Model Image of the Electron Cloud of Hydrogen-like Atoms

D. I. Blokhintsev, Moscow

Although the fundamental concepts of wave mechanics, which decisively break with the last elements of mechanistic notions, have become deeply rooted in modern physics, in those cases where one wishes to form a visual image of the peculiar form of motion performed by an electron moving in the field of an atomic nucleus, one often resorts to the image of the sharply outlined elliptical orbits of the Bohr atom, although by now everyone is fully aware of all its inadequacy. This fact finds its explanation in the circumstance that, while remaining on the ground of wave mechanics, one had to confine oneself to a graphical representation of the electron-distribution function around the atomic nucleus. The recently published works of H. E. White* successfully overcome this difficulty and at the same time make the features of the laws of wave mechanics more comprehensible and familiar; moreover, the results of these works can be successfully used in the teaching of the modern theory of the atom.

According to wave mechanics, the motion of the electron is represented by the wave function $\psi$, whose physical meaning is revealed in the fact that the product

\[ P=\psi\cdot\bar{\psi}\cdot dV, \tag{1} \]

where $\bar{\psi}$ is the function conjugate to $\psi$, and $dV$ is an element of volume, gives the probability that the electron is located in the region of space determined by the volume element $dV$. If the electron is represented by a droplet of fog and in each region of space a large number of such droplets, proportional to $\psi\bar{\psi}$, is placed, then we obtain a cloud whose concentration at each point will represent the probability $P$. The external appearance of such a cloud gives a visual image—in the literal sense of the word—of the distribution of electrons in the space around the atomic nucleus. This distribution is realized, on average over time, in every atom or over short intervals of time if a collection of many atoms is considered simultaneously. The image of such an “electron cloud” de-

* See the literature at the end of the article.

replaces in wave mechanics the classical trajectories of electrons, similar to the orbits of planets. The wave function \(\psi\) itself is found from the Schrödinger wave equation. If we restrict ourselves to the consideration of the simplest atom, the hydrogen atom, which has only one electron, or of ions similar to it, \(\mathrm{He}^{+}\), \(\mathrm{Li}^{++}\), etc. (to this may also be referred, in a first approximation, the motion of a valence electron in atoms of the alkali metals, if the action of the inner electron shells is neglected), then this equation is written in the following form [2]:

\[ \frac{d^{2}\psi}{dx^{2}}+\frac{d^{2}\psi}{dy^{2}}+\frac{d^{2}\psi}{dz^{2}}+\frac{8\pi^{2}\mu}{h^{2}}\left(W+\frac{Ze^{2}}{r}\right)\psi=0 \tag{2} \]

Here \(\mu\) is the mass of the electron, \(W\) is the total energy of the electron’s motion, and \(-\dfrac{Ze^{2}}{r}\) is the potential energy of the electron in the field of the atomic nucleus.

It is known that in this case, just as in classical mechanics, the variables in equation (2) separate if it is transformed to polar coordinates \(r,\theta,\varphi\) (see Fig. 1),* i.e., in other words, \(\psi\) can be found as the product of three functions, each of which depends only on one coordinate:

\[ \psi=\Phi_m(\varphi)\cdot\Theta_{ml}(\theta)\cdot R_{nl}(r). \tag{3} \]

Fig. 1.

Here the indices \(n,l,m\) represent integers by which the physically admissible solutions are indicated (i.e. continuous, single-valued, and finite ones), and which determine the quantum state of the electron. The physical meaning of these numbers is very simple. The number \(n\), the principal quantum number, determines the total energy of the motion and gives the well-known Balmer term:

\[ W_n=-\frac{2\pi^{2}\mu e^{4}Z^{2}}{h^{2}n^{2}}\quad n=1,2,3,\ldots; \]

the number \(l\) determines the orbital angular momentum of the electron \(M\), so that

\[ M^{2}=\left(\frac{h}{2\pi}\right)^{2}l(l+1) \]

* See, for example, the article by Darrrow, Introduction to Schrödinger Wave Mechanics, Uspekhi fizicheskikh nauk, vol. IX, p. 437, 1929.

