Ehrenfest dynamics is a widely used mixed quantum--classical approach for nonadiabatic molecular dynamics, whereas thawed Gaussian wavepacket dynamics provides an efficient semiclassical description of adiabatic nuclear quantum dynamics. Here we describe thawed Gaussian Ehrenfest dynamics (TGED), which unifies and generalizes these two methods to capture both electronic nonadiabaticity and nuclear quantum effects within a single framework. The fully variational formulation of TGED is derived by applying the time-dependent variational principle to a Hartree product of electronic and Gaussian nuclear wavepackets. Replacing the effective locally quadratic molecular potential obtained from this variational treatment by alternative effective locally quadratic potentials yields an infinite family of TGED methods, of which we present several members. We analyze the limiting cases of the general formalism and show, in particular, that it reduces to conventional Ehrenfest dynamics in the classical limit for the nuclei and to thawed Gaussian wavepacket dynamics in the absence of electronic coupling. Finally, we present explicit geometric integrators for the entire family of methods and identify the conditions under which the different approximations become exact.
Ehrenfest dynamics is a widely used mixed quantum--classical approach for nonadiabatic molecular dynamics, whereas thawed Gaussian wavepacket dynamics provides an efficient semiclassical description of adiabatic nuclear quantum dynamics. Here we describe thawed Gaussian Ehrenfest dynamics (TGED), which unifies and generalizes these two methods to capture both electronic nonadiabaticity and nuclear quantum effects within a single framework. The fully variational formulation of TGED is derived by applying the time-dependent variational principle to a Hartree product of electronic and Gaussian nuclear wavepackets. Replacing the effective locally quadratic molecular potential obtained from this variational treatment by alternative effective locally quadratic potentials yields an infinite family of TGED methods, of which we present several members. We analyze the limiting cases of the general formalism and show, in particular, that it reduces to conventional Ehrenfest dynamics in the classical limit for the nuclei and to thawed Gaussian wavepacket dynamics in the absence of electronic coupling. Finally, we present explicit geometric integrators for the entire family of methods and identify the conditions under which the different approximations become exact.
We introduce two non-adiabatic semiclassical methods that employ two coupled Gaussian wavepackets, each one traveling on a separate diabatic potential energy surface. The wavepackets take the form of thawed Gaussians and are driven by classical equations of motion which account for the diabatic coupling. The classical equations of motion are derived in one case by enforcing the thawed Gaussian ansatz, while in the other the time-dependent variational principles to the thawed Gaussian ansatz. After a sanity check where both approximations reproduce Rabi oscillations, the methods are applied to two non-adiabatic potential energy scenarios. The first one involves two coupled displaced harmonic oscillators, as in a typical electron transfer reaction. The second one comprises a Morse potential coupled to an upper dissociative state, modeling a photo-dissociation process. In both scenarios, the variational thawed Gaussian approach is quite accurate, while the standard thawed Gaussian one fails to fully capture the non-adiabatic effects. Ultimately, non-adiabatic molecular dynamics is reproduced by means of two classical trajectories without introducing any artificial jump or other ad-hoc non-classical effects.
We present a novel quantum-classical approach to nonadiabatic dynamics, deduced from the coupled electronic and nuclear equations in the framework of the exact factorization of the electron-nuclear wave function. The method is based on the quasiclassical interpretation of the nuclear wave function, whose phase is related to the classical momentum and whose density is represented in terms of classical trajectories. In this approximation, electronic decoherence is naturally induced as an effect of the coupling to the nuclei and correctly reproduces the expected quantum behavior. Moreover, the splitting of the nuclear wave packet is captured as a consequence of the correct approximation of the time-dependent potential of the theory. This new approach offers a clear improvement over Ehrenfest-like dynamics. The theoretical derivation presented in this Letter is supported by numerical results that are compared to quantum mechanical calculations.
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.
An approach to nonadiabatic dynamics of atoms in molecular and condensed matter systems under general nonequilibrium conditions is proposed. In this method interaction between nuclei and electrons is considered explicitly up to the second order in atomic displacements defined with respect to the mean atomic trajectory, enabling one to consider movement of atoms beyond their simple vibrations. Both electrons and nuclei are treated fully quantum-mechanically using a combination of path integrals applied to nuclei and nonequilibrium Green's functions to electrons. Our method is partitionless: initially, the entire system is coupled and assumed to be at thermal equilibrium. Then, the exact application of the Hubbard-Stratonovich transformation in mixed real and imaginary times enables us to obtain, without doing any additional approximations, an exact expression for the reduced density matrix for nuclei and hence an effective quantum Liouville equation for them, both containing Gaussian noises. It is shown that the time evolution of the expectation values for atomic positions is described by an infinite hierarchy of stochastic differential equations for atomic positions and momenta and their various fluctuations. The actual dynamics is obtained by sampling all stochastic trajectories. It is expected that applications of the method may include photoinduced chemical reactions (e.g., dissociation), electromigration, and atomic manipulation in scanning tunneling microscopy, to name just a few.
We introduce a novel approach for a fully quantum description of coupled electron–ion systems from first principles. It combines the variational quantum Monte Carlo solution of the electronic part with the path integral formalism for the quantum nuclear dynamics. On the one hand, the path integral molecular dynamics includes nuclear quantum effects by adding a set of fictitious classical particles (beads) aimed at reproducing nuclear quantum fluctuations via a harmonic kinetic term. On the other hand, variational quantum Monte Carlo can provide Born–Oppenheimer potential energy surfaces with a precision comparable to the most-advanced post-Hartree–Fock approaches, and with a favorable scaling with the system size. In order to cope with the intrinsic noise due to the stochastic nature of quantum Monte Carlo methods, we generalize the path integral molecular dynamics using a Langevin thermostat correlated according to the covariance matrix of quantum Monte Carlo nuclear forces. The variational parameters of the quantum Monte Carlo wave function are evolved during the nuclear dynamics, such that the Born–Oppenheimer potential energy surface is unbiased. Statistical errors on the wave function parameters are reduced by resorting to bead grouping average, which we show to be accurate and well-controlled. Our general algorithm relies on a Trotter breakup between the dynamics driven by ionic forces and the one set by the harmonic interbead couplings. The latter is exactly integrated, even in the presence of the Langevin thermostat, thanks to the mapping onto an Ornstein–Uhlenbeck process. This framework turns out to be also very efficient in the case of noiseless (deterministic) ionic forces. The new implementation is validated on the Zundel ion (H 5 O 2 + ) by direct comparison with standard path integral Langevin dynamics calculations made with a coupled cluster potential energy surface. Nuclear quantum effects are confirmed to be dominant over thermal effects well beyond room temperature, giving the excess proton an increased mobility by quantum tunneling.
Coupled quantum electron-nuclear dynamics is often associated with the Born-Huang expansion of the molecular wave function and the appearance of nonadiabatic effects as a perturbation. On the other hand, native multicomponent representations of electrons and nuclei also exist, which do not rely on any a priori approximation. However, their implementation is hampered by prohibitive scaling. Consequently, quantum computers offer a unique opportunity for extending their use to larger systems. Here, we propose a quantum algorithm for simulating the time-evolution of molecular systems and apply it to proton transfer dynamics in malonaldehyde, described as a rigid scaffold. The proposed quantum algorithm can be easily generalized to include the explicit dynamics of the classically described molecular scaffold. We show how entanglement between electronic and nuclear degrees of freedom can persist over long times if electrons do not follow the nuclear displacement adiabatically. The proposed quantum algorithm may become a valid candidate for the study of such phenomena when sufficiently powerful quantum computers become available.