On Orbital Drawings Carl W. David
University of Connecticut Storrs, CT 06268 The hydrogen atom wavefunctions form the basis for our intuition about orbitals in chemistry. We recognize that hydrogenic orbitals do not exist in their pristine form in real molecules, but we assume that the nodal properties and electron densities of real orbitals are somehow related to them. Since we can comoute thinns about-and can draw pictures of-hydrogen orbitals (whose mathematical form is simple), we can exercise our imazinations with representations of these orhitals which can helpus mentally to decompose molecular orhitals into contrihutions from various contributing orhitals. It would appear ohvious, therefore, that the better honed our understanding of hydrogen orhitals the more likely we are to understand more complex orbitals. Most representations of hydrogen orhitals depend on a dual representation based on a product of radial wavefunctions and angular wavefunctions. The radial wavefunctions are of the form +radial = e-P (a polynomial in r ) where p is related to the radial coordinate ( r ) and the principal quantum number. Angular wavefunctions are spherical harmonics, or perhaps linear combinations thereof. A typical angular wavefunction (real) is
Figure 1. The basic coordinate system showing positive +, positive x , and positive zdirections (y = 0 throughout)
= sin@) cod#)
which is a moderately difficult function to draw. Since the angular part of the wavefunction is difficult to draw, the total wavefunction is almost imoossihle even to visualize. The difficulty lies in t h e wavefunction being a function of the 3 variables x...v.. and 2. This means that in order to plot the wavefunction properly, one must use a four dimensional space, three for the independent variables x, y, and z with the fourth dimension used fur the dependent variable J. (the wavefunction). Most people find it very difficult to visualize functions of three variables in four dimensional spaces. The need to invent representations for wavefunctions which can be drawn on a blackboard or piece of paper is obvious. Traditionally, two plots are made for each wavefunction, a radial plot, and an angular (polar) plot. The mental problem of multiplying these two together is left to the infamous "interested" reader. For a p orbital, the wavefunction
+ = r e c r I 2 eos(R) is decomposed into R(r) = re-'I2
and
a maximum along the +z-axis, and a minimum along Figure 2, $,,showing the -z-axis. The positive lobe is the upper one; the negative, the lower.
S(8, #) = cos(0)
there is nod, dependence in this particular function so that the polar plot of S(0, 4) is the familiar figure 8 with its signed lobes. However, we show helow another, complementary way of picturing the wavefunction. What if we assumed that either x , y , or z were to be forced to he constant for the duration of the plot. Then the four dimensional plot could be thought of as a set of three dimensional plots with different constant values for the "frozen" coordinate. It turns out that there are well-developed methods for plotting functions of two variables (a pseudo-three dimensional plot) in two dimensions. Such a figure can he quite beautiful. Although Macomber ( 1 ) gives an inexpensive, but
acceptable method for plotting 3 dimensional functions in 2 dimensions, we here choose to use a quite sophisticated version (2) of that scheme employing a hidden line algorithm. The drawines we construct are analonous to those of Streitwieser and owens ( 3 ) ,hut the emphasis Kere is on construction rather than interpretation. It should he noted that an alternative graphical method of "viewing" atomic and molecular orbitals exists, one based on successive contours of constant wavefunction in three dimensions. The best exemplar of this method is due to Jorgensen and Salem ( 4 ) ,and we can here cite an algorithm (5) which has been used to create such plots (although it has not heen tested by the author). Volume 58 Number 5
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377
Figure 5 . The basic cowdinate system with construction lines to show moiecular orbitals.
Figure 3. A contour map (not a polar plot) of the function J/%* 2.
shown in Figure
Figure 4. J/2px. The positive lobe is the upper one; the negative, the lower.
In Figure 1,the general coordinate system for use in atomic orbital drawings is shown. The $ axis is vertical, while the +z-axis runs diagonally to the right, and the +x-axis runs diagonally off to the left. For all these plots, the value of y is zero. I t would be simple to substitute any other positive (or negative) value in for y, and obtain the wavefunction plots at other y values. But the general nature of the parametric dependence of the figures shown on the y-coordinate is such that the shape relations hold regardless of the actual value of y. Figure 2 shows a p, orbital (in they = 0 plane). The function being plotted is
The computer first plots lines of constant x, with x starting at some (arbitrary) negative value, the plot location of this starting x value being somewhere in the foreground (close to US).The positive direction of x is away fromus, i.e., deeper into the page. Since y is zero throughout, J/Z~, = z e x p ( - M / 2 ) and one can now see how to proceed. All one does is fix a value 378
Journal of Chemical Education
Figure 6. The LCAO-MO known as a, formed from 2 different p, orbitals. one located on the +=axis, the other onthe -z-axis. me positive lobe is the upper one: the negative, the lower.
of say 2, and compute $ for a range of x . Then changing z to a new fixed value, we recompute $ over the same range of x . Keeping up this procedure allows us to compute (and plot) values of $ (x,O,a) as functions of a. To achieve cross-hatching, just reverse the procedure and set x equal to various constant values while plotting $ (b,O,z) with z varying. The latter case is illustrated in Figure 1,for the case b = 0. These simple exercises in plotting and visualization help to show, as clearly as possible, that $ (x,y,z) for a 2p, orbital is negative when z is less than zero, and that the maximum value of il/ (x,y,z) for this orbital is in the direction of +z. The phase ~rouertv . . . of orbitals is not easy to understand from polar plots, a l ~ c r cm+ e mu reasm 11x11 one writes signs inside t h e h b e s of polar representations of wavefunctions. To emphasize the point, Figure 3 is a contour map of the same Zp, orbital. For all positive 2, the contours are of positive wavefunction, while for all negative z , the contours are of negative wavefunction. This contour map is not a polar plot! It represents the intersection of the wavefundion surface with constant $ (x,O,z) planes, and, therefore, it represents the locus of x, y = 0, and z values of constant $.
