Hybrids Involving d-Orbitals

Notice that in the trigonal bipyramid, the axial and equatorial (in-plane) atoms are not equivalent as evidenced by the shorter lengths of the equatorial bonds. 

This same type of hybridization can be invoked to explain the structures of many transition metal cations the hexamminezinc(ll) cation depicted below is typical. 

For most of organic chemistry the description in terms of spa hybrids is sufficient for a qualitative picture. However, if the coordination numbers involved are greater than 4, as is the case for the majority of compounds involving transition metals, d-hybridization has to be taken into account. Since the m-quantum number of a d-function influences not 

The first class takes the following form (for the n = 3 shell) in momentum space  

In the above equation, the hybrid is clearly broken down into a real part (second line), and an imaginary part (third line). We have found it convenient to analyze these two parts of the hybrid separately because of the earlier mentioned property that the real part of a hybrid will not mix with the imaginary part when computing expectation values and densities. 

The real part of Eqn. (11) arises from the mixing of s- and d-contributions. It has roots on either side of the arp-plane, on two closed -rotation-symmetric surfaces, that are 

To show these holes more clearly, we have plotted in Fig. (7) circular sections through the densities, passing through the xz-plane with radius 1/3. The graphs show the momentum density as a function of the polar angle 9 in units of n. Note that the density is maximal in the z-direction (9 = tn t = 0,1,2), and in -direction (9 = tir t = 1/2,3/2), 

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Although in its simplest form, the LE model of bonding invokes the use of hybrids involving d orbitals for trigonal bipyramidal and octahedral structures, there is considerable disagreement about the actual participation of d orbitals in such molecules. For a discussion, see A. E. Reed and F. Weinhold, J. Am. Chem. Soc.,... 

Hybridization is not limited to s and p orbitals, but may, in general, involve the mixing of all types of atomic orbitals. Hybrids involving d orbitals... 

For most ceramic materials we will not need to consider hybridization involving d orbitals. However, even when they are not involved in bonding the d orbitals can be extremely important in determining the properties of materials (particularly magnetism). 

It is interesting that a straight line drawn through the tetrahedral radii passes through the metallic radius for calcium this suggests that the metallic bonding orbitals for calcium are sp orbitals, and that those for scandium begin to involve d-orbital hybridization. 

In Section 7.2.1, we have seen that many hybridization schemes involve d orbitals. In fact, we do not anticipate any technical difficulty in the construction of hybrids that have d orbital participation. Let us take octahedral d2sp3 hybrids, directed along Cartesian axes (Fig. 7.1.10), as an example. From Table 7.1.5,... 

R (May also involve d orbital or hybrid orbital on metal)... 

The type of orbitals involved in sulfur-bond formation varies considerably by structure and can be represented as involving orbitals which have p, sp hybrid or d character. The role of the sulfur d orbitals has been a matter of some debate but at least in compounds containing unsaturated bonds and/or electron-withdrawing groups the d orbitals are credited with conferring enhanced stability on the ground state of the molecules as a result of resonance interaction. In photochemical rearrangements of thiophene derivatives for example (vide infra) the transition states proposed all involve d-orbitals participation by sulfur. 

For molecular species with other than linear, trigonal planar or tetrahedral-based structures, it is usual to involve d orbitals within valence bond theory. We shall see later that this is not necessarily the case within molecular orbital theory. We shall also see in Chapters 14 and 15 that the bonding in so-called hypervalent compounds such as PF5 and SFg, can be described without invoking the use of J-orbitals. One should therefore be cautious about using sp"d hybridization schemes in compounds of />-block elements with apparently expanded octets around the central atom. Real molecules do not have to conform to simple theories of valence, nor must they conform to the sp"d" schemes that we consider in this book. Nevertheless, it is convenient to visualize the bonding in molecules in terms of a range of simple hybridization schemes. 

Fig. 3-6. Five important hybridization schemes involving d orbitals. Heavy arrows show the directions in which the lobes point.

The remaining p-orbitals (one on each carbon) form the pi orbital. In ethyne, sp hybridization occurs to give two hybrid orbitals on each atom with lobes pointing along the axis. The two remaining p-orbitals on each carbon form two pi orbitals. Hybrid atomic orbitals can also involve d-orbitals. For instance, square-planar complexes use sp d hybrids octahedral complexes use sp if. 

Hybridization models involving d orbitals, such as dsp and d sp, have been used to explain molecules with an expanded octet of electrons, for example PCI5 and SFg. This is one model used to explain the observed shapes of certain molecules. 

Despite the difficulty posed by hybridization schemes involving d orbitals, the sp, sp, and sp hybridization schemes are well established and very commonly encountered, particularly among the second-period elements. 

For elements adjacent to the noble gases the principal orbitals used in bond formation are those formed by hybridisation of the s and p orbitals. For the transition elements there are nine stable orbitals to be taken into consideration, which in general are hybrids of five d orbitals, one s orbital, and three p orbitals. An especially important set of six bond orbitals, directed toward the comers of a regular octahedron, are the d2sps orbitals, which are involved in most of the Werner octahedral complexes formed by the transition elements. 

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