Herman Kluk Research Articles

Sampling strategies for the Herman–Kluk propagator of the wavefunction

When the semiclassical Herman–Kluk propagator is used for evaluating quantum-mechanical observables or time-correlation functions, the initial conditions for the guiding trajectories are typically sampled from the Husimi density. Here, we employ this propagator to evolve the wavefunction itself. We investigate two grid-free strategies for the initial sampling of the Herman–Kluk propagator applied to the wavefunction and validate the resulting time-dependent wavefunctions evolved in harmonic and anharmonic potentials. In particular, we consider Monte Carlo quadratures based either on the initial Husimi density or on its square root as possible and most natural sampling densities. We prove analytical convergence error estimates and validate them with numerical experiments on the harmonic oscillator and on a series of Morse potentials with increasing anharmonicity. In all cases, sampling from the square root of Husimi density leads to faster convergence of the wavefunction.

Non-linear correlation functions and zero-point energy flow in mixed quantum-classical semiclassical dynamics.

Mixed quantum classical (MQC)-initial value representation (IVR) is a recently introduced semiclassical framework that allows for selective quantization of the modes of a complex system. In the quantum limit, MQC reproduces the semiclassical Double Herman-Kluk IVR results, accurately capturing nuclear quantum coherences and conserving zero-point energy. However, in the classical limit, although MQC mimics the Husimi-IVR for real-time correlation functions with linear operators, it is significantly less accurate for non-linear correlation functions with errors even at time zero. Here, we identify the origin of this discrepancy in the MQC formulation and propose a modification. We analytically show that the modified MQC approach is exact for all correlation functions at time zero, and in a study of zero-point energy (ZPE) flow, we numerically demonstrate that it correctly obtains the quantum and classical limits as a function of time. Interestingly, although classical-limit MQC simulations show the expected, unphysical ZPE leakage, we find that it is possible to predict and even modify the direction of ZPE flow through selective quantization of the system, with the quantum-limit modes accepting energy but preserving the minimum quantum mechanically required energy.

Scaling asymptotics of spectral Wigner functions* *SZ was partially supported by NSF Grant DMS-1810747. BH was funded by NSF CAREER Grant DMS-2143754 as well as NSF Grants DMS-1855684, DMS-2133806 and an ONR MURI on Foundations of Deep Learning.

We prove that smooth Wigner–Weyl spectral sums at an energy level E exhibit Airy scaling asymptotics across the classical energy surface Σ E . This was proved earlier by the authors for the isotropic harmonic oscillator and the proof is extended in this article to all quantum Hamiltonians −ℏ 2Δ + V where V is a confining potential with at most quadratic growth at infinity. The main tools are the Herman–Kluk initial value parametrix for the propagator and the Chester–Friedman–Ursell normal form for complex phases with a one-dimensional cubic degeneracy. This gives a rigorous account of Airy scaling asymptotics of spectral Wigner distributions of Berry, Ozorio de Almeida and other physicists.

The Van Vleck Formula on Ehrenfest Time Scales and Stationary Phase Asymptotics for Frequency-Dependent Phases

The Van Vleck formula is a semiclassical approximation to the integral kernel of the propagator associated to a time-dependent Schrödinger equation. Under suitable hypotheses, we present a rigorous treatment of this approximation which is valid on Ehrenfest time scales, i.e. \(\hbar \)-dependent time intervals which most commonly take the form \(|t| \le c|\log \hbar |\). Our derivation is based on an approximation to the integral kernel often called the Herman–Kluk approximation, which realizes the kernel as an integral superposition of Gaussians parameterized by points in phase space. As was shown by Robert (Rev Math Phys 22(10):1123-1145, 2010) , this yields effective approximations over Ehrenfest time intervals. In order to derive the Van Vleck approximation from the Herman–Kluk approximation, we are led to develop stationary phase asymptotics where the phase functions depend on the frequency parameter in a nontrivial way, a result which may be of independent interest.

Discretising the Herman–Kluk propagator

The Herman---Kluk propagator is a well-known semi-classical approximation of the unitary evolution operator in quantum molecular dynamics. In this paper we formulate the Herman---Kluk propagator as a phase space integral and discretise it by Monte Carlo and quasi-Monte Carlo quadrature. Then, we investigate the accuracy of a symplectic time discretisation by combining backward error analysis with Fourier integral operator calculus. Numerical experiments for two- and six-dimensional model systems support our theoretical results.

