Path-integral Monte Carlo technique

In this section we review several studies of phase transitions in adsorbed layers. Phase transitions in adsorbed (2D) fluids and in adsorbed layers of molecules are studied with a combination of path integral Monte Carlo, Gibbs ensemble Monte Carlo (GEMC), and finite size scaling techniques. Phase diagrams of fluids with internal quantum states are analyzed. Adsorbed layers of H2 molecules at a full monolayer coverage in the /3 X /3 structure have a higher transition temperature to the disordered phase compared to the system with the heavier D2 molecules this effect is... 

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. 

Sprik M, Klein M L and Chandler D 1985 Staging—a sampling technique for the Monte-Carlo evaluation of path-integrals Phys. Rev. B 31 4234-44... 

The computation of quantum many-body effects requires additional effort compared to classical cases. This holds in particular if strong collective phenomena such as phase transitions are considered. The path integral approach to critical phenomena allows the computation of collective phenomena at constant temperature — a condition which is preferred experimentally. Due to the link of path integrals to the partition function in statistical physics, methods from the latter — such as Monte Carlo simulation techniques — can be used for efficient computation of quantum effects. 

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. 

The probability PXqr q ) fo move the reaction coordinate centroid variable from the reactant configuration to the transition state is readily calculated [108] by PIMC or PIMD techniques [17-19] combined with umbrella sampling [77,108,123] of the reaction coordinate centroid variable. In the latter computational technique, a number of windows are set up which confine the path centroid variable of the reaction coordinate to different regions. These windows connect in a piecewise fashion the possible centroid positions in going from the reactant state to the transition state. A series of Monte Carlo calculations are then performed, one for each window, and the centroid probability distribution in each window is determined. These individual window distributions are then smoothly joined to calculate the overall probability function in Eq. (4.11). An equivalent approach is to calculate the centroid mean force and integrate it from the reactant well to barrier top (i.e., a reversible work approach for the calculation of the quantum activation free energy [109,124]). 

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