and, finally, \(m\) (the magnetic quantum number) determines the projection of \(M\) on the axis \(Z\) \((\theta = 0)\), which can physically be singled out, for example, by a magnetic field applied exactly along this axis:

\[ M_z=\frac{h}{2\pi}m,\quad -l<m<l. \]

The functions \(\Phi\), \(\Theta\), and \(R\) have the following form:

\[ \Phi_m(\varphi)=N_m l^{im\varphi} \]

\[ \Theta_{ml}(\theta)=N_{ml}\sin_m\theta\cdot P_l^m(\cos\theta) \tag{4} \]

\[ R_{nl}^{m}(r)=N_{nl}\left(\frac{2Zr}{na_0}\right)^l\cdot l_{\,n+l}^{\,\frac{2r}{na_0},\,2l+1}\left(\frac{2Zr}{na_0}\right); \]

\(P_l^m\) and \(l_{n+l}^{2l+1}\) are, respectively, the Legendre polynomials and the derivatives of Laguerre polynomials, well known in mathematics.^3 The numbers \(N\) are the so-called normalizing factors, which are usually chosen so that

\[ \int \psi\bar{\psi}\,dV=1, \]

which means that the probability of finding the electron anywhere in all space is equal to unity.

If (3) and (4) are used, the probability is represented in the form:

\[ P=\Phi_m\bar{\Phi}_m\cdot|\Theta_{ml}|^2\cdot|R_{nl}|^2; \tag{5} \]

since the variables are separated, we can study the role of each factor separately. First, from (4) it is evident that \(\Phi\bar{\Phi}_m\) does not depend on \(\varphi\); thus, for given \(\theta\) and \(r\), it is equally probable to find the electron in the region of any \(\varphi\) \((0\leqslant\varphi\leqslant2\pi)\), and this means that the electron cloud has the symmetry of a body of revolution with respect to the axis \(Z\) \((\theta=0)\). Of special interest is the study of the factor \(|\Theta_{ml}|^2\), which can be calculated from the known formulas for spherical functions. Let us mentally draw from the center of the atom (from the nucleus) a ray in some direction \((\theta,\varphi)\) and ask ourselves: what is the probability that somewhere on this ray there will be an electron? Obviously, in this case the factor \(R_{nl}^2\) plays no role, and since the atom has, as we saw above, rotational symmetry, this probability will be determined exclusively by the inclination of this ray to the axis \(Z\), i.e., by the factor \(|\Theta_{ml}|^2\).

In Fig. 2 these factors are shown in polar coordinates for various values of \(m\) and \(l\), and beneath them the corresponding Bohr orbits are drawn. In this case the \(z\)-axis is everywhere directed upward from the bottom of the drawing and lies in its plane. The radius vector of the curves shown gives the factor \(|\Theta_{ml}|^2\), i.e., the relative probability of finding the electron in the given direction of the ray \((\theta,\varphi)\). In reality, here one would have had to imagine not the shaded curves, but a surface of revolution; but since this probability does not depend on \(\varphi\), it is possible and sufficient to restrict oneself ...

limit ourselves to considering the section of these surfaces by any meridional plane, which is what has been done here. The first drawing gives the probability curve for electrons with \(l=m=0\), i.e.—according to spectral systematics—

Fig. 2. Angular distribution of probabilities [the factor \((\Theta_{ml})^2\)] for \(s\)-, \(p\)-, \(d\)-, \(f\)-, \(g\)-, and \(h\)-electrons and the corresponding classical orbits.

for electrons belonging to an \(S\)-term (\(S\)-electrons). The curve has the form of a circle; in space this corresponds to a sphere. The factor \((\Theta_{00})^2\) therefore does not depend on \(\theta\), and the probability of finding

the electron, moving along any ray from the center of the atom, is the same, i.e., in other words, the electron cloud for an \(S\)-electron has spherical symmetry. We shall see below that this property is also retained in the case when there is not one but several electrons forming an \(S\)-shell. No classical orbit corresponding to this case can be constructed. The closest to this state would be “pendulum-like” trajectories passing through the nucleus and oriented arbitrarily; however, in Bohr’s theory they always belonged to the number of “forbidden” orbits.

