The internal localization methods

Ruedenberg method the maximum interaction energy of the eiectrons occupying a MO 

The basic concept of this method was given by Lennard-Jones and Pople, and applied by Edmiston and Ruedenberg. It may be easily shown that for a given geometry of the molecule the functional ij is invariant with respect to any 

The proof is very simple and similar to the one on p. 340, where we derived the invariance of the Coulombic and exchange operators in the Hartree-Fock method. Similarly, we can prove another invariance 

Millie, J. Ridard, J. Vinh,/. Electr. Spectr. 4 (1974) 13. 

Lennard-Jones, J.A. Pople,Proc. Jtoy. Soc. (London) A202 (1950) 166. Edmiston, K. Ruedenberg, Rev. Modem Phys. 34 (1962) 457. 

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The External Localization Methods The Internal Localization Methods Examples of Eocalization... 

When a non-centrosymmetric solvent is used, there is still hyper-Rayleigh scattering at zero solute concentration. The intercept is then determined by the number density of the pure solvent and the hyperpolarizability of the solvent. This provides a means of internal calibration, without the need for local field correction factors at optical frequencies. No dc field correction factors are necessary, since in HRS, unlike in EFISHG, no dc field is applied. By comparing intercept and slope, a hyperpolarizability value can be deduced for the solute from the one for the solvent. This is referred to as the internal reference method. Alternatively, or when the solvent is centrosymmetric, slopes can be compared directly. One slope is then for a reference molecule with an accurately known hyperpolarizability the other slope is for the unknown, with the hyperpolarizability to be determined. This is referred to as the external reference method. If the same solvent is used, then no field correction factor is necessary. When another solvent needs to be used, the different refractive index calls for a local field correction factor at optical frequencies. The usual Lorentz correction factors can be used. 

Now let us consider some practical methods of localization. There are two categories internal and extemal. In the external localization methods we plan where the future MOs will be localized, and the localization procedure only slightly alters our plans. This is in contrast with the internal methods, where certain general conditions are imposed that induce automatically localization of the orbitals. 

Thus, the second part of the force Fba is the force over the interface when substructure B is constrained with an enforced displacement at the interface (o 0) and the condition no external force is acting on nodes b . This second part of the force f A is indirectly dependent on stiffiiess changes of submodel A. As a result, a stiffness modification in structure A will change the displacement m, and this will change the internal force. Consequently, any stiffness changes in submodel A will lead to an inherent inaccuracy in the local analysis of submodel A if the internal force method is used. A graphical representation of the submodel A is given in O Fig. 26.11 where in addition the internal force J a is indicated. 

Recently an alternative approach for the description of the structure in systems with self-assembling molecules has been proposed in Ref. 68. In this approach no particular assumption about the nature of the internal interfaces or their bicontinuity is necessary. Therefore, within the same formahsm, localized, well-defined thin films and diffuse interfaces can be described both in the ordered phases and in the microemulsion. This method is based on the vector field describing the orientational ordering of surfactant, u, or rather on its curlless part s defined in Eq. (55). 

The second use of Equations (2.36) is to eliminate some of the composition variables from rate expressions. For example, 0i-A(a,b) can be converted to i A a) if Equation (2.36) can be applied to each and every point in the reactor. Reactors for which this is possible are said to preserve local stoichiometry. This does not apply to real reactors if there are internal mixing or separation processes, such as molecular diffusion, that distinguish between types of molecules. Neither does it apply to multiple reactions, although this restriction can be relaxed through use of the reaction coordinate method described in the next section. 

To control the step size adaptively we need an estimate of the local truncation error. With the Runge - Kutta methods a good idea is to take each step twice, using formulas of different order, and judge the error from the deviation between the two predictions. Selecting the coefficients in (5.20) to give the same a j and d values in the two formulas at least for some of the internal function evaluations reduces the overhead in calculation. For example, 6 function evaluations are required with an appropriate pair of fourth-order and fifth-order formulas (ref. 5). 

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