Density-independent pair potentials

Figure 6 shows the potential of mean force (PMF) between a sodium ion and a chloride ion in water, at infinite dilution of the two ions, obtained from classical atomistic simulations [75]. The first minimum of the potential corresponds to the contact ion pair (CIP) distance, the second minimum corresponds to the solvent-shared ion pair (SIP) distance, and the third minimum to the solvent-separated ion pair (2SIP) distance. Figure 7a shows an example of a SIP in aqueous NaCl [75]. The infinite dilute potential of mean force in Fig. 6 can be used as an effective pah-potential in implicit solvent simulations. The osmotic coefficient (j) (ps) = nilpJc- T (with n the osmotic pressure and ps the salt number density) can be obtained through the virial route. For the case of a binary mixture of components i and j and pairwise additive, density-independent pair potentials, the virial equation can be expressed as... 

Of special interest for the topic of the present chapter is the observation of Weaver that while the double-layer-corrected AS quantities are ligand sensitive, they are found to be independent of potential. This is not the case for the atom and electron transfer process involved in the hydrogen evolution reaction at Hg studied by Conway, et where an appreciable potential dependence of AS is observed, corresponding to conventionally anomalous variation of the Tafel slope with temperature. Unfortunately, in the work with the ionic redox reactions, as studied by Weaver, it is only possible to evaluate the variation of the transfer coefficient or symmetry factor with temperature with a limited variety of redox pairs since Tafel slopes, corresponding to any appreciable logarithmic range of current densities, are not always easily measurable. Alternatively, evaluation of a or /3 from reaction-order determination requires detailed double-layer studies over a range of temperatures. 

The VDW treatment of Ai/NkT has a better foundation. This term is not sensitive to the precise form of the pair potential. Thus, Equation 32 is satisfactory for illustrating the form of Ax. Combining Equations 21, 26, and 32 shows that, except for potentials of an unphysically short range (i.e., z oo), Ai/NkT is independent of T and nearly linear in the density. Thus... 

Being independent of long-wavelength limit, this "pair potential" V(z.C.u) is thus short-ranged in the x-y planes (its spatial "reach" can be larger than the cell parameter a but much less than the typical electronic "size" ir/kp in the low-density limit.) It is,... 

The NFE theory describes a simple metal as a collection of ions that are weakly coupled through the electron gas. The potential energy is volume-dependent but is independent of the position of the electrons. This is valid for both solids and dense liquids. At densities well above that of the MNM transition, we can use effective pair potentials and find the thermophysical properties of metallic liquids with the thermodynamic variational methods usually employed in theoretical treatments of normal insulating liquids. One approach is a variational method based on hard sphere reference systems (Shimoji, 1977 Ashcroft and Stroud, 1978). The electron system is assumed to be a nearly-free-electron gas in which electrons interact weakly with the ions via a suitable pseudopotential. It is also assumed that the Helmholtz free energy per atom can be expressed in terms of the following contributions ... 

As an example of the practical utility of MEMO simulations of the thermal conductivity, a summary of a calculation by Ravi et al. (1992) is given. The object was to explore the contribution of internal degrees of freedom to the conductivity of a polyatomic molecule. This contribution has not been quantitatively demonstrated until very recently (Murad et al. 1991). In fact, it is usually assumed that this contribution is independent of number density and is given by the dilute-gas value at the corresponding temperature. In the paper of Ravi et al. (1992), heat flow is discussed for a model benzene-like liquid with a six-centered Lennard-Jones pair potential. 

Since DFT calculations are in principle only applicable for the electronic ground state, they cannot be used in order to describe electronic excitations. Still it is possible to treat electronic exciations from first principles by either using quantum chemistry methods [114] or time-dependent density-functional theory (TDDFT) [115,116], First attempts have been done in order to calculate the chemicurrent created by an atom incident on a metal surface based on time-dependent density functional theory [117, 118]. In this approach, three independent steps are preformed. First, a conventional Kohn-Sham DFT calculation is performed in order to evaluate the ground state potential energy surface. Then, the resulting Kohn-Sham states are used in the framework of time-dependent DFT in order to obtain a position dependent friction coefficient. Finally, this friction coefficient is used in a forced oscillator model in which the probability density of electron-hole pair excitations caused by the classical motion of the incident atom is estimated. 

Let US consider an isolated pair of reactants A and B whose positions in space are r and re at time t. For simplicity, these reactants will be taken as uncharged and there is no potential of mean force consideration to be included either. Such factors may always be included later. The probability that species A is in the small volume cIta at r and species B is in the small volume drg at rg at time t is n rp, r, t) dr drs- Both species can diffuse independently of one another. They can even diffuse through each other, in principle, though in practice this is improbable because A and B will react before they b in to pass through each other. This can always be prevented by including a hard core repulsive term in the interaction evergy, U, which tums-on just inside the reaction distance, R. The probability density rs, t) can change with time due to diffusion of either A or B or both species. The rate of loss of A from is and, similarly, the loss of B from tg is D V n. In these terms, the subscript A and B refer to the co-ordinates of A or B, respectively, and the diffusion coefficients of A and B are, respectively, I>a The dif-... 

The field F(r) (see Eq. 59) depends on the wavefunction 4 through the density p(r), spinless single-particle density matrix y(r, r ), and the pair-correlation density g(r, r ). Furthermore, this work is path-independent since the field F(r) is conservative. The path of the inverse map C, whereby for every ground-state wavefunction P there corresponds a potential v(r), is now well defined. 

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