Irregular time-dependent perturbations of quantum Hamiltonians

Our main goal in this paper is to prove existence (and uniqueness) of the quantum propagator for time-dependent quantum Hamiltonians \widehat H(t) when this Hamiltonian is perturbed with a quadratic white noise \dot{\beta}\widehat K . \beta is a continuous function in time t , \dot \beta its time derivative and K is a quadratic Hamiltonian. \widehat K is the Weyl quantization of K . For time-dependent quadratic Hamiltonians H(t) we recover, under less restrictive assumptions, the results obtained in [3, 9, 10]. In our approach we use an exact Herman–Kluk formula [20] to deduce a Strichartz estimate for the propagator of \widehat H(t) +\dot \beta K . This is applied to obtain local and global well posedness for solutions for non-linear Schrödinger equations with an irregular time-dependent linear part.

Mixed semiclassical-classical propagators for the Wigner phase space representation.

We formulate mixed semiclassical-classical (SC-Cl) propagators by adding a further approximation to the phase-space SC propagators, which have been formulated in our previous paper [S. Koda, J. Chem. Phys. 143, 244110 (2015)]. We first show that the stationary phase approximation over the operation of the phase-space van Vleck propagator on initial distribution functions results in the classical mechanical time propagation. Then, after dividing the degrees of freedom (DOFs) of the total system into the semiclassical DOFs and the classical DOFs, the SC-Cl van Vleck propagator and the SC-Cl Herman-Kluk (HK) propagator are derived by performing the stationary phase approximation only with respect to the classical DOFs. These SC-Cl propagators are naturally decomposed to products of the phase-space SC propagators and the classical mechanical propagators when the system does not have any interaction between the semiclassical and the classical DOFs. In addition, we also numerically compare the original phase-space HK (full HK) propagator and the SC-Cl HK propagator in terms of accuracy and efficiency to find that the accuracy of the SC-Cl HK propagator can be comparable to that of the full HK propagator although the latter is more accurate than the former in general. On the other hand, we confirm that the convergence speed of the SC-Cl HK propagator is faster than that of the full HK propagator. The present numerical tests indicate that the SC-Cl HK propagator can be more accurate than the full HK propagator when they use a same and finite number of classical trajectories due to the balance of the accuracy and the efficiency.

Dynamics of interacting bosons using the Herman–Kluk semiclassical initial value representation

Recent experimental progress in monitoring the dynamics of ultracold gases in optical lattices necessitates a quantitative theoretical description for a significant number of bosons. In the present paper, we investigate if time-dependent semiclassical initial value methodology, with propagators expressed as integrals over phase space and using classical trajectories, is suitable to describe interacting bosons, concentrating on a single mode. Despite the nonlinear contribution from the self-interaction, the corresponding classical dynamics allows for a largely analytical treatment of the semiclassical propagator. We find that application of the Herman–Kluk (HK) propagator conserves unitarity in the semiclassical limit, but a decay of the norm is seen for low particle numbers n. The frozen Gaussian approximation (FGA) (HK with unit prefactor) is explicitly shown to violate unitarity in the present system for non-vanishing interaction strength, even in the semiclassical limit. Furthermore, we show by evaluating the phase space integral in steepest descent approximation, that the HK propagator reproduces the exact spectrum correctly in the semiclassical limit (). An error is, however, incurred in next-to-next-to-leading order (small parameter ), as seen upon numerical evaluation of the integral and confirmed analytically by considering finite n corrections to the steepest descent calculations. The FGA, in contrast, is only accurate to lowest order, and an erroneous next-to-leading order term in the energy spectrum was found analytically. Finally, as an example application, we study the dynamics of wave packets by computing the time evolution of the Wigner function. While the often-used truncated Wigner approximation cannot capture any interferences present in the exact quantum mechanical solution (known analytically), we find that the HK approach, despite also using classical information only, reproduces the salient features of the exact solution correctly.

Frozen Gaussian approximation based domain decomposition methods for the linear Schrödinger equation beyond the semi-classical regime

The paper is devoted to develop efficient domain decomposition methods for the linear Schrödinger equation beyond the semiclassical regime, which does not carry a small enough rescaled Planck constant for asymptotic methods ( e.g. geometric optics) to produce a good accuracy, but which is too computationally expensive if direct methods ( e.g. finite difference) are applied. This belongs to the category of computing middle-frequency wave propagation, where neither asymptotic nor direct methods can be directly used with both efficiency and accuracy. Motivated by recent works of the authors on absorbing boundary conditions (Antoine et al. (2014) [13] and Yang and Zhang (2014) [43] ), we introduce Semiclassical Schwarz Waveform Relaxation methods (SSWR), which are seamless integrations of semiclassical approximation to Schwarz Waveform Relaxation methods. Two versions are proposed respectively based on Herman–Kluk propagation and geometric optics, and we prove the convergence and provide numerical evidence of efficiency and accuracy of these methods.

