W h e n one meets the Schroedinger orbi- tals for the first time the following question is often asked, "How does a p electron get from one lobe to the other if i t cannot pass through the nuclear plane?" This question can be dismissed in many ways. One can say that the node is just an infinitely thin mathe- matical plane having no physical significance (1). Another way is to show that it is an improper question (2) using the Heisenberg uncertainty principle: it is known that AEAt - h so that in a stationary state, where the energy is known exactly, time is completely indeterminate, and hence one cannot talk about an electron going from one place to the next since this in- volves the concept of time. The most satisfactory way to answer the question is to say that in the relativistic theory of the hydrogen atom there are no restrictive nodes a t all. This fact does not allow one to discuss the electron's motion classically, but it does show that one should not take pictorial representations of orbitals (in- cluding the ones in this article) too seriously since the shapes can he quite different in other, more complete, theory. This aspect of relativistic orbitals was dis- cussed in a recent article in THIS JOURNAL by Powell (8). The purpose of this brief note is to amplify Powell's article by presenting contour diagrams for various relativistic hydrogen orbitals.

It should be notcd that the Dirac theory is not the definitive theory of the hydrogen atom. Levels with the same , j but different 1 are not quite degenerate, hut the level with lower 1 lies slightly higher in energy (4). This is called the Lamb shift; its explanation lies in

Presented in part a t the Canadian Undergraduate Physics Conference, November, 1967.

Attila Szabo Harvard University

Cambridge, Mossachusetts 02138

quantum electrodynamics, but it can be appreciated by noting that the relativistic electron undergoes a type of motion called Zitterleweyung (trembling-motion) (8). Since the electron is oscillating about its non-relativistic position (with all amplitude of about em) the electron is not seeing as small a potential as it should. The effect on the energy is most important when the electron is near the nucleus, and since an s electron spends more time near the nucleus than a p electron, this trembling-motion would raise its energy slightly above that of the p electron (5). Although this argu- ment is not quite correct, i t can be made quantitative (6) and accounts for a large portion of the energy dif- ference.

Contour Diagrams

for Relativistic Orbitals

Method of Computation

We write the probability density 1 $ 1 as P,Pe where P, and Ps are functions of the polar coordinates r and 8, respectively. In order to conceptualize relativistic orbitals more easily, the sequence of energy levels of the hydrogen atom with a rough polar plot of Po, is given in Figure 1. The contour diagrams for orbitals which differ markedly from those of Schroedinger were constructed as follows (Fig. 2): a certain value of I $ I 2 was chosen, and a given angle Pe was evaluated using formulas from White's paper (7), and hence P, was calculated. From a plot of P, versus r, the values of r corresponding to the particular value of 8 were found and plotted as indicated. The entire procedure was then repeated using different values of 8 until enough points were obtained. The resulting graphs consist of contours of constant electron density. Although we used the relativistic form for P,, essentially the same

678 / Journal of Chemical Education

n = 2 I- -- - t

-1200 cm- ' (3/21 11/21

p, /r - 0 4 em-'

LAMB ' P , , ~ SHIFT

1 Figure 3. A 2p orbital: j = a/n; m = +'/,. In this and subsequent diwrams the value of lJ l l X 1OS is written n e w the corresponding con- tour. Tho Bohr radius is token as the unit of length.

Figwe 1. Schemotis diogram of the energy levels of hydrogen according to tho Dimc theory. The Lamb shift is l o w n for the n = 2 levels. The value of tho quantum number m is bracketed.

Figure 2. Illvstration of the method of colcvlation for o 2 p orbital.

contours could have been obtained using the Schroe- dinger form.

Results

Contour plots for Z P ~ / , and 2S./, are spherically sym- metrical, while those for n = 2 and 3,1 = l , j = 3/2, and m = 3/2 look very similar to those for p orbitals al- ready published in THIS JOURNAL (8) : the maximum of the contour lies in the plane perpendicular to the verti- cal axis and since there is rotational symmetry about this axis, the charge cloud is doughnut-shaped. It should be pointed out that the orbitals in this article represent states of definite angular momentum; hence they possess only one well-defined axis, and are anal- ogous to the imaginary Schroedinger wave functions. The contour plot of a 2p orbital, j = m = is

Figure 4. A 3p orbitoh j = %; m = +l/r. The pseudo-node is indicoted with !he dashed line.

shown in Figure 3. I t is seen that the contour for 1 # 1.2 = 0.002 looks very much like the Schroedinger orbltal. However, as I J. 1 decreases the density begins to leak through the Schroedinger node. This effect is

Volume 46, Number 10, Odober 1969 / 679

I i

Figure 6. A 3d orbital: i = sf2/,; ='+a/%

I i

Figure 5. A 3d orbital: i = '1%; m = +Ifz

observed on all subsequent diagrams. In Figure 4, the 3p, j = a/2; m = *I/,, orbital is shown. The pseudo-node does not restrict the electron's motion from from one radius to another. The only 3d orbitals that are substantially different from the Schroedinger orbi- tals are the 2Dc/, with m = * and m = * 3/2 (Figs. 5 and 6). Again the contours illustrate that there are no restrictive nodes.

I would like to thank Dr. D. A. I. Goring for his kind cooperation in the preparation of the diagrams.

Literature Cited

(1) COMPANION, A. L., "Chemical Bonding," McGraw-Hill Co., Inc., New York, 1964, p. 23.

(2) HUME-ROTHERY, W., '(Electrons, Atoms, Metals and Alloys," Dover Publiccttions, New York, 1963, p. 80.

(3) POWELL, R. E., J. CHEM. EDUC., 45, 558 (1968). (4) BETHE, H. A,, SALPETER, E. E., "Quantum Mechanics of

One and Two-Electron Systems" in "Encyclopedia of Physics," (Editor: FLUGGE, S.), Springer-Verlag, Berlin, 1957, Val. XXV, p. 189.

(5) Bonow~m, S., "Fundamentals of Quantum Mechanics," W. A. Benjamin, Inc., New York, 1967, p. 349.

(6) SIKOMV, A. A,, KOSKUMV, Y. M., TERNOV, I. M., "Quan- tum Mechanics." Holt, Rinehart and Winston. Inc.. New York, 1966, p. 353.

(7) WHITE, H. E., Phy8. Rar., 38, 513 (1931). (8) COHEN, I., J. CHEM. EDUC., 38, 20 (1961).

680 / Journal of Chemical Educofion