Wavepacket dynamics and the multi-configurational time-dependent Hartree approach

QuTree: A tree tensor network package

We present QuTree, a C++ library for tree tensor network approaches. QuTree provides class structures for tensors, tensor trees, and related linear algebra functions that facilitate the fast development of tree tensor network approaches such as the multilayer multiconfigurational time-dependent Hartree approach or the density matrix renormalization group approach and its various extensions. We investigate the efficiency of relevant tensor and tensor network operations and show that the overhead for managing the network structure is negligible, even in cases with a million leaves and small tensors. QuTree focuses on providing simple, high-level routines while retaining easy access to the backend to facilitate novel developments. We demonstrate the capabilities of the package by computing the eigenstates of coupled harmonic oscillator Hamiltonians and performing random circuit simulations on a virtual quantum computer.

Surface Hopping with Reliable Wave Function by Introducing Auxiliary Wave Packets to Trajectory Branching

It is well-known that the widely utilized fewest switches surface hopping method suffers from the severe overcoherence problem, and thus adiabatic populations calculated by wave functions are generally inferior to those based on active states. More importantly, to achieve a complete description of nonadiabatic dynamics, the density matrix is essential. In this paper, we present an auxiliary branching corrected surface hopping (A-BCSH) method that introduces auxiliary wave packets (WPs) on the adiabatic potential energy surfaces for trajectory branching. Both rapid and gradual separation of WP components on different surfaces are characterized, and thus the correct decoherence time along each trajectory is captured. As demonstrated in the three standard Tully models, A-BCSH exhibits excellent internal consistency. Namely, close adiabatic populations are obtained based on both wave functions and active states. In particular, A-BCSH successfully obtains a reliable time-dependent spatial distribution of the density matrix, which relies only on electronic wave functions. Due to its high performance, our A-BCSH method provides a new and highly promising perspective on further development of more consistent surface hopping with reliable wave function.

Finite temperature dynamics of the Holstein-Tavis-Cummings model

By employing the numerically accurate multiple Davydov Ansatz (mDA) formalism in combination with the thermo-field dynamics (TFD) representation of quantum mechanics, we systematically explore the influence of three parameters-temperature, photonic-mode detuning, and qubit-phonon coupling-on population dynamics and absorption spectra of the Holstein-Tavis-Cummings (HTC) model. It is found that elevated qubit-phonon couplings and/or temperatures have a similar impact on all dynamic observables: they suppress the amplitudes of Rabi oscillations in photonic populations as well as broaden the peaks and decrease their intensities in the absorption spectra. Our results unequivocally demonstrate that the HTC dynamics is very sensitive to the concerted variation of the three aforementioned parameters, and this finding can be used for fine-tuning polaritonic transport. The developed mDA-TFD methodology can be efficiently applied for modeling, predicting, optimizing, and comprehensively understanding dynamic and spectroscopic responses of actual molecular systems in microcavities.

Eigenstate calculation in the state-averaged (multi-layer) multi-configurational time-dependent Hartree approach

A new approach for the calculation of eigenstates with the state-averaged (multi-layer) multi-configurational time-dependent Hartree (MCTDH) approach is presented. The approach is inspired by the recent work of Larsson [J. Chem. Phys. 151, 204102 (2019)]. It employs local optimization of the basis sets at each node of the multi-layer MCTDH tree and successive downward and upward sweeps to obtain a globally converged result. At the top node, the Hamiltonian represented in the basis of the single-particle functions (SPFs) of the first layer is diagonalized. Here p wavefunctions corresponding to the p lowest eigenvalues are computed by a block Lanczos approach. At all other nodes, a non-linear operator consisting of the respective mean-field Hamiltonian matrix and a projector onto the space spanned by the respective SPFs is considered. Here, the eigenstate corresponding to the lowest eigenvalue is computed using a short iterative Lanczos scheme. Two different examples are studied to illustrate the new approach: the calculation of the vibrational states of methyl and acetonitrile. The calculations for methyl employ the single-layer MCTDH approach, a general potential energy surface, and the correlation discrete variable representation. A five-layer MCTDH representation and a sum of product-type Hamiltonian are used in the acetonitrile calculations. Very fast convergence and order of magnitude reductions in the numerical effort compared to the previously used block relaxation scheme are found. Furthermore, a detailed comparison with the results of Avila and Carrington [J. Chem. Phys. 134, 054126 (2011)] for acetonitrile highlights the potential problems of convergence tests for high-dimensional systems.

