Reconciling semiclassical and Bohmian mechanics: IV. Multisurface dynamics

In previous articles (J. Chem. Phys. 2004, 121, 4501; 2006, 124, 034115; 2006, 124, 034116) a bipolar counter-propagating wave decomposition, Psi = Psi+ + Psi-, was presented for stationary states Psi of the one-dimensional Schr?dinger equation, such that the components Psi+/- approach their semiclassical WKB analogs in the large action limit. The corresponding bipolar quantum trajectories are classical-like and well-behaved, even when Psi has many nodes or is wildly oscillatory. In this paper, the method is generalized for multisurface scattering applications and applied to several benchmark problems. A natural connection is established between intersurface transitions and (+ <--> -) transitions.     

Reconciling semiclassical and Bohmian mechanics. I. Stationary states

The semiclassical method is characterized by finite forces and smooth, well-behaved trajectories, but also by multivalued representational functions that are ill behaved at caustics. In contrast, quantum trajectory methods--based on Bohmian mechanics (quantum hydrodynamics)--are characterized by divergent forces and erratic trajectories near nodes, but also well-behaved, single-valued representational functions. In this paper, we unify these two approaches into a single method that captures the best features of both, and in addition, satisfies the correspondence principle. Stationary eigenstates in one degree of freedom are the primary focus, but more general applications are also anticipated.     

Reconciling semiclassical and Bohmian mechanics. II. Scattering states for discontinuous potentials

In a previous paper [B. Poirier, J. Chem. Phys. 121, 4501 (2004)] a unique bipolar decomposition, psi = psi1 + psi2, was presented for stationary bound states Psi of the one-dimensional Schrodinger equation, such that the components psi1 and psi2 approach their semiclassical WKB analogs in the large action limit. Moreover, by applying the Madelung-Bohm ansatz to the components rather than to Psi itself, the resultant bipolar Bohmian mechanical formulation satisfies the correspondence principle. As a result, the bipolar quantum trajectories are classical-like and well behaved, even when psi has many nodes or is wildly oscillatory. In this paper, the previous decomposition scheme is modified in order to achieve the same desirable properties for stationary scattering states. Discontinuous potential systems are considered (hard wall, step potential, and square barrier/well), for which the bipolar quantum potential is found to be zero everywhere, except at the discontinuities. This approach leads to an exact numerical method for computing stationary scattering states of any desired boundary conditions, and reflection and transmission probabilities. The continuous potential case will be considered in a companion paper [C. Trahan and B. Poirier, J. Chem. Phys. 124, 034116 (2006), following paper].     

Reconciling semiclassical and Bohmian mechanics. III. Scattering states for continuous potentials

In a previous paper [B. Poirier, J. Chem. Phys. 121, 4501 (2004)] a unique bipolar decomposition psi = psi1 + psi2 was presented for stationary bound states Psi of the one-dimensional Schrodinger equation, such that the components psi1 and psi2 approach their semiclassical WKB analogs in the large-action limit. The corresponding bipolar quantum trajectories, as defined in the usual Bohmian mechanical formulation, are classical-like and well behaved, even when Psi has many nodes or is wildly oscillatory. A modification for discontinuous potential stationary scattering states was presented in a second, companion paper [C. Trahan and B. Poirier, J. Chem. Phys.124, 034115 (2006), previous paper], whose generalization for continuous potentials is given here. The result is an exact quantum scattering methodology using classical trajectories. For additional convenience in handling the tunneling case, a constant-velocity-trajectory version is also developed.     

Wavepacket propagation using time-sliced semiclassical initial value methods

A new semiclassical initial value representation (SC-IVR) propagator and a SC-IVR propagator originally introduced by Kay [J. Chem. Phys. 100, 4432 (1994)], are investigated for use in the split-operator method for solving the time-dependent Schrodinger equation. It is shown that the SC-IVR propagators can be derived from a procedure involving modified Filinov filtering of the Van Vleck expression for the semiclassical propagator. The two SC-IVR propagators have been selected for investigation because they avoid the need to perform a coherent state basis set expansion that is necessary in other time-slicing propagation schemes. An efficient scheme for solving the propagators is introduced and can be considered to be a semiclassical form of the effective propagators of Makri [Chem. Phys. Lett. 159, 489 (1989)]. Results from applications to a one-dimensional, two-dimensional, and three-dimensional Hamiltonian for a double-well potential are presented.     

