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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Numerical test of approximate single-step propagators: Harmonic power series expansions versus system-specific split operator representations

Alexander N. Drozdov, Shigeo Hayashi

Published Feb 1, 1999• 5 citations
Publisher

Abstract

This paper compares the efficiency and accuracy of different second-order approximations for the single-step propagator obtained by means of the power series expansion formalism and the split operator method. The former is based on a harmonic reference system, while the latter allows for the use of physically motivated (anharmonic) zeroth-order representations. Three typical examples---a system-bath Hamiltonian, a H\'enon-Heiles anharmonic resonating system, and a Fokker-Planck chaotic model---are considered in the present testing. The examples cover a variety of situations with strongly anharmonic coupling. Although no system-specific reference system is involved in the power series representation of the propagator, it quite accurately describes the dynamics of very anharmonic processes in the entire time domain even though the coupling is strong. Another appealing feature of the approach is that it is essentially analytical and therefore does not require any computational effort to implement. This makes the power series expansion technique particularly attractive for efficiently treating many-body problems. In contrast, numerical implementation of the split operator method can be arduous as a general multidimensional calculation, while its utility is in general restricted to the separable limit, when the coupling is almost turned off. In that limit the method indeed provides accurate results over a broad range of t. With increasing coupling, however, the efficiency of the method deteriorates very rapidly regardless of the particular choice of the zeroth-order representation.

