Repulsive pair potentials

Here the first term is a pair-potential repulsive term and the second term corresponds to the band-structure term. N is the total number of atoms, rM is the distance between atoms i and j of the species a and p, respectively, (a and p stand for A1 and Ni) and the sum j is extended up to fifth neighbours. Numerical values of the constants A, , p, q and r0 are given in [16], MMC simulations of this type have been carried out in the past to investigate the surface configurations of isolated nano-clusters of NixAly alloys [17],... 

Figure A3.1.1. Typical pair potentials. Illustrated here are the Lennard-Jones potential, and the Weeks-Chandler- Anderson potential, which gives the same repulsive force as the Leimard-Jones potential.

Ghrayeb R, Purushotham M, Hou M and Bauer E 1987 Estimate of repulsive interatomic pair potentials by low-energy alkalimetal-ion scattering and computer simulation Phys. Rev. B 36 7364-70... 

Quite recently, Pini et al. [56] have reported a new, thermodynamically self-consistent approximation to the OZ relation for a fluid of spherical particles for a pair potential given by a hard-core repulsion and a Yukawa attractive tail (Eq. (6)). The closure to the OZ equation they have proposed has the form... 

The first step towards the development of appropriate expressions is the decomposition of the nonassociative pair potential into repulsive and attractive terms. In this work we apply the Weeks-Chandler-Andersen separation of interactions [117], according to which the attractive part of the Lennard-Jones potential is defined by... 

The atomic interactions of the system are derived from a many-body empirical potential, the attractive part of which is expressed within the SMA of the TB theory ", while the repulsive term is a pair-potential of Bom-Mayer type. Accordingly, the total energy of the system is written as ... 

Fig. 1. The Lennard Jones 12 6 pair potential plotted for a pair of CH2 united atoms using the OPTS united force field. Enonbond = 4e((o /r) (o /r) ), where s is the well depth for the potential and cr is the distance at which the repulsive energy exactly cancels the attractive energy...

Instead of the hard-sphere model, the Lennard-Jones (LJ) interaction pair potential can be used to describe soft-core repulsion and dispersion forces. The LJ interaction potential is... 

The cation-anion pair potential for a symmetrical salt may be written as the sum of the core repulsion and the coulomb interaction. For example, if the core repulsion is an inverse power potential, then ... 

Experimental data on the compressibilities of solid salts appear to be consistent with the general form of the core repulsion part of the pair potential (1 /i)/(r/rf) even for the exponential form.8 Although for simplicity the equations are written for monovalent ions in a vacuum, they may easily be extended to other valence types and to ions contained in a dielectric. 

Because of coulomb repulsions, the core repulsive potential between ions of like sign may be assumed to be negligibly small in the most significant configurations (see Section II-A). Consequently, in these configurations, the interaction potential between ions of like sign is a function only of the distance between the ions and is independent of the type of ions. The cation-anion pair potentials may be written as... 

The pair potential of colloidal particles, i.e. the potential energy of interaction between a pair of colloidal particles as a function of separation distance, is calculated from the linear superposition of the individual energy curves. When this was done using the attractive potential calculated from London dispersion forces, Fa, and electrostatic repulsion, Ve, the theory was called the DLVO Theory (from Derjaguin, Landau, Verwey and Overbeek). Here we will use the term to include other potentials, such as those arising from depletion interactions, Kd, and steric repulsion, Vs, and so we may write the total potential energy of interaction as... 

Figure 2.16 The pair potential calculated for rutile particles with a radius of 100 nm, background electrolyte concentration of 10 4moldm 3 and a -potential of —50 mV. Curve a was calculated for an isolated pair of particles and curve b corresponds to the potential for a pair ofparticles in a dispersion at a volume fraction of 0.45. Note how the increased electrolyte content due to the counterions introduced with the particles shorten the range of the repulsion enough for a small secondary minimum to be found at h 70 nm...

The term pair potential that contains only the attractive potential, because the repulsion effects have been allowed for by the effective volume fraction and hard sphere diameter. The new potential can be defined as... 

Figure 9.4. Schematic description of the solute-solvent pair potential. The double-arrowed line indicates the hard (repulsive) interaction between a and a water molecule. The dashed lines indicate the interaction between groups on the surface of a and a water molecule, the sum of which is the last term on the rhs of Eq. (9.4.1).

Following our discussion of the dimeric bond centred on eqn (3.46), we will divide up the pair potential into a repulsive and a bonding contribution, namely... 

Fig. 4.2 (a) The full curve shows the normalized pair potential, / , versus the normalized interatomic distance, R/Rh, for the degree of normalized hardness, = 2 corresponding to X = 2. The two dashed curves show the repulsive and attractive contributions respectively. The shaded region delineates the hard-core potential with = 1 corresponding to X = oo. The two vertical arrows mark the equilibrium nearest-neighbour distances for = 1 and respectively, (b) The normalized pair potential, / , versus the normalized interatomic distance for different values of the degree of normalized hardness, . Note that = 0 corresponds to a totally soft potential, ah = 1 to a totally hard potential. 

The resultant pair potentials for sodium, magnesium, and aluminium are illustrated in Fig. 6.9 using Ashcroft empty-core pseudopotentials. We see that all three metals are characterized by a repulsive hard-core contribution, Q>i(R) (short-dashed curve), an attractive nearest-neighbour contribution, 2( ) (long-dashed curve), and an oscillatory long-range contribution, 3(R) (dotted curve). The appropriate values of the inter-atomic potential parameters A , oc , k , and k are listed in Table 6.4. We observe that the total pair potentials reflect the characteristic behaviour of the more accurate ab initio pair potentials in Fig. 6.7 that were evaluated using non-local pseudopotentials. We should note, however, that the values taken for the Ashcroft empty-core radii for Na, Mg, and Al, namely Rc = 1.66, 1.39, and... 

In fact, the origin of the (8-N) rule resides in the delicate balance between the repulsive overlap forces and the attractive covalent bond forces. The bond lengths are not invariant as drawn in Fig. 8.1, since the atoms do not behave as hard spheres with fixed nearest-neighbour distances. Assuming a repulsive pair potential... 

Further, assuming that the repulsive pair potential varies as the square of the bond integrals, as in eqn (8.4), we have... 

The discussion of Kapral s kinetic theory analysis of chemical reaction has been considered in some detail because it provides an alternative and intrinsically more satisfactory route by which to describe molecular scale reactions in solution than using phenomenological Brownian motion equations. Detailed though this analysis is, there are still many other factors which should be incorporated. Some of the more notable are to consider the case of a reversible reaction, geminate pair recombination [286], inter-reactant pair potential [454], soft forces between solvent molecules and with the reactants, and the effect of hydrodynamic repulsion [456b, 544]. Kapral and co-workers have considered some of the points and these are discussed very briefly below [37, 285, 286, 454, 538]. 

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