STRIPED SPECTRA, II*

E. I. Rabinovich, Göttingen

III. THE ELECTRONIC SPECTRUM AND THE THEORY OF MOLECULAR STRUCTURE

1. Relation between electronic and rotational terms. 2. Vector model and eigenfunctions:
a) Vector model. b) Eigenfunctions and symmetry properties.
c) Momentum vectors in wave mechanics. 3. Vector model and eigenfunctions of an individual electron: a) Electrons in the atom. b) Electrons in the molecule. c) Correspondence between electrons in the molecule and electrons in the atom. 4. Interaction between electrons. 5. Systems of electronic terms.

1. Relation between electronic and rotational terms

In the first part of this article the division of the system of terms of polyatomic molecules into electronic, vibrational, and rotational terms was established; in the second part we became acquainted in greater detail with the vibrational structure of striped spectra. The reasons why we began the discussion precisely with vibrational terms are as follows: 1) the vibrational structure is relatively simple (at least for diatomic molecules), 2) the study of this vibrational structure leads directly to a number of conclusions important for chemistry (for example, it makes it possible to calculate heats of dissociation). In the study of vibrational terms there is no need to enter more closely into consideration of the nature of those electronic states to which the separate sequences of vibrational terms belong. There is, however, one exception: in considering the relative intensity of transitions between different vibrational states we arrived at the so-called Franck–Condon principle—a rule relating the probability of a definite change in the vibrational state to a simultaneous change in the electronic state. Yet for this connection, among the properties of the electronic term, only the electronic energy as a function

* See Uspekhi Fizicheskikh Nauk, XI, 4, 554, 1931.

distances, i.e. the potential energy curve, which in each particular case we may regard as given by experiment. On the other hand, we may have no interest at all—for the purpose of estimating transition probabilities—in a more detailed quantum-mechanical characterization of the corresponding electronic states: their multiplicity, symmetry properties, the quantum numbers of the individual electrons—in short, all the properties that constitute the main content of the section “Electronic terms”; in applying the Franck–Condon principle they may remain unknown.

The connection between rotation and the electronic structure is considerably closer. Rotation is such a motion of the constituent parts of a molecule as is characterized by a definite rotational angular momentum (moment of quantity of motion). But the motions of electrons about nuclei also belong, for the most part, to the same type. As is known from the theory of atomic spectra, each electron forming part of an atom is characterized by two angular momenta: the orbital momentum \(l\) and the “spin” momentum \(s\) (the angular momentum for rotation about its own axis). The quantum numbers \(l\) and \(s\) determine these moments in quantum units \(\dfrac{h}{2\pi}\). Among all the angular momenta of the separate constituent parts of an atom or molecule there exist couplings (caused, for example, by magnetic forces arising from the motion of electric charges). These interactions lead to the result that the individual moments cease to retain an unchanged direction in space and begin to precess about the direction of the resultant moment.

In molecules, in the formation of the total angular momentum, besides the orbital moment and the spin moment of the electrons, there also takes part the rotational moment of the molecule as a whole. It is true that there exists one especially simple case, when all the moments (orbital moments and spins) of the individual elements mutually cancel and give a resultant equal to zero. In this case the electronic shell as a whole possesses neither orbital moment nor spin moment; therefore the rotation in this case—at least in the first approximation—does not depend at all on the electronic motions, and the total rotational impulse \(J\) is identical with the rotational impulse of the molecule as a whole. Such a case is realized for many molecules in their unexcited ground states; similar states, devoid of impulse, are denoted for atoms by the symbol \({}^{1}S_{0}\), and for molecules by the symbol \({}^{1}\Sigma\) (singlet sigma). However, this case is nevertheless a special one; for the explanation of the rotational structure of band spectra in the general case we

must take into account the moments of electronic states. This circumstance prescribes for us the order of the further exposition: first it is necessary to consider the electronic terms, and only then—in Part IV—the rotational terms.

2. Vector model and eigenfunctions

The theory of the electronic terms of a diatomic molecule is closely connected with the theory of atomic terms. In what follows we assume that the reader has some familiarity with the field of the theory of atomic spectra; nevertheless, we shall briefly recall the foundations of this theory.

As is known, there exist two methods for considering the electronic states of an atom: the qualitative, intuitive method of constructing a “vector model,” and the less intuitive, but quantitative, method of quantum and wave mechanics. The vector-model method is especially convenient for determining the number of possible states of an atomic system; a rigorous quantum-mechanical treatment is particularly necessary when one needs to know the probability of transitions from one state to another.

a) Vector model. The idea of the vector model arose from the original form of the Bohr–Sommerfeld quantum theory of atomic structure; however, it turned out that this part of the old theory retains its significance even in the new quantum-mechanical theory. Bohr’s theory proceeded from the fact that the motions of the component parts of the atom can be found by the methods of classical mechanics; in this way, for example, ellipses are obtained as the paths of electrons in the field of attraction of the nucleus. This—often very complex—motion is then decomposed into a number of simple periodic motions (for example, motion in a circular orbit and motion of the plane of the orbit relative to some fixed axis), and each of the component motions is separately “quantized,” i.e., the requirement is imposed that a certain constant characteristic of the motion (the “action variable”), measured in quantum units \(h\), can take only integral values. In this way, from the infinite variety of classically possible paths, a discrete series of “allowed” orbits was selected.

