W. T. Bordass and J. W. Linnett Univers~tyof Cambridge Cambridge, England
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b\ Mew way off Presenting- Atomic Orbita
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M a n y students experience difficulty in appreciating fully how the radial and angular components combine to give a complete atomic wave function or electron distribution. The radial part is easy to present because 9 or P2 (or 4rr2P2) can be plotted against r. The angular part is more difficult because 9 or P2is a function of two coordinates and ideally three dimensions are required for the representation. To show the total function completely in three dimensional space, four dimensions would be required, so presentation is inevitably difficult. Various methods of partial and of pictorial representation have been used, and some authors have combined two diagrams1 (e.g., Fig. 1for a 2p, orbital). Many of
Figure 1. Diagram showing the relationship between a graph o f Q (2pJ against z and a contour diagram of the magnitude of Q (2p) in the planes which contain the x axis.
these are valuable but sometimes the student does not grasp fully the relationship between the various presentations and may even be confused by them. A particular difficulty arises with the 2p orbital. The angular variation is often represented by the diagram in Figure 2. But all too frequently this is presented as if it gives a total representation of a 2p orbital, and a student easily gets into the habit of not thinking beyond the diagram of Figure 2 when he is considering a 2p orbital. More-
BROWN, G. I., "A New Guide to Modern Valency Theory," Longmans, London, 1967, pp. 71-4. OGRYZLO, E. A., AND PORTER, G. B., J. CHEM.EDUC.,40, 256 (1963).
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Figure 2. Variation of the angular part measured b y the distance from the origin of the 2p-function with 8, the angle from the x axis. This is often used to depict the whole 2p-wave function.
over, he may even believe that Figure 2 gives a good idea of the total form of the 3 p orbital as well. I n this paper we are suggesting another method of presentation which we think may be of some help to students. Contour maps2 probably provide the best method of portraying an orbital which has been used so far. The reason for this is that, on a sheet of paper, the variation of \k or \k2 with two spatial coordinates can be shown. However, to appreciate relative magnitudes, and signs (for q),they require close scrutiny. This paper attempts, in a different way, to show the variation of *or P2 with two spatial coordinates. For example, the variation of P (2p,) in the xy plane (x = 0) is shown in Figure 3. This and all subsequent diagrams were obtained using the Cambridge University Titan Computer. Several methods of plotting were attempted, but the conventional isometric projection was thought to be the most generally satisfactory. I n this the x, IJ and x axes are drawn a t 120' to each other (i.e., one is assumed to be looking down the body diagonal of a cube whose edges lie along the x, y, and x axes). The advantage of this projection is that there is no distortion: the magnitude of the function a t any point in the plane displayed can be determined by locating that point in the projection of the plane and measuring the vertical distance to the appropriate trace. A transparent grid would help to locate particular points in the plane. I t is not advisable to include this grid in the main diagram because its presence so complicates the figure that the traces representing the function are obscured. The patterns obtained are not pictures, but provide an objective representation of a function. I n the diagrams that follow the magnitude of the function is drawn vertically along the z axis, while the x and y axes are upwards to the right and left, respectively. Initially, functions were drawn by scanning parallel to x and incrementing parallel to y after each scan. However, using a square grid and removing lines that would be hidden if the surface representing the function were opaque gave a more satisfactory portrayal.
Figure 3. Graphs of the value of 2p, in the xy plane at z = 0. Traces are drawn, in isometric projection, along lines for which y is constant every a.u. in the x direction and also along lines for which x is constant every a.u. in they direction (a.u. = Bohr radius = 52.9 pm). The diagram resembles (but is not fhe same as, because it is without perspective) the view of a surface marked with a grid that is square in the base plane, displaced a distance proportional to 9 from that plane.
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do have the disadvantage that it is next to impossible to draw them on a board. However, if these figures are used following the presentation of the radial and angular components, we feel that they could be a great help. The integrated importance of the lower "hills" for large 7. relative to the higher "mountains" at small r (in say 2s, 3s, 3pz) is not, of course, obvious from these diagrams. This feature is made clear by using 47rr2\k2in an ordinary graph. We did consider the possibility of including here diagrams for r\k and y\k2 but decided not to do so. The reason for this decision was that we felt that a great merit of Figures 5-16 resides in the fact that they present directly the function \k (or \k2) as it varies in a plane. By so doing they stress that we are talking of a function which varies in space, a simple point which is
The projection shown in Figure 3 is of a section of the 2pz hydrogen wave function in the xy plane a t x = 0. I n this and all subsequent diagrams, the scale on the right is in atomic units of distance (1 a.u. = 1 Bohr radius = 52.9 pm) and the vertical scale on the left refers to the peak height of the 1s orbital as 1000 for both f and f2.The traces forming the grid have been interrupted on either side of any point where the function is identically zero and the nodal plane is therefore readily identified. This method was preferred to drawing solid curves though the nodal points as the heavier lines added detracted greatly from the appearance of the surface. Similarly, the addition of axes and the strengthening of traces passing through the center (which correspond to the 1-dimensional representations commonly displayed but, except for the s-orbitals, are not identically the radial part of the wave function), as illustrated in Figure 4, also obscure the surface itself.
