E. A. Ogryzlo
On the Shapes of
University of British Columbia Vancouver, B.C., Canada
Two papers dealing with the "shapes of What f orbitals" recently appeared in this Journal.', they actually presented, however, were plots of "polar functions" that show the direction (from the nucleus) of the maximum electron density. As Cohen? and Ogryzlo and Porter4 have indicated, the shape of an orbital can only properly be given by a contour surface of constant Y (or W). The difference between a "contour plot", and a "polar plot" is illustrated in Figures 1 and 2, which show the 3po orbital plotted both ways. Figure 1 can quite correctly be identified with the shape of the orbital. The dots indicate maxima in electron density and the limes join points where the electron density has decreased to 3.16% of the maximum value. This is comparable to describing a mountain by the positions of its peaks and a contour 0.0316 of the way up. The picture can obviously be improved by the addition of other contour lines.3.
I z axis
Figvre 1. Contour. of a 3ps orbital. Dots indicete moximo in B or W The lines are drown for W/W-.= 0.0316.
of maxinlunl "electron density", we feel that it is misleading to draw it with a fuzzy line, or to make a styrofoam model of it. Figure 3 shows the contours of a 4 6 orbital calculated by the graphical method described by Ogryzlo and P ~ r t e r . The ~ dots give the points of maximum "electron density" (W) and the lines represent points where V is 0.1 of the maximum value. This diagram can be compared with the polar plot in Figure 2 by B e ~ k e r . ~ Though the difference between the two plots is not as striking as it is when spherical nodes are present (Fig. I), several points are worth noting. The "electon density" maxima occur a t the same distance from the nucleus in both the small and large lobes. This is to be contrasted with the impression gained from the polar p10ts.~ Because of the intersection of a number I z axis of nodal planes a t the n nucleus, the electron density is very small in a large region around the nucleus. This is evident from Figure 3 but not from the polar plot.3 + Finally, it should be noted that the contour lines for the wave function (*)and the probability function (W) are identical, though theyare more tightly packed for W. This difference in gradient results in a difV. i plots" ofin Ttheand"polar ference
8 i
Figure 2. A "polar plot" of the ongulor dependence of the wave function (Vl811for a 3 s orbital.
The "polar plot" of *2 as a function of 0 shown in Figure 2 simply gives the angular dependence of the probability function; i.e., it shows how W varies along a curve such as the dotted one in Figure 1. We feel that such a plot should not be identified with the shape of the orbital. Although it does indicate the direction
f Orbitals
u
Q
0
OOQ
3. Contourr of a 4fo orbital. H~~~~~~it doesnot mean Figure Dots indicate maxima in V or V. there is a diierence in ~h~ line, are drown for .p/wm.,= shape in the way the polar 0.1. plots Suggest. With information available from a standard textbook5 the contours of s,, .v,. d,. and f orbitals can be calculated for hydrogenic atoms. For many-electron atoms electron interaction will, no doubt, distort the contours. Nevertheless we feel that the polar plot does not give tjrue orbital shape in either case and furthermore does not display the interesting nodal between orbitals.
IFRIEDMAN, H. G. JK.,CHOPPIN, G. R., AND FEUERBACKEK, W G ~ Y Z E. L OA.,, AND PORTER G. B., J. CHEM.EDUC., 40,256 D. G., J. CHEM.EDUC., 41, 354 (1964). (1963).
BECKEK, C., . I CHEM. . EDUC., 41,358 (1964). COHEN,I., J. C H ~ MEDUC., . 38, 20 (1962).
150
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Journwl of Chemicol Educofion
PA~JLING, L., "The Nature of the Chemical Bond," Cornell University Press, Ithrtca, New York, 1960.
To the Editor: In his note, Dr. Ogryzlol makes the point that a "contour plot" is a more appropriate representation of an f orbital than is the "polar plot" published by us1 and by C. Becker.% What he says is generally quite true, in theory. The principal difficulty with this approach, however, is that it assumes that the radial part of the f orbital is reasonably well known. As we have discus~ed,~ this is not the case. Indeed, in presenting our "polar plots" (as Dr. Ogryzlo calls them), we stated that we were presenting only the angular part of the wave function, and that our pictures would be distorted (i.e., changed somewhat in shape) if the radial part of the orbital were also taken into account. We heartily agree with Dr. Ogryzlo that if the radial part of the f orbitals were known with good accuracy, contour plots would be the best pictorial representations of the f orbitals. The choice of the 3po orbital as an example of the differences between a contour plot and a polar plot is perhaps somewhat unfair. As Dr. Ogryzlo points out, the differences between the two types of plot is much less striking when the orbitals contain no spherical nodes. The 2p, orbital would have been a more appropriate choice of an example.
Dr. Ogryzlo states that "it is misleading to draw it [i.e., a polar plot of an orbital] with a fuzzy line, or to make a styrofoam model of it." We agree that the "fuzzy l i e " picture may be misleading. But we do not feel that the styrofoam models of 3ecker3 are any more misleading than the three dimensional pictures which we presenLz The only real difference between them is that Becker plotted the angular part of J,%,while we plotted the angular part of ~. The styrofoam models are not comparable to the "fuzzy lme" picture, as Dr. Ogryzlo implies. One question basic to this discussion is, what is the purpose of a picture of an orbitaI? It seems to us that the principal purpose is to give the student an understandmg, an intuitive idea, of the shape of the orbital. We feel that, so long as the orbital under consideration contains no spherical nodes, the same result is achieved, whether a "contour plot," a "polar plot," or even a " fuzzy picture" is used. If for some reason a more precise picture of an orbital is required, then the contour plot seems to be best, if the entire wave function, both radial and angular parts, can be written down accurately.
H. G. FRIEDMAN, JR.,G. R. CHOPPIN, AND D. G. FEUERRACKER FLon1oA STATEUNIVERSITY, TALLAHASSEE
with CLIFFORD BECKER
' ~ G R Y Z W ,E. A,, THIS JOURNAL, 42,150 (1965).
H. G., JR., CHOPPIN, G . R., FRIEDMAN, D. G., THIS JOURNAL, 41, 354 (1964).
AND
Onro STATEUNIVERSITY, COLUMBU~
FEUERBACKER, a
BECKER, C.,
TKIS
JOURNAL, 41, 358 (1964).
Volume 42, Number 3, March 1965
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