Wavepacket quantum dynamics

The article reviews the use of wavepackets in molecular quantum dynamics. The basic theory concerned with their use in both reactive molecular scattering and photodissociation dynamics is outlined. The great advantage of using wavepackets is that the full S matrix for the scattering problem need not be evaluated, and the numerical effort can be concentrated on those initial molecular quantum states which are of interest. Wavepackets may be used within a time-dependent or a time-independent framework, both are discussed and compared. Some examples of calculations from both reactive scattering and photodissociation theory are given.     

Quantum dynamics with Bohmian trajectories

We describe the advantages and disadvantages of numerical methods when Bohmian trajectory grids are used for numerical simulations of quantum dynamics. We focus on the crucial noncrossing property of Bohmian trajectories, which, numerically, must be given careful attention. Failure to do so causes instabilities or leads to false simulations.     

Bohmian mechanics with complex action: a new trajectory-based formulation of quantum mechanics

In recent years there has been a resurgence of interest in Bohmian mechanics as a numerical tool because of its local dynamics, which suggest the possibility of significant computational advantages for the simulation of large quantum systems. However, closer inspection of the Bohmian formulation reveals that the nonlocality of quantum mechanics has not disappeared-it has simply been swept under the rug into the quantum force. In this paper we present a new formulation of Bohmian mechanics in which the quantum action, S, is taken to be complex. This leads to a single equation for complex S, and ultimately complex x and p but there is a reward for this complexification-a significantly higher degree of localization. The quantum force in the new approach vanishes for Gaussian wave packet dynamics, and its effect on barrier tunneling processes is orders of magnitude lower than that of the classical force. In fact, the current method is shown to be a rigorous extension of generalized Gaussian wave packet dynamics to give exact quantum mechanics. We demonstrate tunneling probabilities that are in virtually perfect agreement with the exact quantum mechanics down to 10(-7) calculated from strictly localized quantum trajectories that do not communicate with their neighbors. The new formulation may have significant implications for fundamental quantum mechanics, ranging from the interpretation of non-locality to measures of quantum complexity.     

A coherent state approach to semiclassical nonadiabatic dynamics

A semiclassical (SC) approximation to the quantum mechanical propagator for nonadiabatic systems is derived. Our derivation starts with an exact path integral expression that uses canonical coherent states for the nuclear degrees of freedom and spin coherent states for the electronic degrees of freedom. A stationary path approximation (SPA) is then applied to the path integral to obtain the SC approximation. The SPA results in complex classical trajectories of both nuclear and electronic degrees of freedom and a double ended boundary condition. The root search problem is solved using the previously proposed "real trajectory local search" algorithm. The SC approximation is tested on three simple one dimensional two-state systems proposed by Tully [J. Chem. Phys. 93, 1061 (1990)], and the SC results are compared to Ehrenfest and surface hopping predictions. Excellent agreement with quantum results is reached when the SC trajectory is far away from caustics. We discuss the origin of caustics in this SC formalism and the strengths and weaknesses of this approach.     

Optimized Monte Carlo sampling in forward-backward semiclassical dynamics

Forward-backward semiclassical dynamics (FBSD) provides a rigorous and powerful methodology for calculating time correlation functions in condensed phase systems characterized by substantial quantum mechanical effects associated with zero-point motion, quantum dispersion, or identical particle exchange symmetries. The efficiency of these simulations arises from the use of classical trajectories to capture all dynamical information. However, full quantization of the density operator makes these calculations rather expensive compared to fully classical molecular dynamics simulations. This article discusses the convergence properties of various correlation functions and introduces an optimal Monte Carlo sampling scheme that leads to a significant reduction of statistical error. A simple and efficient procedure for normalizing the FBSD results is also discussed. Illustrative examples on model systems are presented.     

Phase-space dynamics and quantum mechanics

Ann-dimensional system with a classical HamiltonianH(p, q, t) may be described by a phase-space distribution function D(q, p, t). The dynamical equation for D(q, p, t) is postulated to be

Phase-space dynamics and quantum mechanics

Summary In a recent paper Deal has postulated a new dynamical equation for quantum mechanical phase-space distribution functions. We analyze the new equation and show that it may be related to the traditional standard and antistandard phase-space representations of quantum mechanics. A brief review of these and other representations is also given.     

