn-Sham correlation kernel through the contracted BSE (cBSE) method improves the results for NMR coupling constants. This leads to remarkable results when combined with the eigenvalue-only self-consistent variant (evGW) and Becke's half and half functional (BH&HLYP) or the CAM-QTP family. The developed methodology is used to calculate the Karplus curve of tin molecules, illustrating its applicability to extended chemically relevant molecules. Here, the GW-cBSE method improves upon the chosen BH&HLYP Kohn-Sham starting points.

Calculated linear and nonlinear optical absorption spectra of phosphine-ligated gold clusters

Although prediction of optical excitations of ligated gold clusters by time-dependent density functional theory (TDDFT) is relatively well-established, limitations still exist, for example in the choice of the exchange-correlation functional....

Accelerating molecular property calculations with semiempirical preconditioning

Computing ab initio molecular linear response properties, e.g., electronic excitation energies and transition dipole moments, requires the solution of large eigenvalue problems or large systems of equations. These large eigenvalue problems or large systems of equations are commonly solved iteratively using Krylov space algorithms, such as the Davidson algorithm for eigenvalue problems. A critical ingredient in Krylov space algorithms is the preconditioner, which is used to generate optimal update vectors in each iteration. We propose to use semiempirical approximations as preconditioners to accelerate the calculation of ab initio properties. The crucial advantage to improving the preconditioner is that the converged result is unchanged, so there is no trade-off between accuracy and speedup. We demonstrate our approach by accelerating the calculation of electronic excitation energies and electric polarizabilities from linear response time-dependent density functional theory using the simplified time-dependent density functional theory semiempirical model. For excitation energies, the semiempirical preconditioner reduces the number of iterations on average by 37% and up to 70%. The semiempirical preconditioner reduces the number of iterations for computing the polarizability by 15% on average and up to 33%. Moreover, we show that the preconditioner can be further improved by tuning the empirical parameters that define the semiempirical model, leading to an additional reduction in the number of iterations by about 20%. Our approach bridges the gap between semiempirical models and ab initio methods and charts a path toward combining the speed of semiempirical models with the accuracy of ab initio methods.

Understanding Hyperporphyrin Spectra: TDDFT Calculations on Diprotonated Tetrakis(p-aminophenyl)porphyrin

A detailed TDDFT study (with all-electron STO-TZ2P basis sets and the COSMO solvation model) has been carried out on the effect of diprotonation on the UV-vis-NIR spectra of free-base tetraphenylporphyrin and tetrakis(p-aminophenyl)porphyrin. The diprotonated forms have been modeled as their bis-formate complexes, i.e., as so-called porphyrin diacids. The dramatic redshift of the Q-band of the TAPP diacid has been explained in terms of an elevated "a2u" HOMO and lowered LUMOs, both reflecting infusion of aminophenyl character into the otherwise classic Gouterman-type frontier MOs. The exercise has also yielded valuable information on the performance of different exchange-correlation functionals. Thus, the hybrid B3LYP functional was found to yield a substantially better description of key spectral features, especially the diprotonation-induced redshifts, than the pure OLYP functional. Use of the range-separated CAMY-B3LYP functional, on the other hand, did not result in improvements relative to B3LYP.

In Silico Ultrafast Nonlinear Spectroscopy Meets Experiments: The Case of Perylene Bisimide Dye

Spectroscopy simulations are of paramount importance for the interpretation of experimental electronic spectra, the disentangling of overlapping spectral features, and the tracing of the microscopic origin of the observed signals. Linear and nonlinear simulations are based on the results drawn from electronic structure calculations that provide the necessary parameterization of the molecular systems probed by light. Here, we investigate the applicability of excited-state properties obtained from linear-response time-dependent density functional theory (TDDFT) in the description of nonlinear spectra by employing the pseudowavefunction approach and compare them with benchmarks from highly accurate RASSCF/RASPT2 calculations and with high temporal resolution experimental results. As a test case, we consider the prediction of femtosecond transient absorption and two-dimensional electronic spectroscopy of a perylene bisimide dye in solution. We find that experimental signals are well reproduced by both theoretical approaches, showing that the computationally cheaper TDDFT can be a suitable option for the simulation of nonlinear spectroscopy of molecular systems that are too large to be treated with higher-level RASSCF/RASPT2 methods.

