Unphysical divergences in response theory

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.

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.

Oscillator strengths and excited-state couplings for double excitations in time-dependent density functional theory

Although useful to extract excitation energies of states of double-excitation character in time-dependent density functional theory that are missing in the adiabatic approximation, the frequency-dependent kernel derived earlier [Maitra et al., J. Chem. Phys. 120, 5932 (2004)] was not designed to yield oscillator strengths. These are required to fully determine linear absorption spectra, and they also impact excited-to-excited-state couplings that appear in dynamics simulations and other quadratic response properties. Here, we derive a modified non-adiabatic kernel that yields both accurate excitation energies and oscillator strengths for these states. We demonstrate its performance on a model two-electron system, the Be atom, and on excited-state transition dipoles in the LiH molecule at stretched bond-lengths, in all cases producing significant improvements over the traditional approximations.

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.

Robust relativistic many-body Green's function based approaches for assessing core ionized and excited states

A two-component contour deformation (CD) based GW method that employs frequency sampling to drastically reduce the computational effort when assessing quasiparticle states far away from the Fermi level is outlined. Compared to the canonical CD-GW method, computational scaling is reduced by an order of magnitude without sacrificing accuracy. This allows for an efficient calculation of core ionization energies. The improved computational efficiency is used to provide benchmarks for core ionized states, comparing the performance of 15 density functional approximations as Kohn-Sham starting points for GW calculations on a set of 65 core ionization energies of 32 small molecules. Contrary to valence states, GW calculations on core states prefer functionals with only a moderate amount of Hartree-Fock exchange. Moreover, modern ab initio local hybrid functionals are also shown to provide excellent generalized Kohn-Sham references for core GW calculations. Furthermore, the core-valence separated Bethe-Salpeter equation (CVS-BSE) is outlined. CVS-BSE is a convenient tool to probe core excited states. The latter is tested on a set of 40 core excitations of eight small inorganic molecules. Results from the CVS-BSE method for excitation energies and the corresponding absorption cross sections are found to be in excellent agreement with those of reference damped response BSE calculations.

Curing the Divergence in Time-Dependent Density Functional Quadratic Response Theory

The adiabatic approximation in time-dependent density functional theory is known to give an incorrect pole structure in the quadratic response function, leading to unphysical divergences in excited state-to-state transition probabilities and hyperpolarizabilties. We find the form of the exact quadratic response kernel and derive a practical and accurate approximation that cures the divergence. We demonstrate our results on excited state-to-state transition probabilities of a model system and of the LiH molecule.

Exact non-adiabatic coupling vectors for the time-dependent density functional based tight-binding method

We report on non-adiabatic coupling vectors between electronic excited states for the time-dependent-density functional theory based tight-binding (TD-DFTB) method. The implementation includes orbital relaxation effects that have been previously neglected and covers also the case of range-separated exchange-correlation functionals. Benchmark calculations with respect to first principles TD-DFT highlight the large dependence of non-adiabatic couplings on the functional. Closer investigations of the topology around a conical intersection between excited states show that TD-DFTB delivers near-exact values of the Berry phase, which paves the way for consistent non-adiabatic molecular dynamics simulations for large systems.

Exciton dispersion and exciton-phonon interaction in solids by time-dependent density functional theory

Understanding, predicting, and ultimately controlling exciton band structure and exciton dynamics are central to diverse chemical and materials problems. Here, we have developed a first-principles method to determine exciton dispersion and exciton-phonon interaction in semiconducting and insulating solids based on time-dependent density functional theory. The first-principles method is formulated in planewave bases and pseudopotentials and can be used to compute exciton band structures, exciton charge density, ionic forces, the non-adiabatic coupling matrix between excitonic states, and the exciton-phonon coupling matrix. Based on the spinor formulation, the method enables self-consistent noncollinear calculations to capture spin-orbital coupling. Hybrid exchange-correlation functionals are incorporated to deal with long-range electron-hole interactions in solids. A sub-Hilbert space approximation is introduced to reduce the computational cost without loss of accuracy. For validations, we have applied the method to compute the exciton band structure and exciton-phonon coupling strength in transition metal dichalcogenide monolayers; both agree very well with the previous GW-Bethe-Salpeter equation and experimental results. This development paves the way for accurate determinations of exciton dynamics in a wide range of solid-state materials.

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.

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.

Simulation and Analysis of the Transient Absorption Spectrum of 4-(N,N-Dimethylamino)benzonitrile (DMABN) in Acetonitrile

4-(N,N-Dimethylamino)benzonitrile (DMABN) is a well-known model compound for dual fluorescence-in sufficiently polar solvents, it exhibits two distinct fluorescence emission bands. The interpretation of its transient absorption (TA) spectrum in the visible range is the subject of a long-standing controversy. In the present study, we resolve this issue by calculating the TA spectrum on the basis of nonadiabatic molecular dynamics simulations. An unambiguous assignment of spectral signals to specific excited-state structures is achieved by breaking down the calculated spectrum into contributions from twisted and nontwisted molecular geometries. In particular, the much-discussed excited-state absorption band near 1.7 eV (ca. 700 nm) is attributed to the near-planar locally excited (LE) minimum on the S1 state. On the technical side, our study demonstrates that the second-order approximate coupled cluster singles and doubles (CC2) method can be used successfully to calculate the TA spectra of moderately large organic molecules, provided that the system in question does not approach a crossing between the lowest excited state and the singlet ground state within the time frame of the simulation.