The second drawing belongs to \(P\)-electrons, for which \(l=1\) and, consequently, \(m=+1,0,-1\). From this drawing it is seen that the direction of the rays for which the probability has a maximum value coincides with the plane of the corresponding classical orbit, and it is equal to zero for the direction perpendicular to it. In this one may see the grain of relative truth contained in Bohr’s representations. This, however, is where the correspondence is limited, since it is obvious that in the classical understanding the electron could be found only in the plane of the orbit; according to wave mechanics, however, this plane is only the most probable one, and the electron in reality may also turn out to be in other planes. The orbit of the electron loses its definiteness, and here the inadequacy of a mechanistic understanding of the motion of the electron is clearly visible, in particular the inadequacy of the mechanical concepts of momentum and coordinate for representing the new peculiar form of motion. Passing to consideration of the following drawings, let us note that the values \(m=\pm l\) always correspond to an orbit whose plane is perpendicular to the \(z\)-axis in such a way that the angular momentum, represented in the drawing by an arrow, is completely projected onto this axis, and the difference consists only in the direction of motion: clockwise or counterclockwise. Further, for \(m=0\) the angular momentum is perpendicular to the \(z\)-axis; there is no motion in the direction of the angle \(\varphi\), but all positions of the orbits in the meridional planes are equally probable. Therefore something new in the following drawings is found only for \(0<|m|\,(|m|)<l\). For example, for \(d\)-electrons (\(l=2\)) with \(m=\pm1\) we have, as classical analogues, a set of orbits whose planes are tangent to one and the same cone (in the drawings only two of these planes are shown); this set may also be regarded as a single orbit precessing about the direction of the external magnetic field (the \(z\)-axis). As is seen, in this case too a correspondence is found between the directions of the rays of maximum probability and the position of the classical orbits. The coincidence, however, is not complete; indeed, one could hardly expect more. For still higher values of \(l\), additional maxima are observed (see, for example, \(g\)-electrons \(m=\pm2,\pm1,0\)), which, although not large, nevertheless have no analogues in classical theory. The remaining drawings require no special explanation. Let us note also the following remarkable property of the factor \(|\Theta_{ml}|^2\):

If we form the sum

\[ \sum_{m=-l}^{m=+l} [\Theta_{ml}]^{2}, \]

then it does not depend on \(\theta\). This means that if, from our drawings, for some term one takes the radius vectors of the curves for all \(m\) from \(-l\) to \(+l\), and adds them, then the resulting radius vector will be the same for all \(\theta\); the sum of the curves gives a circle (we note that in the drawings the scale for curves with \(m=0\) is reduced by \((l+1)\) times). This is easy to see, for example, for the \(p\)-term, where

\[ [\Theta_{01}]^{2}=\frac{3}{2}\cos^{2}\theta,\quad [\Theta_{-1,1}]^{2}=\frac{3}{4}\sin^{2}\theta, \]

the sum is equal to \(\dfrac{3}{2}\). Thus, if there are in the atom three \(p\)-electrons, or, for example, five \(d\)-electrons, differing only in the values of \(m\), then, in the first approximation, if their interaction is not taken into account, the electron cloud formed possesses spherical symmetry and gives an \(S\)-term, the total angular momentum in which will be equal to zero [1]. If we now raise the question of the probability that the electron will be found somewhere at a distance \(r, r+dr\), i.e., in other words, between spheres of the indicated radii, then it is obvious that this probability will be determined by the quantity

\[ D=[R_{nl}(r)]^{2}\cdot 4\pi r^{2}\cdot dr . \]

This quantity, calculated from Laguerre polynomials, is shown in Fig. 3, the unit of distance being taken as the radius of the first circular orbit of Bohr, \(a_{0}=0.53\ \text{\AA}\). In attempts to reconcile the spectral data with Bohr’s theory, different definitions of the azimuthal number \(k\) (our \(l\)) were adopted for one and the same \(n\), namely—four models are known: model (a) \(k=l\), model (b) \(k=[l(l+1)]^{1/2}\), model (c) \(k=l+\dfrac{1}{2}\), and, finally, the usual definition—model (d)—\(k=l+1\). None of these models, in essence, can be reconciled with the entire body of experimental data. For each of these models the mean radius vectors of the electron have been calculated; the values obtained are marked in the drawing for models (a), (c), and (d) respectively by triangular, circular, and square symbols [1]. The mean radius calculated for model (b) is marked by a vertical stroke and coincides exactly with the mean value calculated by wave mechanics. However, for this model there is a difficulty for \(S\)-electrons, since for them in this case \(k=0\), i.e., the orbit passes through the nucleus. Therefore the drawing gives the orbits for model (c); practically the discrepancy is insignificant, and the general result is evident without further discussion: the electron cloud is concen-