Nonadiabatic dynamics with spin-flip vs linear-response time-dependent density functional theory: A case study for the protonated Schiff base C5H6NH2. J Chem Phys 2021;155:124111. [PMID: 34598550 DOI: 10.1063/5.0062757]

Nonadiabatic trajectory surface hopping simulations are reported for trans-C₅H₆NH₂ ⁺, a model of the rhodopsin chromophore, using the augmented fewest-switches algorithm. Electronic structure calculations were performed using time-dependent density functional theory (TDDFT) in both its conventional linear-response (LR) and its spin-flip (SF) formulations. In the SF-TDDFT case, spin contamination in the low-lying singlet states is removed by projecting out the lowest triplet component during iterative solution of the TDDFT eigenvalue problem. The results show that SF-TDDFT qualitatively describes the photoisomerization of trans-C₅H₆NH₂ ⁺, with favorable comparison to previous studies using multireference electronic structure methods. In contrast, conventional LR-TDDFT affords qualitatively different photodynamics due to an incorrect excited-state potential surface near the Franck-Condon region. In addition, the photochemistry (involving pre-twisting of the central double bond) appears to be different for SF- and LR-TDDFT, which may be a consequence of different conical intersection topographies afforded by these two methods. The present results contrast with previous surface-hopping studies suggesting that the LR-TDDFT method's incorrect topology around S₁/S₀ conical intersections is immaterial to the photodynamics.

Minimizing the Time-Dependent Density Functional Error in Ehrenfest Dynamics

Simulating electron-ion dynamics using time-dependent density functional theory within an Ehrenfest dynamics scheme can be done in two ways that are in principle exact and identical: propagating time-dependent electronic Kohn-Sham equations or propagating electronic coefficients on surfaces obtained from linear-response. We show here that using an approximate functional leads to qualitatively different dynamics in the two approaches. We argue that the latter is more accurate because the functionals are evaluated on domains close to the ground state where currently used approximations perform better. We demonstrate this on an exactly solvable model of charge transfer and discuss implications for time-resolved spectroscopy.

Nonadiabatic Molecular Dynamics Study of the Relaxation Pathways of Photoexcited Cyclooctatetraene

In the current study, we present nonadiabatic (NAMD) and adiabatic molecular dynamics simulations of the transition-state dynamics of photoexcited cyclooctatetraene (COT). The equilibrium-state structure and absorption spectra are analyzed using the semiempirical Austin Model 1 potential. The NAMD simulations are obtained by a surface-hopping algorithm. We analyzed in detail an active excited to ground state relaxation pathway accompanied by an S₂/S₃(D_2d) → S₁(D_8h) → S₀(D_4h) → S₀(D_2d) double-bond shifting mechanism. The simulated excitation lifetime is in good agreement with experiment. The first excited singlet state S₁ plays a crucial role in the photochemistry. The obtained critical molecular conformations, energy barrier, and transition-state lifetime results will provide a basis for further investigations of the bond-order inversion and photoswitching process of COT.

An Ab Initio Multiple Cloning Method for Non-Adiabatic Excited-State Molecular Dynamics in NWChem

