Test particle method

Simulations in the Gibbs ensemble attempt to combine features of Widom s test particle method with the direct simulation of two-phase coexistence in a box. The method of Panagiotopoulos et al [162. 163] uses two fiilly-periodic boxes, I and II. 

The chemical potential of particles belonging to species a and (3 is measured by using the classical test particle method (as proposed by Fischer and Heinbuch [166]) in parts II and IV of the system i.e., we calculate the average value of (e) = Qxp[—U/kT]), where U denotes the potential energy of the inserted particles. 

The results for the chemical potential determination are collected in Table 1 [172]. The nonreactive parts of the system contain a single-component hard-sphere fluid and the excess chemical potential is evaluated by using the test particle method. Evidently, the quantity should agree well with the value from the Carnahan-Starling equation of state [113]... 

Fig. 20 shows the density profiles in the reactive and nonreactive parts of the system. The number density in the reactive part is very high (a one-component density at the center of this part is 0.596, so the number density of two components is twice as high). However, the density in the nonreactive part is much lower and equal to 0.404. The application of the test particle methods is therefore easy. There is a well-established density plateau in the nonreactive part consequently, the determination of the bulk density in this part is straightforward and accurate. 

Probe methods like particle insertion and test particle methods (29-32) are quite useful for computing chemical potentials of constituent particles in systems with low densities. Test particles are randomly inserted the average Boltzmann factor of the insertion energy yields the free energy. For dense systems these methods work poorly because of the poor statistics obtained. 

Equations (2) and (3) relate intermolecular interactions to measurable solution thermodynamic properties. Several features of these two relations are worth noting. The first is the test-particle method, an implementation of the potential distribution theorem now widely used in molecular simulations (Frenkel and Smit, 1996). In the test-particle method, the excess chemical potential of a solute is evaluated by generating an ensemble of microscopic configurations for the solvent molecules alone. The solute is then superposed onto each configuration and the solute-solvent interaction potential energy calculated to give the probability distribution, Po(AU/kT), illustrated in Figure 3. The excess... 

We note that the calculation of At/ will depend primarily on local information about solute-solvent interactions i.c., the magnitude of A U is of molecular order. An accurate determination of this partition function is therefore possible based on the molecular details of the solution in the vicinity of the solute. The success of the test-particle method can be attributed to this property. A second feature of these relations, apparent in Eq. (4), is the evaluation of solute conformational stability in solution by separately calculating the equilibrium distribution of solute conformations for an isolated molecule and the solvent response to this distribution. This evaluation will likewise depend on primarily local interactions between the solute and solvent. For macromolecular solutes, simple physical approximations involving only partially hydrated solutes might be sufficient. 

Beck, T. L., Quantum path integral extension of Widom s test particle method for chemical potentials with application to isotope effects on hydrogen solubilities in model solids, J. Chem. Phys. 1992, 96, 7175-7177... 

Potential distribution methods are conventionally called test particle methods. Because the assertions above outline a general and basic position for the potential distribution theorem, it is appropriate that the discussion below states the potential... 

As noted in the Introduction, the PDT is widely recognized with the moniker test particle method. This name reflects a view of how calculations of ((e l3AU° ))0 might be tried solute conformations are sampled, solvent configurations are sampled, and then the two systems are superposed the energy change is calculated, and... 

Direct test particle methods are expected to be inefficient, compared to other possibilities, for molecular systems described with moderate realism. Successful placements of a test particle may be complicated, and placements with favorable Boltzmann factor scores may be rare. Fortunately, the tools noted above are generally available to design more-specific approaches for realistic cases. 

Most free energy and phase-equilibrium calculations by simulation up to the late 1980s were performed with the Widom test particle method [7]. The method is still appealing in its simplicity and generality - for example, it can be applied directly to MD calculations without disturbing the time evolution of a system. The potential distribution theorem on which the test particle method is based as well as its applications are discussed in Chap. 9. 

The NPT + test particle method [8, 9] aims to determine phase coexistence points based on calculations of the chemical potentials for a number of state points. A phase coexistence point is determined at the intersection of the vapor and liquid branches of the chemical potential versus pressure diagram. The Widom test particle method [7] of the previous paragraph or any other suitable method [10] can be used to obtain the chemical potentials. Corrections to the chemical potential of the liquid and vapor phases can be made, using standard thermodynamic relationships, for deviations... 

In contrast to the Gibbs ensemble discussed later in this chapter, a number of simulations are required per coexistence point, but the number can be quite small, especially for vapor-liquid equilibrium calculations away from the critical point. For example, for a one-component system near the triple point, the density of the dense liquid can be obtained from a single NPT simulation at zero pressure. The chemical potential of the liquid, in turn, determines the density of the (near-ideal) vapor phase so that only one simulation is required. The method has been extended to mixtures [12, 13]. Significantly lower statistical uncertainties were obtained in [13] compared to earlier Gibbs ensemble calculations of the same Lennard-Jones binary mixtures, but the NPT + test particle method calculations were based on longer simulations. 

Widom test-particle method. Provides the chemical potential in various ensembles. Relatively easy to implement and can be used as an additional measurement in standard MC ensembles (and also MD). Computational overhead is small. Yields good accuracy in simple systems, although less reliable in very dense or complex systems (i.e., chain molecules). 

Moller, D. Fischer, J., Vapour liquid equilibrium of a pure fluid from test particle method in combination with npt molecular dynamics simulations, Mol. Phys. 1990, 69, 463 173... 

Fotfi, A. Vrabec, J. Fischer, J., Vapour liquid equilibria of the Lennard-Jones fluid from the NPT plus test particle method, Mol. Phys. 1992, 76, 1319-1333... 

Boda, D. Fiszi, J. Szalai, I., An extension of the NPT plus test particle method for the determination of the vapour-liquid equilibria of pure fluids, Chem. Phys. Lett. 1995, 235, 140-145... 

Here, we report some basic results that are necessary for further developments in this presentation. The merging process of a test particle is based on the concept of cavity function (first adopted to interpret the pair correlation function of a hard-sphere system [75]), and on the potential distribution theorem (PDT) used to determine the excess chemical potential of uniform and nonuniform fluids [73, 74]. The obtaining of the PDT is done with the test-particle method for nonuniform systems assuming that the presence of a test particle is equivalent to placing the fluid in an external field [36]. 

Indirect Methods Test particle method Grand canonical ensemble method Biased sampling methods Thermodynamic integration... 

Gibbs method at constant pressure A = test particle method at constant volume. Horizontal and vertical lines are error bars, (b) Experimental (x) and empirical equation of state (-) results for acetone-carbon dioxide mixtures. Reprinted with the permission of Taylor Francis Ltd. from Panagiotopoulos et al. [16] and with permission from A. Z. Panagiotopoulos, U. W. Suter, and R. C. Reid, Ind. Eng. Chem. Fundam. 25, 525 (1986). Copyright 1986 American Chemical Society. 

According to Widom s test particle method for calculating p /feT, the test molecules (benzene) inserted in each simulation do not influence the molecular movements of pure C02 in any sense. Fig. 5 (a)Pore density of C02 and (b)residual chemical Therefore, the stabilization of a test molecule potentials of benzene in fluid and pore phases at... 

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