Figure 7. The ?r; molecular orbital.
Figure 9. A contour map of the
Figure 8. A contour map of then, orbital.
Figure 4 shows a 2p, orbital. This orbital is positive for x
> 0, and negative elsewhere. Note that y is still zero for this
Figure 10. !)2, is plotted in the basic coordinate system but shows only the ~ositivex-axis.
traction of orbitals centered on A and B. In formula we have:
plot. The wavefunction is
++
= x exp(-(x2
+ 0%+ z2)/2)
+
= p,(o" A) pz(onB, ( o r z orbital) "1 - p,(n"B) (an*, orbital)
+- = p,(O"
= re-'I2 sin(8) eos(6)
Before proceeding to higher order orbitals, it is worthwhile to suggest that plotting one of these functions is highly instructive. Assuming that an elaborate computer with on-line plotting facilities is not available, one can still obtain a creditable three-dimensional representation of wavefunction for visualization purposes. This can be achieved by plotting individual slices of $ (x,O,z) a t various fixed x values, over a symmetric domain of z. These graphs can be cut out from the paper, and mounted by gluing them to a piece of cardboard, such that the positive wavefunctions are on one side of the card, and the negative wavefunctions are on the other side. This graphical representation can be extended t o n orbitals, generalizing the coordinate system slightly (see Fig. 5). The plots introduced above highlight chemical bonding nicely. If one nucleus (A) is at z = RI2, and the other (B) is at z = -R/2, two p, orbitals can be combined by either addition or suh-
a; orbital.
which corresponds to peculiar choices c~ and ce in the general LCAO-MO
3 =CAP."
+CBP%~
The equations for these orbitals are being shown in schematic form, hut if we intend to actually plot them, we must rewrite them in a consistent coordinate system. In the previously introduced (x,O,z) space, we have + p x A = x exp(- J(r
+ O2 + (2 - R/2)2)/2)
+pZB = x e w - J ( z 2
+ O2 + (2 + R/2)2)/2)
and The sum of these two functions is ++ = x[exp(- J(x2
+ (z + R/2)2)/2) + exp(-J(~P + (L - R/2)2)/2)]
while the difference is
Volume 58
Number 5
May 1981
379
J.-
= x[exp(-d(x2+
(2
+ R/2)2)/2) - e x p ( - J ( x z +
( z - R/2)2)/2)]
The orhitals are shown in Figures 6 and 7, and their contour maps are shown in Figures 8 and 9. In all these Figures, a high value of R, the internuclear distance has been chosen, so that the functional addition can he highlighted by the figure. At normal internuclear distances, the drawing is not quite so dramatic. We do not see two hot dogs floating one above and one below the internuclear line. But we do see for the rr-orbital, a region of positive wavefunction, for x > 0, and a region of negative wavefunction. Thinking hack to the frankfurters, the positively labelled whatever-it-is must correspond to the positive region being depicted here. But, here there is noneed to artificially imbed a sign into the function. For the r * orbital, we see that the regions of positive and negative wavefnnction appear exactly where the "repulsive" positive and negative lobes appear in normal representations. The phase relationship is clear, and the signs of the wavefunction in various regions of space is easy to understand. In Figure lOwe see a partial view of the 2s orbital. This view has been chosen to accentuate the negative values of the wavefunction for larger, and the nodal surface which, there-
380
Journal of Chemical Education
fore, must exist. T h e drawing shows off this radial node well. Acknowledgment Sara Rogers of the University of Connecticut Computer Center was a constant help in keeping our Calcomp plotter properly working in our IBM 3101168 environment. Her role in making these drawings is gratefully acknowledged. Paul Moews supplied a "home grown" contour plotter for the contour plots, and I thank him for itsuse. Contributions to this paper, many years ago, hy graduate students, namely Nancy True, Dave Audette, Silvio DeGregorio, and Marc Keller is also acknowledged with thanks. Finally, the referee has helped to clarify the presentation, and I thank himlher also. Literature Cited (1) MaeombPr.R S...l.C H ~ M E D U C . 119761. . ~ ~ . ~ ~ ~ (z) watkinsS. L., '"Marked l'hre~~nimenrion~l P l ~Programwith t Rotations; Algorithm 183." Comm ACM. 17.620 11974). (:$I Streitwieser,Jr.. A. and Owens, K.S., "orbitd and liirctron Density Disyrams,"Mae. millanCu., New York. 1973. (4) J o r ~ e n ~ eW n ,. L, and Salcm. L.. "The organicChemist's Bookof olbitals: Academic Press. New York, 1973. (51 Wright, T., "Visible Surrace Plottine Prowam. Aleorithm 175," Comm. ACM 17, 152 11874),Thenubroutinemay he key-puncheddireflly from this%ouree.
