Equilibrium-nonequilibrium ring-polymer molecular dynamics for nonlinear spectroscopy.

Two-dimensional Raman and hybrid terahertz-Raman spectroscopic techniques provide invaluable insight into molecular structures and dynamics of condensed-phase systems. However, corroborating experimental results with theory is difficult due to the high computational cost of incorporating quantum-mechanical effects in the simulations. Here, we present the equilibrium-nonequilibrium ring-polymer molecular dynamics (RPMD), a practical computational method that can account for nuclear quantum effects on the two-time response function of nonlinear optical spectroscopy. Unlike a recently developed approach based on the double Kubo transformed (DKT) correlation function, our method is exact in the classical limit, where it reduces to the established equilibrium-nonequilibrium classical molecular dynamics method. Using benchmark model calculations, we demonstrate the advantages of the equilibrium-nonequilibrium RPMD over classical and DKT-based approaches. Importantly, its derivation, which is based on the nonequilibrium RPMD, obviates the need for identifying an appropriate Kubo transformed correlation function and paves the way for applying real-time path-integral techniques to multidimensional spectroscopy.

Efficient Semiclassical Dynamics for Vibronic Spectroscopy beyond Harmonic, Condon, and Zero-Temperature Approximations.

Understanding light-induced processes in biological and human-made molecular systems is one of the main goals of physical chemistry. It has been known for years that the photoinduced dynamics of atomic nuclei can be studied by looking at the vibrational substructure of electronic absorption and emission spectra. However, theoretical simulation is needed to understand how dynamics translates into the spectral features. Here, we review several recent developments in the computation of vibrationally resolved electronic spectra (sometimes simply called 'vibronic' spectra). We present a theoretical approach for computing such spectra beyond the commonly used zero-temperature, Condon, and harmonic approximations. More specifically, we show how the on-the-fly ab initio thawed Gaussian approximation, which partially includes anharmonicity effects, can be combined with the thermo-field dynamics to treat non-zero temperature and with the Herzberg-Teller correction to include non-Condon effects. The combined method, which can treat all three effects, is applied to compute the S1 ← S0 and S2 ← S0 absorption spectra of azulene.

On-the-Fly Ab Initio Semiclassical Evaluation of Electronic Coherences in Polyatomic Molecules Reveals a Simple Mechanism of Decoherence.

Irradiation of a molecular system by an intense laser field can trigger dynamics of both electronic and nuclear subsystems. The lighter electrons usually move on much faster, attosecond timescale but the slow nuclear rearrangement damps ultrafast electronic oscillations, leading to the decoherence of the electronic dynamics within a few femtoseconds. We show that a simple, single-trajectory semiclassical scheme can evaluate the electronic coherence time in polyatomic molecules accurately by demonstrating an excellent agreement with full-dimensional quantum calculations. In contrast to numerical quantum methods, the semiclassical one reveals the physical mechanism of decoherence beyond the general blame on nuclear motion. In the propiolic acid, the rate of decoherence and the large deviation from the static frequency of electronic oscillations are quantitatively described with just two semiclassical parameters-the phase space distance and signed area between the trajectories moving on two electronic surfaces. Because it evaluates the electronic structure on the fly, the semiclassical technique avoids the "curse of dimensionality" and should be useful for preselecting molecules for experimental studies.

On-the-fly ab initio semiclassical evaluation of vibronic spectra at finite temperature.

