Yukawa function

A crude estimation of the charge-density distribution on simple metal surfaces can be made by assuming that the electron charge for each atom is spherical. Especially, as shown by Cabrera and Goodman (1972), by representing the atomic charge distribution with a Yukawa function. 

The central potential can be a simple harmonic oscillator potential/(r) kr2 or more complicated such as a Yukawa function f(r) (e a,/r) 1 or the Woods-Saxon function that has a flat bottom and goes smoothly to zero at the nuclear surface. The Woods-Saxon potential has the form... 

In the final version of the Paris potential, also known as the parametrized Paris potential [14], each component (there is a total of 14 components, 7 for each isospin) is parametrized in terms of 12 local Yukawa functions of multiples of the pion mass. This introduces a very large number of parameters, namely 14 x 12 = 168. Not all 168 parameters are free. The various components of the potential are required to vanish at r = 0 (implying 22 constraints [14]). One parameter in each component is the nNN coupling constant, which may be taken from other sources (e.g., nN scattering). The 2n-exchange contribution is derived from dispersion theory. The range of this... 

We now discuss the g-matrix interaction computed by von Geramb and collaborators [Ge 79, Ge 83a, Ge 83b, Ri 84a, Ri 84b]. This interaction, referred to as the Paris-Hamburg g-matrix , has been used in numerous analyses of data and application of it has been facilitated by its parametrization in terms of sums of Yukawa functions. 

Reference ) also treated fermion systems which model neutron and nuclear matter interacting by simplified pair potentials having the form of a linear combination of Yukawa functions ... 

The calculations that have been carried out [56] indicate that the approximations discussed above lead to very good thermodynamic functions overall and a remarkably accurate critical point and coexistence curve. The critical density and temperature predicted by the theory agree with the simulation results to about 0.6%. Of course, dealing with the Yukawa potential allows certain analytical simplifications in implementing this approach. However, a similar approach can be applied to other similar potentials that consist of a hard core with an attractive tail. It should also be pointed out that the idea of using the requirement of self-consistency to yield a closed theory is pertinent not only to the realm of simple fluids, but also has proved to be a powerful tool in the study of a system of spins with continuous symmetry [57,58] and of a site-diluted or random-field Ising model [59,60]. 

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

Figure 12. The diffusive modes of the periodic Yukawa-potential Lorentz gas represented by their cumulative function depicted in the complex plane ReFk,hnFk) for two different nonvanishing wavenumbers k. The horizontal straight line is the curve corresponding to the vanishing wavenumber k = 0 at which the mode reduces to the invariant microcanonical equilibrium state.

For large r, G(f, Fo) must vanish, which requires that A = 0. For small distances, where kf < < 1, it should be identical to the Coulomb potential, which requires that 5=1. Finally, we find that the Green s function of the Schrodinger equation in vacuum is the Yukawa potential. 

For many practically important interaction functions, the Fourier coefficients in Eq. (D.9) have finite analytic forms, for example, the Lennard-Jones potential, the Yukawa potential, the Morse potential, and functions that can be derived from those functions. For a power-law interaction... 

Figure 11. The difference between the free-energy densities of fee and bcc phases of particles interacting through a Yukawa potential, as a function of temperature, determined through the FG methods discussed in Section V.C. The error bars reflect the difference between the upper and lower bounds provided by FG switches between the phases (along the Bain path [85]) in the two directions.

One shortcoming of the benzoic acid system is the extent of coupling between the car-bo l group and certain lone-pair donors. Insertion of a methylene group between the core (benzene ring) and the functional group (COOH moiety) leads to phenylacetic acids and the establishment of scale from the ionization of X-phenylacetic acids. A flexible method of dealing with the variability of the resonance contribution to the overall electronic demand of a reaction is embodied in the Yukawa-Tsuno equation (86). It includes nor-nial d enhanced resonance contributions to an LFER. 

Figure 3. A. A — 2) as a function of X for the ground state of the Yukawa potential for...

The additional Yukawa interactions also lead to a fluctuating internal energy E(t) that makes it possible to determine the rate at which the large (and slower) particles decorrelate. We consider the integrated autocorrelation time r obtained from the energy autocorrelation function [28],... 

The results obtained from the analytic solution are similar to those obtained previously using purely numerical methods. One of the advantages of the analytic solution is that it is possible to make changes to the closure relation which systematically improve the accuracy of the results. One method of doing this is by adding a Yukawa tail to the site-site direct correlation function. The two parameters in the Yukawa tail (that is, its amplitude and decay rate) can be chosen to give thermodynamic consistency and improve the accuracy of the site-site correlation function g y. This approach has been pursued Cummings and Morriss. " ... 

DFT has been much less successful for the soft repulsive sphere models. The definitive study of DFT for such potentials is that of Laird and Kroll [186] who considered both the inverse power potentials and the Yukawa potential. They showed that none of the theories existing at that time could describe the fluid to bcc transitions correctly. As yet, there is no satisfactory explanation for the failure of the DFTs considered by Laird and Kroll for soft potentials. However, it appears that some progress with such systems can be made within the context of Rosenfeld s fundamental measures functionals [130]. 

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