Equation-of-Motion Methods for the Calculation of Femtosecond Time-Resolved 4-Wave-Mixing and N-Wave-Mixing Signals

Femtosecond nonlinear spectroscopy is the main tool for the time-resolved detection of photophysical and photochemical processes. Since most systems of chemical interest are rather complex, theoretical support is indispensable for the extraction of the intrinsic system dynamics from the detected spectroscopic responses. There exist two alternative theoretical formalisms for the calculation of spectroscopic signals, the nonlinear response-function (NRF) approach and the spectroscopic equation-of-motion (EOM) approach. In the NRF formalism, the system-field interaction is assumed to be sufficiently weak and is treated in lowest-order perturbation theory for each laser pulse interacting with the sample. The conceptual alternative to the NRF method is the extraction of the spectroscopic signals from the solutions of quantum mechanical, semiclassical, or quasiclassical EOMs which govern the time evolution of the material system interacting with the radiation field of the laser pulses. The NRF formalism and its applications to a broad range of material systems and spectroscopic signals have been comprehensively reviewed in the literature. This article provides a detailed review of the suite of EOM methods, including applications to 4-wave-mixing and N-wave-mixing signals detected with weak or strong fields. Under certain circumstances, the spectroscopic EOM methods may be more efficient than the NRF method for the computation of various nonlinear spectroscopic signals.

Matrix Product State Formulation of the MCTDH Theory in Local Mode Representations for Anharmonic Potentials

The matrix product state formulation of the multiconfiguration time-dependent Hartree theory, MPS-MCTDH, reported previously [Kurashige, J. Chem. Phys. 2018, 19, 194114] is extended to realistic anharmonic potentials with n-mode representations beyond the linear vibronic coupling model. For realistic vibrational potentials, the local mode representation should give a more compact representation of the potentials, i.e., lowering the dimensionality of the entanglements, than the normal coordinates, and the MPS-MCTDH formulation should work more efficiently and maintain the accuracy with a small bond dimension of the MPS ansatz. In fact, it was confirmed that the use of the local coordinates made the interaction matrices diagonal dominant and the number of terms in the n-body expansion of the potentials was significantly reduced. The method was applied to the IR spectrum of the CH₂O molecule, the zero-point energies, and the vibrational energy redistribution dynamics of polyenes C_2nH_2n+2. The results showed that the efficiency of the MPS-MCTDH method is significantly accelerated by the use of local coordinates even if the long-range interactions are included in the potential.

A non-hierarchical correlation discrete variable representation

The correlation discrete variable representation (CDVR) facilitates (multi-layer) multi-configurational time-dependent Hartree (MCTDH) calculations with general potentials. It employs a layered grid representation to efficiently evaluate all potential matrix elements appearing in the MCTDH equations of motion. The original CDVR approach and its multi-layer extension show a hierarchical structure: the size of the grids employed at the different layers increases when moving from an upper layer to a lower one. In this work, a non-hierarchical CDVR approach, which uses identically structured quadratures at all layers of the MCTDH wavefunction representation, is introduced. The non-hierarchical CDVR approach crucially reduces the number of grid points required, compared to the hierarchical CDVR, shows superior scaling properties, and yields identical results for all three representations showing the same topology. Numerical tests studying the photodissociation of NOCl and the vibrational states of CH₃ demonstrate the accuracy of the non-hierarchical CDVR approach.

