Equivalent Real Angular ( I # 0) Atomic Orbitals A Particular Transformation of the Spherical Harmonics J. J. C. Mulder The University of Leiden, P.O. Box 9502, 2300 RA Leiden, The Netherlands
For many students, the step from p-orbitals to the functions of higher angular momentum quantum number is a difficult one. The natural association of the axes of the three-dimensional world with the three D-functions and the imoossihilitv of repeating this in any analogous fashion for the other orhitah creates an artificial distinction. It would be most heloful if a method could he found that would deliver the degenerate sets for everv 1-value in exactlv the same manner, the hieher number providing a natural extensionof the lower one. I t is the ournose of this contribution to show the aeneral analysis of this problem, starting from the particular solution which is in the literature. Some time ago there appeared, in THIS JOURNAL, a short communication in the series Textbook Errors, entitled "The Five Equivalent d-Orbitals." 1 With reference to earlier work by Kimbal12,it was shown by Powell that the traditional, cubic symmetry-adapted d-orbitals could be transformed into a set of equivalent orbitals, appropriate t o the symmetry of the pentagonal antiprism. I t was furthermore argued that the same procedure could be applied to the p - and the f-functions, in fact for every 1 # 0 value, and it was mentioned that there is one equivalent p-set, two equivalent d-sets, four f-sets, and so forth. Some remarks about the shape of these orhitals were also made, but the nature of the nodal surfaces, for example, was not discussed. Subsequently Pauling and McClure published a note, "Five Equivalent d-Orhitals." I t was demonstrated that these orbitals are "far from cylindrically symmetrical" instead of merely "not auite cvlindricallv symmetrical" about their axis of makimumbrobability. M O importantly ~ the shape of the orbitals was clarified by showing three cross sections through the polar diagrams of both d-sets. Also the fact that any d orbital can he written as a linear combination of a d,n-orbital and a d,z-porbital was used most profitably. We will return to this point later on. In the meantime textbooks, currently in use, have not made much use of this information. Probably the reason for this is to he found in the rather awkward appearance of the transformation from one set of real orbitals to another. In the followine it will be shown that. if the comolex s~hericalharmonics are taken as the reference set, most elegant and completely general unitary transformation can he obtained that leads directly to the equivalent angular function sets. The normalized soherical harmonics are eiven hv the following expression:
a
We define a new set of functions MI,*(-1 5 k 5 +1) that will be linear combinations of the Yi,,. The unitary transformation is characterized by a phase-factor w = e2*'/21+1 and a normalization constant N = (21
+
'
Powell, R. E., J. CHEM.EDUC., 45, 45 (1968). 2Kimball,G. E., J. Chem. Phys., 8, 188 (1940). Pauling, L., and McClure, V., J. CHEM.EDUC., 47, 15 (1970). 376
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These functions are of course real and the proof of orthonormality runs as follows: (MI~IMI,~)
The structure of the transformation can be made more transparent in matrix form:
The freedom of sien entails 2'-I ~ossibletransformations and thus lead* to the number of diFferent equivalent orbital sets. The fact that this transformation for I = 1 aives rise to the usual p-functions, only oriented in a different way, was mentioned by Powell.' In this case a change in sign does not produce a different set, and therefore the signs of Y,,-I and YL,+I are always positive. From matrix (3)it follows immediately that all MI,*contain all YI,, with the same weight, which is a sufficient--though not necessary-condition for the functions to he completely equivalent. Therefore, from now on we will only consider the MI,U-functions.The following relation holds in general:
From the fact that the e+"* and e-'"'~ functions are combined in phase, it follows that MI,^ is symmetric with respect to the xz-plane (q = 0).This means that viewed as a combination of the traditional real functions it contains a t most 1 + 1members (no sin mp terms). Furthermore we may define a new x' (or z') axis in this xz-plane to coincide with the axis of maximum probability. This freedom in choice of direction eliminates one additional basis function and we are left with
+
the theorem that the general 1 0 orbital may he written as a linear combination of a t most 1 real basis orbitals. T o allow for a comparison with the results obtained by Powell and by Pauling and McClure we treat the case of the d-orhitals in more detail.
This function is a symmetric for the xz-plane. T o determine the main axis we set p = 0 and differentiate with respect to 0. The tilt angle a between the z-axis and the axis of maximum probability is then piven by
We now transform to a new coordinate system in which the main axis of the d-orhital is the ~'-axis;at right angles (in the old xz-plane) we have the y'-axis; and finally z' is the old yaxis which is unaffected. After some algebra we arrive a t
Tilt angle:
tg 2
=
2 4 & + a
The main axis in this case is the y"-axis and we obtain
When discussing these functions, Pauling and McClure did comment on the intimate relation that should exist between the tilt angle and the shape parameter (linear combination coefficient of the new d-orbital). We have determined this relation.
, = --1 * 6
-=-.
4 1
2 6
(9)
(10) sin2a(28) The true nature of these functions appears when nodal surfaces are considered.
For the second d-orbital set we ohtain the following results: 3 cos28- 1
T
sin8 C O S ~cosc
- sin28 ros?,~] 1
(6,
Using this expression the nodal surfaces of the two mixed d-functions have been plotted (Figs. 1 and 2). The xg-plane doesnot hdongtothesurfaw but iidrawn into bringout the orientation. I t clearly is the plane that contains the main axis
-1 00
Figure 1. Ncdal surtace of zeros of W
Figure 2. Nodal surface of zeros of M-.
Volume 62 Number 5 May 1985
of the function. The M + - and M--functions can he characterized as deformed d,z and d,s orbitals, respectively. The double conical surface, distorted ellipsoidally, allows for the presence of the familiar toroid, as depicted in cross section by Pauling and McClure. Their set I is less distorted (M-), and set I1 strongly noncylindrically symmetrical (Mf). Although most of these results on the two d-function sets have been obtained previously it may he of interest to see that they are of an analytic nature and follow directly from the properties of the spherical harmonics. Regarding Mz,wfunctions with 1 > 2 it may be appropriate to conclude this contribution with a brief statement. Whereas the pyramidal porbitals, because they are unchanged from the traditional set, have cylindrical symmetry (D-h) and the symmetry is reduced
378
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for the two pyramidal d-sets t o Dzh, all other 1 > 2 functions will have Czh symmetry. We did already discuss the existence of the symmetry plane; all functions must he gerade or ungerade and the i operation together with u induces Cz, so that we obtain the minimal symmetry mentioned. This means that the definition of the tilt angle becomes amhigous, as no axis of the function is defined except for the uninteresting Cz at right angles with the symmetry plane. Of course, the shape is still known completely. The equivalent orbitals may he compared with the s-tram-hutadiene molecule, with regard to their symmetry characteristics. This situation is in complete agreement with the D21+~dsymmetry of the antiprism that describes the (21 1)-orbital set.
+
