On the Topological Phase around Conical Intersections with Tamm-Dancoff Linear-Response Time-Dependent Density Functional Theory

Regions of nuclear-configuration space away from the Franck-Condon geometry can prove problematic for some electronic structure methods, given the propensity of such regions to possess conical intersections, i.e., (highly connected) points of degeneracy between potential energy surfaces. With the likelihood (perhaps even inevitability) for nonadiabatic dynamics simulations to explore molecular geometries in close proximity to conical intersections, it is vital that the performance of electronic structure methods is routinely examined in this context. In a recent paper [Taylor, J. T. J. Chem. Phys. 2023, 159, 214115.], the ability of linear-response time-dependent density functional theory within the adiabatic approximation (AA LR-TDDFT) to provide a proper description of conical intersections, in terms of their topology and topography, was investigated, with particular attention paid to conical intersections between two excited electronic states. For the same prototypical molecules, protonated formaldimine and pyrazine, we herein consider whether AA LR-TDDFT can correctly reproduce the topological phase accumulated by the adiabatic electronic wave function upon traversing a closed path around an excited-to-excited state conical intersection despite not using the appropriate quadratic-response nonadiabatic coupling vectors. Equally, we probe the ability of the ground-to-excited state intersection ring exhibited by AA LR-TDDFT in protonated formaldimine to give rise to a similar topological phase in spite of its incorrect dimensionality.

Highly efficient implementation of analytic nonadiabatic derivative couplings within the pseudospectral method

A pseudospectral implementation of nonadiabatic derivative couplings in the Tamm-Dancoff approximation is reported, and the accuracy and efficiency of the pseudospectral nonadiabatic derivative couplings are studied. Our results demonstrate that the pseudospectral method provides mean absolute errors of 0.2%-1.9%, while providing a significant speedup. Benchmark calculations on fullerenes (Cn, n up to 100) using B3LYP achieved 10- to 15-fold, 8- to 17-fold, and 43- to 75-fold speedups for 6-31G**, 6-31++G**, and cc-pVTZ basis sets, respectively, when compared to the conventional spectral method.

On the description of conical intersections between excited electronic states with LR-TDDFT and ADC(2)

Conical intersections constitute the conceptual bedrock of our working understanding of ultrafast, nonadiabatic processes within photochemistry (and photophysics). Accurate calculation of potential energy surfaces within the vicinity of conical intersections, however, still poses a serious challenge to many popular electronic structure methods. Multiple works have reported on the deficiency of methods like linear-response time-dependent density functional theory within the adiabatic approximation (AA LR-TDDFT) or algebraic diagrammatic construction to second-order [ADC(2)]-approaches often used in excited-state molecular dynamics simulations-to describe conical intersections between the ground and excited electronic states. In the present study, we focus our attention on conical intersections between excited electronic states and probe the ability of AA LR-TDDFT and ADC(2) to describe their topology and topography, using protonated formaldimine and pyrazine as two exemplar molecules. We also take the opportunity to revisit the performance of these methods in describing conical intersections involving the ground electronic state in protonated formaldimine-highlighting in particular how the intersection ring exhibited by AA LR-TDDFT can be perceived either as a (near-to-linear) seam of intersection or two interpenetrating cones, depending on the magnitude of molecular distortions within the branching space.

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.

Ultrafast Electronic Relaxation in 6-Methyluracil and 5-Fluorouracil in Isolated and Aqueous Conditions: Substituent and Solvent Effects

We report ultrafast extreme ultraviolet photoelectron spectroscopy of 6-methyluracil (6mUra) and 5-fluorouracil (5FUra) in the gas phase and 6mUra and 5-fluorouridine in an aqueous environment. In the gas phase, internal conversion (IC) occurs from ¹ππ* to ¹nπ* states in tens of femtoseconds, followed by intersystem crossing to the ³ππ* state in several picoseconds. In an aqueous solution, 6mUra undergoes IC almost exclusively to the ground state (S₀) in about 100 fs, which is essentially the same process as that for unsubstituted uracil, but much faster than that for thymine (5-methyluracil). The different dynamics for C5 and C6 methylation suggest that IC from ¹ππ* to S₀ is facilitated by out-of-plane (OOP) motion of the C5 substituent. The slow IC for C5-substituted molecules in an aqueous environment is ascribed to the solvent reorganization that is required for this OOP motion to occur. The slow rate for 5FUrd may arise in part from an increased barrier height due to C5 fluorination.

