Interaction potentials vibrational averaging

Here t0 is an average vibrational period in a surface potential well having a value of the order of subpicoseconds for a simple molecule, and U() is a depth of the potential, which strongly depends on the type of interaction. For strong interaction, t exceeds 1 s or more, while in the case of no interaction, such as elastic collision, it is less than T(>. If we assume the Lennard-Jones potential for the interaction potential and expand it at the equilibrium position r0, t() is given by relevant molecular parameters as... 

It is clear that the interaction potential is an essential part of the two-particle Hamiltonian and thus of the translational state of the supermolecule the interaction potentials of initial and final state differ if a molecule undergoes vibrational excitation. For example, for a diatom-diatom pair like H2-H2, the translational state is determined by the vibrational average of the potential,... 

Table 6.1. Various vibrational averages, (rit>2 kb(K ri,r2) rii>2), of the H2-H2 interaction potential [281] the dependences on the rotational quantum numbers j], J2 are here suppressed.

Vo(R) is the vibrational average of the interaction potential, Eq. 6.29. Free-state wavefunctions may be energy normalized,... 

The moments of Y are obtained from similar expressions simply by changing the signs of all g /, g f and g f that appear in Eqs. 6.78 through 6.80, so that we need not repeat those expressions here. We note that the reduced mass is m, B is short for Bj.cJ, and the Vv, Vv> are the vibrational averages of the interaction potential. Superscripted Roman numericals I. .. IV mean the first. .. fourth derivatives with respect to R. The radial distribution functions g = g(R) depend on the interaction potentials, Vv, Vv>, and are thus subscripted like the potentials the low-density limit of the distribution function will be sufficient for our purposes. The functions g and g M are defined in Eq. 6.23. The notation f f R)d2R stands for 4n /0°° / (R) R2 dR as usual. 

The fifth term in (4.67) represents an interaction between vibration and rotation, and ae is called a vibration-rotation coupling constant. [Do not confuse ae with a in (4.26).] As the vibrational quantum number increases, the average internuclear distance increases, because of the anharmonicity of the potential-energy curve (Fig. 4.4). This increases the effective moment of inertia, and therefore decreases the rotational energy. We can define a mean rotational constant Bv for states with vibrational quantum number v by... 

By a statistical model of a solution we mean a model which does not attempt to describe explicitly the nature of the interaction between solvent and solute species, but simply assumes some general characteristic for the interaction, and presents expressions for the thermodynamic functions of the solution in terms of an assumed interaction parameter. The quasi-chemical theory is of this type, and we have noted that a serious deficiency is its failure to consider the vibrational effects in the solution. It is of interest, therefore, to consider briefly the average-potential model which does include the effect of vibrations. 

Whereas the quasi-chemical theory has been eminently successful in describing the broad outlines, and even some of the details, of the order-disorder phenomenon in metallic solid solutions, several of its assumptions have been shown to be invalid. The manner of its failure, as well as the failure of the average-potential model to describe metallic solutions, indicates that metal atom interactions change radically in going from the pure state to the solution state. It is clear that little further progress may be expected in the formulation of statistical models for metallic solutions until the electronic interactions between solute and solvent species are better understood. In the area of solvent-solute interactions, the elastic model is unfruitful. Better understanding also is needed of the vibrational characteristics of metallic solutions, with respect to the changes in harmonic force constants and those in the anharmonicity of the vibrations. 

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