Screened Coulombic potential

The solutions to this approximation are obtained numerically. Fast Fourier transfonn methods and a refomuilation of the FINC (and other integral equation approximations) in tenns of the screened Coulomb potential by Allnatt [M are especially useful in the numerical solution. Figure A2.3.12 compares the osmotic coefficient of a 1-1 RPM electrolyte at 25°C with each of the available Monte Carlo calculations of Card and Valleau [ ]. 

Let us discuss some of the results obtained so far. We present the screened potentials of Coulomb interaction between point ions in the... 

The first density correction to the rate constant depends on the square root of the volume fraction and arises from the fact that the diffusion Green s function acts like a screened Coulomb potential coupling the diffusion fields around the catalytic spheres. 

Mehler EL, Guarnieri F (1999) A self-consistent, microenvironment modulated screened coulomb potential approximation to calculate pH-dependent electrostatic effects in proteins. Biophys J 75 3-22. 

Of course, this result has to come out of the calculation and it will be obtained in Sections V-C and V-E. However, it is intuitively clear that a qualitatively correct result should come out of the static approximation using a screened coulomb potential (see... 

We shall, however, keep here the screened Coulomb potential (314) for describing the interionic interaction the most general case will be discussed in Section V-E. 

In the simplest model, the interactions with the solvent were completely neglected and the collective Coulomb interactions between the ions were imitated by a screened Coulomb potential the result is (see Eq. (321))... 

In conclusion, the repulsive interactions arise from both a screened coulomb repulsion between nuclei, and from the overlap of closed inner shells. The former interaction can be effectively described by a bare coulomb repulsion multiplied by a screening function. The Moliere function, Eq. (5), with an adjustable screening length provides an adequate representation for most situations. The latter interaction is well described by an exponential decay of the form of a Bom-Mayer function. Furthermore, due to the spherical nature of the closed atomic orbitals and the coulomb interaction, the repulsive forces can often be well described by pair-additive potentials. Both interactions may be combined either by using functions which reduce to each interaction in the correct limits, or by splining the two forms at an appropriate interatomic distance . 

Despite the fact that formalism of the standard chemical kinetics (Chapter 2) was widely and successfully used in interpreting actual experimental data [70], it is not well justified theoretically in fact, in its derivation the solution of a pair problem with non-screened potential U (r) = — e2/(er) is used. However, in the statistical physics of a system of charged particles the so-called Coulomb catastrophes [75] have been known for a long time and they have arisen just because of the neglect of the essentially many-particle charge screening effects. An attempt [76] to use the screened Coulomb interaction characterized by the phenomenological parameter - the Debye radius Rd [75] does not solve the problem since K(oo) has been still traditionally calculated in the same pair approximation. 

Camus et a/.34 explained their observations by a picture which has sometimes been called the frozen planet model. Qualitatively, the relatively slowly moving outer electron produces a quasi-static field at the inner electron given by l/rc2, and this field leads to the Stark effect in Ba+. The field allows the transitions to the n >n0Z and ,f 0 states and leads to shifts of the ionic energies. The presence of the njpn0f and n in0t resonances in the spectrum of Fig. 23.12 is quite evident. Camus et al. compared the shifts to those calculated in a fashion similar to a Bom-Oppenheimer calculation. With the outer electron frozen in place at ra they calculated the Ba+ energies, W,(rQ), and wavefunctions. They then added the energy W0(r0) to the normal screened coulomb potential seen by the outer electron. This procedure leads to a phase shift in the outer electron wavefunction... 

Mott describes the localized center in terms of an electron hole pair with a screened Coulombs potential... 

The electrical conductivities of several alkali metals dissolved in liquid ammonia are shown in Figure 1 (7, 119 15). The strong variation of the conductivity, a, with concentration has been most difficult to explain. This difficulty can be assessed by referring to a simple model of conductance, the Thomas-Fermi model of a screened Coulomb potential (19). This model has been used in describing semiconductors as well as in theories of metal-ammonia solutions (1). 

There are a couple of attempts to use the WK.B approximation to the two component radial Dirac problem for screened Coulomb fields. Kosaka and Yonei(17) review these and apply the form offered by Goldberg and Pratt(18) for calculations of bound state orbitals in the potentials suggested by Tomishima(16). Their procedure is rather more involved than the present one. 

If we only consider the static case w = 0we can distinguish two limiting cases First, if the typical range rjj of the screened Coulomb potential Vc is much smaller than the mean electron distance, the potential can be assumed to be a delta distribution and Ttc can be approximated by... 

An early calculation on the effect of the screened Coulomb potential on the energy of the He atom was due to Rogers [183]. Lam and Varshni [184] calculated the ground state energy of the He atom as a function of the... 

The Smith group has also developed the methodology for making high precision calculations for small systems without invoking the Born-Oppenheimer approximation and have made calculations for two-electron atomic ions, small muonic molecules, and potentials of the screened Coulomb form. Their method for determining nonlinear parameters is now referred to as random tempering.169... 

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