Coherent State-Based Path Integral Methodology for Computing the Wigner Phase Space Distribution

Abstract

The accurate evaluation of the Wigner phase space density for multidimensional system remains a challenging task. Path integral Monte Carlo methods offer a numerically exact approach for obtaining the Boltzmann density in coordinate space, but the Fourier-type integral required to construct the Wigner distribution generally leads to poor convergence. This paper describes a path integral method for constructing the Wigner density which substantially mitigates the Monte Carlo sign problem and thus is applicable to systems with many degrees of freedom. The starting point is the path integral representation of the coherent state density, which does not involve a Fourier integral and thus converges rapidly. We then use the relation between the coherent state and Wigner densities to construct the Wigner function, taking advantage of destructive phase cancellation to truncate the infinite series and thus confine the integrand, avoiding highly oscillatory regions. We also describe the use of information-guided noise reduction (IGNoR) to improve the Monte Carlo statistics in the most challenging regimes. The method is applied to strongly anharmonic one-dimensional models, a system-bath Hamiltonian, as well as the formamide molecule within an ab initio quartic potential, and the results are compared to those obtained by various approximate methods. These calculations suggest that the coherent state-based path integral method described in this paper offers an efficient, numerically exact approach for constructing the Wigner phase space density in systems of many degrees of freedom, and thus will be useful for quantizing the initial condition in classical trajectory-based simulations of dynamical properties.

Quantum‐classical path integral evaluation of reaction rates with a near‐equilibrium flux formulation

“…To proceed, we take advantage of the invariance of the rate (in the exponential decay regime) with respect to details of the initial preparation of the system and its environment. In a recent paper [62] we exploited this independence by replacing the full Boltzmann operator by that corresponding to the solvent Hamiltonian. This procedure allowed the use of QCPI with a simple factorized initial condition corresponding to the solvent equilibrated with respect to the reactants.…”

“…Figure 2 shows the time evolution of the reactive flux at T = 300 K for three values of the system‐bath coupling strength, ξ = 0.1, 0.5, and 1.5. Results obtained with the current near‐equilibrium flux approach, where the time evolution was obtained through the QCPI methodology, are compared to those obtained through the simpler non‐equilibrium flux scheme [62] with the propagation of the initial density was obtained with the quasiadiabatic propagator path integral [76, 77] (QuAPI) algorithm. At small values of the system‐bath coupling the two side‐flux correlation functions do not exhibit notable differences.…”

“…These include local [51] or variationally optimized [52] Gaussian wavepacket approaches, and the thermal Gaussian approximation [53, 54] (which employ frozen Gaussian dynamics [55] in imaginary time), along with extensions that capture quantum corrections [56]. We recently introduced [57, 58] a simple, trajectory‐based approximate procedure that makes use of the classical adiabatic theorem to slowly convert the Wigner density of a harmonic reference system to that of the target Hamiltonian. We also described a path integral representation of the Wigner density which exploits the coherent state representation to circumvent the numerical issues associated with the oscillatory Fourier phase [59].…”

“…We recently introduced [57, 58] a simple, trajectory‐based approximate procedure that makes use of the classical adiabatic theorem to slowly convert the Wigner density of a harmonic reference system to that of the target Hamiltonian. We also described a path integral representation of the Wigner density which exploits the coherent state representation to circumvent the numerical issues associated with the oscillatory Fourier phase [59]. Other recent work [60] has used the quasi‐adiabatic propagator path integral methodology [61] to obtain the Wigner distribution of the bath in case of a system interacting with a bath of independent harmonic oscillators.…”

“…In recent work [62] we circumvented this difficulty by introducing a non‐equilibrium flux formulation which employs a simple initial condition that is easy to construct. As an additional benefit, the non‐equilibrium formulation automatically generates the reactant population through the plateau time, offering a unified approach to the dynamics of slow as well as fast reactive processes.…”