Monitoring vibronic coherences and molecular aromaticity in photoexcited cyclooctatetraene with an X-ray probe: a simulation study

Understanding conical intersection (CI) dynamics and subsequent conformational changes is key for exploring and controlling photo-reactions in aromatic molecules. Monitoring of their time-resolved dynamics remains a formidable experimental challenge. In this study, we simulate the photoinduced S₃ to S₁ non-adiabatic dynamics of cyclooctatetraene (COT), involving multiple CIs with relaxation times in good agreement with experiment. We further investigate the possibility to directly probe the CI passages in COT by off-resonant X-ray Raman spectroscopy (TRUECARS) and time-resolved X-ray diffraction (TRXD). We find that these signals sensitively monitor key chemical features during the ultrafast dynamics. First, we distinguish two CIs by using TRUECARS signals with their appearances at different Raman shifts. Second, we demonstrate that TRXD, where X-ray photons scatter off electron densities, can resolve ultrafast changes in the aromaticity of COT. It can further distinguish between planar and non-planar geometries explored during the dynamics, as e.g. two different tetraradical-type CIs. The knowledge gained from these measurements can give unique insight into fundamental chemical properties that dynamically change during non-adiabatic passages.

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.

Nonadiabatic Excited State Dynamics of Organic Chromophores: Take-Home Messages

Nonadiabatic excited state dynamics are important in a variety of processes. Theoretical and experimental developments have allowed for a great progress in this area, while combining the two is often necessary and the best approach to obtain insight into the photophysical behavior of molecules. In this Feature Article we use examples of our recent work combining time-resolved photoelectron spectroscopy with theoretical nonadiabatic dynamics to highlight important lessons we learned. We compare the nonadiabatic excited state dynamics of three different organic molecules with the aim of elucidating connections between structure and dynamics. Calculations and measurements are compared for uracil, 1,3-cyclooctadiene, and 1,3-cyclohexadiene. The comparison highlights the role of rigidity in influencing the dynamics and the difficulty of capturing the dynamics accurately with calculations.

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.

Nonadiabatic Molecular Dynamics by Multiconfiguration Pair-Density Functional Theory

We present the first implementation of multiconfiguration pair-density functional theory (MC-PDFT) ab initio molecular dynamics. MC-PDFT is a multireference electronic structure method that in many cases has a similar accuracy (or even better accuracy) the complete active space second-order perturbation theory (CASPT2) at a significantly lower computational cost. In this study, we introduced MC-PDFT analytical gradients into the SHARC molecular dynamics program for ab initio, nonadiabatic molecular dynamics simulations. We verify our implementation by examining the intersystem crossing dynamics of thioformaldehyde, and we observe excellent agreement with recent CASPT2 and experimental findings. Moreover, with MC-PDFT, we could perform dynamics simulations with the 12 electron in 10 orbitals active space that was computationally too expensive for direct dynamics with CASPT2.