To compute and analyze vibrationally resolved electronic spectra at zero temperature, we have recently implemented the on-the-fly ab initio extended thawed Gaussian approximation [A. Patoz et al., J. Phys. Chem. Lett. 9, 2367 (2018)], which accounts for anharmonicity, mode-mode coupling, and Herzberg-Teller effects. Here, we generalize this method in order to evaluate spectra at non-zero temperature. In line with thermo-field dynamics, we transform the von Neumann evolution of the coherence component of the density matrix to the Schrödinger evolution of a wavefunction in an augmented space with twice as many degrees of freedom. Due to the efficiency of the extended thawed Gaussian approximation, this increase in the number of coordinates results in nearly no additional computational cost. More specifically, compared to the original, zero-temperature approach, the finite-temperature method requires no additional ab initio electronic structure calculations. At the same time, the new approach allows for a clear distinction among finite-temperature, anharmonicity, and Herzberg-Teller effects on spectra. We show, on a model Morse system, the advantages of the finite-temperature thawed Gaussian approximation over the commonly used global harmonic methods and apply it to evaluate the symmetry-forbidden absorption spectrum of benzene, where all of the aforementioned effects contribute.

On-the-fly ab initio semiclassical evaluation of third-order response functions for two-dimensional electronic spectroscopy.

Ab initio computation of two-dimensional electronic spectra is an expanding field, whose goal is improving upon simple, few-dimensional models often employed to explain experiments. Here, we propose an accurate and computationally affordable approach, based on the single-trajectory semiclassical thawed Gaussian approximation, to evaluate two-dimensional electronic spectra. Importantly, the method is exact for arbitrary harmonic potentials with mode displacement, changes in the mode frequencies, and inter-mode coupling (Duschinsky effect), but can also account partially for the anharmonicity of the involved potential energy surfaces. We test its accuracy on a set of model Morse potentials and use it to study anharmonicity and Duschinsky effects on the linear and two-dimensional electronic spectra of phenol. We find that in this molecule, the anharmonicity effects are weak, whereas the Duschinsky rotation and the changes in the mode frequencies must be included in accurate simulations. In contrast, the widely used displaced harmonic oscillator model captures only the basic physics of the problem but fails to reproduce the correct vibronic lineshape.

Single-Hessian thawed Gaussian approximation.

To alleviate the computational cost associated with on-the-fly ab initio semiclassical calculations of molecular spectra, we propose the single-Hessian thawed Gaussian approximation in which the Hessian of the potential energy at all points along an anharmonic classical trajectory is approximated by a constant matrix. The spectra obtained with this approximation are compared with the exact quantum spectra of a one-dimensional Morse potential and with the experimental spectra of ammonia and quinquethiophene. In all cases, the single-Hessian version performs almost as well as the much more expensive on-the-fly ab initio thawed Gaussian approximation and significantly better than the global harmonic schemes. Remarkably, unlike the thawed Gaussian approximation, the proposed method conserves energy exactly, despite the time dependence of the corresponding effective Hamiltonian, and, in addition, can be mapped to a higher-dimensional time-independent classical Hamiltonian system. We also provide a detailed comparison with several related approximations used for accelerating prefactor calculations in semiclassical simulations.

On-the-fly ab initio semiclassical evaluation of time-resolved electronic spectra.

We present a methodology for computing vibrationally and time-resolved pump-probe spectra, which takes into account all vibrational degrees of freedom and is based on the combination of the thawed Gaussian approximation with on-the-fly ab initio evaluation of the electronic structure. The method is applied to the phenyl radical and compared with two more approximate approaches based on the global harmonic approximation-the global harmonic method expands both the ground- and excited-state potential energy surfaces to the second order about the corresponding minima, while the combined global harmonic/on-the-fly method retains the on-the-fly scheme for the excited-state wavepacket propagation. We also compare the spectra by considering their means and widths, and show analytically how these measures are related to the properties of the semiclassical wavepacket. We find that the combined approach is better than the global harmonic one in describing the vibrational structure, while the global harmonic approximation estimates better the overall means and widths of the spectra due to a partial cancellation of errors. Although the full-dimensional on-the-fly ab initio result seems to reflect the dynamics of only one mode, we show, by performing exact quantum calculations, that this simple structure cannot be recovered using a one-dimensional model. Yet, the agreement between the quantum and semiclassical spectra in this simple, but anharmonic model lends additional support for the full-dimensional ab initio thawed Gaussian calculation of the phenyl radical spectra. We conclude that the thawed Gaussian approximation provides a viable alternative to the expensive or unfeasible exact quantum calculations in cases, where low-dimensional models are not sufficiently accurate to represent the full system.