Quantum and semiclassical phase-space dynamics of a wave packet in strong fields using initial-value representations

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.

Optimal control of wave-packets: a semiclassical approach

We studied the optimal quantum control of a molecular rotor in tilted laser fields using the time-sliced Herman–Kluk propagator for the evaluation of the optimal pulse and the light–dipole interaction as the control mechanism. The proposed methodology was used to study the effects of an optimal pulse on the evolution of a wave-packet in a double-well potential and in the effective potential of a molecular rotor in a collinear tilted fields setup. The amplitude and frequency of the control pulse were obtained in such a way that the transition probability between two rotational wave-packets was maximised.

Herman–Kluk allows analysis of quantum discrete breathers in higher dimensional systems

Discrete breathers (DBs) provide an important example of spatial localization of energy encountered in non-linear many-body systems. Studying the quantum equivalent of DBs has always been restricted to small or simple systems. We show that Herman–Kluk (HK), a well studied semi-classical method, can be used successfully to observe the dynamics of a quantum state for systems that are too large for conventional quantum mechanical methods. HK reduces the growth of computational cost that comes with increase of system size. We show that HK performs very well for a quartic potential, when compared to exact quantum dynamics. As an application of the approach, we compute and analyse the spectrum of quantum DBs in a seven-dimensional potential. This result advances our ability to model experimental systems in many fields, including solar energy harvesting, quantum computing, nano-science, medicine, etc.

Frozen Gaussian Approximation for General Linear Strictly Hyperbolic Systems: Formulation and Eulerian Methods

The frozen Gaussian approximation, proposed in [J. Lu and X. Yang, Commun. Math. Sci., 9 (2011), pp. 663–683], is an efficient computational tool for high frequency wave propagation. We continue in this paper the development of frozen Gaussian approximation. The frozen Gaussian approximation is extended to general linear strictly hyperbolic systems. Eulerian methods based on frozen Gaussian approximation are developed to overcome the divergence problem of Lagrangian methods. The proposed Eulerian methods can also be used for the Herman–Kluk propagator in quantum mechanics. Numerical examples verify the performance of the proposed methods.

Spectra of harmonium in a magnetic field using an initial value representation of the semiclassical propagator

For two Coulombically interacting electrons in a quantum dot with harmonic confinement and a constant magnetic field, we show that time-dependent semiclassical calculations using the Herman–Kluk initial value representation of the propagator lead to eigenvalues of the same accuracy as WKB calculations with a Langer correction. The latter are restricted to integrable systems, however, whereas the time-dependent initial value approach allows for applications to high-dimensional, possibly chaotic dynamics and is extendable to arbitrary shapes of the potential.

Semiclassical initial value representation study of internal conversion rates

Internal conversion is an inherently quantum mechanical process. To date, "on the fly" computation of internal conversion rates is limited to harmonic approximations, which would seem to be especially unsuitable, given that the typical transition to the ground electronic state occurs at energies which are far from the harmonic limit. It is thus of interest to study the applicability of the semiclassial initial value representation (SCIVR) approach which is in principle amenable to on the fly studies even with "many" degrees of freedom. In this paper we study the applicability of the Herman-Kluk (HK) SCIVR to a model system with two coupled and anharmonic degrees of freedom. We find that (a) the HK SCIVR is a good approximation to the exact quantum dynamics; (b) computation of the first order correction to the HK-SCIVR approximation corroborates the accuracy; (c) by studying a large parameter range, we find that the harmonic approximation is mostly unsatisfactory; and (d) for the specific model used, the coupling between the modes was found to be relatively unimportant. These results imply that the HK-SCIVR methodology is a good candidate for on the fly studies of internal conversion processes of "large" molecules.

