Potential function inversion

A monolayer can be regarded as a special case in which the potential is a square well however, the potential well may take other forms. Of particular interest now is the case of multilayer adsorption, and a reasonable assumption is that the principal interaction between the solid and the adsorbate is of the dispersion type, so that for a plane solid surface the potential should decrease with the inverse cube of the distance (see Section VI-3A). To avoid having an infinite potential at the surface, the potential function may be written... 

Equation XVII-78 turns out to ht type II adsorption isotherms quite well—generally better than does the BET equation. Furthermore, the exact form of the potential function is not very critical if an inverse square dependence is used, the ht tends to be about as good as with the inverse-cube law, and the equation now resembles that for a condensed him in Table XVII-2. Here again, quite similar equations have resulted from deductions based on rather different models. 

There have been several suggested forms of the potential function which reproduce the way in which the potential energy V(Q) depends on the vibrational coordinate Q which relates to the inversion motion. Perhaps the most successful form for fhe pofenfial function is... 

Torsional barriers are referred to as n-fold barriers, where the torsional potential function repeats every 2n/n radians. As in the case of inversion vibrations (Section 6.2.5.4a) quantum mechanical tunnelling through an n-fold torsional barrier may occur, splitting a vibrational level into n components. The splitting into two components near the top of a twofold barrier is shown in Figure 6.45. When the barrier is surmounted free internal rotation takes place, the energy levels then resembling those for rotation rather than vibration. 

In the chapter on vibrational spectroscopy (Chapter 6) 1 have expanded the discussions of inversion, ring-puckering and torsional vibrations, including some model potential functions. These types of vibration are very important in the determination of molecular structure. 

We consider a general inverse power potential function of the form... 

The work of Melander and Carter (1964) on 2,2 -dibromo-4,4 -di-carboxybiphenyl-6,6 -d2 (1) has been referred to above in the introductory and theoretical sections, where it was pointed out that the availability of two detailed theoretical computations of the inversion barrier (Westheimer and Mayer, 1946, Westheimer, 1947 Hewlett, 1960) made this system especially attractive for the study of steric isotope efifects. Furthermore, in the preferred initial-state conformation the two bromines are probably in van der Waals contact (cf. Hampsoii and Weissberger, 1936 Bastiansen, 1950), and thus initial-state steric effects are unaffected by deuterium substitution in the 6 and 6 positions. The barrier calculations provided two different theoretical values for the non-bonded H Br distance in the transition state which, together with the corresponding H Br potential function, could be inserted in equation (10) to yield values for A AH. For... 

H H non-bonded interactions are of great importance in organic compoimds, and thus it was of interest to attempt to investigate H H non-bonded potential functions via the determination of a steric isotope effect in the configurational inversion of an unsubstituted biaryl. In view of the extensive work of Harris and her co-workers in the 1,1 -binaphthyl series (see, for example, Badar et al., 1965 Cooke and Harris, 1963), and since the parent compound is one of the simplest hydrocarbons that may be obtained in enantiomeric forms, the determination of the isotope effect in the inversion of l,l -binaphthyl-2,2 -d2 (9) was... 

Tbginteraction between the two particles in this system is described by Coulomb s law, in which the force is proportional to the inverse-square of the distance between the particles and —e2 is the product of the charges on the electron and the proton. The corresponding potential function is then of the form... 

One may therefore wish to know what are the potential functions V(r) that correspond to a given algebraic model. The general answer to this question is provided by the solution of the inverse Schrodinger problem Since one knows the spectrum of the algebraic model, one finds the potential that reproduces the spectrum.1 A simple approach consists in expanding the potential V(r) into a set of functions with unknown coefficients, say... 

The Direct Lattice Sum. Dispersion forces between two atoms can be described by a potential function expressed in terms containing inverse powers of the internuclear separations, s. The simplest function of this sort includes a potential energy of attraction proportional to the inverse sixth power of the separation and a repulsion that is zero at distances of separation greater than a particular value se and infinite at separations less than sc. This is the so-called hard sphere or van der Waals model. Such an approximate potential function can be improved in two respects. Investigations of the second virial coefficient have revealed that the potential energy of repulsion is best described as proportional to the inverse twelfth power of the separation and the term in sr9, which accounts for the greater part of the total attraction potential, due to the attraction of mutually induced dipoles, should have added to it the dipole-quadrupole and quadrupole-quadru-pole attractions, expressed as terms in sr8 and s-10, respectively. The complete potential function for the forces between two atoms is, therefore ... 

Inversion of experimental data to calculate the potential function (RKR)... 

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