Comparison of the propagation of semiclassical frozen Gaussian wave functions with quantum propagation for a highly excited anharmonic oscillator

Abstract

Quantum and semiclassical description of a triply degenerate anharmonic oscillatorThe propagation of an initially highly excited localized wave packet in an anharmonic oscillator potenti~ is stu~ied within th~ frozen Gaussian approximation. Comparison is made to quantum mec.h~lcal basIS set ~alc~lat1ons. The frozen Gaussian approximation involves the expansion of the ID1t1al wave function 10 terms of an overcomplete Gaussian basis set. The wave function evolution is ~v.aluated by allowing eac~ Gaussian to travel along a classical trajectory with its shape held n81d. A Monte Carlo algonthm is employed in the selection of the initial Gaussian basis functions. The frozen Gaussian results are very good for times on the order of a few vibrational periods of the oscillator and remain qualitatively correct for the entire length of the calculations which is 12 vibrational periods. The dependence of the calculations on the width of the Gaussian basis functions is investigated and the effect of a simplifying approximation for the prefactor of the Gaussians is tested.

Gaussian-based techniques for quantum propagation from the time-dependent variational principle: Formulation in terms of trajectories of coupled classical and quantum variables

“…Beginning with the work of Heller [3], FG wave packets have a long history of use in quantum and semiclassical simulations. CS provide a very convenient analog of the classical phase space point, following classical trajectory motion, and ensembles of CS have been extensively used within the semiclassical Herman-Kluk propagator formulation [4,5] (see for example the reviews [6][7][8][9][10]). Recently a number of quantum methods have emerged, such as multiple spawning (MS) by Martinez [11][12][13][14] and Coupled Coherent States (CCS)…”

Long-time and unitary properties of semiclassical initial value representations

“…(9) is done numerically by the Monte Carlo integration procedure described in Ref. [16]. To test the numerical performance of the different methods we chose two different nonlinear potentials which we now introduce.…”

Communication: HK propagator uniformized along a one-dimensional manifold in weakly anharmonic systems

“…[1][2][3][4][5] In brief, this method is a stationary phase approximation of the quantum coherent state propagator evaluated in an uniformized initial value representation. When applied to finding autocorrelations and from the perspective of phase space, every timestep requires the calculation of the overlap of a swarm of "frozen" Gaussians, often selected by Monte Carlo-based sampling dependent on the initial state.…”