(
GUEST AUTHOR Irwin Cohen
I
Youngstown University Youngstown, Ohio
Textbook Errors, 28
The Shape of the 2p and Related Orbitals
M a n y recent textbooks include diagrams of atomic and molecular orbitals. Most of the texts which do this mistake graphs of the angular variation of the 2p and related wave functions for actual geometrical descriptions of these orbitals.' The result is a distorted and illogical picture of the orbital. The formula for the wave function, $, for a 2 p orbital may be written as follows2 in polar coordinates (1): +(r, +) =
(yw
7 COB
*
(1)
where the wave function $(r, 4) may be considered for our purpose as being the square root of the probability of fmding the electron at the coordinates (7, 4), k is one half the effective nuclear charge, r is the distance from the nucleus expressed in units of Bohr radii (0.53 A), and 6 is the angle between r and an axis. Any axis may be chosen. p-Orbitals a t axes which are a t right angles to each other are orthogonal (i.e.. noninterfering), so that the three Cartesian axes provide a complete set of orthogonal 2p orbitals. It is sometimes convenient to consider fi as being made up of radial and angular components, e-"r and cos 4, respectively. We may then express J. either as a function of the distance a t a constant angle or as a function of the angle a t a constant distance: +(r)+ = Aeck'r +(b). = R cos
+
(2) (3)
where A and R are expressions whose values are given by equation (1). A graph of equation (2) is sketched in Figure 1 and shows how J. varies radially in any given direction from the nucleus. A graph of equation (3) is the familiar cosine curve, which in a polar graph (8, 3) is two circles tangent a t the origin. I t is this polar graph which is commonly represented as being the 2p orbital. I n three dimensions, the graph is two Suggestions of material ~uitahlefor this column and guest columns suitable for publication directly are eagerly solicited. They should he sent with as many details as possible, and particularly with references to modern texthonks, to Karol J. Mysels, Department of Chemistry, University of Southern California, Los Angeles 7, California. I Since the purpose of this column is to prevent the spread and continuation of errors and not the evaluation of individual texts, the source of errors discussed will not be cited. The error must occur in a t least two independent standard books to be presented. 'The formula is mitten in different forms in many sources The form used here is that of Coulson (I),although he uses C/2 for k, where C i s the effective nuelcar charge. a The earliest use of these graphs is prohably Pauling's (8,5). However, Panling explicitly strates that t h e e gmphs do not actually ahow the shape of the orbital.
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Jovrnd o f Chemicol Education
t,angent spheres. Sometimes the spheres are given fuzzy outlines, which is not very meaningful. Such graphs show precisely the angular variation of the orbital and should not he used to pictnre the orbital itself . 3
Figure 1.
Groph showing rodiol vclridion d .zlJ
To describe the shape of the orbital, one would need t o draw a contour line, i.e., a line following a path of constant J.. To do this, equation (1) may be transformed into Cartesian coordinates by use of r cos .$ = x and r = (x2 y2 z2)'/*, where the X-axis is taken as the axis from which the angle 4 is measured. This yields the formula for the 2p, orbital (formulas for the 2p, and 2p, orbitals are analogous) :
+ +
+(z, y, I) =
(F)'"
ze
-k d ; i T i T 2
(4)
Now a contour line in 'the XY plane can be drawn by setting z = 0 and solving for y: =
[(i 1" k6/g -)= - q/: z
*'/a+
(5)
This equation is somewhat simplified by expressing $ in terms of percentage of J.,,,. J.mC,,. may be found by differentiating equation (4) with respect to x where y = z = 0, and setting d$/dx = 0. The value so found, $-, = ka/zs-'/2e-', may then he combined with equation (5) to yield: y = [(k-' In 8-' k e z p -
z2I1/?
(6)
where 0 = J./$,,. Contour lines are shown in Figures 2 and 3 for p = 0.316 and 0 = 0.10 (i.e., J. = 0.316$-, and $ = 0.10&,.,). These lines are therefore also the contours for $% = 0.10J.2,., and fi2 = 0.01$2,.,, re~pectively.~The dots show the The notion that equation (3) or its polar graph represents the 2p orbital hsa led to the apparently widespread belief that the contours for must be different (narrower) from those for This oonclusion results because RPcos2 6 is 11 different curve from R cos +, if R is held constant. A more thorough analysis, however, considers the variation in R, and it must conclude that if +is constant along a. contour line, then +*must also be constant.
+.
positions of the maxima. The value chosen for k was 1.625, half the estimated effective nuclear charge for a 2p electron of carbon (4). The probability that the electron may be found within a contour is not given by the above figures but may be calculated by integrating pb2 over the volume enclosed by the contour. The shape of a 2p orbital may therefore roughly be described as two separated spheres, ellipsoids, or distorted ellipsoids, in increasing order of precision. However, the orbital should not he described as two tangent lobes of any kind since that leads directly to the following logical impasse:
For a 2p orbital, the angular variation in pb is given by equation (3), and therefore +2(4), = R2 cos2 4. For a second 2p orbital disposed at right angles to the fist, the angular variation in the plane of the two axes, in terms of the angle 4 a t the first axis, is = R cos (4 90") = i R sin 4, and pb280~(4), = R2 sin2 4. Therefore the sum of the angular variations of +Z in t,he plane of the two axes is R2 cos2 4 R2 sin2 4 = R2, which is constant a t any fixed distance from the nucleus but which varies radially according to equation (2).
