Accuracy model potential approximations

In this section we establish simple model expressions that approximately describe narrow EELS resonances near a critical point in the binary approximation [8]. In this analysis we terminate the correlational series, taking into account only the pairwise contributions (the accuracy of this approximation is increased from the band of moderate densities). We use a Lennard-Jones potential... 

In the case when there is no appreciable overlap between atomic cores there is every reason to believe that the simulation of the Pauli principle in the region of each atomic core would be approximated to a high degree of accuracy by the simple sum of the separate-atom core potentials. This statement can be made more precise by saying that, if the core-core overlap is negligible, then the product of the corresponding core projectors will be zero, which is equivalent to saying that the electrons of one core have no Pauli-principle driven interaction with those of a remote core. In this case the molecular case is just the same as the above many-electron atomic case with the model potential replaced by a sum of model potentials for each atomic core ... 

Thus, a need remains to know, with sufficient accuracy (i) whether the uniform electric field approximation is relevant, i.e. whether quantum chemistry based calculation may be meaningfully compared to experiment. This issue has to do with the fact that we are trying to connect macroscopic (electrode potential) and microscopic (local electrostatic field) quantities (ii) correct orders of magnitude for Eq, to be injected in a model potential of transcient radical anions chemisorbed on polarized surfaces. The methodology and computational details are presented in the next section. 

One of the difficulties in performing calculations of intermolecular potentials is that it is very difficult to evaluate their accuracy. The calculation of measurable properties from a potential often involves approximations, and generally requires an integration over a large region of the potential surface, so that inaccuracies are smeared out. Errors may cancel, so an inaccurate potential may give better predictions than it deserves to. When there is disagreement between a calculated property and the experimental measurements it is often difficult to know what feature of the potential is at fault. It has often been assumed that supermolecule calculations provide a benchmark for model potentials, but we have seen that for weak interactions this assumption is untenable. 

The model potential V is the key to accuracy here. The potential may come from a molecular orbital or crystal orbital calculation, in which case the derivatives must be computed numerically. In another approximation, the potential may consist of a sum of infra- and intermolecular terms in the form of the empirical force fields described in Section 2.2. This is particularly convenient because all the derivatives of equation 6.17 can be computed analytically. In an even coarser approximation, the molecule may be considered as a rigid unit, without allowing for internal deformations. In this case the displacement coordinates are just three coordinates for the center of mass and three coordinates for rotation around the inertial axes. Equation 6.17 is rewritten in terms of these coordinates, the potential is just the intermolecular part and there is no need to define an intramolecular force field, and the problem is reduced from a 3ZAat X 3ZAIat one to 6Z x 6Z one [14]. 

The accuracy of the CSP approximation is, as test calculations for model. systems show, typically very similar to that of the TDSCF. The reason for this is that for atomic scale masses, the classical mean potentials are very similar to the quantum mechanical ones. CSP may deviate significantly from TDSCF in cases where, e.g., the dynamics is strongly influenced by classically forbidden regions of phase space. However, for simple tunneling cases it seems not hard to fix CSP, by running the classical trajectories slightly above the barrier. In any case, for typical systems the classical estimate for the mean potential functions works extremely well. 

The effect of the disturbance on the controlled variable These models can be based on steady-state or dynamic analysis. The performance of the feedforward controller depends on the accuracy of both models. If the models are exac t, then feedforward control offers the potential of perfect control (i.e., holding the controlled variable precisely at the set point at all times because of the abihty to predict the appropriate control ac tion). However, since most mathematical models are only approximate and since not all disturbances are measurable, it is standara prac tice to utilize feedforward control in conjunction with feedback control. Table 8-5 lists the relative advantages and disadvantages of feedforward and feedback control. By combining the two control methods, the strengths of both schemes can be utilized. 

Equation (9.2) can be used to calculate the metal s surface potential. The value of the electron work function X can be determined experimentally. The chemical potential of the electrons in the metal can be calculated approximately from equations based on the models in modem theories of metals. The accuracy of such calculations is not very high. The surface potential of mercury determined in this way is roughly -F2.2V. 

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