Scalable Ehrenfest Molecular Dynamics Exploiting the Locality of Density-Functional Tight-Binding Hamiltonian

To explore the science behind excited-state dynamics in high-complexity chemical systems, a scalable nonadiabatic molecular dynamics (MD) technique is indispensable. In this study, by treating the electronic degrees of freedom at the density-functional tight-binding level, we developed and implemented a reduced scaling and multinode-parallelizable Ehrenfest MD method. To achieve this goal, we introduced a concept called patchwork approximation (PA), where the effective Hamiltonian for real-time propagation of the electronic density matrix is partitioned into a set of local parts. Numerical results for giant icosahedral fullerenes, which comprise up to 6000 atoms, suggest that the scaling of the present PA-based method is less than quadratic, which yields a significant advantage over the conventional cubic scaling method in terms of computational time. The acceleration by the parallelization on multiple nodes was also assessed. Furthermore, the electronic and structural dynamics resulting from the perturbation by the external electric field were accurately reproduced with the PA, even when the electronic excitation was spatially delocalized.

Internal Conversion between Bright (11Bu+) and Dark (21Ag-) States in s-trans-Butadiene and s-trans-Hexatriene

Internal conversion (IC) between the two lowest singlet excited states, 1¹B_u⁺ and 2¹A_g^-, of s-trans-butadiene and s-trans-hexatriene is investigated using a series of single- and multi- reference wave function and density functional theory (DFT) methodologies. Three independent types of the equation-of-motion coupled-cluster (EOMCC) theory capable of providing an accurate and balanced description of one- as well as two-electron transitions, abbreviated as δ-CR-EOMCC(2,3), DIP-EOMCC(4h2p){N_o}, and DEA-EOMCC(4p2h){N_u} or DEA-EOMCC(3p1h,4p2h){N_u}, consistently predict that the 1¹B_u⁺/2¹A_g^- crossing in both molecules occurs along the bond length alternation coordinate. However, the analogous 1¹B_u⁺ and 2¹A_g^- potentials obtained with some multireference approaches, such as CASSCF and MRCIS(D), as well as with the linear-response formulation of time-dependent DFT (TDDFT), do not cross. Hence, caution needs to be exercised when studying the low-lying singlet excited states of polyenes with conventional multiconfigurational methods and TDDFT. The multistate many-body perturbation theory methods, such as XMCQDPT2, do correctly reproduce the curve crossing. Among the simplest and least expensive computational methodologies, the DFT approaches that incorporate the contributions of doubly excited configurations, abbreviated as MRSF (mixed reference spin-flip) TDDFT and SSR(4,4), accurately reproduce our best EOMCC results. This is highly promising for nonadiabatic molecular dynamics simulations in larger systems.

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.

Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package

This article summarizes technical advances contained in the fifth major release of the Q-Chem quantum chemistry program package, covering developments since 2015. A comprehensive library of exchange-correlation functionals, along with a suite of correlated many-body methods, continues to be a hallmark of the Q-Chem software. The many-body methods include novel variants of both coupled-cluster and configuration-interaction approaches along with methods based on the algebraic diagrammatic construction and variational reduced density-matrix methods. Methods highlighted in Q-Chem 5 include a suite of tools for modeling core-level spectroscopy, methods for describing metastable resonances, methods for computing vibronic spectra, the nuclear-electronic orbital method, and several different energy decomposition analysis techniques. High-performance capabilities including multithreaded parallelism and support for calculations on graphics processing units are described. Q-Chem boasts a community of well over 100 active academic developers, and the continuing evolution of the software is supported by an "open teamware" model and an increasingly modular design.

Electronic Structure Methods for the Description of Nonadiabatic Effects and Conical Intersections

Nonadiabatic effects are ubiquitous in photophysics and photochemistry, and therefore, many theoretical developments have been made to properly describe them. Conical intersections are central in nonadiabatic processes, as they promote efficient and ultrafast nonadiabatic transitions between electronic states. A proper theoretical description requires developments in electronic structure and specifically in methods that describe conical intersections between states and nonadiabatic coupling terms. This review focuses on the electronic structure aspects of nonadiabatic processes. We discuss the requirements of electronic structure methods to describe conical intersections and nonadiabatic couplings, how the most common excited state methods perform in describing these effects, and what the recent developments are in expanding the methodology and implementing nonadiabatic couplings.

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.

