Using Bennett’s Monte Carlo (MC) method, we calculate the quantum partition functions of path integral formulations. First, from numerically exact results for a harmonic oscillator and a double-well potential, we discuss how fast each approximate partition function converges to the exact value as the number of integral variables involved in the formulation is increased. It turns out that most effective and most suitable for the MC simulation is Takahashi and Imada’s path integral fomulation based on a modified Trotter formula in which the original potential is replaced with an effective one. This formulation is well balanced between the following two factors: the effect of zero potential energy is underestimated, resulting in an improper increase in the partition function; and, on the other hand, effective potential restricts the motion of fictitious particles born in the formulation so that the partition function value tends to be smaller. Fictitious particles can be treated as classical ones. We therefore can apply Bennett’s MC method to calculating the ratio of two quantum partition functions (of a system under consideration and a reference system). As the number of fictitious particles N is increased, choice of reference system becomes less and less important and multistage sampling becomes dispensable. This, to some extent, compensates for the expense that N is larger than the real particle number. The tunneling mechanism of fictitious particles in the simulation is discussed.
In this paper we calculate the sorption characteristics (isotherm and isosteric heat) of Ar and N2 in dehydrated zeolite 4A on the basis of classical statistical mechanics. It is demonstrated that the canonical partition functions for various numbers of sórbate molecules, which are involved in the calculations, can be evaluated by the Monte Carlo method proposed by Bennett and Voter. In contrast to most statistical mechanics calculations of sorptions previously done, which were intended for the low sórbate concentration case (i.e., the sorbate-sorbate interaction is not considered), this Monte Carlo method provides the way of evaluating the partition functions even if the sórbate concentration is considerably high (i.e., even if the gas in contact with the crystal is of high pressure). In this method, sorbate-sorbate interaction can be fully taken into account and one can predict how many sórbate molecules can be accommodated in the zeolite. The combination of the London approximation for dispersion energy and the point charge model for polarization energy (the assumption that the lattice atoms in the zeolite have a point charge) is applied to calculating the Ar-zeolite 4A interaction potential. The isotherm and isosteric heat calculated from this potential agree with the corresponding experimental ones. Close examinations, however, reveal that the coincidence is due to the overestimation of polarization energy. This originates from the inadequate point charge distribution and the fault inherent in the point charge model. The effect of the three-body interaction among two Ar atoms and zeolite 4A is examined, indicating that this effect is not significant for the Ar-zeolite 4A sorption system. The three-body interaction among Ar atoms is of no importance. It is shown for N2-zeolite 4A that the quadrupole of N2 interacting with the electric field in the zeolite fills the role of rendering energetic heterogeneity to the N2-zeolite 4A interaction potential. Furthermore, the models for partition functions presented by Ruthven et al. and based on the lattice gas model are compared, for various numbers of sorba tes, with the Monte Carlo simulation values, showing that for the calculated Ar-zeolite 4A interaction potential the lattice gas model is preferable to the Ruthven model. An adjustable parameter in the lattice gas model, namely, the number of trap sites in a cavity, is determined as 6-7, which is in accord with the number of sites in the calculated interaction potential.
We offer a way of determining the temperature range in which a path integral (PI) formulation of the quantum partition function works well and a way of calculating the ground state properties without employing extremely low temperatures (in order to elude the awkward problem that the quantities calculated by the PI formulation become inaccurate with decreasing temperature owing to unavoidable truncation of an infinite number of path integral variables). The fact that the PI energy, specific heat, etc. behave in a low temperature range against physical laws makes it possible to locate the ‘‘marginal’’ temperature at which the PI specific heat begins to grow infinitely and to estimate the lowest temperature at which the PI formulation functions well (the ‘‘threshold temperature’’). Whether or not the threshold temperature is low enough to extract only the ground state properties can be answered by either checking if the PI energy and free energy are equal at the threshold temperature or checking if the PI specific heat is relatively small thereat. If the system is in the ground state at the threshold temperature obtained, it is recommended to calculate the ground state properties at this temperature. This scheme can be executed by Monte Carlo methods, being open to many-particle systems. Using the discretized PI formulations, we apply the above procedure to a harmonic oscillator and a double-well potential. It is concluded that this scheme is successful at least as long as the potential is a slowly varying function of coordinates.
The adsorption process of methanol vapour on porous materials such as molecular sieves 3A, 4A, 5A, natural mordenite, natural clinoptilolite, activated alumina, silica gel and active carbon was investigated by using a ceramic gas sensor. The measurement of methanol vapour adsorption in the range from the initial relative vapour pressure of 0.6 to the final relative vapour pressure of below 0.35 was found to be suitable for the estimation of specific surface area of hydrophobic porous materials. The results obtained were compared with the data based on the nitrogen adsorption method. The measurement by this method is considerably simple and reproducible so that the study of methanol vapour adsorption process and the estimation of specific surface area of hydrophobic porous materials become easy. Thus, the application of this method seems feasible for such line processes as solor heating and cooling, and solvent recovery from active carbon.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.