In the simplest cases the simple periodic components of the electron’s motion are linear oscillations or rotations. Thus an elliptical motion is represented, for example, as a superposition of a linear oscillation along the radius vector and a rotation with the same period. The oscillation corresponds to a radial quantum

the number \(n_r\), and for rotation—the azimuthal quantum number \(n_\varphi\). Between oscillation and rotation there is an important difference: an oscillation does not create a magnetic moment, whereas rotation, like any circulation of electric charge in a closed orbit, leads to the appearance of a definite magnetic moment. This difference entails the consequence that the oscillatory component of the motion of an electron does not depend on the motion of the other electrons, whereas all rotational motions of the constituent parts of the atom are connected with one another by magnetic interactions and, in addition, are subject to the action of external magnetic fields. In mechanics, rotation is symbolized by a vector perpendicular to the plane in which the rotation takes place, and equal in magnitude to the moment of momentum. To the two possible directions of rotation (clockwise and counterclockwise) there correspond two opposite directions of the moment vector, the correspondence between the direction of rotation and the direction of this vector being established by a definite convention. In considering the interactions of different intra-atomic motions one may often neglect the oscillations, symbolize the rotations by vectors, and speak briefly of the “mutual influence of the moment vectors.” Since the characteristic “magnitude of the action” for rotation is the integral of the moment \(p_\varphi\) over the whole cycle of rotation, i.e.

\[ \int_{0}^{2\pi} p_\varphi\, d\varphi = 2\pi p_\varphi \]

and since this quantity must be equal to an integral multiple of \(h\), the moment \(p_\varphi\) itself must be an integral multiple of \(\dfrac{h}{2\pi}\).

If the quantity \(\dfrac{h}{2\pi}\) is used as the unit of moment, then one may speak of moments of \(1, 2, 3 \ldots\) units. At the same time, there exist only angular momenta of rotation with absolute magnitudes \(1, 2, 3 \ldots\) (only the angular momentum of the electron about its own axis—the spin—may have half-integral values).

The moment of momentum will be constant only when the rotation is free. For the construction of the vector model it is of decisive importance that, when external influences on the rotation are weak, in first approximation one may still speak of a moment-of-momentum vector constant in time. The action of such a weak perturbation consists only in the fact that the plane of rotation—and with it also the moment vector—begin slowly to precess-

rate. If the perturbation is caused by an external field, then the axis of precession is the direction of the field; if, however, what is involved is the interaction of two rotations with free axes, then both vectors precess about the direction of their resultant. The true constant of the motion is now no longer the initial moment, but, in the case of the action of an external field, the component of this moment in the direction of the field (all the other components mutually cancel on the average over time); in the case of the interaction of two rotations, such a constant is the resultant of both moments. Quantum theory requires that these new vectors, in turn, be able to assume only integral values. However, so long as the interactions are weak, the old quantum numbers, as has been said, still retain, at least approximately, their significance. As the perturbation becomes stronger and stronger, the precession becomes ever faster and, finally, we arrive at the limiting case in which the initial moments completely lose their independence and, properly speaking, only their resultant (or its component in a definite direction) can be quantized. If, in addition to the case considered, there appears a second perturbation of a smaller order of magnitude, then with the “resultant of the first order” the same thing is repeated that happened with the initial vectors—namely, the formation of a quantized component or of a new quantized resultant. At the same time, the possibility of constructing a vector model of the atom or molecule rests on a gradation of interactions. In the majority of cases it is possible to regard certain rotations as “almost free” and to determine their moment vectors; then certain interactions are taken into account, and their influence is described by forming a quantized resultant; then second-order actions are taken into account, which entail the precession of the first-order resultant and the formation of a second-order resultant, and so on, until we arrive at the full angular momentum of the system under consideration (or at the component of this moment in the direction of the field), which, finally, will be constant and strictly quantized. The required gradation of interactions does not always actually exist; sometimes one has to deal with several forces of the same order of magnitude; in constructing the vector model one must, from case to case, work with different idealizations.

The state of the electronic system of an atom or molecule is characterized in the vector model by a system of integral (in part also half-integral) numbers, which measure the separate moments and their resultants in quantum units.

particles \(h/2\pi\). To this are further added the “radial” quantum numbers of the individual electrons*; practically, however, instead of the radial quantum number of an electron one indicates its principal quantum number \(n\), which in the new theory is equal to the sum, increased by one, of the azimuthal and radial quantum numbers. The reason why preference is given to the quantum number \(n\) is that, in some simple cases (for example in the case of the H atom), the binding energy of the electron is a simple function of the principal quantum number thus defined.

