Discretization integration path

Coalson, R. D., On the connection between Fourier coefficient and discretized Cartesian path integration, J. Chem. Phys. 1986, 85, 926... 

The techniques of discretized Feynman path integrals make the use of Eq. 41 practical for the more general case of quantized nuclear motion which is not restricted to harmonic behavior [36, 94, 99b[. Applications of this approach are discussed in Section 1.5 of this chapter. 

Figure 2. Calculation of an operator equilibrium average A) in the discretized Feynman path-integral picture. The operator [ (t)] can be evaluated anywhere on the path-integral ring, or thermal loop, 0st< /3, due to the cyclic invariance of the trace operation. The centroid variable, which is off the loop, is also shown.

Figure 5.6 Illustration of the discretization of the integration path into Mr = 7 overlapping intervals. A configuration (replica) is asscxriated with each interval. Each interval, in turn, is comprised of four states of an expanded ensemble. Expanded-ensemble Monte Carlo moves that change the values of XN and % N within one interval are indicated by horizontal...

In order to calculate the free energy difference between the disordered state, and the lamellar ordered structure at Xfinai N = 20, we discretize both branches of the integration path such that the distributions of the integrands, ord and ext. overlap for neighboring points along the path of integration. 

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. 

Fiber and Karplus [38] presented an effective set of numerical methods for computing the reaction paths based on this approximation. First the path is discretized—it is expressed as a chain of intermediate configurations of the system rj,. The line integrals of Fq. (19) are then written as... 

Let us now turn to the case T -> 0. First of all, somewhat suspicious is the combination of the continuous integral (2.1) with the discrete partition function Zq (2.13), usual for CLTST. This serious deficiency cannot be circumvented in the framework of this CLTST-based formalism, and a more rigorous reasoning is needed to describe the quantum situation. Introduction of adequate methods will be the objective of the next sections devoted to the path-integral formalism, so here we... 

Lee s discretization of nonrclativistic quantum mechanics is almost as straightforward as the discretization of classical mechanics discussed above he uses Feynman s path integral formalism [feyn65b]. 

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... 

To determine A we need to implement the path integration as a numerical path summation. The path integral in Eqs. 23-24 is in fact isomorphic to the configuration integral of a flexible polymer which interacts with the external potential V(r(P)). This analogy can be made more explicit by considering a discrete approximation to the path integral [71]. If the path is cut up into P... 

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