Simple overlap model

The effective CF models, intended to include covalence effects via effective charges and shielding parameters [46] (superposition model [47], effective charge model [48], simple overlap model [49, 50]), keep the radial (M-L distance) dependence of the CF parameters as in the simple (point charge) electrostatic model. Dedicated studies have shown, however, that the radial dependence of these parameters deviates strongly from the latter for the whole series of lanthanide ions [51, 52]. 

The point-charge electrostatic model is useful in illustrating how symmetry influences the signs of the crystal field parameters B. However, it does not usually result in accurate determinations of their magnitude and therefore other methods have been developed that lead to a better estimation. One such approach is based on the angular overlap model AOM developed and expanded to the/elements by Jprgensen [45]. Another approach is the simple overlap model SOM, proposed by Malta [46]. 

The Lewis stmcture of a molecule shows how its valence electrons are distributed. These stmctures present simple, yet information-filled views of the bonding in chemical species, hi the remaining sections of this chapter, we build on Lewis stmctures to predict the shapes and some of the properties of molecules. In Chapter 10. we use Lewis stmctures as the starting point to develop orbital overlap models of chemical bonding. 

In this chapter, we develop a model of bonding that can be applied to molecules as simple as H2 or as complex as chlorophyll. We begin with a description of bonding based on the idea of overlapping atomic orbitals. We then extend the model to include the molecular shapes described in Chapter 9. Next we apply the model to molecules with double and triple bonds. Then we present variations on the orbital overlap model that encompass electrons distributed across three, four, or more atoms, including the extended systems of molecules such as chlorophyll. Finally, we show how to generalize the model to describe the electronic structures of metals and semiconductors. 

We cannot generate a tetrahedron by simple overlap of atomic orbitals, because atomic orbitals do not point toward the comers of a tetrahedron. In this section, we present a modification of the localized bond model that accounts for tetrahedral geometry and several other common molecular shapes. 

Many of the Lewis structures in Chapter 9 and elsewhere in this book represent molecules that contain double bonds and triple bonds. From simple molecules such as ethylene and acetylene to complex biochemical compounds such as chlorophyll and plastoquinone, multiple bonds are abundant in chemistry. Double bonds and triple bonds can be described by extending the orbital overlap model of bonding. We begin with ethylene, a simple hydrocarbon with the formula C2 H4. 

In the linear methylene ( >ooii-s5mimetry) binding results from the overlapping of these orbitals, to give the 2 Og and 1 ffa-orbitals. The two remaining 7t -orbitals (2px and 2py) are degenerate. Linear methylene should be a triplet Hg ] this follows from the simple MO-model. 

In situations where maximum overlap criteria are important, it is often possible to construct simple perturbation theory arguments to account for broad geometric trends. Burdett has illustrated in a very neat fashion how the geometries of binary metal carbonyls M(CO) may be rationalized using such an angular overlap model (55). The same problem has also been tackled by Elian and Hoffmann using more complete extended Hiickel calculations (78). 

The fundamental vibrations have been assigned for the M-H-M backbone of HM COho, M = Cr, Mo, and W. When it is observable, the asymmetric M-H-M stretch occurs around 1700 cm-1 in low temperature ir spectra. One or possibly two deformation modes occur around 850 cm l in conjunction with overtones that are enhanced in intensity by Fermi resonance. The symmetric stretch, which involves predominantly metal motion, is expected below 150 cm l. For the molybdenum and tungsten compounds, this band is obscured by other low frequency features. Vibrational spectroscopic evidence is presented for a bent Cr-H-Cr array in [PPN][(OC)5Cr-H-Cr(CO)5], This structural inference is a good example of the way in which vibrational data can supplement diffraction data in the structural analysis of disordered systems. Implications of the bent Cr-H-Cr array are discussed in terms of a simple bonding model which involves a balance between nuclear repulsion, M-M overlap, and M-H overlap. The literature on M-H -M frequencies is summarized. 

Model correlation functions. Certain model correlation functions have been found that model the intracollisional process fairly closely. These satisfy a number of physical and mathematical requirements and their Fourier transforms provide a simple analytical model of the spectral profile. The model functions depend on the choice of two or three parameters which may be related to the physics (i.e., the spectral moments) of the system. Sears [363, 362] expanded the classical correlation function as a series in powers of time squared, assuming an exponential overlap-induced dipole moment as in Eq. 4.1. The series was truncated at the second term and the parameters of the dipole model were related to the spectral moments [79]. The spectral model profile was obtained by Fourier transform. Levine and Birnbaum [232] developed a classical line shape, assuming straight trajectories and a Gaussian dipole function. The model was successful in reproducing measured He-Ar [232] and other [189, 245] spectra. Moreover, the quantum effect associated with the straight path approximation could also be estimated. We will be interested in such three-parameter model correlation functions below whose Fourier transforms fit measured spectra and the computed quantum profiles closely see Section 5.10. Intracollisional model correlation functions were discussed by Birnbaum et a/., (1982). 

In columnar stacked one-dimensional complexes, such as the tetracyanoplatinates, with relatively short intrachain Pt—Pt separations, it is possible to postulate a simple band model by considering overlap of the filled 5dzi orbitals on individual atoms in the chain to produce a full 5dp band, and overlap of the empty 6pz orbitals to produce an empty 6pz band (Figure 1). 

The arrangement of atoms in a molecule is based on attractive and repulsive forces (see Fig. 2.2), as well as the directionality of the bonds, which is determined by the orientation of the bonding orbitals and their desire for maximum overlap. At first glance, the simple mechanical model (see Fig. 2.1) does not explicitly include specific electronic interactions. However, in developing a model that reproduces experimentally derived structural and thermodynamic data, it is inevitable that electronic factors are included implicitly to account for electronic effects responsible for some of the structural and thermodynamic variation present in the data used in the parameterization. Depending on the model used, the electronic effects may not be directly attributable to specific parameters. 

It is possible to reformulate the CF approach by considering the ligand field in terms of explicit o and n interactions. This approach, the angular overlap model, is as simple as the old CF approach and can yield results of... 

The way to remove entanglements, viz. the manner in which topological constraints limit the drawability, is seemingly well understood and crystallization from semi-dilute solution is an effective and simple route to make disentangled precursors for subsequent drawing into fibers and tapes [ 17,18]. A simple 2D model visualizing the entanglement density is shown in Fig. 3. Here 0 is the polymer concentration in solution and 0 is the critical overlap concentration for polymer chains. 

Fig. 3 A simple 2D model envisaging how the entanglement density varies upon crystallization at decreasing polymer concentration, . is the critical overlap concentration for polymer chains...

The Angular Overlap Model is a ligand field model which uses parameters20 (A = a, n, 6...) for expressing the orbital energies. For d orbitals X can be only a, n and <5, but it is customary to use as parameters e = e a - ej and e = e - ej. In octahedral complexes a simple correlation exists between these parameters and the usual crystal field parameters21 ... 

The rock salt nitrides are formed only if there are three or less d electrons on the formally ionic cation, so that the e orbitals are empty and the t2g orbitals are half or less filled. In a simple ionic model, there is an effective energy gap due to an overlapping of the energy... 

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