Impact of the Dynamic Electron Correlation on the Unusually Long Excited-State Lifetime of Thymine

Non-radiative relaxation of the photoexcited thymine in the gas phase shows an unusually long excited-state lifetime, and, over the years, a number of models, i.e., S₁-trapping, S₂-trapping, and S₁&S₂-trapping, have been put forward to explain its mechanism. Here, we investigate this mechanism using non-adiabatic molecular dynamics (NAMD) simulations in connection with the recently developed mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT) method. We show that the previously predicted S₂-trapping model was due to an artifact caused by an insufficient account of the dynamic electron correlation. The current work supports the S₁-trapping mechanism with two lifetimes, τ₁ = 30 ± 1 fs and τ₂ = 6.1 ± 0.035 ps, quantitatively consistent with the recent time-resolved experiments. Upon excitation to the S₂ (ππ*) state, thymine undergoes an ultrafast (ca. 30 fs) S₂→S₁ internal conversion and resides around the minimum on the S₁ (n_Oπ*) surface, slowly decaying to the ground state (ca. 6.1 ps). While the S₂→S₁ internal conversion is mediated by fast bond length alternation distortion, the subsequent S₁→S₀ occurs through several conical intersections, involving a slow puckering motion.

Experimental and theoretical study of energy transfer in a chromophore triad: What makes modeling dynamics successful?

Simulation of electronic dynamics in realistically large molecular systems is a demanding task that has not yet achieved the same level of quantitative prediction already realized for its static counterpart. This is particularly true for processes occurring beyond the Born-Oppenheimer regime. Non-adiabatic molecular dynamics (NAMD) simulations suffer from two convoluted sources of error: numerical algorithms for dynamics and electronic structure calculations. While the former has gained increasing attention, particularly addressing the validity of ad hoc methodologies, the effect of the latter remains relatively unexplored. Indeed, the required accuracy for electronic structure calculations to reach quantitative agreement with experiment in dynamics may be even more strict than that required for static simulations. Here, we address this issue by modeling the electronic energy transfer in a donor-acceptor-donor (D-A-D) molecular light harvesting system using fewest switches surface hopping NAMD simulations. In the studied system, time-resolved experimental measurements deliver complete information on spectra and energy transfer rates. Subsequent modeling shows that the calculated electronic transition energies are "sufficiently good" to reproduce experimental spectra but produce over an order of magnitude error in simulated dynamical rates. We further perform simulations using artificially shifted energy gaps to investigate the complex relationship between transition energies and modeled dynamics to understand factors affecting non-radiative relaxation and energy transfer rates.

Fast Nonadiabatic Molecular Dynamics via Spin-Flip Time-Dependent Density-Functional Tight-Binding Approach: Application to Nonradiative Relaxation of Tetraphenylethylene with Locked Aromatic Rings

Nonadiabatic dynamics around conical intersections between ground and excited states are crucial to understand excited-state phenomena in complex chemical systems. With this background in mind, we present an approach combining fewest-switches trajectory surface hopping and spin-flip (SF) time-dependent (TD) density-functional tight binding (DFTB), which is a simplified version of SF-TD density functional theory (DFT) with semiempirical parametrizations, for computationally efficient nonadiabatic molecular dynamics simulations. The estimated computational time of the SF-TD-DFTB approach is several orders of magnitude lower than that of SF-TD-DFT. In addition, the proposed method reproduces the time scales and quantum yields in photoisomerization reactions of azobenzene at a level comparable with conventional ab initio approaches, demonstrating reasonable accuracy. Finally, we report a practical application of the developed technique to explore the nonradiative relaxation processes of tetraphenylethylene and its derivative with torsionally locked aromatic rings and discuss the effect of locking the rings on the excited-state lifetime.

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.

Excitation energy transport in DNA modelled by multi-chromophoric field-induced surface hopping

Revealing the extended excited state lifetime due to excitation energy transport in DNA by multi-chromophoric field-induced surface-hopping (McFISH).