The recently developed ab initio multiple cloning (AIMC) approach based on the multiconfigurational Ehrenfest (MCE) method provides a powerful and accurate way of describing the excited-state dynamics of molecular systems. The AIMC method is a controlled approximation to nonadiabatic dynamics with a particular strength in the proper description of decoherence effects because of the branching of vibrational wavepackets at a level crossing. Here, we report a new implementation of the AIMC algorithm in the open source NWChem computational chemistry program. The framework combines linear-response time-dependent density functional theory with Ehrenfest mean-field theory to determine the equations of motion for classical trajectories. The multidimensional wave function is decomposed into a superposition of Gaussian coherent states guided by Ehrenfest trajectories (i.e., MCE approach), which can clone with fully quantum mechanical amplitudes and phases. By using an efficient time-derivative based nonadiabatic coupling approach within the AIMC method, all observables are calculated on-the-fly in the nonadiabatic molecular dynamics process. As a representative example, we apply our implementation to study the ultrafast photoinduced electronic and vibrational energy transfer in a pyridine molecule. The effects of the cloning procedure on electronic and vibrational coherence, relaxation and unidirectional energy transfer are discussed. This new AIMC implementation provides a high-level nonadiabatic molecular dynamics framework for simulating photoexcited dynamics in complex molecular systems and experimentally relevant ultrafast spectroscopic probes, such as nonlinear coherent optical and X-ray signals.

Resonant Inelastic X-ray Scattering Calculations of Transition Metal Complexes Within a Simplified Time-Dependent Density Functional Theory Framework

We present a time-dependent density functional theory (TDDFT) approach to compute the light-matter couplings between two different manifolds of excited states relative to a common ground state in the context of 4d transition metal systems. These quantities are the necessary ingredients to solve the Kramers-Heisenberg (KH) equation for resonant inelastic X-ray scattering (RIXS) and several other types of two-photon spectroscopies. The procedure is based on the pseudo-wavefunction approach, where the solutions of a TDDFT calculation can be used to construct excited-state wavefunctions, and on the restricted energy window approach, where a manifold of excited states can be rigorously defined based on the energies of the occupied molecular orbitals involved in the excitation process. Thus, the present approach bypasses the need to solve the costly TDDFT quadratic-response equations. We illustrate the applicability of the method to 4d transition metal molecular complexes by calculating the 2p4d RIXS maps of three representative ruthenium complexes and comparing them to experimental results. The method can capture all the experimental features in all three complexes to allow the assignment of the experimental peaks, with relative energies correct to within ∼0.6 eV at the cost of two independent TDDFT calculations.

Nonlinear light absorption in many-electron systems excited by an instantaneous electric field: a non-perturbative approach

Applications of low-cost non-perturbative approaches in real time, such as time-dependent density functional theory, for the study of nonlinear optical properties of large and complex systems are gaining increasing popularity. However, their assessment still requires the analysis and understanding of elementary dynamical processes in simple model systems. Motivated by the aim of simulating optical nonlinearities in molecules, here exemplified by the case of the quaterthiophene oligomer, we investigate light absorption in many-electron interacting systems beyond the linear regime by using a single broadband impulse of an electric field; i.e. an electrical impulse in the instantaneous limit. We determine non-pertubatively the absorption cross section from the Fourier transform of the time-dependent induced dipole moment, which can be obtained from the time evolution of the wavefunction. We discuss the dependence of the resulting cross section on the magnitude of the impulse and we highlight the advantages of this method in comparison with perturbation theory by working on a one-dimensional model system for which numerically exact solutions are accessible. Thus, we demonstrate that the considered non-pertubative approach provides us with an effective tool for investigating fluence-dependent nonlinear optical excitations.

Two-Photon Absorption Activity of BOPHY Derivatives: Insights from Theory

We present a theoretical study of a two-photon absorption (2PA) process in dipolar and quadrupolar systems containing two BF₂ units. For this purpose, we considered 13 systems studied by Ponce-Vargas et al. [J. Phys. Chem. B2017, 121, 10850−1085829136383] and performed linear and quadratic response theory calculations based on the RI-CC2 method to obtain the 2PA parameters. Furthermore, using the recently developed generalized few-state model, we provided an in-depth view of the changes in 2PA properties in the molecules considered. Our results clearly indicate that suitable electron-donating group substitution to the core BF₂ units results in a large red-shift of the two-photon absorption wavelength, thereby entering into the desired biological window. Furthermore, the corresponding 2PA strength also increases significantly (up to 30-fold). This makes the substituted systems a potential candidate for biological imaging.

