Smoothed potential model

Starting with the cell partition function (16.3.17) and the lattice energy (16.4.2) we may easily obtain the explicit expressions for the thermodynamic properties. The main difference is that the cell partition function depends only on the voliune but not on the temperature. The calculations are the same as for the harmonic oscillator model and will not be repeated. Instead of (16.5.4) we obtain the equation of state 

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Thus, we obtain two different series of expressions. We usually use the smoothed potential model we link the properties of gases to those of liquids, and the harmonie oscillator model to link the properties of liquids to those of solids. 

The complexity of the excess functions in mixtures make an analytical discussion highly desirable. For this purpose the Lennard-Jones and Devonshire model is unfortunately not suitable because of the complicated form of the mean potential model valid in a restricted range of density. For hig densities, as in the solid state, we may use a harmonic potential approximation (cf. Fig. 7.1.2). We shall develop this approximation in more detail in the next paragraph. On the other hand, for the range of densities corresponding to the liquid state we may use the smoothed potential model (Prigogine and Mathot [1952]) (cf. Fig. 7.1.2). This is however an oversimplification and the conclusions have to be used with some caution. 

Equations of state (a) "exact" model (Hirschfeldkr et al. [1954]) (b) smoothed potential model (c) harmonic oscillator model (cf. 4)... 

In paragraph 6 we shall briefly consider the smoothed potential model. We shall see that these simplified versions give very similar results. 

When particles are exchanged between QM and region, a smoothing treatment is invoked to prevent discontinuities of forces. The solvent potential model describing the interaction of solute particles in the MM zone should account for molecular flexibility, as all molecular vibrations are accessible in the QM part. The use of rigid models is not advised, as molecules would freeze in unfavorable conformations whenever a QM to MM transition takes place. 

Doll64 has applied classical S-matrix theory to the collinear A + BC collision where atoms A and B interact via a hard sphere collision this is the model studied quantum mechanically by Shuler and Zwanzig.65 Doll treats classically allowed and forbidden processes and finds good agreement between semiclassical and quantum mechanical transition probabilities. This is a remarkable achievement for the semiclassical theory, for the hard sphere interaction is far from the smooth potential that one normally assumes to be necessary for the dynamics to be classical-like. 

Lange, A. W., and Herbert, J. M. (2010). Polarizable continuum reaction-field solvation models affording smooth potential energy surfaces, J. Phys. Chem. Lett. 1, pp. 556-561. 

Su, R, and Li, H. (2009). Continuous and smooth potential energy surface for conductor-like screening solvation model using fixed points with variable areas,/. Chem. Phys. 130, pp. 074109 1-13. 

The conditions required by (ii) are not allowed quantum mechanically the sharp discontinuity would be smoothed. Indeed, what few examinations there have been of the nature of P(fe, ), either by analysis of reactive scattering using the optical potential model or by trajectory calculation, show a different dependence than that required by (ii). This is merely support for the familiar idea that, even in the absence of energy barriers, reactivity is dependent upon the total angular momentum. 

Restricting the quadrupole deformation to a mixing of closest unperturbed states is exact for harmonic confinement, but leads us to underestimate (2n by 20% for a smooth potential j8 = 0.1. The latter model, with the parameters (11.19), gives = 0.004 fm. ... 

Model That Yields Intrinsically Smooth Potential-Energy Surfaces. 

The model of the liquid chosen is Guggenheim s smoothed potential... 

Fig. 7.3.1 compares the equation of state deduced from the smoothed potential cell model with that deduced from the exact Lennard-Jones model. Especially in the region between 20 > ze >15 the orders of magnitude are the same. 

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