is dragged out into the region of classical orbits. In his work X. Urey sets himself the goal of giving an image of the electron cloud as a whole, so as to avoid the necessity of considering the graphical representation of each of the factors. For this purpose he constructed the following apparatus (see Fig. 4). On the axis of a motor \(M\) a spindle \(SC\) was fastened of such a form that the profile of its longitudinal section coincides as accurately as possible

Fig. 3. Distribution of probabilities as a function of distance from the nucleus (factor \(R_{nl}^{2}\)). The hatched curves represent the distribution of the electron density \(D=4\pi r^{2}R_{nl}^{2}\). The orbits are drawn for the model with azimuthal number \(k=l+1\); (model \(C\)).

with the curves of Fig. 3 (of course, for each curve, for each state, there is its own spindle; the figure shows the spindle for the state \(3d\)). By means of the cord \(SHR\) the spindle was secured in an inclined position and was set into rapid rotation about a vertical axis, which here plays the role of the \(z\)-axis. If one now photographs a spindle rotating in this way, then, evidently, on the plate one obtains a photograph of the cloud in the region of the given \(\theta\), since the action on the plate may be considered proportional to the width

spindle; in order to photograph the cloud for all values of the angle $\theta$ and to take into account the dependence of the “concentration” of the cloud on $\theta$, the tilt of the spindle was changed, by means of a cord, from $\theta = 0$ to $\theta = \frac{\pi}{2}$ (the horizontal position), and in each position the spindle remained for a time proportional to $|\Theta_{ml}|^2$, which was achieved by the slow and uniform motion of a board $A$ with a special profile, which, in its motion, pulled the cord by means of a roller $R$ at the required speed. The profile of the board was chosen so that, apart from certain necessary corrections, the angular velocity of the end of the spindle in the meridional plane, $\frac{d\theta}{dt}$, was equal to $|\Theta_{ml}|^2$.

Fig. 4. Mechanical apparatus used for reproducing the model of the electron cloud. The spindle shown corresponds to a 3d electron.

The required symmetry with respect to the $z$ axis was evidently already ensured by rotating the spindle about this axis with sufficient rapidity. The photographs obtained in this way obviously coincide with those which we would obtain by photographing a cloud whose concentration in each region is proportional to the probability $P$. Some slight distortions are inevitably introduced by imperfections of the apparatus, for example owing to the impossibility of making the spindle sufficiently thin at the ends and in the middle, as required by the curves in Fig. 3. Photographs of the electron cloud obtained in this way for various quantum states of the electron are shown in Fig. 5. The first photograph (1S) shows the electron cloud of the unexcited $S$ term ($n = 1$). The cloud has spherical symmetry; this same property of the $S$ term is also seen in the photographs for the excited states ($n = 2, n = 3$). None of the other terms possesses this property, but all of them have a cloud symmetric as a body of revolution.

Comparing these photographs with the perspective drawings of the orbits in Fig. 2, one can see that the electron cloud, by its shape, is as it were a misty phantom of the Bohr orbits. The last drawing shows a photograph of the spindle used for the model of the state $3d$ ($n = 3, l = 2$).

The sharpness of the falloff of the electron cloud with distance from the center of the atom turned out in the photographs to be apparently somewhat exaggerated, as can be seen from comparing the photographs with the distribution function.

