Discretized path-integral

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... 

A single calculation of the discrete path integral with a fixed length of time t can be employed to compute the state conditional probability at many other times. It is possible to use segments of the path of time length At, 2At,..., NAt sampled in trajectories of total length of NAt and to compute the corresponding state conditional probabilities. The result of the calculations will make it possible to explore the exponential relaxation of P Ao B,t) for times between 0 and t. 

The p point discretized path integral for Z is obtained by inserting complete sets of states p times ... 

Discretized Path Integral Exact Quantum Result Quadrature Points... 

R. M. Levy, P. Zhang and R. A. Friesner, Variable Quadratic Reference System for Evaluating Discretized Path Integrals. Chem. Phys. Lett., submitted. 

There is considerable interest in the use of discretized path-integral simulations to calculate free energy differences or potentials of mean force using quantum statistical mechanics for many-body systems [140], The reader has already become familiar with this approach to simulating with classical systems in Chap. 7. The theoretical basis of such methods is the Feynmann path-integral representation [141], from which is derived the isomorphism between the equilibrium canonical ensemble of a... 

In the so-called primitive representation of the discretized path-integral approach [141], the canonical partition function for finite P has the form... 

Schweizer, K.S. Stratt, R.M. Chandler, D. Wolynes, P.G., Convenient and accurate discretized path integral methods for equilibrium quantum mechanical calculations, J. Chem. Phys. 1981, 75, 1347-1364... 

Friesner, R.A. Levy, R.M., An optimized harmonic reference system for the evaluation of discretized path integrals, J. Chem. Phys. 1984, 80, 4488-4495... 

By representing the operator containing the potential energy in position state space and the one containing the kinetic energy in momentum space, one obtains the following phase space discretized path integral representation ... 

There have been two principal methods developed to evaluate the kinetic energy using path integral methods. One method, based on Eq. (3.5), has been termed the T-method and the other, based on Eq. (4.1), has bwn termed the //-method. In discretized path integral calculations the T-method and the //-method have similar properties, but in the Fourier method the expressions and the behavior of the kinetic energy evaluated by Monte Carlo techniques are different. 

We next introduce a discretized path integral representation for the nuclear part of the propagators, and choose to do so in a hybrid momentum-coordinate representation [18,41]. This can be accomplished, for example for the forward propagator, by first using the identity... 

Here we review well-known principles of quantum statistical mechanics as necessary to develop a path-integral representation of the partition function. The equations of quantum statistical mechanics are, like so many equations, easy to write down and difficult to implement (at least, for interesting systems). Our purpose here is not to solve these equations but rather to write them down as integrals over configuration space. These integrals can be seen to have a form that is isomorphic to the discretized path-integral representation of the kernel developed in the previous section. 

By comparing analogous terms in ( , x) and Q, we see that we can think of the partition function as a path integral over periodic orbits that recur in a complex time interval equal to i s flh/i = — ifih. There is no claim here that the closed paths used to generate Q correspond to actual quantum dynamics, but simply that there is an isomorphism. We therefore can refer to the equation above as the discretized path-integral (DPI) representation of the partition function. Using Feynman s notation, we have in the infinite-P limit... 

We shall be concerned with the computation of real-time quantities (e.g., correlation functions or time-dependent occupation probabilities). The standard approach in a QMC simulation of such dynamical quantities consists of constructing a suitably discretized path integral expression for the quantum mechanical propagator of the system... 

To compute the two-state dynamics numerically, we employ a discretized path-integral representation of the dynamical quantities [23, 24, 38]. The correlation function (cr (O)cr (f)) can be regarded as the probability amplitude for a sequence of steps in the complex-time plane. In particular, one propagates along the Kadanoff-Baym contour y defined in Fig. 1 z = -1 /3 and measures cr at z = 0 and z = t. Of... 

Path integrals are particularly useful for describing the quantum mechanics of an equilibrium system because the canonical distribution for a single quantum particle in the path integral picture becomes isomorphic with that for a classical ring polymer of quasiparticles [17-19, 26] (cf. Fig. 1). In the discretized path-integral representation, the partition function for a quantum particle is given by the expression... 

In the discretized path-integral picture (i.e., for finite P), the path centroid variable is equivalent to the center of mass of the isomorphic polymer of classical quasiparticles (cf. Fig. 1) such that... 

In order to obtain an expression suitable for numerical calculations, we express the density matrix using the discretized path integral as [1]... 

R. Friesner and R. M. Levy, /. Chem. Phys., 80, 4488 (1984). An Optimized Harmonic Reference System for the Evaluation of Discretized Path Integrals. 

Use of this result in equation (6) leads to the following discretized path integral expression for the propagator. 

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