The significance of the vector model consists in the fact that it makes it possible at once to indicate the number of possible states of the system—states which, for given “primary” moments, can arise as a result of interaction or under the influence of an external force. This problem reduces to the purely geometrical question of the possible number of integer components of a vector of given magnitude, or of the possible number of integer resultants of two or several integer vectors (or also of the possible ways of decomposing a vector into integer components). Thus, for example, two electrons with orbital moments \(l_1 = l_2 = 2\) can give five different resultant orbital momenta \(L = 0, 1, 2, 3, 4\); a moment \(J = 1\) can, in an external field, give three integer components in the direction of the field, \(M = -1, 0, 1\). To each allowed structure of the vector model (to each possible combination of quantum numbers) there corresponds its own special possible state (term) of the system.

b) Eigenfunctions and symmetry properties. The new quantum mechanics entirely rejects the space-time description of the motion of the constituent parts of an atom or molecule. Instead of posing questions about the shape of orbits and the temporal course of motions along these orbits, the new quantum mechanics confines itself to considering the question of the probability that each individual particle (when its position is determined) is found at any given instant in a definite place. If the particles are, to one degree or another, connected with one another, then the results to which the new quantum mechanics leads reduce to establishing the probability that the whole system will be in a known configuration. This probability, in wave-mechanical form—

* In an electric field (applied externally or caused by the presence of another nucleus of the molecule), the “oscillatory part” of the electron’s motion is also affected; instead of the radial quantum number \(n_r\), an “ellipsoidal” quantum number \(n_\rho\) appears.

MODULATION THEORY is determined from the “wave equation” (Schrödinger’s equation), namely: the probability is characterized by the square of the local amplitude of the “matter wave” described by this equation (in exactly the same way as the intensity of an optical wave is determined by the square of its amplitude). In the case of systems that do not change with time (atoms in their stationary states), the wave must be stationary, and the amplitude depends only on the coordinates \(q_1, q_2\), but not on the time \(t\). The function \(\Psi(q_1, q_2, q_3,\ldots)\) describing the distribution of amplitudes in a stationary state is called the eigenfunction of this state. It may be real or complex. In the latter case the probability—which must always be real—is determined not by \(\Psi^2\), but by \(\Psi\bar{\Psi}\), where \(\bar{\Psi}\) is the function complex-conjugate to \(\Psi\). If \(\Psi=a+bi\), then \(\bar{\Psi}=a-bi\) and \(\Psi\bar{\Psi}=a^2+b^2\).

In considering systems with several degrees of freedom, attempts are made to “separate” the wave equation, i.e. to represent the function \(\Psi(q_1, q_2, q_3,\ldots)\) in the form of a product \(\Psi_{q_1}\cdot \Psi_{q_2}\cdot \Psi_{q_3}\cdots\), where each of these functions \(\Psi_{q_n}\ldots\) depends on only one variable. This operation presents a known analogy with the decomposition of a complex motion into periodic component motions—an operation used in the old quantum theory. The success of the separation operation means that the probability of finding the particle with coordinate \(q_1\) by itself does not depend on the remaining coordinates of this and other particles (the theorem of multiplication of probabilities).

Complete separation is not always possible. Sometimes it is possible to “separate off” only one definite particle, or only part of the coordinates of one particle. In other cases separation can be carried out only approximately; this corresponds to those cases in which, in the old theory, the periodic motion is perturbed by weak forces, so that periodicity and quantization are preserved only in the first approximation.

We shall first assume that the eigenfunction for one particle is completely separable; for only for three coordinates can an eigenfunction be represented visually in three-dimensional space. For the purpose of such a representation one usually indicates the spatial distribution of the nodes, i.e. the points, lines, or surfaces at which the eigenfunction vanishes. Further, if the eigenfunction \(\Psi(q_1, q_2, q_3)\) separates into three functions \(\Psi_{q_1}, \Psi_{q_2}, \Psi_{q_3}\), then each of them has its own nodes, which, however, will also be nodes of the original eigenfunction; for the vanishing of one factor is equivalent to the vanishing of the product. Let, for exam-

for example, the position of the particles is specified by three spatial polar coordinates \(r, \vartheta, \varphi\) (Fig. 1). (The choice of the coordinate system in which separation is best achieved is suggested, from case to case, by the very symmetry of the problem; if the field has spherical symmetry, then polar coordinates are the most suitable.) The eigenfunction \(\Psi(r,\vartheta,\varphi)\) splits into three functions \(\Psi_r\), \(\Psi_\vartheta\), \(\Psi_\varphi\). The nodes of the “radial” function \(\Psi_r\) must correspond to definite values \(r=\mathrm{const}\), i.e. they will be concentric spheres. The nodes of the function \(\Psi_\vartheta\) will be cones \(\vartheta=\mathrm{const}\) with vertex at the origin of coordinates (cf. Fig. 8); the nodes of the function \(\Psi_\varphi\) are half-planes \(\varphi=\mathrm{const}\), bounded by the axis \(\vartheta=0\). In order to establish the characteristic properties of the eigenfunctions, we may at first restrict ourselves to one of the three functions. For this purpose we shall choose the function \(\Psi_\varphi\), since a change in \(\varphi\) corresponds to a simple rotation about the fixed axis \(\vartheta=0\), i.e. to the type of motion which, as indicated above, is of especially great importance for the structure of the atom and the molecule.