Variational Quantum Simulation of Chemical Dynamics with Quantum Computers

Classical simulations of real-space quantum dynamics are challenging due to the exponential scaling of computational cost with system dimensions. Quantum computers offer the potential to simulate quantum dynamics with polynomial complexity; however, existing quantum algorithms based on the split-operator techniques require large-scale fault-tolerant quantum computers that remain elusive in the near future. Here, we present variational simulations of real-space quantum dynamics suitable for implementation in noisy intermediate-scale quantum (NISQ) devices. The Hamiltonian is first encoded onto qubits using a discrete variable representation and binary encoding scheme. We show that direct application of a real-time variational quantum algorithm based on the McLachlan's principle is inefficient as the measurement cost grows exponentially with the qubit number for general potential energy, and an extremely small time-step size is required to achieve accurate results. Motivated by the insights that many chemical dynamics occur in the low-energy subspace, we propose a subspace expansion method by projecting the total Hamiltonian, including the time-dependent driving field, onto the system low-energy eigenstate subspace using quantum computers, and the exact quantum dynamics within the subspace can then be solved classically. We show that the measurement cost of the subspace approach grows polynomially with dimensionality for general potential energy. Our numerical examples demonstrate the capability of our approach, even under intense laser fields. Our work opens the possibility of simulating chemical dynamics with NISQ hardware.

Surface Hopping Dynamics on Vibronic Coupling Models

The simulation of photoinduced non-adiabatic dynamics is of great relevance in many scientific disciplines, ranging from physics and materials science to chemistry and biology. Upon light irradiation, different relaxation processes take place in which electronic and nuclear motion are intimately coupled. These are best described by the time-dependent molecular Schrödinger equation, but its solution poses fundamental practical challenges to contemporary theoretical chemistry. Two widely used and complementary approaches to this problem are multiconfigurational time-dependent Hartree (MCTDH) and trajectory surface hopping (SH). MCTDH is an accurate fully quantum-mechanical technique but often is feasible only in reduced dimensionality, in combination with approximate vibronic coupling (VC) Hamiltonians, or both (i.e., reduced-dimensional VC potentials). In contrast, SH is a quantum–classical technique that neglects most nuclear quantum effects but allows nuclear dynamics in full dimensionality by calculating potential energy surfaces on the fly. If nuclear quantum effects do not play a central role and a linear VC (LVC) Hamiltonian is appropriate—e.g., for stiff molecules that generally keep their conformation in the excited state—then it seems advantageous to combine the efficient LVC and SH techniques. In this Account, we describe how surface hopping based on an LVC Hamiltonian (SH/LVC)—as recently implemented in the SHARC surface hopping package—can provide an economical and automated approach to simulate non-adiabatic dynamics. First, we illustrate the potential of SH/LVC in a number of showcases, including intersystem crossing in SO₂, intra-Rydberg dynamics in acetone, and several photophysical studies on large transition-metal complexes, which would be much more demanding or impossible to perform with other methods. While all of the applications provide very useful insights into light-induced phenomena, they also hint at difficulties faced by the SH/LVC methodology that need to be addressed in the future. Second, we contend that the SH/LVC approach can be useful to benchmark SH itself. By the use of the same (LVC) potentials as MCTDH calculations have employed for decades and by relying on the efficiency of SH/LVC, it is possible to directly compare multiple SH test calculations with a MCTDH reference and ponder the accuracy of various correction algorithms behind the SH methodology, such as decoherence corrections or momentum rescaling schemes. Third, we demonstrate how the efficiency of SH/LVC can also be exploited to identify essential nuclear and electronic degrees of freedom to be employed in more accurate MCTDH calculations. Lastly, we show that SH/LVC is able to advance the development of SH protocols that can describe nuclear dynamics including explicit laser fields—a very challenging endeavor for trajectory-based schemes. To end, this Account compiles the typical costs of contemporary SH simulations, evidencing the great advantages of using parametrized potentials. The LVC model is a sleeping beauty that, kissed by SH, is fueling the field of excited-state molecular dynamics. We hope that this Account will stimulate future research in this direction, leveraging the advantages of the SH/VC schemes to larger extents and extending their applicability to uncharted territories.

