A global spacetime optimization approach to the real-space time-dependent Schrödinger equation

ConspectusSimulating molecular dynamics (MD) within a comprehensive quantum framework has been a long-standing challenge in computational chemistry. An exponential scaling of computational cost renders solving the time dependent Schrödinger equation (TDSE) of a molecular Hamiltonian, including both electronic and nuclear degrees of freedom (DOFs), as well as their couplings, infeasible for more than a few DOFs. In the Born-Oppenheimer (BO), or adiabatic, picture, electronic and nuclear parts of the wave function are decoupled and treated separately. Within this framework, the nuclear wave function evolves along potential energy surfaces (PESs) computed as solutions to the electronic Schrödinger equation parametrized in the nuclear DOFs. This approximation, together with increasingly elaborate numerical approaches to solve the nuclear time dependent Schrödinger equation (TDSE), enabled the treatment of up to a few dozens of degrees of freedom (DOFs). However, for particular applications, such as photochemistry, the BO approximation breaks down. In this regime of non-adiabatic dynamics, solving the full molecular problem including electron-nuclear couplings becomes essential, further increasing the complexity of the numerical solution. Although valuable methods such as multiconfigurational time-dependent Hartree (MCTDH) have been proposed for the solution of the coupled electron-nuclear dynamics, they remain hampered by an exponential scaling in the number of nuclear DOFs and by the difficulty of finding universal variational forms.In this Account, we present a perspective on novel quantum computational algorithms, aiming to alleviate the exponential scaling inherent to the simulation of many-body quantum dynamics. In particular, we focus on the derivation and application of quantum algorithms for adiabatic and non-adiabatic quantum dynamics, which include efficient approaches for the calculation of the BO potential energy surfaces (PESs). Thereafter, we study the time-evolution of a model system consisting of two coupled PESs in first and second quantization. In a first application, we discuss a recently introduced quantum algorithm for the evolution of a wavepacket in first quantization and exploit the potential quantum advantage of mapping its spatial grid representation to logarithmically many qubits. For the second demonstration, we move to the second quantization framework and review the scaling properties of two alternative time-evolution algorithms, namely, a variational quantum algorithm (VQA) (based on the McLachlan variational principle) and conventional Trotter-type evolution (based on a Lie-Trotter-Suzuki formula). Both methods clearly demonstrate the potential of quantum algorithms and their favorable scaling compared to the available classical approaches. However, a clear demonstration of quantum advantage in the context of molecular quantum dynamics may require the implementation of these algorithms in fault-tolerant quantum computers, while their application in near-term, noisy quantum devices is still unclear and deserves further investigation.

We study numerical methods for time-dependent partial differential equations describing wave propagation, primarily applied to problems in quantum dynamics governed by the time-dependent Schrodinger equation (TDSE). We consider both methods for spatial approximation and for time stepping. In most settings, numerical solution of the TDSE is more challenging than solving a hyperbolic wave equation. This is mainly because the dispersion relation of the TDSE makes it very sensitive to dispersion error, and infers a stringent time step restriction for standard explicit time stepping schemes. The TDSE is also often posed in high dimensions, where standard methods are intractable.The sensitivity to dispersion error makes spectral methods advantageous for the TDSE. We use spectral or pseudospectral methods in all except one of the included papers. In Paper III we improve and analyse the accuracy of the Fourier pseudospectral method applied to a problem with limited regularity, and in Paper V we construct a matrix-free spectral method for problems with non-trivial boundary conditions. Due to its stiffness, the TDSE is most often solved using exponential time integration. In this thesis we use exponential operator splitting and Krylov subspace methods. We rigorously prove convergence for force-gradient operator splitting methods in Paper IV. One way of making high-dimensional problems computationally tractable is low-rank approximation. In Paper VI we prove that a splitting method for dynamical low-rank approximation is robust to singular values in the approximation approaching zero, a situation which is difficult to handle since it implies strong curvature of the approximation space.

