Semiclassical propagator

Campolieti G and Brumer P 1994 Semiclassical propagation phase indices and the initial-value formalism Phys. Rev. A 50 997... 

Kay K G 1994 Semiclassical propagation for multidimensional systems by an initial value method J. Chem. Phys. 101 2250... 

Thompson K and Makri N 1999 Influence functionals with semiclassical propagators in combined fonA/ard-backward time J. Chem. Phys. 110 1343... 

Bolte J. and Glaser R. Semiclassical propagation of coherent states with spin-orbit interaction, to appear in Ann. H. Poincare. 

Tunneling Splitting from a Real Time Semiclassical Propagation. 

Since the birth of quantum theory, there has been considerable interest in the transition from quantum to classical mechanics. Because the two formulations are given in a different theoretical framework (nonlinear classical trajectories versus expectation values of linear operators), this transition is far more involved than the naive limit —> 0 suggests. By exploring the classical limit of quantum mechanics, new theoretical concepts have been developed, including path integrals [1], various phase-space representations of quantum mechanics [2], the semiclassical propagator and the trace formula [3], and the notion of quantum... 

Due to the development of efficient initial-value representations of the semiclassical propagator, recently there has been considerable progress in the semiclassical description of multidimensional quantum processes [104—111,... 

Second, the mapping approach to nonadiabatic quantum dynamics is reviewed in Sections VI-VII. Based on an exact quantum-mechanical formulation, this approach allows us in several aspects to go beyond the scope of standard mixed quantum-classical methods. In particular, we study the classical phase space of a nonadiabatic system (including the discussion of vibronic periodic orbits) and the semiclassical description of nonadiabatic quantum mechanics via initial-value representations of the semiclassical propagator. The semiclassical spin-coherent state method and its close relation to the mapping approach is discussed in Section IX. Section X summarizes our results and concludes with some general remarks. 

A semiclassical description is well established when both the Hamilton operator of the system and the quantity to be calculated have a well-defined classical analog. For example, there exist several semiclassical methods for calculating the vibrational autocorrelation function on a single excited electronic surface, the Fourier transform of which yields the Franck-Condon spectmm [108, 109, 150, 244]. In particular, semiclassical methods based on the initial-value representation of the semiclassical propagator [104-111, 245-248], which circumvent the cumbersome root-search problem in boundary-value-based semiclassical methods, have been successfully applied to a variety of systems (see, for example, Refs. 110, 111, 161, and 249 and references therein). The mapping procedure introduced in Section VI results in a quantum-mechanical Hamiltonian with a well-defined classical limit, and therefore it... 

Following a brief introduction of the basic concepts of semiclassical dynamics, in particular of the semiclassical propagator and its initial value representation, we discuss in this section the application of the semiclassical mapping approach to nonadiabatic dynamics. Based on numerical results for the... 

To introduce the basic concept of a semiclassical propagator, let us consider an n-dimensional quantum system with Hamiltonian H, which is assumed to possess a well-defined classical analog. In order to obtain the semiclassical approximation to the transition amplitude between the initial... 

As a consequence, the semiclassical propagator is given as a phase-space integral over the initial conditions qo and Po, which is amenable to a Monte Carlo evaluation. For this reason, semiclassical initial-value representations are regarded as the key to the application of semiclassical methods to multidimensional systems. 

In the past two decades, a variety of semiclassical initial-value representations have been developed [105-111], which are equivalent within the semiclassical approximation (i.e., they solve the Schrodinger equation to first order in H), but differ in their accuracy and numerical performance. Most of the applications of initial-value representation methods in recent years have employed the Herman-Kluk (coherent-state) representation of the semiclassical propagator [105, 108, 187, 245, 252-255], which for a general n-dimensional system can be written as... 

The trajectories in the semiclassical propagator (129) are determined by the equations of motion... 

Another possibility to introduce a semiclasscial initial value representation for the spin-coherent state propagator is to exploit the close relation between Schwinger s representation of a spin system and the spin-coherent state theory [100, 133-135]. To illustrate this approach, we consider an electronic two-level system coupled to Vvib nuclear DoF. Within the mapping approach the semiclassical propagator for this system is given by... 

The semiclassical propagator for a single siuface problem has the form... 

L. Malegat, H. Bachau, A. Hamido, B. Piraux, Analysing a two-electron wavepacket by semiclassically propagating its Fourier components in space, J. Phys. B At. Mol. Opt. Phys. 43 (2010) 245601. 

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