Figure 5. Diagram for nential decay with r.
9 (1s) showing
the peak at r = 0 and the expo-
Figure 4. The same graphs as in Figure 3 with the following additions: (a) the x and y axes are shown; (b) the line giving the variation of '@ along the x axis (y = 0, z = 0) i s strengthened (cf., Fig. 1).
However, these additions do improve quantitative appreciation of the functions, and it is felt that either duplicate diagrams or perhaps tranparent overlays are really needed to bring home the point. Figures 5-16 show graphs of f and f in the x y plane, the sample plane, for Is, 2s, 2pz, 3s, 3pz and 3dzP-yz hydrogenic wave functions. The wave function graphs are all on the same scale, but it is not practicable to use the same scale for \k2. I t is hoped that Figures 5-16, and others like them, can be used by teachers to take students further towards a complete appreciation of the form of atomic wave functions. Clearly they must be used as an addition to other simple, though also more limited, methods. They
Figure 6. Diagram for '@ (2s) showing the peak at r = 0. The sign of '@ changes where the spherical node cuts the sample plane (z = 0).
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' (2p) showing the line where the nodal plane Figure 7. Diagram for 2 (x = 0) cuts the sample plane. Figure 11. Diagram for \k2 (1 s) showing the concentration of electron probability at small r. Note the change in mesh size from Figure 5. The exceptionally long spike has been cut off. The arrow indicates its height in vertical units.
Figure 8. Diagram of \k (3s)showing the two spherical nodes and the change in sign of P ' across both.
Figure 12. Diagram for \k2 (2s). The node is now at a minimum in \k2. The arrow indicates the height, in vertical units of the spike.
Figure 9. nodes.
Diagram of \k (3pJ showing both the spherical and the planar
Figure 13. Diagram for \k2 (2p,). trated about the x-axis.
showing the two planar nodes. Figure 10. Diagram of \k (3d,2-,2) two peaks match the two depressions identically.
Note the way in which \k2 is concen-
The
sometimes not fully realized by the beginner who may be shown a particular diagram representing an orbital before he is even told anything about a function. I n these circumstances, the diagram can assume a greater importance to him than the function. This method can, of course, be extended to display hybrid and molecular orbitals.
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Appendix on Computation
Most computers with a plotter can readily be programmed t o produce these drawings. The program used, which was written in Autocode for the Cambridge Titan computer, contains analytic
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Figure 14. Diagram of \k2 (3s). Note the very small amplitude in the outer region. The arrow indicates the height, in vertical units, of the spike.
Figure 15.
with 'P2 ( 2 ~ in~ Figure ) 13. Diagram of q 2( 3 ~ ~ )Compare .
functions for the orbitals required and evaluates the function a t suitable intervals and t o a scale specified by a data stream. I n the plots presented here, the evaluation increment was half the distance between grid points. Scanning is started along +x from the lower apex of the enclosing box. The plotter coordinates (X, Y ) are determined from the position (x, y ) and the magnitude of the function a t the point, and the Y values stored in the subscripted variable C. For the very first scan, a line joining all the evaluated points is plotted. The coordinate y is then incremented and the next scan is performed in the -x direction, to save plotter time. Each value of Y determined is compared with the previous one a t the same X coordinates: if it is greater, a line is drawn to the point ( X , Y) and the previous value of C is overwritten by the new Y; if not, the point a t which the present scan line crosses the previous one is calculated and the pen lifted when this point is reached.
Figure 16.
Diagram of q 2(3d&
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showing the four equivalent lobes.
I n the latter instance the value of Cis not overwritten. While the pen is raised, each value of I is checked against the appropriate C and when again Y is greater than C the crossing point is determined and the pen is lowered when it is reached. The values of C overwritten are retained for a few increments unless they are required for interpolation. Scanning then continues, backwards and forwards, until the top of the box is reached. Scans parallel to y are then carried out in the same way. The program can be speeded up by exploiting the symmetry of the functions. On all scans, except a t the borders, the behavior of the function is tested for the presence of nodes by comparing the magnitude of the function a t the presently-considered point with the ones immediately before and after it. If a node is encountered, its position is interpolated quadratically if it lies a t a minimum and cubically if a t a point of inflection, and the pen is lifted for a distance either side specified by the data stream.
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