Discretized path-integral representation

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 ... 

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. 

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... 

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... 

Fig. 10. Examples of world fine configurations in (a) a path-integral representation where the time direction is continuous and (b) the stochastic series expansion (SSE) representation where the time direction is discrete. Since the SSE representation perturbs not only in offdiagonal terms but also in diagonal terms, additional diagonal terms are present in the representation, indicated by dashed lines...

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... 

Here i, sf,.., and iQ, if,... denote discretizations of the forward and backward path employed in the path integral representation of the forward and reverse time evolution operators, respectively. The influence functional has the structure... 

Equilibrium properties can be determined from the partition function Zq and this quantity can, in turn, be computed using Feynman s path integral approach to quantum mechanics in imaginary time [86]. In this representation of quantum mechanics, quantum particles are mapped onto closed paths r(f) in imaginary time f, 0 f )8ft. The path integral expression for the canonical partition function of a quantum particle is given by the P 00 limit of the quantum path discretized into P segments. 

In the above equations, E, and are the thermal energies of the reactants and of the transition state, respectively. Such a thermodynamic integration was used within a discrete variable representation of QI approximation to compute the rate constant for several collinear triatomic reactions [33]. In Ref. [46], it is generalized and presented in a form suitable for a path integral evaluation. Unlike gr and Caa, the energies are normalized quantities because they can be written as logarithmic derivatives ... 

An alternative way to obtain a semiclassically exact description for systems involving discrete quantum degrees of freedom is based on the spin-coherent state representation. In particular, the spin-coherent state path integral has been used to investigate the semiclassical description of spin systems. As has been discussed in detail in Ref. 70 there exists a... 

In discussing the configurational statistics of individual chains, we have already considered taking the limit as a discrete chain (or path) approaches a continuous one. So this limit is performed on (10.4). Provided it is independent of the manner in which the limit is taken (thus the use of an arbitrary lattice), the definition of a functional integral is obtained. In this limit the lattice point i is the representation of the point r in real space and becomes continuous. Writing... 

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