Fig. 1. \(\vartheta=0\) to \(\pi\); \(\varphi=2\pi\).

The nodal half-planes of the function \(\Psi_\varphi\) divide space into regions which have the form of orange slices; on the surface of each sphere \(r=\mathrm{const}\) these half-planes create a network of traces resembling meridians on the surface of the terrestrial globe. The number and distribution of the nodal half-planes depends on the special form of the eigenfunction. However, in any case, in order that the function \(\Psi_\varphi\) may have the physical significance of an eigenfunction, it is necessary to require that \(\Psi_\varphi\) have period \(2\pi\); for, when \(\varphi\) increases by \(2\pi\), we arrive at the same point, and the eigenfunction must be single-valued. Examples of functions satisfying this requirement are the functions \(\sin n\varphi\), \(\cos n\varphi\), as well as all their linear combinations \(A\sin n\varphi+B\cos n\varphi\), where \(n\) is any integer, and \(A\) and \(B\) are real or complex coefficients. The function \(\Psi_\varphi=\cos n\varphi\) has, for example, the form shown in Fig. 2; for every integer \(n\) it has period \(2\pi\); the number of nodal half-planes is \(2n\), and they pairwise complement one another to whole infinite planes, whose number will thus be \(n\). At

Fig. 2a shows \(\cos n\varphi\) plotted as a function of \(\varphi\); in Fig. 2b the quantity \(\Psi_\varphi^2\) is given (always positive), which, according to wave mechanics, characterizes the probability. In Fig. 2c the same quantity \(\Psi_\varphi^2\) is plotted on a circle, its different values being symbolized by different thicknesses of the black strip; here the nodal planes are visible in cross-section, while in Fig. 6 they are again represented in perspective.

When passing through a node, the function \(\cos n\varphi\) changes its sign; if, as in Fig. 2c II, it is positive on the left, then on the right it will be negative, and conversely. If \(\varphi\) is increased by \(\pi\), then the function retains its absolute value; as for the sign, it is either retained or changed to the opposite, depending on whether an even or odd number of nodal planes is traversed when the argument is increased by \(\pi\). This number is even for even \(n\) and odd for odd \(n\). Here we encounter the first of the symmetry properties of \(\psi\)-functions—properties that play a large role in the theory of the structure of atoms and molecules. The eigenfunctions \(\Psi_\varphi\) also possess this symmetry property, although these eigenfunctions are somewhat more complicated than the simple function \(\cos n\varphi\). The symmetry property mentioned consists in the following: upon reflection in the origin of coordinates (it is easy to verify that such a reflection amounts to increasing the argument by \(\pi\)), they do not change their absolute value and either change or retain their sign depending on whether a certain characteristic integer entering into the function is even or odd. The eigenfunctions of atomic or molecular electrons that do not change under reflection in the origin of co-

Fig. 2. Function \(\cos n\varphi\).

of coordinates, are called “even”; functions which in doing so change their sign are called “odd.” In the case of molecules, besides symmetry with respect to the electrons there also exists symmetry with respect to the nuclei. Later we shall return again to this problem of symmetry.

c) The connection between the angular momenta and the eigenfunctions. In point a) we indicated that in the old quantum theory a series of allowed states of rotation of a particle was characterized by a series of quantum angular momenta (Fig. 3):

\[ J=n\cdot\frac{h}{2\pi}\quad(n=0,\,1,\,2\ldots). \]

In Fig. 2 we were presented with a series of functions \(\Psi_\varphi\), which in turn are characterized by a series of integers; we further pointed out that these functions have a great similarity to those encountered in wave mechanics in the description of the electron distribution with respect to a fixed axis of rotation. In order to have the right, in wave mechanics as well, to operate with the simple means of the vector model, it must be shown that the series of eigenfunctions with increasing characteristic number \(n\) actually correspond to moments \(n\cdot \frac{h}{2\pi}\). The stationary state in wave mechanics is characterized by the complete uncertainty of the position of the electron in the atom. There exists a certain number of chances of finding the electron at any place in space (although the probability of this varies very greatly from place to place). According to Heisenberg’s “uncertainty relation,” complete uncertainty of the coordinate of position corresponds to exact definiteness of the corresponding momentum. If the coordinate of position is the angle \(\varphi\), then the corresponding coordinate of momentum will be the angular momentum \(p_\varphi\). This momentum, consequently, in every stationary state must have an exactly determined value. This value can be determined from the eigenfunctions and, moreover, by a formula which is completely analogous to the hydrodynamic equation for determining the angular momentum of a liquid for a given distribution of density. (As is known from Schrödinger,

Fig. 3.

many problems of atomic structure can be treated as if \(\Psi\overline{\Psi}\) were not a probability distribution, but the actual distribution of a “smeared-out” electron; under such a distribution the electron forms something like a liquid cloud with local density \(\Psi\overline{\Psi}\), to which the hydrodynamic equations can be directly applied.)

