Potentials of the average force

Thus, non-Ising behavior may be expected in systems determined by Coulomb and charge dipole interactions. However, due to the screening by counter ions the potential of the average force becomes short range. Therefore, Ising-like criticality may be restored as in liquid metals, where the electrons screen the interactions of the Coulomb interactions of the cores [84],... 

V. Potential of the Average Force between Two and Three Ions in a... 

Both the potential of the average force and the thermodynamic quantities related to solute particles at infinite dilution can in principle be calculated within the framework of classical statistical mechanics. Formal relations have been derived by McMillan and Mayer and discussed by Hill. We shall discuss them briefly in Section III. 

The first term, Pt ch. can be called the free energy of charging the set of ions, held in fixed positions PT(0) is the potential of the average force for discharged ions. We limit our discussion to Wch. 

The total potential of the average force can be decomposed into a sum of contributions of single ions, pairs, triplets, etc. If we denote for one ion by... 

In this way we have shown that the Coulomb macroscopic law (1) can be obtained as an asymptotic approximation to the potential of the average force between two ions in a non-polar solvent. 

V. POTENTIAL OF THE AVERAGE FORCE BETWEEN TWO AND THREE IONS IN A DIPOLAR SOLVENT... 

The equilibrium thermodynamic and structural properties of coUoidal dispersions may be treated in the same way as in the case of simple liquids by considering the colloidal particles as "supramolecules dispersed in a continuous (but fluctuating) back-ground. The potential which for the case of fluctuating forces replaces the interaction potential between molecules (in vacuo) is the potential of the average forces which act between the dispersed particles. This effective interaction is the input for statistical mechanical theories. Therefore statistical mechanical theories developed for simple fluids can be applied to colloidal dispersions. The theoretical basis for such a treatment was given by Onsager and Me Millan and Mayer. In recent years concepts of liquid state theory have been applied successfully to understand the behavior of concentrated colloidal dispersions. ... 

Another primitive hydrophobic effect is the effect of water on the association of simple hydrocarbon molecules in aqueous solution. We will below give some discussion and an example of association of pairs of inert gas atoms, focusing on the function that gives the mean forces between such a pair upon differentiation, i.e, the pair potential of the average forces. These issues broaden the topic from hydration of solitary hydrophobic solutes to the interactions associated with disruption of the water structure by more than one hydrophobic solute in proximity. A further extension of the study of hydrophobic interactions is the treatment of conformational equilibrium of simple, flexible non-polar molecules in aqueous solution. 

The rate constant can be expressed in terms of the potential of mean force at the activated complex. This potential may, e.g., be defined such that the gradient of the potential gives the average force on an atom in the activated complex due to the solvent molecules, Boltzmann averaged over all configurations. 

However, in many applications the essential space cannot be reduced to only one degree of freedom, and the statistics of the force fluctuation or of the spatial distribution may appear to be too poor to allow for an accurate determination of a multidimensional potential of mean force. An example is the potential of mean force between two ions in aqueous solution the momentaneous forces are two orders of magnitude larger than their average which means that an error of 1% in the average requires a simulation length of 10 times the correlation time of the fluctuating force. This is in practice prohibitive. The errors do not result from incorrect force fields, but they are of a statistical nature even an exact force field would not suffice. 

If there are no reactions, the conservation of the total quantity of each species dictates that the time dependence of is given by minus the divergence of the flux ps vs), where (vs) is the drift velocity of the species s. The latter is proportional to the average force acting locally on species s, which is the thermodynamic force, equal to minus the gradient of the thermodynamic potential. In the local coupling approximation the mobility appears as a proportionality constant M. For spontaneous processes near equilibrium it is important that a noise term T] t) is retained [146]. Thus dynamic equations of the form... 

The function W(X) is called the potential of mean force (PMF). The fundamental concept of the PMF was first introduced by Kirkwood [4] to describe the average structure of liquids. It is a simple matter to show that the gradient of W(X) in Cartesian coordinates is related to the average force. 

A chemical will be a solvent for another material if the molecules of the two materials are compatible, i.e. they can co-exist on the molecular scale and there is no tendency to separate. This statement does not indicate the speed at which solution may take place since this will depend on additional considerations such as the molecular size of the potential solvent and the temperature. Molecules of two different species will be able to co-exist if the force of attraction between different molecules is not less than the forces of attraction between two like molecules of either species. If the average force of attraction between dissimilar molecules A and B is and that between similar molecules of type B Fbb and between similar molecules of type A F a then for compatibility Fab - bb and AB - P/KA- This is shown schematically in Figure 5.5 (a). 

Whereas the quasi-chemical theory has been eminently successful in describing the broad outlines, and even some of the details, of the order-disorder phenomenon in metallic solid solutions, several of its assumptions have been shown to be invalid. The manner of its failure, as well as the failure of the average-potential model to describe metallic solutions, indicates that metal atom interactions change radically in going from the pure state to the solution state. It is clear that little further progress may be expected in the formulation of statistical models for metallic solutions until the electronic interactions between solute and solvent species are better understood. In the area of solvent-solute interactions, the elastic model is unfruitful. Better understanding also is needed of the vibrational characteristics of metallic solutions, with respect to the changes in harmonic force constants and those in the anharmonicity of the vibrations. 

Among the functions one can, at least in principle, calculate at the Schroedinger level is the Born-Oppenheimer (BO) potential surface, the potential of the forces among the nuclei assuming that at each nuclear configuration the time-independent Schroedinger equation is satisfied. We may think of this as the electron-averaged potential. Such an N-body potential Ujj often may be adequately represented as a sum of pair potentials... 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

---