Initial value representation method

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... 

Most of the applications of initial-value representation methods in recent years have employed the Herman-Kluk (coherent-state) representation of the semiclassical propagator,which for a general n-dimensional system can be written as... 

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... 

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. 

Applications of exact quantum mechanics to systems with four atoms involved is still a major numerical burden, not to mention application to even bigger systems, and so approximate methods based on the solution of the classical equations of motion are nevertheless highly desired. In this context the semiclassical initial value representation (IVR), a precursor of which had already been derived in Miller s 1970 paper [2], may turn out to become a significant tool in molecular dynamics (see Ref. [16] for a comprehensive list of recent references). In the IVR approach the quantum mechanical 5-matrix elements are approximated by a (multi-dimensional) integral over the initial classical phase space and the integrand contains only ingredients from clas-... 

This transfomi also solves the boundary value problem, i.e. there is no need to find, for an initial position x and final position a ", tlie trajectory that coimects the two points. Instead, one simply picks the initial momentum and positionp, x and calculates the classical trajectories resulting from them at all times. Such methods are generally referred to as initial variable representations (IVR). 

The remaining aspect of a trajectory simulation is choosing the initial momenta and coordinates. These initial conditions are chosen so that the results from an ensemble of trajectories may be compared with experiment and/or theory, and used to make predictions about the chemical system s molecular dynamics. In this chapter Monte Carlo methods are described for sampling the appropriate distributions for initial values of the coordinates and momenta. Trajectories may be integrated in different coordinates and conjugate momenta, such as internal [7], Jacobi [8], and Cartesian. However, the Cartesian coordinate representation is most general for systems of any size and the Monte Carlo selection of Cartesian coordinates and momenta is described here for a variety of chemical processes. Many of... 

Meshless methods belong to a class of techniques for solving boundary/initial value partial differential equations in which both geometry representation and numerical discretization are principally performed based on nodes or particles. In meshless methods, there is no inherent reliance on a particular mesh topology, meaning that no element connectivity is required. In practice, however, in many meshless methods, recourse must be taken to some kind of background meshes at least in one stage of the implementation. 

Fig. 1.7 Schematic representation of the three-isotope exchange method. Natural samples plotted on the primary mass fractionation hne (PF). Initial isotopic composition are mineral (Mo) and water (Wo) which is well removed from equilibrium with Mq in 8 0, but very close to equUibrium with Mo in 5 0. Complete isotopic equihbrium is defined by a secondary mass fractionation hne (SF) parallel to PF and passing through the bulk isotopic composition of the mineral plus water system. Isotopic compositions of partially equilibrated samples are Mf and Wf and completely equilibrated samples are Mg and Wg. Values for Me and W. can be determined by extrapolation from the measured values of M , Mf, Wo, and Wf (after Matthews et al. 1983a...

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