Ultrafast excited-state dynamics of tetraphenylethylene studied by semiclassical simulation

Detailed simulation study is reported for the excited-state dynamics of photoisomerization of cis-tetraphenylethylene (TPE) following excitation by a femtosecond laser pulse. The technique for this investigation is semiclassical dynamics simulation, which is described briefly in the paper. Upon photoexcitation by a femtosecond laser pulse, the stretching motion of the ethylenic bond of TPE is initially excited, leading to a significant lengthening of ethylenic bond in 300 fs. Twisting motion about the ethylenic bond is activated by the energy released from the relaxation of the stretching mode. The 90 degrees twisting about the ethylenic bond from an approximately planar geometry to nearly a perpendicular conformation in the electronically excited state is completed in 600 fs. The torsional dynamics of phenyl rings which is temporally lagging behind occurs at about 5 ps. Finally, the twisted TPE reverts to the initial conformation along the twisting coordinate through nonadiabatic transitions. The simulation results provide a basis for understanding several spectroscopic observations at molecular levels, including ultrafast dynamic Stokes shift, multicomponent fluorescence, viscosity dependence of the fluorescence lifetime, and radiationless decay from electronically excited state to the ground state along the isomerization coordinate.     

Herman-Kluk semiclassical dynamics of molecular rotations in laser fields

The action-angle mapping algorithm [R. Saha and M. Ovchinnikov, J. Chem. Phys. 124, 204112 (2006)] is utilized to provide a Herman-Kluk semiclassical initial value representation (SC-IVR) treatment of quantum dynamics of systems with non-Cartesian degrees of freedom. The non-Cartesian system under investigation is a linear rotor molecule in static electric and pulsed laser field. The results demonstrate that the SC-IVR procedure described in this work provides an accurate representation of quantum rotational dynamics of the system.     

A semiclassical hybrid approach to many particle quantum dynamics

We analytically derive a correlated approach for a mixed semiclassical many particle dynamics, treating a fraction of the degrees of freedom by the multitrajectory semiclassical initial value method of Herman and Kluk [Chem. Phys. 91, 27 (1984)] while approximately treating the dynamics of the remaining degrees of freedom with fixed initial phase space variables, analogously to the thawed Gaussian wave packet dynamics of Heller [J. Chem. Phys. 62, 1544 (1975)]. A first application of this hybrid approach to the well studied Secrest-Johnson [J. Chem. Phys. 45, 4556 (1966)] model of atom-diatomic collisions is promising. Results close to the quantum ones for correlation functions as well as scattering probabilities could be gained with considerably reduced numerical effort as compared to the full semiclassical Herman-Kluk approach. Furthermore, the harmonic nature of the different degrees of freedom can be determined a posteriori by comparing results with and without the additional approximation.     

Detailed dynamics of the nonradiative deactivation of adenine: a semiclassical dynamics study

A realistic dynamics simulation study is reported for the ultrafast radiationless deactivation of 9H-adenine. The simulation follows two different excitations induced by two 80 fs (fwhm) laser pulses that are different in energy: one has a photon energy of 5.0 eV, and the other has a photon energy of 4.8 eV. The simulation shows that the excited molecule decays to the electronic ground state from the (1)pipi* state in both excitations but through two different radiationless pathways: in the 5.0 eV excitation, the decay channel involves the out-of-plane vibration of the amino group, whereas in the 4.8 eV excitation, the decay strongly associates with the deformation of the pyrimidine at the C 2 atom. The lifetime of the (1) npi* state determined in the simulation study is 630 fs for the 5.0 eV excitation and 1120 fs for the 4.8 eV excitation. These are consistent with the experimental values of 750 and 1000 fs. We conclude that the experimentally observed difference in the lifetime of the (1) npi* state at various excitations results from the different radiationless deactivation pathways of the excited molecule to the electronic ground state.     

Wavepacket dynamics on dynamically adapting grids: application of the equidistribution principle

A moving grid approach to wavepacket dynamics is described that enables grid points to be used efficiently in regions where high resolution of the wavepacket is required. The grid movement is based on the principle of equidistribution and by using a grid smoothing technique the grid points trace a path that continuously adapt to reflect the dynamics of the wavepacket. The technique is robust and allows accurate computations to be obtained for long wavepacket propagation times. Results are presented for two systems: tunnelling dynamics in a double well potential and scattering of a wavepacket from a repulsive Eckart barrier.