Forces short range

The Hamiltonian considered above, which connmites with E, involves the electromagnetic forces between the nuclei and electrons. However, there is another force between particles, the weak interaction force, that is not invariant to inversion. The weak charged current mteraction force is responsible for the beta decay of nuclei, and the related weak neutral current interaction force has an effect in atomic and molecular systems. If we include this force between the nuclei and electrons in the molecular Hamiltonian (as we should because of electroweak unification) then the Hamiltonian will not conuuiite with , and states of opposite parity will be mixed. However, the effect of the weak neutral current interaction force is mcredibly small (and it is a very short range force), although its effect has been detected in extremely precise experiments on atoms (see, for... 

It is usefiil to classify various contributions to intennolecular forces on the basis of the physical phenomena that give rise to them. The first level of classification is into long-range forces that vary as inverse powers of the distance r , and short-range forces that decrease exponentially with distance as m exp(-ar). 

The first part of the method involves sorting all the atoms into their appropriate cells. This sorting is rapid, and may be perfonned at every step. Then, within the force routine, pointers are used to scan tlirough the contents of cells, and calculate pair forces. This approach is very efficient for large systems with short-range forces. A certain amount of unnecessary work is done because the search region is cubic, not (as for the Verlet list) spherical. 

Fig. 1. Motion of a material point on the body over time (left, short time interval right, long interval). The rigid body swings repeatedly toward the plane where it is repelled by the strong short-range force.

Lattice vibrations are calculated by applying the second order perturbation theory approach of Varma and Weber , thereby combining first principles short range force constants with the electron-phonon coupling matrix arising from a tight-binding theory. 

In Chapter 2 the curve of Fig. 7 was introduced, to show the mutual potential energy arising from short-range forces in contrast to that arising from long-range electrostatic forces. To account for the existence of molecules and molecular ions in solution, we need the same curve with the scale of ordinates reduced so as to be comparable with those of Fig. 

We see that we can attach a definite physical meaning both to the existence of a neutral molecule in solution, and to the dissociation of this molecule into a pair of ions. Consider points near P and near Q in Fig. 27c. A point on the curve near P corresponds to the situation where the distance between the nuclei of the two ions has, say, the value OA, while a point on the curve near Q corresponds to the separation OB. If the separation of the nuclei is increased from OA to OB, a considerable amount of work is done against the short-range forces of attraction, in order to go from P to Q. But at Q the short-range forces are no longer operative and the neutral molecule has been dissociated into a pair of ions, between which there is the usual electrostatic attraction. 

Incomplete Dissociation into Free Ions. As is well known, there are many substances which behave as a strong electrolyte when dissolved in one solvent, but as a weak electrolyte when dissolved in another solvent. In any solvent the Debye-IIiickel-Onsager theory predicts how the ions of a solute should behave in an applied electric field, if the solute is completely dissociated into free ions. When we wish to survey the electrical conductivity of those solutes which (in certain solvents) behave as weak electrolytes, we have to ask, in each case, the question posed in Sec. 20 in this solution is it true that, at any moment, every ion responds to the applied electric field in the way predicted by the Debye-Hiickel theory, or does a certain fraction of the solute fail to respond to the field in this way In cases where it is true that, at any moment, a certain fraction of the solute fails to contribute to the conductivity, we have to ask the further question is this failure due to the presence of short-range forces of attraction, or can it be due merely to the presence of strong electrostatic forces ... 

Forces Superimposed on the Coulomb Forces. The discussion has been based on the idea that, superimposed on the electrostatic forces between a pair of ions, there are rather short-range forces of other origin, which may be attractive or repulsive. Consider now what the situation will be if these forces cause the mutual potential energy to fall at short distances, below the value — e2/er that is assumed in the Debye-Hlickel theory. In Fig. 74 let the broken curve be a plot of — e2/er, while the full curve gives the actual potential energy between a certain pair of... 

In contrast to the foregoing, consider now the case depicted in Fig. 75, where the presence of moderately short-range forces of repulsion... 

The contribution of short-range forces to the activity coefficient can be described much better and in greater detail by the methods of the statistical thermodynamics of liquids, which has already created several models of electrolyte solutions. However, the procedures employed in the statistical... 

In a homogeneous medium of an electrolyte solution, an ionic liquid or a solid electrolyte under conditions of constant pressure and temperature, mechanical, electrostatic and short-range forces act on the individual particles in solution, but these forces average out in time. The effect of these forces is reflected in the activity values of the individual components of the system. 

The mobility of ions in melts (ionic liquids) has not been clearly elucidated. A very strong, constant electric field results in the ionic motion being affected primarily by short-range forces between ions. It would seem that the ionic motion is affected most strongly either by fluctuations in the liquid density (on a molecular level) as a result of the thermal motion of ions or directly by the formation of cavities in the liquid. Both of these possibilities would allow ion transport in a melt. 

Equation (3.1.2) would imply separation of the effect of short-range forces (also including dipole interactions) and of the individual ionic atmospheres, related to piy from the long-range forces related to 0, identical with purely coulombic interaction between excess charges. It will be seen later that such splitting, although arbitrary, is very useful. 

A more general relation between potential and electronic pressure for a density-functional treatment of a metal-metal interface has been given.74) For two metals, 1 and 2, in contact, equilibrium with respect to electron transfer requires that the electrochemical potential of the electron be the same in each. Ignoring the contribution of chemical or short-range forces, this means that —e + (h2/ m)x (3n/7r)2/3 should be the same for both metals. In the Sommerfeld model for a metal38 (uniformly distributed electrons confined to the interior of the metal by a step-function potential), there is no surface potential, so the difference of outer potentials, which is the contact potential, is given by... 

Since the forces giving rise to the formation of chemical bonds are very short-range forces, reactions in liquid solutions will require some sort of encounter or collision between re-... 

---