Fast and converged classical simulations of evidence for the utility of quantum computing before fault tolerance.

A recent quantum simulation of observables of the kicked Ising model on 127 qubits implemented circuits that exceed the capabilities of exact classical simulation. We show that several approximate classical methods, based on sparse Pauli dynamics and tensor network algorithms, can simulate these observables orders of magnitude faster than the quantum experiment and can also be systematically converged beyond the experimental accuracy. Our most accurate technique combines a mixed Schrödinger and Heisenberg tensor network representation with the Bethe free entropy relation of belief propagation to compute expectation values with an effective wave function-operator sandwich bond dimension >16,000,000, achieving an absolute accuracy, without extrapolation, in the observables of <0.01, which is converged for many practical purposes. We thereby identify inaccuracies in the experimental extrapolations and suggest how future experiments can be implemented to increase the classical hardness.

Two-dimensional infrared-Raman spectroscopy as a probe of water's tetrahedrality.

Two-dimensional spectroscopic techniques combining terahertz (THz), infrared (IR), and visible pulses offer a wealth of information about coupling among vibrational modes in molecular liquids, thus providing a promising probe of their local structure. However, the capabilities of these spectroscopies are still largely unexplored due to experimental limitations and inherently weak nonlinear signals. Here, through a combination of equilibrium-nonequilibrium molecular dynamics (MD) and a tailored spectrum decomposition scheme, we identify a relationship between the tetrahedral order of liquid water and its two-dimensional IR-IR-Raman (IIR) spectrum. The structure-spectrum relationship can explain the temperature dependence of the spectral features corresponding to the anharmonic coupling between low-frequency intermolecular and high-frequency intramolecular vibrational modes of water. In light of these results, we propose new experiments and discuss the implications for the study of tetrahedrality of liquid water.

Computational Design of Molecular Probes for Electronic Preresonance Raman Scattering Microscopy.

Recently developed electronic preresonance stimulated Raman scattering (epr-SRS) microscopy, in which the Raman signal of a dye is significantly boosted by setting the incident laser frequency near the electronic excitation energy, has pushed the sensitivity of SRS microscopy close to that offered by confocal fluorescence microscopy. Prominently, the maintained narrow line-width of epr-SRS also offers high multiplexity that breaks the "color barrier" in optical microscopy. However, detailed understanding of the fundamental mechanism in these epr-SRS dyes still remains elusive. Here, we combine experiments with theoretical modeling to investigate the structure-function relationship, aiming to facilitate the design of new probes and expanding epr-SRS palettes. Our ab initio approach employing the displaced harmonic oscillator (DHO) model provides a consistent agreement between simulated and experimental SRS intensities of various triple-bond bearing epr-SRS probes with distinct scaffolds. We further review two popular approximate expressions for epr-SRS, namely the short-time and Albrecht A-term equations, and compare them to the DHO model. Overall, the theory allows us to illustrate how the observed intensity differences between molecular scaffolds stem from the coupling strength between the electronic excitation and the targeted vibrational mode, leading to a general design strategy for highly sensitive next-generation vibrational imaging probes.

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.

Applicability of the Thawed Gaussian Wavepacket Dynamics to the Calculation of Vibronic Spectra of Molecules with Double-Well Potential Energy Surfaces.

Simulating vibrationally resolved electronic spectra of anharmonic systems, especially those involving double-well potential energy surfaces, often requires expensive quantum dynamics methods. Here, we explore the applicability and limitations of the recently proposed single-Hessian thawed Gaussian approximation for the simulation of spectra of systems with double-well potentials, including 1,2,4,5-tetrafluorobenzene, ammonia, phosphine, and arsine. This semiclassical wavepacket approach is shown to be more robust and to provide more accurate spectra than the conventional harmonic approximation. Specifically, we identify two cases in which the Gaussian wavepacket method is especially useful due to the breakdown of the harmonic approximation: (i) when the nuclear wavepacket is initially at the top of the potential barrier but delocalized over both wells, e.g., along a low-frequency mode, and (ii) when the wavepacket has enough energy to classically go over the low potential energy barrier connecting the two wells. The method is efficient and requires only a single classical ab initio molecular dynamics trajectory, in addition to the data required to compute the harmonic spectra. We also present an improved algorithm for computing the wavepacket autocorrelation function, which guarantees that the evaluated correlation function is continuous for arbitrary size of the time step.

