Hybridization of s, p, and d Orbitals

Valence-shell expansion can be explained by assuming hybridization of s, p, and d orbitals. 

In valence bond theory, hybridized atomic orbitals are formed by the combination and rearrangement of orbitals of the same atom. The hybridized orbitals are all of equal energy and electron density, and the number of hybridized orbitals is equal to the number of pure atomic orbitals that combine. Valence-sheU expansion can be explained by assuming hybridization of s,p, and d orbitals. 

We now use a Pauling-like approach to show how hybrid orbitals for a variety of combinations of s, p, and d orbitals may be formulated.10 We assume that the radial dependences of the s, p, d orbitals are similar so that they can be neglected. The angular parts of the orbital wavefunctions are given by the following expressions (in the usual spherical coordinates 9, ) ... 

The VSEPR theory is only one way in which the molecular geometry of molecules may be determined. Another way involves the valence bond theory. The valence bond theory describes covalent bonding as the mixing of atomic orbitals to form a new kind of orbital, a hybrid orbital. Hybrid orbitals are atomic orbitals formed as a result of mixing the atomic orbitals of the atoms involved in the covalent bond. The number of hybrid orbitals formed is the same as the number of atomic orbitals mixed, and the type of hybrid orbital formed depends on the types of atomic orbital mixed. Figure 11.7 shows the hybrid orbitals resulting from the mixing of s, p, and d orbitals. 

The difference between d2sp3 and sp3d2 hybrids lies in the principal quantum number of the d orbitals. In d2sp3 hybrids, the principal quantum number of the d orbitals is one less than the principal quantum number of the s and p orbitals. In sp3d2 hybrids, the s, p, and d orbitals have the same principal quantum number. To determine which set of hybrids is used in any given complex, we must know the magnetic properties of the complex. 

Since most covalent bonds are formed from hybridized orbitals, their directional characteristics are different from those of the atomic orbitals that comprise the hybrid. Deciding the directional characteristics for the most stable bonds formed from a given set of orbitals is a wave mechanical task for most chemists it is easier to accept the results than to learn how to do the calculations. The chemical physicist G. E. Kimball, however, has worked out the spatial arrangements for over forty sets of bonds that presumably can be formed from different combinations of s, p, and d orbitals. Only about eight of these types occur frequently a few of the combinations occur only in exceptional cases many have not as yet been observed. These bond types are most frequently classed according to the coordination number of the central atom. 

Table 8.7 shows the variety of hybrid orbitals that can be constructed from various combinations of s, p, and d orbitals, the shapes of the molecules that result, and selected examples. 

VB concept of hybridization proposes the mixing of particular combinations of s, p, and d orbitals to give sets of hybrid orbitals, which have specifie geometries. Similarly, for eoordination eompounds, the model proposes that the number and type of metal-ion hybrid orbitals occupied by ligand lone pairs determine the geometry of the complex ion. Let s diseuss the orbital eombinations that lead to octahedral, square planar, and tetrahedral geometries. 

The most common—and perhaps most important—hybrid orbitals are the tetrahdral ones formed by adding one s-, and three p- type orbitals. These can be arranged to form various crystal structures diamond, zincblende, and wurtzite. Combinations of the s-, p-, and d- orbitals allow 48 possible symmetries (Kimball, 1940). 

The Ni octahedra derive their stability from the interactions of s, p, and d electron orbitals to form octahedral sp3d2 hybrids. When these are sheared by dislocation motion this strong bonding is destroyed, and the octahedral symmetry is lost. Therefore, the overall (0°K) energy barrier to dislocation motion is about COCi/47r where = octahedral shear stiffness = [3C44 (Cu - Ci2)]/ [4C44 + (Cu - C12)] = 50.8 GPa (Prikhodko et al., 1998), and the barrier = 4.04 GPa. The octahedral shear stiffness is small compared with the primary stiffnesses C44 = 118 GPa, and (Cn - C12)/2 = 79 GPa. Thus elastic as well as plastic shear is easier on this plane than on either the (100), or the (110) planes. 