The problem of integrating the time-dependent Schr\odinger equation (TDSE) describing the interaction of a polyelectronic atom with a laser pulse is treated by expanding the time-dependent wave function \ensuremath{\Psi}(r,t) in terms of wave functions ${\mathrm{\ensuremath{\Phi}}}_{\mathit{n},}$E computed for discrete, autoionizing, and scattering states separately. The TDSE is transformed into a system of coupled first-order differential equations with time-dependent coefficients, whose number (in the thousands), necessary for convergence to be reliable, depends mainly on the degree of the contribution of the continuous spectrum, as a function of the frequency and strength of the field. This approach allows the systematic incorporation of the significant electronic structure, electron correlation, and spectral characteristics of each N-electron system under investigation. Furthermore, since the free-electron function is computed numerically in the polarized core potential of the remaining (N-1)-electron atom, properties such as the angular distribution and partial above-threshold ionization (ATI) of the photoelectron are directly computable. We present results from the application of our methods to H and ${\mathrm{Li}}^{\mathrm{\ensuremath{-}}}$. For the applications to H, which served as testing grounds for the method, the state-specific wave functions for discrete and continuum states were obtained numerically, for n and l up to 12 and 5, respectively, and for positive energies up to \ensuremath{\varepsilon}=34 eV with l up to 6. When comparisons with other time-independent and time-dependent results are possible, very good agreement is observed. On the other hand, our calculations do not confirm recent experimental results on absolute ionization rates for laser pulses of 248 nm. For ${\mathrm{Li}}^{\mathrm{\ensuremath{-}}}$, our results on ATI for photon energy \ensuremath{\Elzxh}\ensuremath{\omega}=1.36 eV demonstrate the effects of initial-state electron correlation and of final-state field-induced coupling of open channels (the Li 1${\mathit{s}}^{2}$2s $^{2}$S and 1${\mathit{s}}^{2}$2p $^{2}$${\mathit{P}}^{\mathit{o}}$), as a function of field intensity.

We present an efficient and accurate grid method to study the strong field dynamics of planar H(2)(+) under Born-Oppenheimer approximation. After introducing the elliptical coordinates to the planar H(2)(+), we show that the Coulomb singularities at the nuclei can be successfully overcome so that both bound and continuum states can be accurately calculated by the method of separation of variables. The time-dependent Schrödinger equation (TDSE) can be accurately solved by a two-dimensional discrete variable representation (DVR) method, where the radial coordinate is discretized with the finite-element discrete variable representation for easy parallel computation and the angular coordinate with the trigonometric DVR which can describe the periodicity in this direction. The bound states energies can be accurately calculated by the imaginary time propagation of TDSE, which agree very well with those computed by the separation of variables. We apply the TDSE to study the ionization dynamics of the planar H(2)(+) by short extreme ultra-violet (xuv) pulses, in which case the differential momentum distributions from both the length and the velocity gauge agree very well with those calculated by the lowest order perturbation theory.

PCTDSE: A parallel Cartesian-grid-based TDSE solver for modeling laser–atom interactions

Optimally-shaped electromagnetic fields have the capacity to coherently control the dynamics of quantum systems and thus offer a promising means for controlling molecular transformations relevant to chemical, biological, and materials applications. Currently, advances in this area are hindered by the prohibitive cost of the quantum dynamics simulations needed to explore the principles and possibilities of molecular control. However, the emergence of nascent quantum-computing devices suggests that efficient simulations of quantum dynamics may be on the horizon. In this article, we study how quantum computers could be employed to design optimally-shaped fields to control molecular systems. We introduce a hybrid algorithm that utilizes a quantum computer for simulating the field-induced quantum dynamics of a molecular system in polynomial time, in combination with a classical optimization approach for updating the field. Qubit encoding methods relevant for molecular control problems are described, and procedures for simulating the quantum dynamics and obtaining the simulation results are discussed. Numerical illustrations are then presented that explicitly treat paradigmatic vibrational and rotational control problems, and also consider how optimally-shaped fields could be used to elucidate the mechanisms of energy transfer in light-harvesting complexes. Resource estimates, as well as a numerical assessment of the impact of hardware noise and the prospects of near-term hardware implementations, are provided for the latter task.

Understanding the elementary steps involved in a chemical reaction forms the cornerstone of physical chemistry research. One way to deepen this understanding is by studying chemical and physical processes using linear and nonlinear spectroscopic techniques. However, the outcomes of such experiments can be difficult to decipher due to the interweaving of several effects. Therefore, in order to help experimentalists to disentangle such spectra, the role of theorists is to develop efficient tools that are able to accurately describe molecular systems. The starting point of such tools is solving the time-dependent Schrödinger equation. In this thesis, after implementing geometric integrators, which are based on a combination of the split-operator algorithm and Magnus expansion, for the exact nonadiabatic quantum dynamics of a molecule interacting with a time-dependent electromagnetic field, we derive and implement these geometric integrators for the time-dependent perturbation theory, the Condon, rotating-wave, and ultrashort-pulse approximations, as well as every possible combination thereof. As verified in several model systems, these integrators exactly preserve the geometric invariants, and achieve an arbitrary prescribed order of accuracy in the time step and an exponential convergence in the grid spacing. We also explore in more detail the ultrashort-pulse approximation and derive an analytical expression for the combination with the time-dependent perturbation theory; this expression significantly accelerates numerical calculations. We show that in the limit of the zero pulse width, the d-pulse approximation is recovered. We illustrate the performance of the introduced approximations, using a three-dimensional model of pyrazine, in which it is essential to go beyond the d-pulse limit in order to describe the dynamics correctly. The high-order algorithms are also applied to the photodissociation dynamics of iodomethane (CH3I), following its excitation to the A band. We implement a general split-operator with both discrete-variable and finite-basis representations that can treat one non-Cartesian, such as angular coordinate. To test the effect of various degrees of freedom and of the nonadiabatic dynamics, we apply these algorithms to one-, two-, and three-dimensional models of iodomethane, both in the presence and in the absence of nonadiabatic couplings. A full quantum calculation is, however, limited to problems with low dimensionality (approximately ten degrees of freedom). Beyond this, one must seek an affordable balance between computational efficiency and physical accuracy and can employ, for example, semiclassical methods that are based on classical trajectories. A simple semiclassical approximation that can treat larger systems and requires only local knowledge of the potential is the on-the-fly ab initio thawed Gaussian approximation. We implement a generalization of the method that goes beyond the Franck-Condon approximation and treats Herzberg-Teller active molecules. Our method is used to compute absorption spectra of phenyl radical and of benzene, for which the Herzberg-Teller contribution is essential.