Machine Learning for Electronically Excited States of Molecules

Electronically excited states of molecules are at the heart of photochemistry, photophysics, as well as photobiology and also play a role in material science. Their theoretical description requires highly accurate quantum chemical calculations, which are computationally expensive. In this review, we focus on not only how machine learning is employed to speed up such excited-state simulations but also how this branch of artificial intelligence can be used to advance this exciting research field in all its aspects. Discussed applications of machine learning for excited states include excited-state dynamics simulations, static calculations of absorption spectra, as well as many others. In order to put these studies into context, we discuss the promises and pitfalls of the involved machine learning techniques. Since the latter are mostly based on quantum chemistry calculations, we also provide a short introduction into excited-state electronic structure methods and approaches for nonadiabatic dynamics simulations and describe tricks and problems when using them in machine learning for excited states of molecules.

Machine Learning for Electronically Excited States of Molecules

Electronically excited states of molecules are at the heart of photochemistry, photophysics, as well as photobiology and also play a role in material science. Their theoretical description requires highly accurate quantum chemical calculations, which are computationally expensive. In this review, we focus on not only how machine learning is employed to speed up such excited-state simulations but also how this branch of artificial intelligence can be used to advance this exciting research field in all its aspects. Discussed applications of machine learning for excited states include excited-state dynamics simulations, static calculations of absorption spectra, as well as many others. In order to put these studies into context, we discuss the promises and pitfalls of the involved machine learning techniques. Since the latter are mostly based on quantum chemistry calculations, we also provide a short introduction into excited-state electronic structure methods and approaches for nonadiabatic dynamics simulations and describe tricks and problems when using them in machine learning for excited states of molecules.

Efficient propagation of the hierarchical equations of motion using the Tucker and hierarchical Tucker tensors

We develop new methods to efficiently propagate the hierarchical equations of motion (HEOM) by using the Tucker and hierarchical Tucker (HT) tensors to represent the reduced density operator and auxiliary density operators. We first show that by employing the split operator method, the specific structure of the HEOM allows a simple propagation scheme using the Tucker tensor. When the number of effective modes in the HEOM increases and the Tucker representation becomes intractable, the split operator method is extended to the binary tree structure of the HT representation. It is found that to update the binary tree nodes related to a specific effective mode, we only need to propagate a short matrix product state constructed from these nodes. Numerical results show that by further employing the mode combination technique commonly used in the multi-configuration time-dependent Hartree approaches, the binary tree representation can be applied to study excitation energy transfer dynamics in a fairly large system including over 10⁴ effective modes. The new methods may thus provide a promising tool in simulating quantum dynamics in condensed phases.

Symmetries in the multi-configurational time-dependent Hartree wavefunction representation and propagation

In multi-configurational time-dependent Hartree (MCTDH) approaches, different multi-layered wavefunction representations can be used to represent the same physical wavefunction. Transformations between different equivalent representations of a physical wavefunction that alter the tree structure used in the multi-layer MCTDH wavefunction representation interchange the role of single-particle functions (SPFs) and single-hole functions (SHFs) in the MCTDH formalism. While the physical wavefunction is invariant under these transformations, this invariance does not hold for the standard multi-layer MCTDH equations of motion. Introducing transformed SPFs, which obey normalization conditions typically associated with SHFs, revised equations of motion are derived. These equations do not show the singularities resulting from the inverse single-particle density matrix and are invariant under tree transformations. Based on the revised equations of motion, a new integration scheme is introduced. The scheme combines the advantages of the constant mean-field approach of Beck and Meyer [Z. Phys. D 42, 113 (1997)] and the singularity-free integrator suggested by Lubich [Appl. Math. Res. Express 2015, 311]. Numerical calculations studying the spin boson model in high dimensionality confirm the favorable properties of the new integration scheme.