The formula for the angular momentum reads:

\[ p_{\varphi}=\frac{h}{4\pi i}\left(\overline{\Psi}\frac{d\Psi}{d\varphi}-\Psi\frac{d\overline{\Psi}}{d\varphi}\right). \tag{1} \]

From (1) it is clear that the momentum becomes zero if the eigenfunction is real \((\Psi=\overline{\Psi})\). Thus no momentum different from zero corresponds to the simple trigonometric functions considered above. The reason is that, in a distribution represented by a real eigenfunction, the direction of rotation is in no way distinguished, which is necessary for the appearance of angular momentum. But such a distinction is established as soon as the eigenfunction is complex. In fact, the eigenfunctions of plane rotation have the complex form (2, 3). To the two conjugate complex functions (2) and (3),

\[ \Psi_{\varphi}=e^{+in\varphi}=\cos n\varphi+i\sin n\varphi, \tag{2} \]

\[ \overline{\Psi}_{\varphi}=e^{-in\varphi}=\cos n\varphi-i\sin n\varphi \tag{3} \]

there correspond opposite directions of rotation. If (2) or (3) is substituted into (1), one obtains:

\[ p_{\varphi}=\frac{nh}{2\pi}. \tag{4} \]

Thus, in fact, the \(n\)-th eigenfunction (2, 3) corresponds to a vector of angular momentum of magnitude \(n\).

A somewhat different result is obtained when one considers not a distribution relative to an axis fixed in space, but the direction of the axis is left undetermined (i.e. in passing from the two-dimensional case to the three-dimensional one). In this case more complicated eigenfunctions are obtained and, substituting them into (1), we find that the moment corresponding to the \(n\)-th eigenfunction is expressed as:

\[ P_{\varphi}=\sqrt{n(n+1)}\,\frac{h}{2\pi} \tag{5} \]

(the transition from the plane problem to the spatial one in wave mechanics often leads to the fact that instead of \(n^2\) an expression of the form \(n(n+1)\) appears). Thus in wave mechanics the absolute values of the angular momentum...

so different from those in the Bohr–Sommerfeld theory. However, for constructing a vector model the distinction between $n$ and $\sqrt{n(n+1)}$ is not essential. A vector of length $\sqrt{n(n+1)}$ may, for example, just like a vector $n$, form $2n+1$ different “quantized” components; two vectors $\sqrt{n_1(n_1+1)}$ and $\sqrt{n_2(n_2+1)}$ may form as many different resultant vectors of the form $\sqrt{n(n+1)}$ as the vectors $n_1$ and $n_2$ form resultants of the form $n$. Therefore, in constructing the vector model, one may henceforth make full use of the convenient integral vectors of the old theory, even though the discussion concerns rotation about free axes in space, rather than about fixed axes.

It should be borne in mind that the eigenfunctions of rotation of the form (2) differ from the simple functions $\cos n\varphi$ represented in Fig. 2 in that these eigenfunctions in fact have no nodes in the exact sense of the word. For a complex function can vanish only on the condition that its real and imaginary parts are separately equal to zero; but the cosine and sine never vanish simultaneously. The probability $\psi\psi'$, corresponding to the functions (2), is in fact constant everywhere:

$$ \psi\psi'=(\cos n\varphi+i\sin n\varphi)(\cos n\varphi-i\sin n\varphi)= =\cos^2 n\varphi+\sin^2 n\varphi=1. \tag{6} $$

However, the graphs of Fig. 2 may also serve to illustrate the functions (2), since these graphs represent the distribution of nodes separately for the real and imaginary parts of the eigenfunction. In this, somewhat inexact, sense the law often used in wave mechanics is also valid for the functions (2), according to which the quantum number of an eigenfunction is equal to the number of its nodal surfaces.

The relation illustrated in the example of Fig. 2 between the quantum number $n$ and the property of symmetry—“even or odd”—is also preserved for the complex functions (2); for when $\varphi$ is increased by $\pi$, both parts of the function—the real and the imaginary—preserve or change their sign according as $n$ is an even or an odd number.

