Spherically symmetrical s orbitals

In the previous section the g value was considered as a scalar quantity, which was indeed a good approximation since the unpaired electron on the hydrogen atom occupies a spherically symmetric s orbital. If the unpaired electron exhibits p or d character the electron possesses both spin and orbital angular momentum. As a result the spin is not quantized exactly along the direction of the external field and the g value becomes a tensor... 

Figure 4.8 shows that the individual bonds in four-atom s-valent clusters are unsaturated, taking bond orders that are much less than unity. The two end bonds of the linear chain provide the only exception. This reduction in bond order compared to the isolated dimer is, of course, due to a given spherically symmetric s orbital forming bonds with all its neighbours. However, we will see later in Chapter 7 that the angular character of hybrid orbitals allows some sp-valent solids to exhibit saturated bond behaviour. Thus, the concept of the bond order is important because it not only... 

Overlap in the manner shown next is nonbonding. Both the positive lobe and the negative lobe of the p orbital overlap with the spherically symmetrical s orbital. The bonding overlap between the s orbital and one lobe of the p orbital is exactly canceled by an antibonding interaction between the s orbital and the lobe of opposite sign. 

A circle with a nucleus is used to represent the spherically symmetrical s orbital. 

On the other hand, the three-center bond model leads straightforwardly to a regular octahedral configuration. The six pairs of bonding electrons are assigned to the three mutually perpendicular 3c-4e bonds formed from the p orbitals of atom A the remaining electron pair is then assigned to the spherically symmetrical s orbital. 

The use of spherically symmetric s orbitals avoids the problems associated with transformations of the axes. The core Hamiltonians (Lf ) are not calculated but are obtained from experimental ionisation energies. This is because it is important to distinguish between s and p orbitals in the valence shell (i.e. the 2s and 2p orbitals for the first-row elements), and without explicit core electrons this is difficult to achieve. The resonance integrals, /3ab/ 3re written in terms of empirical single-atom values as follows ... 

These observations are consistent with a model in which the Pb s and p orbitals hybridize under normal energetic conditions to yield a stereochemically active lone pair and hemidirected structures. [The symmetries of either the nonrelativistic or the relativistic sets of orbitals are consistent with this hybridization (77, 78).] In the holodirected case, on the other hand, no hybridization takes place. The 6s electrons remain in the spherically symmetric s orbital and do not affect the molecular geometry. Holodirected complexes only appear with very large or bulky ligands, in which steric effects outweigh... 

The interaction diagrams in Sects. 2.1 and 2.2 dealt with 5-AOs. Because they are spherically symmetric, s-orbitals have no directional preferences. In contrast, p-AOs interact very differently depending on their relative directions. Two p-AOs can, for instance, form a- and a -MOs, as shown in Fig. 2.6. 

If and are spherically symmetric s orbitals, then a ssa bonding is formed as shown schematically in Figure 3.18a. The areas of the spherical probability density overlap partially. If xpp, and are p orbitals (see Figures 3.4 and 3.5), then the bondings ppa and ppjr occur. If xpp, and are d orbitals whose electronic probability densities are sketched in Figure 3.5 then electronic distributions for... 

The U orbital. The Is orbital is similar to the fundamental vibration of a guitar string. The wave function is instantaneously all positive or all negative. The square of the wave function gives the electron density. A circle with a nucleus is used to represent the spherically symmetrical s orbital. 

Now we must define the value of J 4>ijdz if i j. The overlap of two atomic orbitals on different atoms varies according to the distance between the two atomic nuclei and with the orientation of the two orbitals in space (except for spherically symmetric s orbitals). A calculation by Mulliken determined that the value of Sy for two adjacent parallel p orbitals in ethene (1.34 A apart) is 0.27, while that expected for two adjacent p orbitals in benzene (1.39 A apart) is about 0.21. In general, we will not know the distance between p orbitals before starting a calculation. More important, trying to assign a value of S12 would make the computation more complex. Therefore, we make the obviously incorrect but very convenient assumption that S,y = 0 unless i= . ... 

Soon after the Heitler-London article, Linus Pauling developed two concepts, directed valence and orbital hybridization, that have proved to be highly useful. These concepts were applied to a question of specific concern to organic chemistry, the tetrahedral orientation of the bonds to carbon. Pauling reasoned that since covalent bonds require mutual overlap of orbitals, stronger bonds result from better overlap. Orbitals that possess directional properties such as p-orbitals should therefore be more effective than spherically symmetric s-orbitals. Covalent bond formation between two atoms involving overlap of a p-orbital of one atom with an s- or p-orbital of another is illustrated in Fig. 1.1. The electron distribution that results is cylindrically symmetric with respect to the internuclear axis and defines a or-bond. 

The bonding in phosphorns componnds, as in other componnds, is explicable in terms of the overlap of atomic orbitals in directions of high electron density. The basic geometries of the varions atomic orbitals on the P atom are shown in Fignre 3.5. In addition to the spherically symmetrical s orbital and the three orthogonal p lobes, there are hve sansage-shaped d orbitals with the orientations shown. 

It is now helpful to introduce the concept of a local axis set. When the LGOs for a Y group in an XY molecule involve orbitals other than spherically symmetric. s orbitals, it is often useful to define the axis set on each Y atom so that the z axis points towards X. Diagram 5.9 illustrates this for the Fh fragment. 

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