NAC-TDDFT: Time-Dependent Density Functional Theory for Nonadiabatic Couplings

First-order nonadiabatic coupling (NAC) matrix elements (fo-NACMEs) are the basic quantities in theoretical descriptions of electronically nonadiabatic processes that are ubiquitous in molecular physics and chemistry. Given the large size of systems of chemical interests, time-dependent density functional theory (TDDFT) is usually the first choice of methods. However, the lack of many-electron wave functions in TDDFT renders the formulation of NAC-TDDFT for fo-NACMEs conceptually difficult. Because of this, various variants of NAC-TDDFT have been proposed in the literature from different standing points, including the Hellmann-Feynman-like expression and auxiliary/pseudo-wave function (AWF)-, equation-of-motion (EOM)-, and time-dependent perturbation theory (TDPT)-based formulations. Based on critical analyses, the following conclusions are made here: (1) The Hellmann-Feynman-like expression, which is rooted in exact wave function theory, is hardly useful due to huge demand on basis sets. (2) Although most popular, the AWF variants of NAC-TDDFT are not theoretically founded and become ambiguous particularly for the fo-NACMEs between two excited states, although they do agree with the EOM and TDPT variants under the Tamm-Dancoff approximation. (3) The TDPT variant of NAC-TDDFT is theoretically most rigorous but suffers from numerical instabilities on the one hand and does not differ to a significant extent from the EOM variant on the other hand. (4) As such, the EOM variant of NAC-TDDFT for the fo-NACMEs between the ground and excited states and between two excited states is solely the right choice in practice. These formal analyses are fully supported by numerical experimentations, taking azulene as a showcase. The proper implementation of the EOM variant of NAC-TDDFT is also highlighted, showing that the fo-NACMEs between the ground and excited states and between two excited states are computationally very much the same as the analytic energy gradients of DFT and TDDFT, respectively. Possible future developments of the EOM variant of NAC-TDDFT are also highlighted. Its extensions to spin-adapted open-shell TDDFT and proper treatment of spin-orbit couplings (which are another source of force for electronically nonadiabatic processes) are particularly warranted in the near future.

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.

Combining Graphics Processing Units, Simplified Time-Dependent Density Functional Theory, and Finite-Difference Couplings to Accelerate Nonadiabatic Molecular Dynamics

Starting from our recently published implementation of nonadiabatic molecular dynamics (NAMD) on graphics processing units (GPUs), we explore further approaches to accelerate ab initio NAMD calculations at the time-dependent density functional theory (TDDFT) level of theory. We employ (1) the simplified TDDFT schemes of Grimme et al. and (2) the Hammes-Schiffer-Tully approach to obtain nonadiabatic couplings from finite-difference calculations. The resulting scheme delivers an accurate physical picture while virtually eliminating the two computationally most demanding steps of the algorithm. Combined with our GPU-based integral routines for SCF, TDDFT, and TDDFT derivative calculations, NAMD simulations of systems of a few hundreds of atoms at a reasonable time scale become accessible on a single compute node. To demonstrate this and to present a first, illustrative example, we perform TDDFT/MM-NAMD simulations of the rhodopsin protein.

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.

Nonadiabatic Molecular Dynamics on Graphics Processing Units: Performance and Application to Rotary Molecular Motors

Nonadiabatic molecular dynamics (NAMD) simulations of molecular systems require the efficient evaluation of excited-state properties, such as energies, gradients, and nonadiabatic coupling vectors. Here, we investigate the use of graphics processing units (GPUs) in addition to central processing units (CPUs) to efficiently calculate these properties at the time-dependent density functional theory (TDDFT) level of theory. Our implementation in the FermiONs++ program package uses the J-engine and a preselective screening procedure for the calculation of Coulomb and exchange kernels, respectively. We observe good speed-ups for small and large molecular systems (comparable to those observed in ground-state calculations) and reduced (down to sublinear) scaling behavior with respect to the system size (depending on the spatial locality of the investigated excitation). As a first illustrative application, we present efficient NAMD simulations of a series of newly designed light-driven rotary molecular motors and compare their S1 lifetimes. Although all four rotors show different S1 excitation energies, their ability to rotate upon excitation is conserved, making the series an interesting starting point for rotary molecular motors with tunable excitation energies.