3. Vector model and eigenfunctions of an individual electron

a) Atomic electron. Let us first consider an individual electron in a field with spherical symmetry. Strictly speaking, our reasoning will be applicable only to the H-atom, but pri-

obviously, they are valid also for any electron in an atomic bond, since the influence of the other electrons can be reduced to the “screening” of the nuclear charge to some constant value. Under this assumption it is possible to “separate out” the proper function of an individual electron from the proper function of the whole atom.

The vector model of a single electron in a spherically symmetric field consists of the angular momentum of its orbital motion about the nucleus (briefly, the “orbital moment”), which is denoted by \(l\), and the angular momentum for rotation about its own axis (briefly, the “spin”), which is usually denoted by \(s\). According to the hypothesis of Goudsmit and Uhlenbeck the quantum number \(s\) is always equal to \(\frac{1}{2}\). The magnetic interaction of the two vectors leads to the formation of a quantized resultant—the total moment of the electron \(j\), which is equal to the vector sum of \(l\) and \(s\). The corresponding “quantum number of the total moment” \(j\) obviously can take only the values \(l+\frac{1}{2}\) and \(l-\frac{1}{2}\). Thus each value \(l\) of one electron (with the exception of \(l=0\)) corresponds to two states, differing in the values of the total angular momentum \(j\).

If the atom is placed in an external electric or magnetic field, then (cf. p. 97) precession of the vector \(j\) begins with respect to the direction of the field, with the formation of a quantized component \(m\) in the direction of the field. Obviously, for \(m\) there are \(2j+1\) different values: \(j, j-1, j-2, \ldots, (j-1), -j\). Hence it follows that each atomic state, characterized by definite values of \(l\) and \(j\), in a weak external field is split into \(2j+1\) different states; for the two values of \(j\) possible for a given \(l\), namely \(l+\frac{1}{2}\) and \(l-\frac{1}{2}\), in all there are obtained \(2(2l+1)\) different states. If the field becomes stronger and stronger, then the coupling between the vectors \(l\) and \(s\) undergoes an ever greater perturbation. In the limiting case, when the influence of the field considerably exceeds the magnetic interactions between \(l\) and \(s\), the vector model is formed in a somewhat different sequence: the vectors \(l\) and \(s\) each separately form quantized components in the direction of the field, \(m_l\) and \(m_s\), and only in the last approximation do \(m_l\) and \(m_s\) add up to the total angular momentum \(m\) in the direction of the field. For \(m_l\) there are possible \(2l+1\) different states between \(+l\) and \(-l\); for \(m_s\) two values are possible, \(+\frac{1}{2}\) and \(-\frac{1}{2}\); this again gives, as in the case of a weak field, \(2(2l+1)\) different states.

Figs. 4 and 5 illustrate the vector model for two limiting cases of a weak and a strong field.

The proper function of an electron in a spherically symmetric field can be decomposed into three functions of the polar coordinates \(r,\vartheta,\varphi\):

\[ \Psi(r,\vartheta,\varphi)=\Psi_r\cdot\Psi_\vartheta\cdot\Psi_\varphi . \tag{7} \]

In Fig. 6 the nodal surfaces of these three functions are presented. We denote the corresponding quantum numbers by \(n_r, n_\vartheta, n_\varphi=\lambda\); we use the notation \(\lambda\) in connection with the notations adopted in the spectroscopy of band spectra. We shall not enter here into a discussion of the exact form of the functions \(\Psi_r, \Psi_\vartheta, \Psi_\varphi\) (cf. p. 109); we shall only note that

Fig. 4. Example of the vector model in a weak field:
\(l=2,\)
\(s=\frac12,\)
\(j=1\frac12,\quad m=+\frac12.\)

Fig. 5. Example of the vector model in a strong field:
\(l=2,\quad m_l=1,\)
\(m_s=-\frac12.\)

the quantum number \(n_r\) is equal to the number of nodal spheres of the function \(\Psi_r\). The quantum number \(n_\vartheta\) is equal to the number of nodal cones of the function \(\Psi_\vartheta\). We note that nodal cones \(\vartheta=\mathrm{const}\) occur in pairs, complementing one another to a double cone. If \(n_\vartheta\) is odd, the nodal cone degenerates into the nodal plane \(\vartheta=\frac{\pi}{2}\). As for the quantum number \(\lambda\), then, as was explained on p. 111, it is connected with the number of nodal planes of the real and imaginary parts of the function \(\Psi_\varphi\).