The lecture notes on “Many-body Quantum Dynamics with MCTDH-X”, adapted from the 2023 Heidelberg MCTDH Summer School, provide an in-depth exploration of the Multiconfigurational Time-Dependent Hartree approach for indistinguishable particles. They serve as a comprehensive guide for understanding and utilizing the MCTDH-X software for both bosonic and fermionic systems. The tutorial begins with an introduction to the MCTDH-X software, highlighting its capability to handle various quantum systems, including those with internal degrees of freedom and long-range interactions. The theoretical foundation is then laid out on how to solve the time-dependent and time-independent Schrödinger equations for many-body systems. The workflow section provides practical instructions on setting up and executing simulations using MCTDH-X. Detailed benchmarks against exact solutions are presented, showcasing the accuracy and reliability of the software in ground-state and dynamic simulations. The notes then delve into the dynamics of quantum systems, covering relaxation processes, time evolution, and the analysis of propagation for both bosonic and fermionic particles. The discussion includes the interpretation of various physical quantities such as energy, density distributions, and orbital occupations. Advanced features of MCTDH-X are also explored in the last section, including the calculation of correlation functions and the creation of visualizations through video tutorials. The notes conclude with a Linux/UNIX command cheat sheet, facilitating ease of use for users operating the software on different systems. Overall, these lecture notes provide a valuable resource for researchers and students in the field of quantum dynamics, offering both theoretical insights and practical guidance on the use of MCTDH-X for studying complex many-body systems.

We assess the suitability of quantum and semiclassical initial-value representations (IVRs), exemplified by the coupled coherent states (CCS) method and the Herman–Kluk (HK) propagator, respectively, for modeling the dynamics of an electronic wave packet in a strong laser field, if this wave packet is initially bound. Using Wigner quasiprobability distributions and ensembles of classical trajectories, we identify signatures of over-the-barrier and tunnel ionization in phase space for static and time-dependent fields and the relevant sets of phase-space trajectories to model such features. Overall, we find good agreement with the full solution of the time-dependent Schrödinger equation (TDSE) for Wigner distributions constructed with both IVRs. Our results indicate that the HK propagator does not fully account for tunneling and over-the-barrier reflections. This leads to a dephasing in the time-dependent wave function, which becomes more pronounced for longer times. However, it is able to partly reproduce features associated with the wave packet crossing classically forbidden regions, although the trajectories employed in its construction always obey classical phase-space constraints. We also show that the CCS method represents a fully quantum initial value representation and accurately reproduces the results of a standard TDSE solver. Finally, we show that the HK propagator may be successfully employed to compute the time-dependent dipole acceleration and high-harmonic spectra. Nevertheless, the outcome of the semiclassical computation exhibits disagreements with the TDSE, as a consequence of the previously mentioned dephasing.

We introduce a Python package based on matrix product states (MPS) to simulate both the time-dependent Schrödinger equation (TDSE) and the hierarchical equations of motion (HEOM). The wave function in the TDSE or the reduced density operator/auxiliary density operators in the HEOM are represented using MPS. A matrix product operator (MPO) is then constructed to represent the Hamiltonian in the TDSE or the generalized Liouvillian in the HEOM. The fourth-order Runge-Kutta method and the time-dependent variational principle are used to propagate the MPS. Several examples, including the nonadiabatic interconversion dynamics of the pyrazine molecule, excitation energy transfer dynamics in molecular aggregates and photosynthetic light-harvesting complexes, the spin-boson model, a laser driven two-state model, the Holstein model, and charge transport in the Anderson impurity model, are presented to demonstrate the capability of the package.

