Specification hybrid orbitals

Here the crystallographic indices in the subscripts refer to the hybrid molecular orbital directions. Because the Schrbdinger Equation governing electron motion is linear, any combination of wave functions that solve it will also be a solution. In other words, choosing the hybrid orbitals or the atomic orbitals as a starting point for the calculation must yield identical results. The most flexible and general approaeh is not to be restricted to specific hybrid orbitals but rather to consider all possible orbital-by-orbital interactions of the fundamental atomic states. These states apply to a given atom in any environment. Thus, their use is valid for any material in which the atom occurs. As an example of a specific interaction, one can ask how does the Px orbital on one atom interact with the orbital on another atom. 

PM3/TM is an extension of the PM3 method to include d orbitals for use with transition metals. Unlike the case with many other semiempirical methods, PM3/TM s parameterization is based solely on reproducing geometries from X-ray diffraction results. Results with PM3/TM can be either reasonable or not depending on the coordination of the metal center. Certain transition metals tend to prefer a specific hybridization for which it works well. 

The formation of the BeF2 molecule can be explained by assuming that, as two fluorine atoms approach Be, the atomic orbitals of the beryllium atom undergo a significant change. Specifically, the 2s orbital is mixed or hybridized with a 2p orbital to form two new sp hybrid orbitals. (Figure 7.12). 

We generate hybrid orbitals on inner atoms whose bond angles are not readily reproduced using direct orbital overlap with standard atomic orbitals. Consequently, each of the electron group geometries described in Chapter 9 is associated with its own specific set of hybrid orbitals. Each type of hybrid orbital scheme shares the characteristics described in our discussion of methane ... 

The steric number of an inner atom determines the electron group geometry, each of which is associated with one specific type of hybrid orbital. 

To circumvent problems associated with the link atoms different approaches have been developed in which localized orbitals are added to model the bond between the QM and MM regions. Warshel and Levitt [17] were the first to suggest the use of localized orbitals in QM/MM studies. In the local self-consistent field (LSCF) method the QM/MM frontier bond is described with a strictly localized orbital, also called a frozen orbital [43]. These frozen orbitals are parameterized by use of small model molecules and are kept constant in the SCF calculation. The frozen orbitals, and the localized orbital methods in general, must be parameterized for each quantum mechanical model (i.e. energy-calculation method and basis set) to achieve reliable treatment of the boundary [34]. This restriction is partly circumvented in the generalized hybrid orbital (GHO) method [44], In this method, which is an extension of the LSCF method, the boundary MM atom is described by four hybrid orbitals. The three hybrid orbitals that would be attached to other MM atoms are fixed. The remaining hybrid orbital, which represents the bond to a QM atom, participates in the SCF calculation of the QM part. In contrast with LSCF approach the added flexibility of the optimized hybrid orbital means that no specific parameterization of this orbital is needed for each new system. 

We will explain how to do this by taking the specific example of methane. Methane has a central carbon atom which is a-bonded to four hydrogen atoms with each a-bond pointing to one of the comers of a tetrahedron. We therefore require four hybrid orbitals on the carbon atom which similarly point to the comers of a tetrahedron. Since the four bonds are indistinguishable, the four hybrids must be equivalent, that is to say they must be identical in all respects except for their orientation. For the reasons given in 11-2, they will be taken to be linear combinations of the atomic orbitals of carbon, which are... 

The valence bond picture for six-coordinate octahedral complexes involves dispi hybridization of the metal (Fig. I i.lc. d). The specific d orbitals that meet the symmetry requirements for the metal-ligand o bonds are the four-coordinate d complexes discussed above, the presence of unpaired electrons in some octahedral compounds renders the valence level ( — l)J orbitals unavailable for bonding. This is true, for instance, for paramagnetic [CoFJ3- (Fig. I I.lc). In these cases, the VR model invokes participation of -level dorbitals in the hybridization. 

The other electron-pair geometries that are listed in Table 9-2 are also related to specific hybrid molecular orbitals, but they are more complicated because they involve midp. In every case, the ami/ orbitals are of the same priflfap l( BMlSqis%mfeehySflffMftrofif feg... 

This description, which uses a mixture of MO and valence bond concepts and attributes specific energies to hybrid orbitals, may appall the theoretician. It has the advantage, however, of reducing the problem to a series of two-orbital interactions, thus allowing us to find the CO orbitals very quickly. 

To understand this effect, we need to consider the Si-Si bonding within the bulk crystalline solid. As we discussed earlier, electrons are promoted from valence to conduction bands due to thermal excitation. The valence band of the extended solid is formed from the overlap of sp hybridized orbitals residing on each Si atom. When an electron migrates from valence to conduction bands under normal circumstances, there is no directional preference. However, when a strain is introduced along a specific direction of the lattice, the energies of the hybrid orbitals along this direction are altered. 

Hybrid orbitals are localized in space and are directional, pointing in a specific direction. In general, these hybrids point firom a central atom toward surrounding atoms or lone pairs. Therefore, the symmetry properties of a set of hybrid orbitals will be identical to the properties of a set of vectors with origins at the nucleus of the central atom of the molecule and pointing toward the surrounding atoms. 

The chemist s sketches, which are typically drawn to emphasize directionality of the sp hybrid orbitals, and a contour plot of the actual shape, are shown in Figure 6.44. Each of these contours can be rotated about the x-y plane to produce a three-dimensional isosurface whose amplitude is chosen to be a specific fraction of the maximum amplitude of the wave function. These isosurfaces demonstrate that sp hybridization causes the amplitude of the boron atom to be pooched out at three equally spaced locations around the equator of the atom (see Fig. 6.42). The 2p orbital is not involved and remains perpendicular to the plane of the sp hybrids. The standard chemist s sketches of the sp hybrid orbitals and a contour plot that displays the exact shape and directionality of each orbital are shown in Figure 6.44. The isosurfaces shown in Figure 6.43 were generated from these contour plots. 

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