Path-integral

When quantum effects are large, the PF can be evaluated by path integral methods [H], Our exposition follows a review article by Gillan [12], Starting with the canonical PF for a system of particles... 

Gillan M J 1990 Path integral simulations of quantum systems Computer Modeling of Fluids and Polymers ed C R A Catlow et al (Dordrecht Kluwer)... 

Alavi A 1996 Path integrals and ab initio molecular dynamics Monte Carlo and Molecular Dynamics of Condensed Matter Systems ed K Binder and G Ciccotti (Bologna SIF)... 

Berne B J and Thirumalai D 1986 On the simulation of quantum systems path integral methods Ann. Rev. Rhys. Chem. 37 401... 

Voth G 1996 Path integral centroid methods Advances in Chemical Physics, New methods in Computational Quantum Mechanics vol XCIII, ed I Prigogine and S A Rice... 

As a result of several complementary theoretical efforts, primarily the path integral centroid perspective [33, 34 and 35], the periodic orbit [36] or instanton [37] approach and the above crossover quantum activated rate theory [38], one possible candidate for a unifying perspective on QTST has emerged [39] from the ideas from [39, 40, 4T and 42]. In this theory, the QTST expression for the forward rate constant is expressed as [39]... 

The quantity is the Feynman path integral centroid density [43] that is understood to be expressed asymptotically as... 

It should be noted that in the cases where y"j[,q ) > 0, the centroid variable becomes irrelevant to the quantum activated dynamics as defined by (A3.8.Id) and the instanton approach [37] to evaluate based on the steepest descent approximation to the path integral becomes the approach one may take. Alternatively, one may seek a more generalized saddle point coordinate about which to evaluate A3.8.14. This approach has also been used to provide a unified solution for the thennal rate constant in systems influenced by non-adiabatic effects, i.e. to bridge the adiabatic and non-adiabatic (Golden Rule) limits of such reactions. 

Voth G A, Chandler D and Miller W H 1989 Time correlation function and path integral analysis of quantum rate constants J. Phys. Chem. 93 7009... 

Voth G A 1993 Feynman path integral formulation of quantum mechanical transition state theory J. Phys. Chem. 97 8365... 

Lobaugh J and Voth G A 1994 A path integral study of electronic polarization and nonlinear coupling effects in condensed phase proton transfer reactions J. Chem. Phys. 100 3039... 

Here is a complex time which is given by t = t- hl2kT. Methods for evaluating this equation have included path integrals [45], wavepackets [48, 49] and direct evaluation of the trace in square integrable basis sets [ ]. 

Miller W H 1970 Semiclassical theory of atom-diatom collisions path integrals and the classical S matrix J. Chem. Phys. 53 1949-59... 

In this section we look briefly at the problem of including quantum mechanical effects in computer simulations. We shall only examine tire simplest technique, which exploits an isomorphism between a quantum system of atoms and a classical system of ring polymers, each of which represents a path integral of the kind discussed in [193]. For more details on work in this area, see [22, 194] and particularly [195, 196, 197]. 

This is better understood with a picture see figure B3.3.11. The discretized path-integral is isomorphic to the classical partition fiinction of a system of ring polymers each having P atoms. Each atom in a given ring corresponds to a different imaginary tune point p =. . . P. represents tire interatomic interactions... 

It is necessary to sum over these pemuitations in a path integral simulation. (The same sum is needed for bosons, without the sign factor.) For femiions, odd pemuitations contribute with negative weight. Near-cancelling positive and negative pemuitations constitute a major practical problem [196]. 

Feynman R P and Hibbs A R 1965 Quantum Mechanics and Path Integrals (New York McGraw-Hill)... 

Tuckerman M E and Hughes A 1998 Path integral molecular dynamics a computational approach to quantum statistical mechanics Classical and Quantum Dynamics In Condensed Phase Simulations ed B J Berne, G Ciccotti and D F Coker (Singapore World Scientific) pp 311-57... 

Herman M F, Bruskin E J and Berne B J 1982 On path integral Monte Carlo simulations J. Chem. Phys. 76 5150-5... 

Tuckerman M, Berne B J, Martyna G J and Klein M L 1993 Efficient molecular dynamics and hybrid Monte Carlo algorithms for path integrals J. Chem. Phys. 99 2796-808... 

Sprik M, Klein M L and Chandler D 1985 Staging—a sampling technique for the Monte-Carlo evaluation of path-integrals Phys. Rev. B 31 4234-44... 

In the classical region of space, where the potential is less than the energy, the standard formula leads to an approximate relation between phase and modulus in the form of the following path integral ([237], Section 28)... 

Hi) The use of quantum methods to obtain correct statistical static (but not dynamic) averages for heavy quantum particles. In this category path-integral methods were developed on the basis of Feynman s path... 

It seems that surface hopping (also called Molecular Dynamics with Quantum Transitions, MDQT) is a rather heavy tool to simulate proton dynamics. A recent and promising development is path integral centroid dynamics [123] that provides approximate dynamics of the centroid of the wavefunctions. Several improvements and applications have been published [123, 124, 125, 126, 127, 128). 

Feynman, R.P., Hibbs, A.R. Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965). 

Martyna, G.J. Adiabatic path integral molecular dynamics methods. I. Theory. J. Chem. Phys. 104 (1996) 2018-2027. 

Application of a Stochastic Path Integral Approach to the Computations of an Optimal Path and Ensembles of Trajectories ... 

Abstract. A stochastic path integral is used to obtain approximate long time trajectories with an almost arbitrary time step. A detailed description of the formalism is provided and an extension that enables the calculations of transition rates is discussed. 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

D[X t) is used to denote a path integral. Hence, equation (14) corresponds to a summation of all paths leading from X(0) to X t). The same expression is used for the Brownian trajectories and for Newtonian s trajectories with errors. The action is of course different in both cases. 

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