Numerical path integral methods

In this section we discuss a few representative applications of numerical path integral methods. The presentation that follows is not intended to replace detailed reviews of path integral simulations. The calculations discussed below serve as illustrative examples of the range of chemical questions that can be successfully addressed with path integral methods. For more information and for discussion of numerous other important applications of the path integral in chemistry and physics the reader is referred to comprehensive review articles and recent books. 

The path integral offers an insightful approach to time-dependent quantum mechanics and quantum statistical mechanics. In recent years a host of powerful numerical path integral methods have been developed which have enabled simulations of quantum many-particle systems not treatable by other means. 

As we shall show by numerical example, the convergence is frequently rapid. Furthermore, the convergence can be improved by partial averaging techniques, which we discuss later. In contrast to the classical energy, which was obtained by integration over the 3n coordinates of the system, the dimensionality of Eq. (4.31) is 2n(k + 1). The increase in the dimensionality of the integrals is typical of path integral methods. Fortunately Monte Carlo methods depend only weakly on the dimensionality of the problem, and Eq. (4.31) is about as easy to evaluate as the classical problem. 

We have obtained a broad range of detailed quantitative results relating to universality in the absence of the electron-phonon interaction by various theoretical techniques. The methods include perturbation theory, the coherent potential approximation (CPA), field theory, path integral methods, numerical calculations and the potential well analogy. The results include the density of states, the nature of the wave functions, the mean free path, the energy dependent... 

Path integral methods have proven very useful for studying the equilibrium properties of excess electrons in cluster and classical fluids, and recently spectral properties have begun to be accessible as well. Such studies have revealed a wealth of information about the nature of these electronic states and the corresponding solvent structure. Numerous reviews on this subject have appeared since the late 1980s. 

These methods, which probably deserve more attention than they have received to date, simultaneously optimize the positions of a number of points along the reaction path. The method of Elber and Karpins [91] was developed to find transition states. It fiimishes, however, an approximation to the reaction path. In this method, a number (typically 10-20) equidistant points are chosen along an approximate reaction path coimecting two stationary points a and b, and the average of their energies is minimized under the constraint that their spacing remains equal. This is obviously a numerical quadrature of the integral s f ( (.v)where... 

Following Fey nman s original work, several authors pmsued extensions of the effective potential idea to construct variational approximations for the quantum partition function (see, e g., Refs. 7,8). The importance of the path centroid variable in quantum activated rate processes was also explored and revealed, which gave rise to path integral quantum transition state theory and even more general approaches. The Centroid Molecular Dynamics (CMD) method for quantum dynamics simulation was also formulated. In the CMD method, the position centroid evolves classically on the efiective centroid potential. Various analysis and numerical tests for realistic systems have shown that CMD captures the main quantum effects for several processes in condensed matter such as transport phenomena. 

Basically the perturbative techniques can be grouped into two classes time-local (TL) and time-nonlocal (TNL) techniques, based on the Nakajima-Zwanzig or the Hashitsume-Shibata-Takahashi identity, respectively. Within the TL methods the QME of the relevant system depends only on the actual state of the system, whereas within the TNL methods the QME also depends on the past evolution of the system. This chapter concentrates on the TL formalism but also shows comparisons between TL and TNL QMEs. An important way how to go beyond second-order in perturbation theory is the so-called hierarchical approach by Tanimura, Kubo, Shao, Yan and others [18-26], The hierarchical method originally developed by Tanimura and Kubo [18] (see also the review in Ref. [26]) is based on the path integral technique for treating a reduced system coupled to a thermal bath of harmonic oscillators. Most interestingly, Ishizaki and Tanimura [27] recently showed that for a quadratic potential the second-order TL approximation coincides with the exact result. Numerically a hint in this direction was already visible in simulations for individual and coupled damped harmonic oscillators [28]. 

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