So long as there is no external field, only the choice of the origin of coordinates is prescribed by the position of the nucleus; the choice of the direction \(\vartheta=0\) remains completely arbitrary. If from the function \(\Psi(r,\vartheta,\varphi)\) one separates the radial function \(\Psi_r\), then the angular function \(\Psi(\vartheta,\varphi)\) remains. Its further separation into \(\Psi_\vartheta\) and \(\Psi_\varphi\) will lead to different results, depending on what direction is chosen for the axis \(\vartheta=0\). All these decompositions have in common that for them the sum \(l=\lambda+n_\vartheta\) has a constant value; or, in other words, that the total number of nodal—

of the surfaces \(\vartheta\) and \(\varphi\) remain constant. The numbers \(\lambda\) and \(n_\vartheta\), by themselves, may take all values between \(0\) and \(l\). Since there is no basis for giving preference to any one of these possible decompositions, one must imagine that the true state of the atom is a superposition of all states characterized by the various decompositions. The energy and the angular momentum of rotation depend only on the sum \(l\), and not on the separate terms \(n_\vartheta\) and \(\lambda\). When, however, a field is applied, one of the previously equivalent decompositions turns out to be distinguished; in a vanishingly weak field this singling out entails practically no energetic consequences.

Fig. 6.

As the field increases, the precession \(l\) about the direction of the field becomes noticeable, and the additional energy of this precession depends on which value of \(\lambda\) corresponds to the distinguished decomposition.

Not only the energy and angular momentum, but also the symmetry property “even” or “odd” depends only on the sum \(n_\vartheta+\lambda\), for all nodal surfaces of the functions \(\Psi_\vartheta\) and \(\Psi_\varphi\) pass through the origin of coordinates, and they have to be crossed in order to pass from some point \(A\) (Fig. 2) to the point \(B\), which is the mirror reflection of \(A\). Electrons with even values of \(l\) \((l=0, 2, 4, 6\ldots)\) therefore have even eigenfunctions; electrons with odd \(l\) have odd eigenfunctions.

b) The electron in a molecule. Exactly as in

the preceding arguments were strictly applicable only to the H-atom, and the subsequent ones likewise are strictly applicable to the ion \(H_2^+\). However, as a rough approximation they may also be used for electrons in other diatomic molecules.

The vector model of a “separated” electron (in the sense of the operation of separating the differential equation) strongly resembles the vector model of an atomic electron in an external field. Owing to the replacement of one nucleus by two, a preferred direction arises in the molecule (the “axis of the molecule”), which plays the same role as the direction of the external field. And indeed, along this axis of the molecule there acts a considerable electric force.

So long as the two nuclei are still very close to one another, the axis of the molecule passing through them singles out only one proper function \(\Psi(n_r,\varphi)\) and predetermines the values of \(n_r\) and \(\lambda\); however, the electronic energy and the angular momentum remain practically the same as in the “united” atom (i.e., the atom formed by the coincidence of both nuclei), and the quantum numbers \(l\) and \(n_\varphi\) retain their value. If, conversely, the nuclei are moved somewhat farther apart, then the moment \(I\), under the influence of the increasing axial electric field, begins to precess ever more rapidly about the direction of the molecular axis. On the other hand, \(n_r\) and \(l\) lose their original significance, for the system is deprived of spherical symmetry, and it can no longer be separated in the spherical polar coordinates that underlie the definition of \(n_r\) and \(l\). The consequence of precession about the molecular axis is the formation of the quantized component \(\lambda\) of the moment \(I\); at the same time, \(I\) itself increasingly loses its significance as angular momentum. But the component \(\lambda\), even when the nuclei are separated considerably from one another, retains its significance as the angular momentum with respect to the molecular axis. If, with respect to an electron in a molecule, one speaks of the “orbital moment” \(I\), such an expression is imprecise, and it means: “the orbital moment which the electron would possess if its nuclei were brought into coincidence.”

The quantum numbers 0, 1, 2… in atomic spectroscopy are usually denoted by the letters \(s, p, d, f\). Analogously, in a molecule the quantum numbers \(\lambda\) are replaced by the corresponding Greek letters \(\sigma,\pi,\delta,\ldots\). If one wishes to indicate both quantum numbers \(l\) and \(\lambda\), one speaks of \(s\sigma, p\sigma, p\pi\ldots\) electrons. Sometimes it is necessary to know only the symmetry properties, but not the exact magnitude of \(l\); then the electron is denoted, for example, as follows: \(\sigma_g\) (gerade— even) or \(\sigma_u\) (ungerade— odd). The number of \(\sigma_g\)-electrons includes \(s\sigma, d\sigma, g\sigma\ldots\) electrons; the \(\sigma_u\)-electrons include \(p\sigma, f\sigma\ldots\) electrons. Since \(\lambda\)

if there is a projection \(l\) on the molecular axis, then the inequality \(\lambda < l\) must hold; consequently, there can be no \(s\pi, p\delta\)-electrons.

We have not yet said anything about the spin angular momentum \(s\). The action of the axial electric field of the molecule on \(l\) is so considerable that it destroys the connection between \(l\) and \(s\) and annihilates the significance of the quantum number \(j\). The molecular axial field acts on the spin \(s\) itself; however, the connection between \(s\) and the axis is very weak. If there are no other perturbing forces, then the spin \(s=\frac{1}{2}\) forms a quantized projection \(\sigma=+\frac{1}{2}\) or \(-\frac{1}{2}\) in the direction of the axis; finally, the two angular momenta \(\lambda\) and \(\sigma\) combine into one resultant \(\omega\)—the total angular momentum of rotation of the molecular electron relative to the molecular axis.