Renormalization of the frozen Gaussian approximation to the quantum propagator

The frozen Gaussian approximation to the quantum propagator may be a viable method for obtaining "on the fly" quantum dynamical information on systems with many degrees of freedom. However, it has two severe limitations, it rapidly loses normalization and one needs to know the Gaussian averaged potential, hence it is not a purely local theory in the force field. These limitations are in principle remedied by using the Herman-Kluk (HK) form for the semiclassical propagator. The HK propagator approximately conserves unitarity for relatively long times and depends only locally on the bare potential and its second derivatives. However, the HK propagator involves a much more expensive computation due to the need for evaluating the monodromy matrix elements. In this paper, we (a) derive a new formula for the normalization integral based on a prefactor free HK propagator which is amenable to "on the fly" computations; (b) show that a frozen Gaussian version of the normalization integral is not readily computable "on the fly"; (c) provide a new insight into how the HK prefactor leads to approximate unitarity; and (d) how one may construct a prefactor free approximation which combines the advantages of the frozen Gaussian and the HK propagators. The theoretical developments are backed by numerical examples on a Morse oscillator and a quartic double well potential.

ON THE HERMAN–KLUK SEMICLASSICAL APPROXIMATION

For a subquadratic symbol H on ℝd× ℝd= T*(ℝd), the quantum propagator of the time dependent Schrödinger equation [Formula: see text] is a Semiclassical Fourier-Integral Operator when Ĥ = H(x, ℏDx) (ℏ-Weyl quantization of H). Its Schwartz kernel is described by a quadratic phase and an amplitude. At every time t, when ℏ is small, it is "essentially supported" in a neighborhood of the graph of the classical flow generated by H, with a full uniform asymptotic expansion in ℏ for the amplitude.In this paper, our goal is to revisit this well-known and fundamental result with emphasis on the flexibility for the choice of a quadratic complex phase function and on global L2estimates when ℏ is small and time t is large. One of the simplest choice of the phase is known in chemical physics as Herman–Kluk formula. Moreover, we prove that the semiclassical expansion for the propagator is valid for [Formula: see text] where δ > 0 is a stability parameter for the classical system.

Semiclassical Quantization in Liouville Space for Vibrational Dynamics

Semiclassical approximations to quantum mechanics can include quantum coherence effects in dynamical calculations based on classical mechanics. The Herman-Kluk (HK) semiclassical propagator has been demonstrated to reproduce quantum effects in nonlinear vibrational response functions of anharmonic oscillators but does not provide a practical numerical route to calculations for multiple degrees of freedom. In an HK calculation of a response function, quantum coherence effects enter through interference between pairs of classical trajectories. We have previously elucidated the mechanism by which the HK approximation reproduces quantum effects in response functions in the regime of quasiperiodic dynamics. We have applied this understanding to significantly simplify the semiclassical calculation of response functions in this dynamical regime. The phase space difference between trajectories is treated perturbatively in anharmonicity, allowing integration over these differences to be performed analytically and leaving integration over mean trajectories to be performed numerically. This mean-trajectory (MT) approximation has been applied to linear and nonlinear vibrational response functions for isolated and coupled anharmonic motions. Here, we derive an MT approximation for the Liouville space time evolution operator or superoperator that propagates the density operator. This analysis provides a form of the MT approximation that is readily applicable to other dynamical quantities besides response functions and clarifies the connection between semiclassical quantization of propagators for the wave function and for the density operator.

Semiclassical nonlinear response functions for coupled anharmonic vibrations

Observables in linear and nonlinear infrared spectroscopy may be computed from vibrational response functions describing nuclear dynamics on a single electronic surface. We demonstrate that the Herman-Kluk (HK) semiclassical approximation to the quantum propagator yields an accurate representation of quantum coherence effects in linear and nonlinear response functions for coupled anharmonic oscillators. A considerable numerical price is paid for this accuracy; the calculation requires a multidimensional integral over a highly oscillatory integrand that also grows without bound as a function of evolution times. The interference among classical trajectories in the HK approximation produces quantization of good action variables. By treating this interference analytically, we develop a mean-trajectory (MT) approximation that requires only the propagation of classical trajectories linked by transitions in action. The MT approximation accurately reproduces coherence effects in response functions of coupled anharmonic oscillators in a regime in which the observables are strongly influenced by these interactions among vibrations.

Tunneling in two-dimensional systems using a higher-order Herman–Kluk approximation

A principal weakness of the Herman-Kluk (HK) semiclassical approximation is its failure to provide a reliably accurate description of tunneling between different classically allowed regions. It was previously shown that semiclassical corrections significantly improve the HK treatment of tunneling for the particular case of the one-dimensional Eckart system. Calculations presented here demonstrate that the lowest-order correction also substantially improves the HK description of tunneling across barriers in two-dimensional systems. Numerical convergence issues either do not arise or are easily overcome, so that the calculations require only a moderate number of ordinary, real, classical trajectories.