Long-range forces energy

Long-range forces are most conveniently expressed as a power series in Mr, the reciprocal of the intemiolecular distance. This series is called the multipole expansion. It is so connnon to use the multipole expansion that the electrostatic, mduction and dispersion energies are referred to as non-expanded if the expansion is not used. In early work it was noted that the multipole expansion did not converge in a conventional way and doubt was cast upon its use in the description of long-range electrostatic, induction and dispersion interactions. However, it is now established [8, 9, 10, H, 12 and 13] that the series is asymptotic in Poincare s sense. The interaction energy can be written as... 

Sharma R D and Brau C A 1969 Energy transfer in near-resonant molecular collisions due to long-range forces with application to transfer of vibrational energy from the mode of CO2 to N2 J. Chem. Phys. 50 924-30... 

Tlierc are two major sources of error associated with the calculation of free energies fi computer simulations. Errors may arise from inaccuracies in the Hamiltonian, be it potential model chosen or its implementation (the treatment of long-range forces, e j lie second source of error arises from an insufficient sampling of phase space. 

For the industrially important class of mixed solvent, electrolyte systems, the Pitzer equation is not useful because its parameters are unknown functions of solvent composition. A local composition model is developed for these systems which assumes that the excess Gibbs free energy is the sum of two contributions, one resulting from long-range forces between ions and the other from short-range forces between all species. 

This chapter comprises two sections. The first describes the most usual techniques to directly measure force versus distance profiles between solid or liquid surfaces. We then describe different long-range forces (range >5 nm) accessible to evaluation via these techniques for different types of surface active species. The second section is devoted to attractive interactions whose strong amplitude and short range are difficult to determine. In the presence of such interactions, emulsion droplets exhibit flat facets at each contact. The free energy of interaction can be evaluated from droplet deformation and reveals interesting issues. 

For a transition of Mott type we shall show in Chapter 4, Section 3, neglecting the discontinuity resulting from long-range forces, that the transition should occur when 2zl = U. Near the transition the energy needed to excite an electron into the upper Hubbard band is U — 2zl. The wave function of an electron then falls off as e-fltr, where a=2m(I7 — 2zI)1/2/fc2. Thus the amount of spin in the sphere surrounding each atom will be made up from electrons on many of the surrounding atoms, and will clearly go to zero as [Pg.88]

These methods have been applied to calculate the polarizabilities of atoms,31 and the long-ranged forces between atoms,33 with a typical calculated accuracy of 10 % or less. Thus, we have been able to estimate successfully the significant features of zero-point fluctuations of atomic dipole moments, without actually solving the quantum equations of motion to obtain all the excited state energies and wave functions. 

Physical properties related to the electron motion in crystals fall essentially into two categories. Some, such as the electrical properties of crystals, arise from long-range interactions in the lattice here long-range forces from the electron--electron or the electron-core interactions play an important role. In these cases, the use of energy band theory is essential. On the other hand, in NLO effects the process of electronic excitation by the incident... 

Dissociation energies vary from a few thousandths of an eV for physically bound van der Waals molecules to several eV for chemically bound molecules. Van der Waals molecules are bound by the weak long-range forces and exist only at very low temperatures, either in a supersonic beam or in the interstellar space (Buckingham, Fowler, and Hutson 1988). Typical examples are t... 

The isotropic interaction energy—long-range forces experienced by the solute, e.g. electrostatic forces, polarization and dispersion energy. 

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