Semiclassical initial value representation versus exact quantum reaction probabilities.

Eli Pollak

Eva Martín-Fierro, Jose M. Gomez Llorente

Shesheng Zhang, Eli Pollak

The recently derived exact representation of the quantum propagator in terms of semiclassical initial value representations (SCIVR) is used to optimize the width parameter in the SCIVR. Minimization of the expectation value of the correction operator related to the SCIVR leads to improved convergence of the representation. A test on a model one-dimensional double-well potential demonstrates that this optimization can give essentially exact results using only the first two terms in the SCIVR expansion of the exact propagator.

Herbert Massmann, K. Möhring, Peter A. Ring

The reflection and transmission of a wave on a two-dimensional barrier is studied quantum mechanically and semiclassically. The simple model consists of two degrees of freedom. One of them contains a barrier (of the Eckart type); the second allows only a bound motion in discrete quantum states. These two degrees of freedom are coupled. The transition probabilities for transitions between the different quantum states can, in the model considered here, be evaluated analytically, both quantum mechanically and semiclassically. Several of the so called uniform semiclassical approximations and also the initial value integral representation for the semiclassical transition amplitudes are considered and compared to the exact quantum mechanical results. Some new insight on the semiclassical treatment of the tunneling process is gained, specially concerning the proper choice of the integration path for the initial value integral representation of the transition amplitudes.

Shesheng Zhang, Eli Pollak

A new class of prefactor free semiclassical initial value representations (SCIVR) of the quantum propagator is presented. The derivation is based on the physically motivated demand, that on the average in phase space and in time, the propagator obey the exact quantum equation of motion. The resulting SCIVR series representation of the exact quantum propagator is also free of prefactors. When using a constant width parameter, the prefactor free SCIVR propagator is identical to the frozen Gaussian propagator of Heller [J. Chem. Phys. 75, 2923 (1981)]. A numerical study of the prefactor free SCIVR series is presented for scattering through a double slit potential, a system studied extensively previously by Gelabert et al. [J. Chem. Phys. 114, 2572 (2001)]. As a basis for comparison, the SCIVR series is also computed using the optimized Herman-Kluk SCIVR. We find that the sum of the zeroth order and the first order terms in the series suffice for an accurate determination of the diffraction pattern. The same exercise, but using the prefactor free propagator series needs also the second order term in the series, however the numerical effort is not greater than that needed for the Herman-Kluk propagator, since one does not need to compute the monodromy matrix elements at each point in time. The numerical advantage of the prefactor free propagator grows with increasing dimensionality of the problem.