Computational approaches to dissociative chemisorption on metals: towards chemical accuracy

We review the state-of-the-art in the theory of dissociative chemisorption (DC) of small gas phase molecules on metal surfaces, which is important to modeling heterogeneous catalysis for practical reasons, and for achieving an understanding of the wealth of experimental information that exists for this topic, for fundamental reasons. We first give a quick overview of the experimental state of the field. Turning to the theory, we address the challenge that barrier heights (E_b, which are not observables) for DC on metals cannot yet be calculated with chemical accuracy, although embedded correlated wave function theory and diffusion Monte-Carlo are moving in this direction. For benchmarking, at present chemically accurate E_b can only be derived from dynamics calculations based on a semi-empirically derived density functional (DF), by computing a sticking curve and demonstrating that it is shifted from the curve measured in a supersonic beam experiment by no more than 1 kcal mol^-1. The approach capable of delivering this accuracy is called the specific reaction parameter (SRP) approach to density functional theory (DFT). SRP-DFT relies on DFT and on dynamics calculations, which are most efficiently performed if a potential energy surface (PES) is available. We therefore present a brief review of the DFs that now exist, also considering their performance on databases for E_b for gas phase reactions and DC on metals, and for adsorption to metals. We also consider expressions for SRP-DFs and briefly discuss other electronic structure methods that have addressed the interaction of molecules with metal surfaces. An overview is presented of dynamical models, which make a distinction as to whether or not, and which dissipative channels are modeled, the dissipative channels being surface phonons and electronically non-adiabatic channels such as electron-hole pair excitation. We also discuss the dynamical methods that have been used, such as the quasi-classical trajectory method and quantum dynamical methods like the time-dependent wave packet method and the reaction path Hamiltonian method. Limits on the accuracy of these methods are discussed for DC of diatomic and polyatomic molecules on metal surfaces, paying particular attention to reduced dimensionality approximations that still have to be invoked in wave packet calculations on polyatomic molecules like CH₄. We also address the accuracy of fitting methods, such as recent machine learning methods (like neural network methods) and the corrugation reducing procedure. In discussing the calculation of observables we emphasize the importance of modeling the properties of the supersonic beams in simulating the sticking probability curves measured in the associated experiments. We show that chemically accurate barrier heights have now been extracted for DC in 11 molecule-metal surface systems, some of which form the most accurate core of the only existing database of E_b for DC reactions on metal surfaces (SBH10). The SRP-DFs (or candidate SRP-DFs) that have been derived show transferability in many cases, i.e., they have been shown also to yield chemically accurate E_b for chemically related systems. This can in principle be exploited in simulating rates of catalyzed reactions on nano-particles containing facets and edges, as SRP-DFs may be transferable among systems in which a molecule dissociates on low index and stepped surfaces of the same metal. In many instances SRP-DFs have allowed important conclusions regarding the mechanisms underlying observed experimental trends. An important recent observation is that SRP-DFT based on semi-local exchange DFs has so far only been successful for systems for which the difference of the metal work function and the molecule's electron affinity exceeds 7 eV. A main challenge to SRP-DFT is to extend its applicability to the other systems, which involve a range of important DC reactions of e.g. O₂, H₂O, NH₃, CO₂, and CH₃OH. Recent calculations employing a PES based on a screened hybrid exchange functional suggest that the road to success may be based on using exchange functionals of this category.

Dynamics of Ultracold Bosons in Artificial Gauge Fields-Angular Momentum, Fragmentation, and the Variance of Entropy

We consider the dynamics of two-dimensional interacting ultracold bosons triggered by suddenly switching on an artificial gauge field. The system is initialized in the ground state of a harmonic trapping potential. As a function of the strength of the applied artificial gauge field, we analyze the emergent dynamics by monitoring the angular momentum, the fragmentation as well as the entropy and variance of the entropy of absorption or single-shot images. We solve the underlying time-dependent many-boson Schrödinger equation using the multiconfigurational time-dependent Hartree method for indistinguishable particles (MCTDH-X). We find that the artificial gauge field implants angular momentum in the system. Fragmentation-multiple macroscopic eigenvalues of the reduced one-body density matrix-emerges in sync with the dynamics of angular momentum: the bosons in the many-body state develop non-trivial correlations. Fragmentation and angular momentum are experimentally difficult to assess; here, we demonstrate that they can be probed by statistically analyzing the variance of the image entropy of single-shot images that are the standard projective measurement of the state of ultracold atomic systems.

