Five Equivalent d Orbitals

The d and f orbitals have more complex shapes there are five equivalent d orbitals and seven equivalent f orbitals for each principal quantum number, each orbital containing a maximum of 2 electrons with opposed spins. 

Powell2 has recently pointed out that in some textbooks the erroneous statement is made that there is no way of choosing the five d orbitals of a subshell so that they are equivalent, and, after mentioning that Kimball3 had discussed five equivalent d orbitals long ago, he has shown that there are in fact two sets of five equivalent d orbitals, and has given expressions for them in terms of the conventional set, in which one differs in shape from the other four. 

The sets of five equivalent d orbitals discussed by Kimball and by Powell have symmetry axes lying along five equivalent directions in space, related by a... 

Figure 2. The principal cross sections of a d orbital of the oblate set of five equivalent d orbitals (set I).

Figure 4. The oblate and prolate pentagonol antiprisms, the axes of which give the direction of the principal axes of the two sets of five equivalent d orbitals.

FIGURE 14.7 Stick models of the staggered arrangement of five-equivalent d-orbital bond directions. The application to the bonding in ferrocene seems obvious (From Powell, R.E., J. Chem. Educ., 45, 45, 1968. With permission.)... 

The H atom solutions have a principal shell number n, and each shell has n different I shapes. Within each I shape there are 21 + 1) sublevels with m quantum numbers as for 4f-orbitals (n = 4,1 = 3) with m = —3, —2, —1, 0, +1, +2, +3. While some orbital shapes have been standardized as canonical Cartesian (x, y, z) shapes in textbooks, later work by Powell shows that in any 1-shell an equivalent shape can be found for aU 21+ 1) members of the shell. In practice this is obvious for p-orbitals and occasionally useful for five equivalent d-orbitals or even seven equivalent f-orbitals. 

Sketch the IS, 2S, 3S, 2P, 3P, and 3D angular and radial functions for the H atom, show the standard 3D orbitals and PoweU/s five-equivalent-D orbitals (see Chapter 14). 

SOLUTION According to Example 3.3, SF4 has a trigonal bipyramidal arrangement of electron pairs. We can construct five equivalent hybrid orbitals by mixing an s-orbital with three p-orbitals and one d-orbital. The result is five sp3d orbitals. 

A holistic molecular orbital theory description of bonding in complexes provides a more sophisticated model of bonding in complexes, leading to ligand field theory (LFT), which deals better with ligand influences. Both CFT and LFT reduce to equivalent consideration of d electron location in a set of five core d orbitals. 

In equation (11) <])/ refers to the function of the cf(z ) atomic orbital and refers to the function of the - /) atomic orbital, taken as a representative of one of the four d orbitals with four major lobes. Two different sets of five orthogonal equivalent d orbitals can be constructed by choosing five orthogonal linear combinations of the d ) and dtj - y )orbitals using... 

A complication arises for functions of d or higher symmetry. There are five real d orbitals, which transform as xy, xz, yz, x —y, and z, that are called pure d functions. The orbital commonly referred to as is actually Iz —x —y. An alternative scheme for the sake of fast integral evaluation is to use the six Cartesian orbitals, which are xy, xz, yz, x, y, and These six orbitals are equivalent to the five pure d fimctions plus one additional spherically symmetric function x +y +z ). Calculations using the six d functions often yield a very slightly lower energy due to this additional function. Some ab initio programs give options to control which method is used, such as Sd, 6d, pure-J, or Cartesian. Pure- f is equivalent to 5d and Cartesian is equivalent to 6d. Similarly, If and 10/ are equivalent to pure-/ and Cartesian / functions, respectively. 

In the early discussions of hybrid orbitals4,5 it was pointed out that the maximum strength (the maximum value in the bond direction) of a bond orbital formed from completed subshells of orbitals is associated with cylindrical symmetry of the orbital. In order to simplify the analysis of spd hybridization Hultgren5 decided to discuss only orbitals with cylindrical symmetry. He pointed out that no more than three d orbitals with cylindrical symmetry can be formed in a set of five d orbitals, and that each of these three is equivalent to the function d2 (see Table 1), except in orientation. 

Recall the splitting of the d orbitals in octahedral environments. The energies of the t2g and g subsets are shown in Fig. 8-4 with respect to their mean energy. We have used the conventional barycentre formalism. In effect, we express the energy of an electron in the t2g or orbitals with respect to the total energy possessed by a set of five electrons equally distributed amongst the five d functions. Alternatively, we say that our reference energy is that of 2l d electron within the equivalent spherical mean field. 

We shall now construct two concentrated spd hybrid orbitals that are equivalent, orthonormal, and have maximum concentration of electron density in their respective bonding directions. We can choose any arbitrary direction for the two bond hybrids that we construct. The judicious choice for the direction of the first bond hybrid is along the z axis, because only one of the five d-orbital functions (dz2) can contribute to a bond in this direction. The largest value of the angular part of the first hybrid function (i.e., the maximum concentration of electron density) along the z axis for a chosen spM hybrid orbital hi is insured by choosing both the p and d components to lie along the z direction,... 

The valency electrons in phosphor are M electrons since the M shell contains s, p and d electrons, it can be assumed that in PF5 not only the s and p electrons are active in bond formation, but that d orbitals may participate, as well. In the gaseous state, where it is shown by electron diffraction that the five bonds in PC15 are almost equivalent and certainly cannot be described as four covalent and one ionic bond, the bond formation by d orbitals must be accepted. It is in agreement with the idea that the d orbitals contribute to the bond formation that the higher fluorides (e.g. NF5 and OF6) of nitrogen and oxygen which have L shells in which d electrons do not occur cannot be obtained. 

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