DOI
10.1103/physreve.59.1386

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This paper compares the efficiency and accuracy of different second-order approximations for the single-step propagator obtained by means of the power series expansion formalism and the split operator method. The former is based on a harmonic reference system, while the latter allows for the use of physically motivated (anharmonic) zeroth-order representations. Three typical examples---a system-bath Hamiltonian, a H'enon-Heiles anharmonic resonating system, and a Fokker-Planck chaotic model---are considered in the present testing. The examples cover a variety of situations with strongly anharmonic coupling. Although no system-specific reference system is involved in the power series representation of the propagator, it quite accurately describes the dynamics of very anharmonic processes in the entire time domain even though the coupling is strong. Another appealing feature of the approach is that it is essentially analytical and therefore does not require any computational effort to implement. This makes the power series expansion technique particularly attractive for efficiently treating many-body problems. In contrast, numerical implementation of the split operator method can be arduous as a general multidimensional calculation, while its utility is in general restricted to the separable limit, when the coupling is almost turned off. In that limit the method indeed provides accurate results over a broad range of t. With increasing coupling, however, the efficiency of the method deteriorates very rapidly regardless of the particular choice of the zeroth-order representation.
This paper presents two perturbation approaches to calculating the quantum statistics of multidimensional nonlinear oscillators in a simple, economic way. These are the power series expansion formalism and the system-specific split operator method. The former involves a harmonic reference system, while the latter allows for the use of physically motivated (anharmonic) zeroth-order representations. Both approaches are outlined with some improvements, and their possible limitations are discussed. The relative efficacy of the approaches is tested on two typical models that are often used by researchers as benchmarks for different numerical methods. Although no system-specific reference system is involved in the power series expansion of the Boltzmann operator, it quite accurately describes the quantum statistics in the entire temperature range, even though the coupling strength is rather large. Yet another important advantage of this approach is that it is essentially analytical, and therefore its numerical implementation does not require any computational effort. This makes the power series expansion technique particularly attractive for treating many-body problems. In contrast, the split operator method is found to be efficient only in the limit of separable motion. Otherwise, this approach results in a less than optimally efficient approximation for the density matrix, or even one which is worse than the standard Trotter approximation.
A simple framework for accurate solution of a general class of one-dimensional Fokker-Planck and/or Schr\"odinger equations is presented. The main idea is representing the propagator in the form P(x,t∣x0)=P0(x,t∣x0)exp[W(x,t∣x0)]P(x, t|{x}_{0})={P}_{0}(x, t|{x}_{0})\mathrm{exp}[W(x, t|{x}_{0})]P(x,t∣x0​)=P0​(x,t∣x0​)exp[W(x,t∣x0​)] and expanding the exponent WWW in a power series in a given function of ttt, where P0{P}_{0}P0​ is the exact solution of a reference harmonic-oscillator problem. The expansion coefficients are analytically evaluated from recursive relations. This approach is shown to be a dramatic improvement over the standard Taylor series expansion for the propagator in that just a few terms of the present expansion are sufficient to attain a very accurate description in the whole time domain.
We describe an efficient path integral scheme for calculating the propagator of an arbitrary quantum system, as well as that of a stochastic system in special cases where the Fokker–Planck equation obeys strict detailed balance. The basic idea is to split the respective Hamiltonian into two exactly solvable parts and then to employ a symmetric decomposition of the time evolution operator, which is exact up to a high order in the time step. The resulting single step propagator allows rather large time steps in a path integral and leads to convergence with fewer time slices. Because it involves no system-specific reference system, the algorithm is amenable to all known numerical schemes available for evaluating quantum path integrals. In this way one obtains a highly accurate method, which is simultaneously fast, stable, and computationally simple. Numerical applications to the real time quantum dynamics in a double well and to the stochastic dynamics of a bistable system coupled to a harmonic mode show our method to be superior over the approach developed by the Makri group in their quasiadiabatic propagator representation, to say nothing about the propagation scheme based on the standard Trotter splitting.
Various numerical integration schemes to calculate the propagation of a state following the time-dependent Schrodinger equation in the one dimensional position representation are presented and compared to each other. Three potentials have been used: a harmonic, a double-well and a zero potential. Eigenstates and a coherent state have been chosen as initial states. Special attention has been given to the long-time stability of the algorithms. These are: kinetic referenced split operator (KRSO), kinetic referenced Cayley (KRC), distributed approximating functions (DAF), Chebysheff expansion (CH), residuum minimization (RES), second-order differencing (SOD), an eigenstate expansion (EE) and a corrected kinetic referenced split operator (CKRSO). In addition, a speedup of the KRC and KRSO methods is presented which is specially suited when very few grid points are used. Numerical results are compared to analytically calculated values. Mixed classical/quantum mechanical simulations require a representation of the quantum state on a limited number of grid points, classical integration time steps of about one femtosecond and compatibility with methods to solve the time-ordering problem. For the considered potentials which differ quite essentially from the potentials used for scattering problems in particle physics, the EE method has been found to be faster, more accurate and more stable than the other methods if only a few grid points are required. Otherwise, good results have been obtained with KRC, KRSO, CH, DAF and RES. SOD has been found to be too slow, and CKRSO is not stable enough for long simulation times.
Because the Schrodinger equation cannot be solved analytically for systems larger than hydrogen, approximate solution methods must be applied to study systems of chemical interest. Quantum Monte Carlo is one of these methods. Quantum Monte Carlo simulations involve the diffusion of walkers through configuration space and necessitate the use of one of two imaginary time propagators, 2nd or 4th order. The choice of the propagator implemented depends on a balance between accuracy and efficiency. Obtaining accurate solutions is the primary objective; once this is accomplished, finding ways to make the simulations efficient is the next intention. Using a model system of a multidimensional harmonic oscillator, this study intends to compare and contrast both 2nd and 4th order propagators' diminished accuracy as the dimensionality of the system is increased. Two phases of the propagation are sampled to determine accuracy: a single propagation step and the asymptotic limit. To test the accuracy after one step, we start a simulation by placing walkers at the origin, moving them forward one time step in imaginary time, creating normalized histograms of their positions and comparing these with the known result for the imaginary time propagation of the delta function centered at x=0. The convergence of the simulation is determined by observing the change in distribution from the known asymptotic limit distribution. This is done by distributing the walkers in the known asymptotic distribution and observing the change in average weight upon further propagation. A converged distribution’s average weight will not change.
The numerical evaluation of coherent-state path integrals for quantum dynamical problems is discussed for one-dimensional examples. To propagate an initial state, we use the normal and antinormal ordered coherent-state path integrals combined with a split-operator technique dividing the Hamiltonian into harmonic and anharmonic parts. For numerical purposes integrations must be approximated by quadrature formulae. This leads to a matrix multiplication scheme which is systematically tested for the double-well and Morse potentials. The method is accurate for propagation times much longer than the natural time scale of the system, and it allows for short as well as long time steps without loss of stability.