Finite-Temperature, Anharmonicity, and Duschinsky Effects on the Two-Dimensional Electronic Spectra from Ab Initio Thermo-Field Gaussian Wavepacket Dynamics.

Accurate description of finite-temperature vibrational dynamics is indispensable in the computation of two-dimensional electronic spectra. Such simulations are often based on the density matrix evolution, statistical averaging of initial vibrational states, or approximate classical or semiclassical limits. While many practical approaches exist, they are often of limited accuracy and difficult to interpret. Here, we use the concept of thermo-field dynamics to derive an exact finite-temperature expression that lends itself to an intuitive wavepacket-based interpretation. Furthermore, an efficient method for computing finite-temperature two-dimensional spectra is obtained by combining the exact thermo-field dynamics approach with the thawed Gaussian approximation for the wavepacket dynamics, which is exact for any displaced, distorted, and Duschinsky-rotated harmonic potential but also accounts partially for anharmonicity effects in general potentials. Using this new method, we directly relate a symmetry breaking of the two-dimensional signal to the deviation from the conventional Brownian oscillator picture.

Semiclassical Approach to Photophysics Beyond Kasha's Rule and Vibronic Spectroscopy Beyond the Condon Approximation. The Case of Azulene.

Azulene is a prototypical molecule with an anomalous fluorescence from the second excited electronic state, thus violating Kasha's rule, and with an emission spectrum that cannot be understood within the Condon approximation. To better understand the photophysics and spectroscopy of azulene and other nonconventional molecules, we developed a systematic, general, and efficient computational approach combining the semiclassical dynamics of nuclei with ab initio electronic structure. First, to analyze the nonadiabatic effects, we complement the standard population dynamics by a rigorous measure of adiabaticity, estimated with the multiple-surface dephasing representation. Second, we propose a new semiclassical method for simulating non-Condon spectra, which combines the extended thawed Gaussian approximation with the efficient single-Hessian approach. S1 ← S0 and S2 ← S0 absorption and S2 → S0 emission spectra of azulene, recorded in a new set of experiments, agree very well with our calculations. We find that accuracy of the evaluated spectra requires the treatment of anharmonicity, Herzberg-Teller, and mode-mixing effects.

On-the-Fly Ab Initio Semiclassical Evaluation of Absorption Spectra of Polyatomic Molecules beyond the Condon Approximation.

To evaluate vibronic spectra beyond the Condon approximation, we extend the on-the-fly ab initio thawed Gaussian approximation by considering the Herzberg-Teller contribution due to the dependence of the electronic transition dipole moment on nuclear coordinates. The extended thawed Gaussian approximation is tested on electronic absorption spectra of the phenyl radical and benzene; calculated spectra reproduce experimental data and are much more accurate than standard global harmonic approaches, confirming the significance of anharmonicity. Moreover, the extended method provides a tool to quantify the Herzberg-Teller contribution; we show that in the phenyl radical, anharmonicity outweighs the Herzberg-Teller contribution, whereas in benzene, the Herzberg-Teller contribution is essential because the transition is electronically forbidden and the Condon approximation yields a zero spectrum. Surprisingly, both adiabatic harmonic spectra outperform those of the vertical harmonic model, which describes the Franck-Condon region better. Finally, we provide a simple recipe for orientationally averaging spectra, valid beyond the Condon approximation, and a relation among the transition dipole, its gradient, and nonadiabatic coupling vectors.