Linur Pauling' and Vance McClure University of California at Son Diego Lo Jollo, California 92037
II
R v t hpivalent
Powell2 has recently pointed out that in some textbooks the erroneous statement is made that there is no way of choosing the five d orhitals of a subshell so that they are equivalent, and, after mentioning that Kimballa had discussed five equivalent d orhitals long ago, he has shown that there are in fact two sets of five equivalent d orhitals, and has given expressions for them in terms of the conventional set, in which one differs in shape from the other four. In this paper we amplify Powell's discussion, which is in some respects misleading. For example, Powell made the following statement: "Unlike the familiar four-lobed cubic d orbital, the pyramidal d orbital has only rather inconspicuous lobes of opposite sign. Each orbital is not quite cylindrically symmetrical about its own axis of maximum probability." In fact, the pyramidal d orhital that he discusses in detail is far from cylindrically symmetrical about its own axis of maximum probability, and the other pyramidal d orbital is also far from cylindrically symmetrical. In t,he equatorial plane about the axis of maximum prohability the functions of Powell's first set (which we shall call 11) vary from -0.3706 in two opposite directions to -1.7247 in the orthogonal directions. Each of these functions has almost the same value (strength) in the latter directions as in the principal directions, for which its value is 2.0950. The functions of the other set (which we call I) vary in this plane from -0.7247 to -1.4696, their value in the principal direction being 2.1943. In the early discussions of hybrid orbitals4P it was pointed out that the maximum strength (the maximum value in the bond direction) of a bond orhital formed from completed subshells of orhitals is associated with cylindrical symmetry of the orbital. In order to simplify the analysis of spd hybridization Hultgrens 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 dz2 (see Table I ) , except in orientation. The conventional set of five d orbitals, given in Table 1, is chosen such that the symmetry elements of the orthorhombic point group (the minimum symmetry of a general d orbital) lie along the x,y, and z axes. All possible shapes of a d orbital can be expressed as a 'Present address: Chemistry Department, Stanford University, Stanford, California 94305. POWELL, R. E., J. CHEM.EDUC.,45, 45 (1968). D X ~G. E.,~ J. Chem. ~ ~Phys.,~ 8, 188 ~ (1940). , PAWLING, LINUS,J. Am. Clum. Soc., 53, 1367 (1931). ' HULTRREN, R., Phys. Rev., 40, 891 (1932).
Table 1.
d Orbitals
The Usual Set of d Orbitals
d,, =
4%sin e w s B cos +
d,, =
43 sin B oos e sin +
d,a_,r = 6
2
5 sine B w s
a
linear function of de2 and d,2-,2, with the coefficient a ranging from 1 to d3/2
It is convenient to introduce the shape parameter 5 = 4(1 - a*). The function d,2 corresponds to 5 = 0 and the function d,2-,2 to E = 1. Values of the general d function in the &x, +y, and +z directions are shown in Figure 1 as a function of a. The functions in each region differ from those in the other two regions by a t most a phase factor, - 1 , and a rotation, which serves to permute the variables z,y, and 2. 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
Set I Set I l
S
region I
region 2
Figure 1. Diagram shewing the valuer of d orbitals in the directions of the throe principal axes, as o function of the shape poremeter o.
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It is seen that the maximum bond strength for set I (Powell's second set) is greater than that for set I1 (Powell's first set), and also that there is for each of the two kinds of functions great deviation from cylindrical symmetry. The values of the functions in the planes of symmetry are shown in Figures 2 and 3. Comparison with the corresponding cross sections for da2 and dS2Lv2shows that the functions I are qualitatively similar to dZ2,and the functions I1 are more similar to dZ2-2. The values for the two functions are indicated in Figure 1. The symmetry axis with maximum strength for each function of set I is 69" 1.35' from the fivefold axisof the pentagonal antiprism, corresponding to the ratio 0.38341 of half height to radius of the antiprism, with the corresponding values for set I1 of 41' 47.65' and 1.1186. The functions I accordingly correspond to an oblate antiprism and I1 to a prolate antiprism. There is a simple explanation for the difference in orientation of the principal axes. The theorem that the sum of the squares of the values of the functions for a complete set (a subshell) is constant requires that the shape parameters vary in a satisfactory way with change in orientation of the principal axes. For the prolate set (11) the maximum value in the plane orthogonal to the principal axis of the function lies in the basal plane of the antiprism, and thus serves to increase the electron 16
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Journal of Chemkol Education
Figure 2. The principal cross sections of a d orbital of the oblate set of equivalent
n
n
redionlof Figure 3. T~~ flve equivalent d orbitals (set Ill.
.
d
of ,he
set of
density in the equatorial region. For the oblate set (I) the electron density in the equatorial region is larger because of the larger contribution of the principal lobes of the function. The maximum value of the function in the plane perpendicular to the principal symmetry axis of the function for this set extends more in the direction of the fivefold axis of the antiprism, and thus serves to fill in this region (see Fig. 4). Our attention was attracted to the considerable deviation from axial symmetry of the Powell orbital through our application of a theorem about the values of the function along the principal axes. This theorem is that for any d orbital the sum of the squares of the values along the six principal directions is equal to 15. (In our discussion all functions are normalized to 4r.) This theorem is proved in the following way. Hultgren5 has shown that the most general d orbital, D , can be written as a linear combination of dZ2and dz2-,2
OBLATE SET I
PROLATE SET
U
Figure 4. The oblate and prolate pentagonol antiprisms, the axes of which give the direction of the principal ares of the two scb of Rve equivalent d orbitals.
+
with a12 m2 = 1 . I n our evaluation of the function D2(x,?/,Z ) in the +x, +y, and *z directions the terms involving products of variables will vanish, because of the orthorhornhic symmetry m m m, and may be omitted. The use of braces, ( ), indicates that this has been done.
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