Non-adiabatic Excited-State Molecular Dynamics: Theory and Applications for Modeling Photophysics in Extended Molecular Materials

Optically active molecular materials, such as organic conjugated polymers and biological systems, are characterized by strong coupling between electronic and vibrational degrees of freedom. Typically, simulations must go beyond the Born-Oppenheimer approximation to account for non-adiabatic coupling between excited states. Indeed, non-adiabatic dynamics is commonly associated with exciton dynamics and photophysics involving charge and energy transfer, as well as exciton dissociation and charge recombination. Understanding the photoinduced dynamics in such materials is vital to providing an accurate description of exciton formation, evolution, and decay. This interdisciplinary field has matured significantly over the past decades. Formulation of new theoretical frameworks, development of more efficient and accurate computational algorithms, and evolution of high-performance computer hardware has extended these simulations to very large molecular systems with hundreds of atoms, including numerous studies of organic semiconductors and biomolecules. In this Review, we will describe recent theoretical advances including treatment of electronic decoherence in surface-hopping methods, the role of solvent effects, trivial unavoided crossings, analysis of data based on transition densities, and efficient computational implementations of these numerical methods. We also emphasize newly developed semiclassical approaches, based on the Gaussian approximation, which retain phase and width information to account for significant decoherence and interference effects while maintaining the high efficiency of surface-hopping approaches. The above developments have been employed to successfully describe photophysics in a variety of molecular materials.

Predicting Phosphorescence Rates of Light Organic Molecules Using Time-Dependent Density Functional Theory and the Path Integral Approach to Dynamics

In this work, we present a general method for predicting phosphorescence rates and spectra for molecules using time-dependent density functional theory (TD-DFT) and a path integral approach for the dynamics that relies on the harmonic oscillator approximation for the nuclear movement. We first discuss the theory involved in including spin–orbit coupling (SOC) among singlet and triplet excited states and then how to compute the corrected transition dipole moments and phosphorescence rates. We investigate the dependence of these rates on some TD-DFT parameters, such as the nature of the functional, the number of roots, and the Tamm–Dancoff approximation. After that, we evaluate the effect of different SOC integral schemes and show that our best method is applicable to a large number of systems with different excited state characters.

Ultrafast carbon monoxide photolysis and heme spin-crossover in myoglobin via nonadiabatic quantum dynamics

Light absorption of myoglobin triggers diatomic ligand photolysis and a spin crossover transition of iron(II) that initiate protein conformational change. The photolysis and spin crossover reactions happen concurrently on a femtosecond timescale. The microscopic origin of these reactions remains controversial. Here, we apply quantum wavepacket dynamics to elucidate the ultrafast photochemical mechanism for a heme-carbon monoxide (heme-CO) complex. We observe coherent oscillations of the Fe-CO bond distance with a period of 42 fs and an amplitude of ∼1 Å. These nuclear motions induce pronounced geometric reorganization, which makes the CO dissociation irreversible. The reaction is initially dominated by symmetry breaking vibrations inducing an electron transfer from porphyrin to iron. Subsequently, the wavepacket relaxes to the triplet manifold in ∼75 fs and to the quintet manifold in ∼430 fs. Our results highlight the central role of nuclear vibrations at the origin of the ultrafast photodynamics of organometallic complexes.

On the theoretical prediction of fluorescence rates from first principles using the path integral approach

In this work, we present and implement the theory for calculating fluorescence rates and absorption and emission spectra from first principles, using the path integral approach. We discuss some approximations and modifications to the full set of equations that improve speed and numerical stability for the case when a large number of modes are considered. New methods to approximate the excited state potential energy surface are also discussed and it is shown that for most purposes, these can be used instead of a full geometry optimization to obtain the rates mentioned above. A few examples are presented and the overall performance of the method is discussed. It is shown that the rates and spectra computed in this way are well within the acceptable range of errors and can be used in future predictions, particularly for screening purposes, with the only limitation on size being that of the electronic structure calculation itself.