Multistate hybrid time-dependent density functional theory with surface hopping accurately captures ultrafast thymine photodeactivation

We report an efficient analytical implementation of first-order nonadiabatic derivative couplings between arbitrary Born–Oppenheimer states in the hybrid time-dependent density functional theory (TDDFT) framework using atom-centered basis functions.

Excited-state absorption in tetrapyridyl porphyrins: comparing real-time and quadratic-response time-dependent density functional theory

Three meso-substituted tetrapyridyl porphyrins (free base, Ni(ii), and Cu(ii)) were investigated for their optical limiting (OL) capabilities using real-time (RT-), linear-response (LR-), and quadratic-response (QR-) time-dependent density functional theory (TDDFT) methods. These species are experimentally known to display a prominent reverse saturable absorption feature between the Q and B bands of the ground-state absorption (GSA), which has been attributed to increased excited-state absorption (ESA) relative to GSA. A recently developed RT-TDDFT based method for calculating ESA from a LR-TDDFT density was utilized with eight exchange-correlation functionals (BLYP, PBE, B3LYP, CAM-B3LYP, PBE0, M06, BHLYP, and BHandH) and contrasted with calculations of ESA using QR-TDDFT with five exchange-correlation functionals (BLYP, B3LYP, CAM-B3LYP, BHLYP, and BHandH). This allowed for comparison between functionals with varying amounts of exact exchange as well as between the ability of RT-TDDFT and QR-TDDFT to reproduce OL behavior in porphyrin systems. The absorption peak positions and intensities for GSA and ESA are significantly impacted by the choice of DFT functional, with the most critical factor identified as the amount of exact exchange in the functional form. Calculating ESA with QR-TDDFT is found to be significantly more sensitive to the amount of exact exchange than GSA and ESA with RT-TDDFT, as well as GSA with LR-TDDFT. An analogous behavior is also demonstrated for the polycyclic aromatic hydrocarbon coronene. This is problematic when using the same approximate functional for calculation of both GSA and ESA, as the LR- and QR-TDDFT excitation energies will not have similar errors. Overall, the RT-TDDFT method with hybrid functionals reproduces the OL features for the porphyrin systems studied here and is a viable computational approach for efficient screening of molecular complexes for OL properties.

Charge transfer in time-dependent density functional theory

Charge transfer plays a crucial role in many processes of interest in physics, chemistry, and bio-chemistry. In many applications the size of the systems involved calls for time-dependent density functional theory (TDDFT) to be used in their computational modeling, due to its unprecedented balance between accuracy and efficiency. However, although exact in principle, in practise approximations must be made for the exchange-correlation functional in this theory, and the standard functional approximations perform poorly for excitations which have a long-range charge-transfer component. Intense progress has been made in developing more sophisticated functionals for this problem, which we review. We point out an essential difference between the properties of the exchange-correlation kernel needed for an accurate description of charge-transfer between open-shell fragments and between closed-shell fragments. We then turn to charge-transfer dynamics, which, in contrast to the excitation problem, is a highly non-equilibrium, non-perturbative, process involving a transfer of one full electron in space. This turns out to be a much more challenging problem for TDDFT functionals. We describe dynamical step and peak features in the exact functional evolving over time, that are missing in the functionals currently used. The latter underestimate the amount of charge transferred and manifest a spurious shift in the charge transfer resonance position. We discuss some explicit examples.

Importance of double-resonance effects in two-photon absorption properties of Au25(SR)18. Chem Sci 2017. [PMID: 28626569 PMCID: PMC5471455 DOI: 10.1039/c7sc00968b]

We show that double-resonance effects for Au₂₅(SR)₁₈^– are less pronounced and do not lead to significantly enhanced two-photon absorption cross-sections.

The two-photon absorption (TPA) cross-sections of small thiolate-protected gold clusters have been shown to be much larger than typical small organic molecules. In comparison with larger nanoparticles, their TPA cross-sections per gold atom are also found to be larger. Theoretical simulations have suggested that the large enhancement of these TPA cross-sections comes from a one-photon double-resonance mechanism. However, it remains difficult to simulate TPA cross-sections of thiolate-protected gold clusters due to their large system size and a high density of states. In this work, we report a time-dependent density functional theory (TDDFT) study of the TPA spectra of the Au₂₅(SR)₁₈^– cluster based on a damped response theory formalism. Damped response theory enables a consistent treatment of on- and off-resonance molecular properties even for molecules with a high density of states, and thus is well-suited for studying the TPA properties of gold clusters. Our results indicate that the one- and two-photon double-resonance effect is much smaller than previously found, and thus is unlikely to be the main cause of the large TPA cross-sections found experimentally. The effect of symmetry breaking of the Au₂₅(SR)₁₈^– cluster due to the ligands on the TPA cross-sections has been studied and was found to only slightly increase the cross-section. Furthermore, by comparing with larger nanoparticles we find that the TPA cross-section per gold atom scales linearly with the diameter of the particles, and that the Kerr non-linear response of the Au₂₅(SR)₁₈^– cluster is on the same order as that of bulk gold films.