More Sophisticated Semi-Empirical Methods

The simple, or Hiickel based, molecular orbital theory (HMO and PPP methods) frequently provides useful qualitative insights but cannot be used reliably in a quantitative manner. For this purpose it is necessary to use a method which takes account of all the electrons as well as their mutual repulsions. A major bottleneck in such calculations is in the computation and storage of the enormous number of electron-repulsion integrals involved. Early efforts to reduce this problem led Hoffmann to the EH approximation (I.N. Levine, Quantum Chemistry, 4-th ed., 1991, Prentice-Hall, Inc., Ch. 16, 17), and Pople and co-workers to the CNDO, INDO and NDDO-approximations (B-70MI40100). 

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Nevertheless, the increasing demands for models which can do more in terms of exploring PE surfaces continues to provide an impetus for the inclusion of TM atoms into the more sophisticated semi-empirical schemes. While the general performance of the models (vide infra) is encouraging, it is equally apparent that a significant amount of work remains before such methods enjoy the same level of accuracy for TM systems as they currently achieve for molecules comprising lighter, non-TM atoms. 

Recently, the MNDO type methods (MNDO [32], AMI [33] and PM3 [34]) have been tested for their ability to produce reliable MEP maps. These semi-empirical methods, just as the CNDO and INDO methods, are ZDO methods, and are based on the more sophisticated NDDO approximation [35]. 

A classical description of the molecule M in Figure 14.9 can be a force field with (partial) atomic charges, while a quantum description involves calculation of the electronic wave function. The latter may be either a semi-empirical model, such as AMI or PM3, or more sophisticated electronic structure methods, i.e. FIF, DFT, MCSCF, MP2, CCSD, etc. When a quantum description of M is employed, the calculated electric moments induce charges in the dielectric medium, which in turn acts back on the molecule, causing the wave function to respond and thereby changing the electric moments, etc. The interaction with the solvent model must thus be calculated by an iterative procedure, leading to various Self-Consistent Reaction Field (SCRF) models. 

More sophisticated procedures involve taking the start MO coefficients from a semi-empirical calculation, such as Extended HUckel Theory (EHT) or Intermediate Neglect of Differential Overlap (INDO) (Sections 3.12 and 3.9). The EHT method has the advantage that it is readily parameterized for all elements, and it can provide start orbitals for systems involving elements from essentially the whole periodic table. An INDO calculation normally provides better start orbitals, but at a price. The INDO... 

Much effort has been expended on models that can be used to predict the solubility behavior of solutes, with good success being attained using a semi-empirical, group contribution approach [75]. In this system, the contributions made by individual functional groups are summed to yield a composite for the molecule, which implies a summation of free energy contributions from constituents. This method has proven to be useful in the prediction of solubility in water and in water-cosolvent mixtures. In addition to the simplest methodology, a variety of more sophisticated approaches to the prediction of compound solubility have been advanced [68]. 

More sophisticated interpretation methods have been developed, e.g. application of semi-empirical complexation isotherms or a priori affinity spectra (Karush and Sonenberg, 1949 Posner, 1966 Perdue and Lytle, 1983a, b Allison and Perdue, 1994 Grzyb, 1995 Ruzic, 1996). All these methods have in common a conceptual approach similar to that used in the study of metal complexation by... 

Non-Lorentzian dielectric functions discussed in Section 1.6.7 cannot be directly applied to treat solvation energies. The poles of e(k) promote numerical instabilities in calculations. They have deep physical roots originating from the interference between polarization and density fluctuations in the vicinity of the solute [37], Attempts to suppress this complication in terms of unusually sophisticated methods have been reported [51,52], However, simple traditional solutions look more expedient and efficient. Restricting the treatment by purely Lorentzian functions s(k) resolves the problem and provide a consistent and satisfactory semi-empirical theory for ordinary practical implementations. 

Early theoretical studies based on a semi-empirical self-consistent tight-binding scheme indicate that the core-level shifts in the Pd/W(l 10) and PtAV(l 10) systems come from initial state effects (d-s,p rehybridisation, for example) [37]. The calculated shift for the Pd core level was 0.7 eV versus the value of 0.8 eV measured experimentally [53]. More sophisticated calculations (fiill-potential linear muffin-tin orbital method with LDF) for the Pd/Mo(110) system also indicate that the Pd 3d core-level shifts reflect initial state effects (substantial polarization of electrons around Pd) [40]. In this case, the calculated Pd 3ds/2 core level (0.9 eV) is identical to the experimental value and most of it (0.77 eV) comes from initial state effects while the rest (0.13 eV) originates in changes in the screening of the core hole [40]. 

In molecules the various methods of estimating percent ionic character give results ranging smoothly from 0 percent to near 100 percent. This includes the kind of semi-empirical model used above, as well as much more sophisticated calculations. But in solids there appears to be a discontinuity in ionicity, just as there is a discontinuity in structure. It may be more meaningful to classify solids as either highly ionic or highly covalent, rather than as positioned in a continuous scale. 

The methods used to estimate the effects of plasticization on Tg fall into two general classes namely, semi-empirical equations based on considerations of "free volume" and the much more sophisticated theoretical treatments based on statistical thermodynamics. Neither type of approach is able to provide accurate predictions on a consistent basis. The equations based on free volume [30,67,121] have generally been favored over the statistical mechanical methods (see [122] for an example) for use in practical applications, because of their greater simplicity. For example, the Tg of a plasticized polymer can be estimated very roughly by using the following two equations ... 

We can thus conclude that our model leads to the same qualitative conclusions as ab initio or semi-empirical met hods with incommensurably less computational effort (all the calculations reported here have been done on a 486DX2 personal computer cadenced at 66 MHz xmder the Windows environment with 32 Mo RAM). On the other hand it correlates very nicely with experiments and has provided very interesting results not attainable by more sophisticated methods. 

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