+
+
First: +must equal zero evm-where on the node and it must not be sera any place else; Second: is constant and finite along a line showing the shape of the orbital (i.e., a contour line); Therefore, if the finite, non-zero contour lines for the two lobes meet at the node, then at the pomt of contact +must he h0t.h zero and not zero at the same time.
+
I
Figure 4. low).
Axial approach of 2p orbitals (obove), ond o o (2pl bond (be-
Figure 5. (right).
Porallel approach of 2p orbitals (lefl),
Figure 2. Contour liner for the Zp, orbital in one qvodrant of the XY.plane. Curve A is the contour for J, = 0.31 6 &.x and 0.1 0 curve B is the contour for J, = 0.1 0 +,.,and $'= 0.01 .,.,2i/l UnitsareBohr radii.
+"
As a bonns to the correct understanding of the wave function, it becomes easy to show that the shape of the charge cloud, or the probability distribution, for two orthogonal 2p orbitals is not something like a fourleaf clover but rather is something like a doughnut, a cross-section of which is Figure 3. This is especially useful in showing the cylindrical symmetry of the triple bond. The following paragraph summarizes this picture in a non-rigorous manner.
Figure 3. Conlow liner for the 2p, orbital in the XY-plane. The 2p, orbit01 may be developed from this b y rotation obout the Y-oxis Contours for total +=for the 2p. and 2p, orbitoh moy b e developed b y rotation obout the X.axis
and o r12p) bond
With the aid of these corrected orhital contours, one can form much better pictures of the covalent bonds derived from the 2p orbitals. Figures 4 and 5 show the overlapping of 2p orbitals to form a 4 p ) and a r(2p) orbital, respectively. The a portion of a triple bond (which may be thought of as a a bond and two a(2p) bonds) is sketched in Figure 6. An improved presentation of the sp3orbital is perhaps even more important for the organic chemist. Elementary texts show this variously as an off-center ellipsoid, two unequal lobes tangent at the center, or two unequal lohes separated with a node at the center. While there is an improvement in the above sequence, a calculation of the contour lines shows that none of these is correct. Figure 7 clearly shows that the node is a curved surface which passes between the center and the more extended lobe (5). There are several available references which offer discussions of these orbital shapes, though most of Volume 38, Number 7, Junuory 1967
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The above analysis shows clearly that the shape of a 2 p orbital is quite different from the picture usually given, which is at best a polar graph of the angular variation alone. The correct shape of the orbit,al can he simply derived in terms of equation (6) and presented graphically through contour lines. This leads not only to an improved understanding of double and triple bonds hut also to a corrected picture of the spqybrid orhit,al. Figure 6. Two pairs of orthogono1 2p orbitals lZpv their r-bond at right ( 0 %in a triple bond).
+ Zp,)
at ieft and
Acknowledgment
This paper was prepared \vith invaluable assistance from Professor Karol J. Mysels. who offered many suggestions and corrections, and Donald R. Lavin, 1-oungstown University, who prepared the drawings. Literalure Cited
Figure 7. Contours for an rpbrbititi, oher COULSON, C.A.. "Vdence," Oxford University Press. 1952, p. 190, with permission of The Clarendon Press, Oxford.
them are in highly mathematical terms. The diag~am.i in (6) and (7) and the phot,ographic simulations in (8) are very helpful, as are the more limited discus&ons and drawings in (9),(lo), and ( I f ) . The contour line equat,ions (5) and (6), however, have not yet been observed any place by this mriter.
22 / Journol of Chemical Educotion
(1) C o u ~ s o h ,C. A., "1-alence," Oxford University Prcsa, London, 1952, pp. 40-1. (2) PAULISG,L. N., J . ;lm.Chem. Soc., 53, 1367 (1931). (3) P A V L ~ NL. G ,N., ''Nat~lreof the Chemical Bond," 2nd ed., Cornell University Press, Ithaca, 1945, p. 77. (-1) SLATER. J. C.. Phw. Rev.. 36. 58 (1930). i j j COULBON, C. A., Op. it., 190. (6) FANO, U.,and FANO,L., "Basic Physics of Atonl~and Molecules," John Wile? & Sons, New York, 1959, pp. 191-216. ( i ) WISIVESSER, W. J., J. CHEII.EDUC.,22, 314 (1945). (8) LEIGHTON~ R. B., "Principles of Modern Physics," NcGraaHill Book Co., Yen- York, 1959, pp. li8-9. cal Chemistry" (tmnsl. (9) H E R ~ N P. S , H.,' i T l ~ e ~ r ~ t iOrganic hy S. Coffey), Elsevirr Publishing Co., Amst~rdam,1954, pp. 42, 73. (10) SOLLER,C. R., J . CHEII.EOUC.,32, 23 (1955). 37, 118-21 (1960). (11) T H O ~ S O A. N ,B., 3. CHEV.EDVC.,
6.