First-principles molecular dynamics simulations, which can reveal the microscopic mechanisms and the macroscopic properties of molecules and condensed systems, have become a significant driving force in the development of physics. In the past decades, the ground state electronic properties of various materials can be accurately described using the first-principles method based on the Born-Oppenheimer approximation. However, in nonadiabatic processes, such as the ultrafast excited-state dynamics where laser interacts with molecules and condensed matters, the Born-Oppenheimer approximation evolving nuclear wave function on the ground state potential energy plane is no longer valid because of the nonnegligible nonadiabatic couplings between electrons and nuclei. In order to investigate the novel physical phenomena in nonadiabatic processes, many nonadiabatic molecular dynamics methods have been proposed, for instance, full quantum dynamics and mixed quantum-classical dynamics. When the nuclear quantum effects are trivial, the mixed quantum-classical dynamics is effective, which solves the time-dependent electron Schrodinger equation and describes the motion of the nuclei in the form of Newtonian mechanics. Two methods based on the mixed quantum-classical dynamics, the fewest-switches trajectory surface-hopping method (FSSH) and Ehrenfest dynamics, have accepted much attention since they could be easily implemented and combined with real-time time-dependent density functional theory (rt-TDDFT) for high-precision calculations. The combination of quantum and classical dynamics reduces the computational cost and allows the applications for real materials. At low temperatures, the nuclear quantum effects, including quantum tunneling and zero-point vibrations, cannot be ignored. Ring-polymer molecular dynamics (RPMD) based on the imaginary-time path integral is widely used to consider the nuclear quantum effects, which describes the quantum behavior of atomic nuclei by sampling a larger number of quantum configurations and paths. By combining RPMD and mixed quantum-classical nonadiabatic dynamics, one can describe quantum motions of nuclei and the excited electrons in quantum nonadiabatic dynamics simulations. Here, we summarize the recent progress of the rt-TDDFT methods based on numerical atomic orbital basis and plane- wave basis set, as well as highly efficient RP-TDAP method combining rt-TDDFT and RPMD towards a quantum description of electronic-nuclear dynamics. Finally, we show several representative applications employing these methods, including high harmonic generation modulated by two-color light, charge density wave dynamics, photocatalytic water decomposition as well as the quantum evolution of nuclear wave packets. These applications are in good agreement with experimental measurements, which demonstrates the reliability of these methods and their advantages in the corresponding research areas. These developments and applications are in a significant step from ground state methods to full quantum simulation of the coupled motion of electrons and nuclei, providing a new perspective for understanding and predicting the quantum interactions and dynamical behavior of condensed materials in the degree of atomic spatial scales and attosecond time scales.

Solving the time-dependent Schrödinger equation (TDSE) is pivotal for modeling nonadiabatic electron dynamics, a key process in ultrafast spectroscopy and laser-matter interactions. However, exact solutions to the TDSE remain computationally prohibitive for most realistic systems, as the Hilbert space expands exponentially with dimensionality. In this work, we propose an approach integrating the stochastic representation framework with a neural network wavefunction ansatz, a flexible model capable of approximating time-evolving quantum wavefunctions. We first validate the method on one-dimensional, single-electron systems, focusing on ionization dynamics under intense laser fields, a critical process in attosecond physics. Our results demonstrate that the approach accurately reproduces key features of quantum evolution, including the energy and dipole evolution during ionization. We further show the feasibility of extending this approach to three-dimensional systems. Due to the increased complexity of real-time simulations in higher dimensions, these results remain at an early stage and highlight the need for more advanced stabilization strategies.

Based on the direct solution of the time-dependent Schrödinger equation (TDSE), we theoretically study two-photon ionization (TPI) of a hydrogen atom by ultrashort vacuum ultraviolet (VUV) laser pulses with a photon energy close to the ionization threshold and a pulse width from 10 fs down to subfemtoseconds, for which the distinction between stepwise and direct processes becomes subtle. Our analysis on TPI by a double pulse reveals that direct processes are classified into two categories: a purely direct process with no real intermediate levels, and the one via Rydberg or continuum states, which rapidly escape from the nucleus. Our results also show that TPI becomes stepwise for subfemtosecond VUV pulses even for a wavelength corresponding to a direct process in the long pulse limit, since the broad spectrum of the pulse overlaps several discrete bound levels and excites them resonantly. This also leads to a phenomenon peculiar to attosecond pulses, namely, a significant red shift of the photoelectron energy spectrum. The Rydberg wave packet generated by an ultrashort near-threshold laser pulse, containing low-lying levels and the continuum, rapidly disintegrates into several parts. Nevertheless, the bound parts come back to the nucleus in fragments, and each fragment returns in the Kepler orbit time corresponding to its central principal quantum number. The lower-energy part of the double-pulse TPI electron energy spectrum exhibits the effect of the interference between the returning fragments and the wave packet excited by the second pulse.

Quantum dynamics of an electron moving in coupled quartic and coupled double-well oscillators under intense laser fields

Symmetries of cyclic work distributions for an isolated harmonic oscillator