Efficient implementation of isotropic cubic response functions for two-photon absorption cross sections within the self-consistent field approximation

Within the self-consistent field approximation, computationally tractable expressions for the isotropic second-order hyperpolarizability have been derived and implemented for the calculation of two-photon absorption cross sections. The novel tensor average formulation presented in this work allows for the evaluation of isotropic damped cubic response functions using only ∼3.3% (one-photon off-resonance regions) and ∼10% (one-photon resonance regions) of the number of auxiliary Fock matrices required when explicitly calculating all the needed individual tensor components. Numerical examples of the two-photon absorption cross section in the one-photon off-resonance and resonance regions are provided for alanine-tryptophan and 2,5-dibromo-1,4-bis(2-(4-diphenylaminophenyl)vinyl)-benzene. Furthermore, a benchmark set of 22 additional small- and medium-sized organic molecules is considered. In all these calculations, a quantitative assessment is made of the reduced and approximate forms of the cubic response function in the one-photon off-resonance regions and results demonstrate a relative error of less than ∼5% when using the reduced expression as compared to the full form of the isotropic cubic response function.

Nonadiabatic molecular dynamics simulation for the ultrafast photoisomerization of dMe-OMe-NAIP based on TDDFT on-the-fly potential energy surfaces

Global switching on-the-fly trajectory surface hopping molecular dynamics simulation was performed on the accurate TD-B3LYP/6-31G* potential energy surfaces for E-to-Z and Z-to-E photoisomerization of dMe-OMe-NAIP up to S1(ππ*) excitation. The present TD(DFT) simulation provides accurate calculation for conical intersections between the first-excited and ground states. Thus, simulated quantum yield and lifetime of 0.23 and 620 fs (0.15 and 600 fs) for E-to-Z (Z-to-E) isomerization are in good (relatively good) agreement with experimental observation of 0.25 and 480 fs (0.24 and 430 fs), respectively. Simulated results reveal that photoisomerization pathways are initially uphill to conical intersection zones on the S1 potential energy surface and then downhill to product zones. Three types of representative conical intersections are found for determining photoisomerization mechanisms: one is the rotation type responsible for reactive isomerization and the other two are close to E and Z configurations, respectively, only for nonreactive isomerization. The present conclusions can be held in general for similar large NAIP systems of photoinduced isomerization based on E and Z configurations.

TURBOMOLE: Modular program suite for ab initio quantum-chemical and condensed-matter simulations

TURBOMOLE is a collaborative, multi-national software development project aiming to provide highly efficient and stable computational tools for quantum chemical simulations of molecules, clusters, periodic systems, and solutions. The TURBOMOLE software suite is optimized for widely available, inexpensive, and resource-efficient hardware such as multi-core workstations and small computer clusters. TURBOMOLE specializes in electronic structure methods with outstanding accuracy-cost ratio, such as density functional theory including local hybrids and the random phase approximation (RPA), GW-Bethe-Salpeter methods, second-order Møller-Plesset theory, and explicitly correlated coupled-cluster methods. TURBOMOLE is based on Gaussian basis sets and has been pivotal for the development of many fast and low-scaling algorithms in the past three decades, such as integral-direct methods, fast multipole methods, the resolution-of-the-identity approximation, imaginary frequency integration, Laplace transform, and pair natural orbital methods. This review focuses on recent additions to TURBOMOLE's functionality, including excited-state methods, RPA and Green's function methods, relativistic approaches, high-order molecular properties, solvation effects, and periodic systems. A variety of illustrative applications along with accuracy and timing data are discussed. Moreover, available interfaces to users as well as other software are summarized. TURBOMOLE's current licensing, distribution, and support model are discussed, and an overview of TURBOMOLE's development workflow is provided. Challenges such as communication and outreach, software infrastructure, and funding are highlighted.

Exciton coupling effects on the two-photon absorption of squaraine homodimers with varying bridge units

Excitonically coupled squaraine dimers show high two-photon absorption cross sections.