Path integral methods examples

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. 

In the preceding sections we have discussed how quantum effects of the electrons are treated first of all within density-functional theory, but we emphasize that, from a conceptual point of view, treatments based on the Hartree-Fock approximation are fairly similar. There exist some few methods where the quantum treatment of the electrons is extended by a quantum treatment of (some of) the nuclei (see, e.g., refs. 60-62) but whereas the electrons still are treated within the standard electronic-strueture approaehes, the path-integral method of Feynman is used for the nuclei. The basie ideas behind these will be briefly outlined here followed by some few examples of their implieations. 

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. 

In this section, we will discuss some examples from the literature, in which the approximation methods derived in this chapter have been used. In several cases, the approximations have been compared with more-accurate path integral simulations to assess their validity. This is not meant as a full review rather, several case studies have been chosen to illustrate the tools we have developed. We will first look at simpler examples and then discuss water models and applications in enzyme kinetics. 

FTIR, TDLAS, and LIF are in situ techniques, whereas DOAS is a long-path method that gives only a path-integrated result. NO, N02, NO,3, and HONO have been successfully measured with DOAS even in rural and remote regions. PAN, HN03, and NH3 have been measured with FTIR in urban areas, but its sensitivity at present is not adequate for levels below a few parts per billion by volume. NO, N02, PAN, HN03, and NH3 have been measured with TDLAS down to sub-ppbv levels. A review with references for applications of these three methods is available (13). The LIF method has been more recently developed, has been applied to the measurement of NO, N02, NH3, and HONO (see reference 14 for an example), and offers sensitivity down to the parts per trillion by volume level. 

Warshel and Chu [42] and Hwang et al. [60] were the first to calculate the contribution of tunneling and other nuclear quantum effects to PT in solution and enzyme catalysis, respectively. Since then, and in particular in the past few years, there has been a significant increase in simulations of quantum mechanical-nuclear effects in enzyme and in solution reactions [16]. The approaches used range from the quantized classical path (QCP) (for example. Refs. [4, 58, 95]), the centroid path integral approach [54, 55], and variational transition state theory [96], to the molecular dynamics with quantum transition (MDQT) surface hopping method [31] and density matrix evolution [97-99]. Most studies of enzymatic reactions did not yet examine the reference water reaction, and thus could only evaluate the quantum mechanical contribution to the enzyme rate constant, rather than the corresponding catalytic effect. However, studies that explored the actual catalytic contributions (for example. Refs. [4, 58, 95]) concluded that the quantum mechanical contributions are similar for the reaction in the enzyme and in solution, and thus, do not contribute to catalysis. 

Several other path integral-based approaches to compute KIE exist, which however do not use QI approximation. These include, for example, approaches using other quantum TSTs [55-60] or the quantized classical path method [7,61,62]. 

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