Exponentially repulsive potential

According to Schwartz, Slawksy, and Herzfeld, when v approaches vj and the energy transferred from translation approaches zero, the transition probability of Jackson and Mott [see equation (25)], for an exponential repulsive potential goes over to the form... 

In the derived equations of this section the limit Qp- 0 can be taken and the results are those for an exponentially repulsive potential. There are two characteristic frequencies of this potential in the limit of classical mechanics. These are the collision bandwidth... 

For K 2kT and /Km, the collinear result of Eq. (5.13) differs from the three-dimensional model of this work by a factor of 3. This difference is interpreted in terms of the projection of forces along the internuclear axis. The slightly different kinematic factors arise, in part, from the definition of the collision frequency that is used to derive, Eq. (5.11). The hard-sphere model gave excellent agreement with simulations for a very steep exponential repulsive potential with exponent 2a = 256h, where b is that of the Morse oscillator. It is to be remembered that Eq. (5.12) was derived from a stochastic model with three major assumptions ... 

It is not always obvious whether a given potential supports or does not support resonance states. The exponential well for instance (i.e. a potential of the form -A exp (-or) with A, o>0) supports only bound and virtual states (17). The exponential repulsive potential on the other hand (i.e. of the form A exp (-or) with A, o>0) supports resonances and virtual states (18). The potential 9 exp (- 2r) (with the kinetic energy operator written as -d2/dr ) supports two (and only two) resonance states with the energies calculated (19) from an analytical treatment to be... 

These results, in turn, then formed the basis for the non-jellium part of the jel-lium model in [73]. Berkowitz [52-54] and later Zhu and Philpott [55] and Spohr [56] followed the approach by Steele [93] and Fourier-expanded the lattice sum of all (pairwise) interactions between the atoms in the solid and one molecule or ion in the liquid. Only the lowest order corrugation terms are kept in the expansion but in principle the summation can be extended to any desired accuracy. The procedure is adequate as long as there is no substantial coupling between liquid and metal motions that could influence the liquid structure and relaxation phenomena. Spohr [56] used a corrugated Morse potential for the oxygen-metal interactions and an exponentially repulsive potential for the hydrogen metal interactions in the form... 

As an alternative to the Lennard-Jones potential, one can also carry out calculations using an exponential repulsive potential... 

In order to verify the ideas presented by Bocchetta and Gerratt [16] and above, we have done some elastic scattering calculations on a simple exponential repulsive potential using both a non-orthogonal sine basis (as in Ref. 16) and a basis of distributed Gaussian functions. The Hamiltonian was that used in Ref. 11 ... 

T and T2 are the kinetic energy operators. The harmonic potential V2 describes the interaction of the heavy particle with the surface. The Morse potential Vi describes the interaction of the lighter particle with the surface. The two particles interact by the exponentially repulsive potential V. Parameters particular to the simulation are provided in Table 1 and are the same as in Ref. [13]. 

The treatment of diatomic recombination and dissociation using the theory of section IV has been previously applied, at least for impulsive collisions [21]. In the low bath density regime, an adiabatic correction factor C has been used [9] based on the form for B for an exponentially repulsive potential [26]. The important result was that the iodine recombination rate could be accounted for over the entire range of bath densities using as input the diatomic potentials and the atomic diffusion... 

Slater and Kirkwood s idea [121] of an exponential repulsion plus dispersion needs only one concept, damping fiinctions, see section Al.5.3.3. to lead to a working template for contemporary work. Buckingham and Comer [126] suggested such a potential with an empirical damping fiinction more than 50 years ago ... 

Buckingham R A and Corner J 1947 Tables of second virial and low-pressure Joule-Thompson coefficients for intermolecular potentials with exponential repulsion Proc. R. Soc. A 189 118... 

The above potential is referred to as a Lennard-Jones or 6-12 potential and is summed over all nonbonded pairs of atoms ij. The first positive term is the short range repulsion and the second negative term is the long range attraction. The parameters of the interaction are Aj and B... The convenient analytical form of the 6-12 potential means that it is often used, although an exponential repulsion term is usually considered to be a more accurate representation of the repulsive forces (as used in MM-t). 

The MMh- van der Waals interactions do not use a Lennard-Jones potential but combine an exponential repulsion with an attractive... 

Very similar to the properties of the free surface are the properties of water near smooth walls, which interact only weakly with water molecules. Many different models have been used, such as hard walls [81-83], exponentially repulsive walls [84-86], and Lennard-Jones potentials of various powers [81,87-96]. 

The vibrational motion of atoms in diatomic molecules and, by extension, in crystals cannot be fully assimilated to harmonic oscillators, because the potential well is asymmetric with respect to Xq. This asymmetry is due to the fact that the short-range repulsive potential increases exponentially with the decrease of interionic distances, while coulombic terms vary with 1/Z (see, for instance, figures 1.13 and 3.2). To simulate adequately the asymmetry of the potential well, empirical asymmetry terms such as the Morse potential are introduced ... 

Similar calculations with use of an exponential form for the repulsive potential have been made by M. L. Huggins and J. E. Mayer, J. Chem. Phys. 1, G43 (1933), and M. L. Huggins, ibid, S, 143 (1937). The problem has been treated also by J. A. Wasastjerna, Soc. Set. Fenn. Comm. Phys. Math. VIII, 21 (1935). 

Potential functions used82 were an exp-6 function with two free repulsive parameters and a modified exp-6 with three repulsive parameters. The modification was introduced as a screened ion-induced dipole attraction, which softens the repulsion at small r. The fitted potentials are shown in Fig- 10. The dashed curves in Fig. 9 show the fits obtained with the simple exponential repulsion and the solid curves with the modified repulsion. 

Schwartz et al.26 deduced values of / from the Lennard-Jones parameters e and r0, which have been derived from viscosity measurements, and analysed and tabulated by Hirschfelder et a/.31. A detailed account of the required calculation has been given by Herzfeld and Litovitz32. The problem is to obtain a satisfactory fit of the exponential interaction potential, with the repulsive region of the Lennard-Jones function... 

The explanation of compressibility follows, apparently, from the features of the intermolecular interaction. If one uses the potential (6-exp) [19], i.e., the sum of dipole-dipole attraction and the exponential repulsion, and connects the parameters of this potential with the equilibrium distance r0 and the depth of the potential well a, we have... 

---