...with respect to distance in Fig. 4. This shortcoming is absent from the later photographs of H. Yuang, obtained by him for models of the electron cloud of hydrogen-like atoms according to Dirac’s theory; to consideration of this second part of the work we shall now turn. Dirac’s wave equation cannot be written in ordinary numbers, but only in matrix numbers [6][1]. Owing to this circumstance the wave function \(\psi\) has four components, \(\psi_1, \psi_2, \psi_3\), and \(\psi_4\), and the probability \(P\) is represented in this theory in the following form:

\[ P=\psi\psi^{*}\,dV=(\psi_1\psi_1^{*}+\psi_2\psi_2^{*}+\psi_3\psi_3^{*}+\psi_4\psi_4^{*})\,dV. \tag{6} \]

The variables in this case are also separated, and in such a way that not only is each \(\psi_i\) represented as a product of functions, each of which depends on only one coordinate, but the product \(\psi\psi^{*}\) (see (6)) is also separated in the same variables \(r, \theta\), and \(\varphi\) as in the case we considered of the Schrödinger equation.

Thus the entire analysis of \(P\) and the method of photographing the cloud model can be applied here as well without any changes. However, the quantum states themselves in this case present special features on which one should dwell. Dirac’s theory is a relativistic theory of the motion of the electron in a given external field, in the particular case—in the field of the positively charged nucleus of an atom of hydrogen-like elements or ions. An essential feature of this theory is the circumstance that it necessarily leads to the conclusion that the electron has angular and magnetic moment and, in this way, provides a theoretical basis for the Goudsmit and Uhlenbeck hypothesis of the spinning electron. It turns out that the projection of the electron’s angular momentum on any axis is equal to

\[ \pm \frac{1}{2}\frac{h}{2\pi}. \]

In the absence of an external field, the motion of the electron around the nucleus for a given energy \(W\) can take place in one of two states, corresponding to two possible values of the electron’s intrinsic angular momentum \(+\frac{1}{2}\) or \(-\frac{1}{2}\). In this case the constant of motion will no longer be \(l\), which determines the angular momentum of the electron’s motion in its orbit, but a new number \(J\), which determines the total angular momentum of the system, i.e. the sum of the orbital and intrinsic angular momentum of the electron, so that \(J=l+\tfrac{1}{2}\) or \(J=l-\tfrac{1}{2}\).

Thus Dirac’s theory leads to a doubling of the number of states as compared with the number of states given by Schrödinger’s theory, and at the same time those difficulties disappear which the latter theory encountered in explaining the “anomalous” Zeeman effect and complex spectra. The state of the electron in Dirac’s theory will be characterized by the following numbers: \(n\) and \(l\), having the same meaning as in Schrödinger’s theory; the number \(J\), taking positive half-integer values; and the magnetic number \(m: -J\leqq m\leqq J\). For the calculation of the probability \(P\)

Fig. 5. Photographs of the electron-cloud model for various states of hydrogen-like atoms. The scale may be taken from Fig. 4, where the radius of the first circular Bohr orbit is taken as unity.

Fig. 7. Photographs of models of the electron cloud for various states of hydrogen-like atoms according to Dirac. The scale is given for each state in angstroms.

... what is essential is that it proves to be the same for electrons having the same \(n\), \(J\), \(l\), and \(m\), whether positive or negative, and that the distribution \(P\) over directions is the same for electrons having

Fig. 6. Angular distribution of probabilities (the factor \(P_\theta\)) according to Dirac’s theory, and the classical orbits when the electron’s rotational moment is taken into account.

equal \(n\), \(J\), \(m\), and \(l = J \pm \tfrac{1}{2}\). Thus, for example, the angular distribution in the states \(m = \pm \tfrac{3}{2}\) for the term \({}^{2}P_{3/2}\) \((l = 1,\ J = \tfrac{3}{2})\) and the term \({}^{2}D_{3/2}\) \((l = 2,\ J = \tfrac{3}{2})\) is completely identical (the index 2 at the upper left indicates the doublet character of the term), whereas the distribution in the radial direction for the—

terms \(^{2}P\) and \(^{2}D\) is different [7]. As has already been mentioned, the probability \(P\) may be represented in the form:

\[ P=|\psi(\varphi)\psi(\varphi)|\cdot P_{\theta}(\theta)\cdot P_{r}(r)\cdot dV; \]

the factor \(|\psi(\varphi)\psi(\varphi)|\), just as in Schrödinger’s theory, does not depend on \(\varphi\), and thus the cloud of hydrogen-like atoms according to Dirac’s theory also possesses the symmetry of a body of revolution. As for the factor \(P_{\theta}(\theta)\), which determines the angular distribution of probability in the direction of the angle \(\theta\), it is formed from spherical functions, but has a different form in comparison with the factor \(|\Theta_{ml}|^{2}\) in Schrödinger’s theory.

