Order out of Chaos: Shapes of Hydrogen Orbitals G. L. Breneman Eastern Washington University, Cheney, WA 99004 Most chemistry students are familiar with the shapes of the s, p, and d hydrogen orhitals. An occasional book will show the shapes o f f orbitals. There appears to be no overall systematic pattern to these shapes, and thus they have to be for the most part memorized. There is, however, an underlying pattern that, once seen, will allow you to describe the shape of any hydrogen orbital including all those well bevond the ones with which we are most familiar. The pattern involves what are called the expunrnrial form of the hydroaen . . orhitals. 'l'he direct general solutions of the Shrodinger equation for the hydrogen atom lead to functions that all contain exp(im4). When m Z 0 these functions are all complex. When the probability is calculated, the 4 variable drops out and all the exponential orbitals are cylindricallv . svmmetrical about the z axis iust like the orbitals with m = 0.They all involve dit't'erenl nice stacks of "doughnuts" (see Figs. 1 and 2). l'he shapes we are familiar with arc made hy taking linear comhinations of these exponential forms. These functions are also solutions to Schrodinrer's equation but are real instead of complex. These funciions contain sin(+) and cos(4) and are referred to as the sine and cosine forms. For example for the angular part of p orbitals,
.
P, =
+ P-,) = sin (8)cos ($1
p, = (p,
- p-,) = sin (8)sin (4)
This mathematical process can be viewed geometrically as slicing up the "doughnuts" in a certain pattern. Refer to Figure 1for this process. The doughnuts are sliced vertically by the number of equally spaced nodes equal to the iml value. Thus the iml = 1d orbital doughnuts are sliced by one
Figure 1. Obtaining cosine form 36 orbitslsfromexponential forms by "slicing up the doughnuts" (raxis is vertical, xexis is horizontal in page).
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Number 1 January 1988
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n
I type
0
0 1
2 3 5 4 FigureI 2. Exponential or "doughnut" fwms of the s. p, d, f, and g orbitals (axes same as in Fig. 1).
node and with the edges rounded off you get the familiar four-lobed d,, orbital (the cosine form). Slice it with a node perpendicular t o this and you have the four-lobed dyzorhital. Thus each set of two doughnuts (+m and -m values give the samedoughnut shape) can hrsliced two ways togive two orbitals that correspond to the real linear combinations. It' Iml = 2, two equallispaced vertical nodes are used to make the slice (see the right half of Fig. 1).This case involves a single doughnut sliced into four lobes giving us the d,z-p orbital. Slicing halfway between these first two slices gives the four-lobed d, orbital. Figure 2 shows the exponential (or doughnut) forms for all of the s, p, d, f, and g orbitals. The orbitals with +m and -m values have the same shape. Notice the pattern for the numher of doughnuts. The number in the first column equals 1 1 and decreases by one as the value of m increases.
+
doughnuts = (I + 1) - lml Figure 3 shows the result of slicing the "doughnuts" into 32
Journal of Chemical Education
lohes using iml nodes in each case. These are the orbitals we usually use, and for each non-zero value of iml we get two orbitals that have the same shape hut with the lobes placed in between each other by rotation around the vertical axis. The number of lobes in each case equals the number of doughnuts times twice the number of slices or lobes = ( ( 1 + 1)- Iml) X 21ml except for m = 0 where we do no slicing. Once this pattern is seen, the common orbitals can be remembered and the not so common can he easily described. For example, take 1 = 6 and iml = 4. This is an i orhital. The m = 0 i orbital will have seven "doughnuts" by extending Figure 2 two more rows. Moving over to the m = 4 row (reducing the number of doughnuts by 1for each row) gives three doughnuts. These are now sliced vertically by four equally spaced nodes dividing each doughnut into eight lobes. So the orbital has 24 lobes stacked vertically in three rings of eight.
n. I type
0
Figure 3. Cosine or "lobe" forms of me s,p, d, f, and g orbitals (axes same as in Fig. I),
Figures 1, 2, and 3 can he used where n - 1 > 1 since increasing the value of n leaving 1 constant only adds radial nodes to the orbitals with the overall shapes staying the same. Figures 2 and 3 have all orbitals scaled to the same size since we are only considering shape. Of course the size really increases as the value of n increases. The plots of the orbitals
were generated using CAD-3D on an Atari 520ST from cross sections plotted on a VAX computer. This process has been described in detail by the author.' Breneman. G. L. ~ o m h p SCI. 1987, 1. 17-22.
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