The one-electron s, p, and d orbitals frequently used to explain observed stereochemistries are a convenient but arbitrary means of decomposing the electron density into spherical harmonics. They represent nothing more than a suitable basis set for a quantum mechanical calculation. When assigned solely on the basis of the observed geometry, they convey no very profound information about the bonding processes at work. It is much simpler and more informative to say that an atom is tetrahedrally coordinated than to say that it is sp hybridized, just as it is easier to say that it forms three equatorial or two axial bonds than to say it is sp or sp hybridized, respectively. Only in the case of the electronically distorted ions discussed in Chapter 8 does an orbital description provide a meaningful rationale for the observed stereochemistry. 

What s, p, and d orbitals of a central atom can be used to form a hybrid orbitals for an AB8 molecule having a square antiprism structure ... 

Some hybrid orbitals containing s, p, and d orbitals are listed in Table 5,2. The structural aspects of various hybrid orbitals will be discussed in Chapter 6, but the bond angles between orbitals of a given hybridization are also listed in Table 5.2 for reference. 

The hybrid orbital type d2sp3 refers to a case in which the d orbitals have a smaller principal quantum number than that of the s and p orbitals (e.g., 3d combined with 4s and 4p orbitals). The sp3d2 hybrid orbital type indicates a case where the s, p, and d orbitals all have the same principal quantum number (e.g., 4s, 4p, and 4d orbitals) in accord with the natural order of filling atomic orbitals having a given principle quantum number. Some of the possible hybrid orbital combinations will now be illustrated for complexes of first-row transition metals. 

This was developed by Linus Pauling in 1931 and was the first quantum-based model of bonding. It is based on the premise that if the atomic s, p, and d orbitals occupied by the valence electrons of adjacent atoms are combined in a suitable way, the hybrid orbitals that result will have the character and directional properties that are consistent with the bonding pattern in the molecule. The rules for bringing about these combinations turn out to be remarkably simple, so once they were worked out it became possible to use this model to predict the bonding behavior in a wide variety of molecules. The hybrid orbital model is most usefully applied to the p-block elements the first two rows of the periodic table, and is especially important in organic chemistry see Page 37. 

In the models of chemical bonding we have discussed up to now, we have assumed that the electrons that interpose themselves between adjacent nuclei (the bonding electrons ) are in orbitals associated with one or the other of the parent atoms. In the simple Lewis and VSEPR models, these were just the ordinary s, p, and d orbitals. The more sophisticated hybridization model recognized that these orbitals will be modified by the interaction with other atoms, and the concept of mixed (hybrid) orbitals was introduced. 

Polyhedra with six coplanar vertices cannot be formed from hybrids using only s, p, and d orbitals. The smallest such polyhedron is the seven-vertex hexagonal pyramid, which requires the/(v(jr - 3y )) orbital with six major lobes pointed toward the vertex of a hexagon (Table 2). 

Hybrid orbitals are atomic orbitals formed by combinations of s, p, and d atomic orbitals, and are useful in describing the bonding in compounds. There are various types. In carbon, for instance, the electron configuration is Is 2s 2p. Carbon, in its outer (valence) shell, has one... 

The exact details of the bonding mechanisms in these ceramics are still controversial, and several different approaches to explain the wide range of observed properties have been suggested. One common feature to all the proposed mechanisms is that of orbital hybridization. Hybridization of the s, p, and d orbitals of the transition metal as well as hybridization of the s and p orbitals of the nonmetal has been proposed. 

There are nine stable orbitals available for the transition elements (one 4j, three 4p, five 3d), and, with one required as the metallic orbital, the metallic valence might be expected to continue to increase, and have the value 7 for manganese and 8 for iron. However, as mentioned above, the physical properties show that the metallic valence remains at the maximum of 6 for manganese, iron, cobalt, and nickel, and then begins to decrease at copper. The maximum value of 6 corresponds to the number of good bond orbitals that can be formed by hybridization of the s, p, and d orbitals. The decrease in metallic valence beginning at copper is caused by the limited number of orbitals, as shown by the example of tin. 

---