In Fig. 6 this factor is represented graphically in polar coordinates for various states, and below are drawn the corresponding orbits with allowance for the electron’s rotational moment. Here the length of the vector of the electron’s orbital angular momentum is measured by the number \(l^{*}=[l(l+1)]^{1/2}\), while the length of the vector of its own rotational moment is measured by the number \(S^{*}=[S(S+1)]^{1/2}\) \((S=1/2)\); these moments are oriented relative to one another in such a way that the total angular momentum, representing their sum, is equal to \(J^{*}=[J(J+1)]^{1/2}\). The first drawing gives the term \(^{2}S_{1/2}\) \((l=0)\); the same distribution simultaneously represents the term \(^{2}P_{1/2}\). The orbits shown below correspond to the term \(^{2}P_{1/2}\) \((l=1)\). In the drawing one can trace the formation of the total angular momentum \(J^{*}\) as the sum of the electron’s rotational moment \(S^{*}\) and the orbital angular momentum \(l^{*}\). For the \(S\)-term itself the corresponding orbit, similarly to the case of Schrödinger’s theory and for the same reasons, cannot be constructed. A distinctive feature of the Dirac model is the fact that here not only the \(S\)-term, but also the term \(^{2}P_{1/2}\), which is the normal state for B, Al, In, and Tl, possesses spherical symmetry. Just as in Schrödinger’s theory, states with intermediate values of \(m\) \((0<(m)<J)\) correspond to precessing orbits (see, for example, the state \(^{2}F_{5/2}\), \(m=\pm 3/2\)). The correspondence between the “classical” orbits and the angular distribution of the electrons here appears still more remarkable than in Schrödinger’s theory. The factor \(P_{\theta}\), when summed over \(m\) from \(m=1/2\) to \(m=J\), gives a constant value. Thus the electron cloud of the set of electrons forming half of a Smith-Stoner subgroup possesses spherical symmetry. The distribution of electrons in the radial direction, determined by the factor \(P_{r}(r)4\pi r^{2}\,dr\), differs extremely little from the same distribution in Schrödinger’s theory, and therefore its consideration may be omitted [7].

In Fig. 7 are given photographs obtained for the model of the electron cloud of hydrogen-like atoms according to Dirac, by the same method as that used for the Schrödinger model. Under each of the photographs, for each term, the scale is given in ångström units. The terms whose designation is enclosed in square brackets have

the electron cloud is so similar to the cloud of simply superposed terms that this difference could not be detected in these photographs. If these photographs are compared with those obtained for the Schrödinger model, taking into account that $m=0$ for Schrödinger corresponds to $m=\pm \frac{1}{2}$ for Dirac, then it is evident that the pictures of the electron cloud in both cases are, in their general features, identical, with the exception of the term ${}^{2}P_{\frac{1}{2}}$, which Schrödinger does not have. Thus, the photographs presented give a visual image of hydrogen-like atoms in the form in which they are represented in the light of the modern theory of the atom; the guarantee that the obtained models of the electron cloud correspond to objective reality is the whole aggregate of brilliant successes in the knowledge of the properties of matter achieved in recent years by wave mechanics.

References

1. H. E. White, Physical Review, 37, 1416, 1931.
   “ ” “ 38, 513, 1931.
2. See, for example, Foundations of the New Quantum Mechanics, a collection edited by A. Ioffe, or the article by Darrow in Advances in the Physical Sciences.
3. See A. Sommerfeld, Atombau und Spektrallinien, Ergänzungsband.
4. Unsöld, Ann. d. Phys., 82, 379, 1927.
5. Pauling, Proc. Roy. Soc., 114, 181, 1927; Waller, Z. f. Physik, 38, 635, 1926.
6. See 3) or P. Dirac, Principles of the Quantum Mechanics.
7. Hartree, Proc. Cam. Phil. Soc., 25, 225, 1929.
8. Roess, Phys. Rev., 37, 532, 1931.