Widom Test Particle Method

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 method is based on the following expression for constant-AVT simulations (modified expressions are available in other ensembles [1]) 

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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... 

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). 

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. 

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... 

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... 

This problem has been brilliantly reviewed by Kumar in a recent book and hence we summarize only the most sahent features here. For small molecule systems, sampling of the chemical potential rests on the Widom test particle insertion method ... 

Gas solubiUties can also be determined by molecular dynamics simulations, using the Widom test particle insertion method to calculate the excess chemical potential fiex. or free energy of the penetrant molecules. The solubility can be obtained using Henry s law. If E is the interaction energy of a virtual penetrant molecule with the polymer inserted at random within the sample (the molecule is invisible to the polymer) then... 

The test particle method of Widom and related particle insertion schemes reviewed recently by Kofke and Cummings have been used extensively to investigate the hydration of small nonpolar molecules. Widom s approach to this problem involves performing NVT-ensemble simulations of pure water. A variety of solution properties may be investigated by measuring the energy of solute molecules randomly inserted into the pure solvent. These test particles probe the system but do not affect the solvent trajectories. The excess chemical potential of a solute molecule is thus calculated as... 

Another important application of the CBMC algorithm is the calculation of the chemical potential using Widom s test particle method [29,32,96], For the parallel CBMC algorithm described here it is straightforward to show that the excess chemical potential can be calculated using... 

Nearly 10 years after Zwanzig published his perturbation method, Benjamin Widom [6] formulated the potential distribution theorem (PDF). He further suggested an elegant application of PDF to estimate the excess chemical potential -i.e., the chemical potential of a system in excess of that of an ideal, noninteracting system at the same density - on the basis of the random insertion of a test particle. In essence, the particle insertion method proposed by Widom may be viewed as a special case of the perturbative theory, in which the addition of a single particle is handled as a one-step perturbation of the liquid. 

The residual chemical potentials of benzene, p f aI,d P2 p> ar d that of C02 in the fluid phase, p[ are calculated by Widom s test particle insertion method, Eq. (6) [6], which has been embedded in all the simulation programs. 

A simple method of improving the efficiency of test particle insertion [106, 107, 108 and 109] involves dividing the simulation box into small cubic regions, and identifying those which would make a negligible contribution to the Widom formula, due to overlap with one or more atoms. These cubes are excluded from the sampling, and a correction applied afterwards for the consequent bias. 

Hybrid, or modified real particle method was proposed by Kumar (53) as a method suitable for calculation of chemical potential at high densities. This technique combines Widom s test particle and the so-called real particle methods and calls for factious removal and subsequent reinsertion of particles afready present in the y stem. The hybrid technique can be classified as a nondestructive one since it does not affect the proper time evolution of the system. It was suggested that this method would be particularly advantageous for simulation of macromolecules when a removal of a whole polymer chain is likely to create substantial free spaee and thus facilitate the reinsertion. This technique was tested in the original paper on Lennard-Jones particles and proven to yield good results at densities up to 1.1 and T down to 0.7. 

Widom s Test Particle Insertion Method. The solubility of small molecules at infinite dilution can be estimated based on the knowledge of the chemical potential of these molecules. Widom s test particle insertion method (483) provides a technique for calculating this quantity. The residual chemical potential (the difference between the chemical potentials of the fluid and the ideal gas at the same temperatin-e and density) of a fluid is given by... 

The method of Widom to calculate the excess chemical potential of a system by random insertion of a test particle, particle insertion, is in essence a special case of the thermodynamic perturbation formula. The excess chemical potential (or free energy per particle), is the chemical poten-... 

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