Globally uniform semiclassical wave functions for multidimensional systems

Abstract

The globally uniform semiclassical approximation for energy eigenstates developed by D. Zor and K. G. Kay ͓Phys. Rev. Lett. 76, 1990 ͑1996͔͒ is derived explicitly for the case of multidimensional systems and is applied to two test cases. The adiabatic switching approximation is used to obtain various quantities that enter the semiclassical expression. Two formulations of the method are examined: one requiring several trajectories for each desired state and another requiring only a single trajectory per state. The multitrajectory version yields accurate results for all states investigated ͑overlap between semiclassical and quantum eigenstates Ͼ0.98͒, including some influenced by classical chaos. The single-trajectory treatment, however, is more efficient, gives accurate results for regular states, and is even applicable for certain chaotic states, although the multiple-trajectory method is preferred in such cases. Despite the substantial resemblance of the present theory to the frozen Gaussian approximation ͑FGA͒, it is a true semiclassical approximation and is found to produce wave functions that are significantly more accurate than those obtained from the FGA for all states examined.

Nonadiabatic molecular dynamics simulations of the photofragmentation and geminate recombination dynamics in size-selected I2−⋅(CO2)n cluster ions

“…At the heart of this new approach is the semiclassical procedure for propagating Gaussian basis functions to describe the nuclear quantum state which is based on the initial value representation as first used by Herman and Kluk,56,57 and subsequently developed by others. [58][59][60][61][62][63][64][65] In Miller's application to the problem of nonadiabatic dynamics, the electronic mixing coefficients are also treated semiclassically in a similar fashion to the earlier ''classical electron'' approach. 66 Implementation of these new methods to study large many-body systems of the kind we consider here is still fraught with considerable difficulty, and it is for this reason that we have chosen to employ a surface hopping approach in the calculations reported here.…”

Resonance-induced spectral tuning

“…This has even been extended recently to computation of a semiclassical wave function for a chaotic system [19]. However, despite these and other advances [20][21][22][23][24][25][26] in semiclassical treatment of chaotic systems, there is as yet no generally used semiclassical theory specifically for eigenstates in the chaotic regime. )…”

Uniform semiclassical IVR treatment of the S-matrix

“…The general theory 64,65 actually permits these quantities to be any complex functions of , , R, and r provided that ϱϾRe ␥ 1 , Re ␥ 2 Ͼ0. ͑4͒ are ''arbitrary parameters.''…”

“…64,65 The present parameterization of the manifold in terms of variables and is especially convenient for a scattering system. ͑4͒ is derived in Refs.…”

“…However, the MM-IVR approach has a number of disadvantages. In analogy to the IVR treatments of the propagator 36,40 and wave function, 64,65 it is not hard to show that caustics prevent the MM-IVR expressions from approaching the accurate quantum S-matrix results uniformly for all transitions and all energies as ប approaches zero. 22,23 A more fundamental problem is that the accuracy of the MM-IVR method is frequently not sufficient to yield quantitative agreement with exact results 22 and is, in fact, sometimes quite poor.…”