With the aid of quantum numbers defined in this way we can undertake the division of molecular electrons into groups or “shells.” According to the Pauli principle, in a closed system there cannot exist two electrons with the same four quantum numbers. Which four quantum numbers should be used to compute the “capacity” of individual electron shells is a question of the coupling conditions. In the atom we are interested in the filling of “shells” with \(n=\text{const}\) and “groups” with \(n=\text{const}\) and \(l=\text{const}\). In the molecule we ask about the maximum number of electrons with given values \(n, l\), and \(\lambda\)—for example, about the filling of the groups \(1s\sigma, 2p\pi\), etc. As the fourth quantum number we may choose \(\sigma\). To every combination \(n,l,\lambda\) there correspond two different possible values of \(\sigma\): \(+\frac{1}{2}\) (\(l\) and \(\sigma\) are directed alike) and \(-\frac{1}{2}\) (\(l\) and \(\sigma\) are directed oppositely). For groups with \(\lambda \ne 0\), moreover, one must take into account the two values \(\pm\lambda\), which, although energetically equal, must nevertheless be counted separately. Thus, for \(\lambda \ne 0\) in each \(nl\lambda\)-group there exist 4 places, while for \(\lambda=0\) only 2. Thus, in a molecule there exist only groups of two and of four electrons.

We now turn to the wave-mechanical description. The coordinates in which the wave equation of an electron in the field of two fixed centers can be separated are obtained by a natural generalization of spherical polar coordinates, by means of which the case of coincident nuclei is treated. The spheres \(r=\text{const}\), when the nuclei are separated, are replaced by ellipsoids \(\rho=r_1+r_2=\text{const}\), in the foci of which the nuclei are located; instead of the cones \(\vartheta=\text{const}\) one obtains hyperboloids \(v=r_1-r_2=\text{const}\), with the nuclei again located in the foci of these hyperboloids;

Finally, the coordinate \(\varphi\) retains its meaning as an azimuth relative to a certain axis, only the direction of this axis will no longer be arbitrary, but is determined by the direction of the line joining the nuclei. At the same time the quantum number \(\lambda\) also retains its meaning; but the place of the quantum number \(n_r\) is taken by the “ellipsoidal” quantum number \(n_\rho\), which is equal to the number of nodal ellipsoids of the function \(\Psi_\rho\). Similarly, the quantum number \(n_\theta\) is transformed into the “hyperboloidal”

In the figure:
\(a\)—section along the axis joining the nuclei; \(b\)—section perpendicular to the axis.
Labels shown include:
\(n=1,\ l=0,\ \lambda=0,\ 1S\sigma,\) even;
\(n=2,\ l=0,\ \lambda=0,\ 2S\sigma,\) even;
\(n=3,\ l=0,\ \lambda=0,\ 3S\sigma,\) even;
\(n=2,\ l=1,\ \lambda=0,\ 2P\sigma,\) odd;
\(n=3,\ l=1,\ \lambda=0,\ 3P\sigma,\) odd;
\(n=2,\ l=1,\ \lambda=1,\ 2P\pi,\) odd;
\(n=3,\ l=1,\ \lambda=1,\ 3P\pi,\) odd;
\(n=3,\ l=2,\ \lambda=0,\ 3d\sigma,\) even;
\(n=3,\ l=2,\ \lambda=1,\ 3d\pi,\) even;
\(n=3,\ l=2,\ \lambda=2,\ 3d\delta,\) even;
\(n=4,\ l=3,\ \lambda=0,\ 4f\sigma,\) odd.

Fig. 7. Nodal surfaces of a molecular electron
(after Weizel).

“quantum number” \(n_\nu\), which is equal to the number of nodal hyperboloids of the function \(\Psi_\theta\). The sum \(n_\rho+n_\nu\) may in what follows be denoted by \(l\); since, however, of the two coordinates \(\varphi\) and \(\nu=r_1-r_2\) only one, namely \(\varphi\), is a purely angular coordinate, a change in which signifies rotation, it is clear that, as has already been indicated, strictly speaking no constant angular momentum of rotation will correspond to the quantum number \(l\); the latter can be spoken of only with regard to

...the degree of approximation with which the hyperboloids \(\nu=\mathrm{const}\) can be replaced by the double cones \(\theta=\mathrm{const}\).

In Fig. 7, according to Weigel, the nodes of the functions \(\Psi_\rho\), \(\Psi_\nu\), and \(\Psi_\varphi\) are presented in sections perpendicular and parallel to the molecular axis for some of the simplest combinations of quantum numbers. For the function \(\Psi_\varphi\), what was said on p. 264 is valid: it has no nodes by itself—the real and imaginary parts separately have nodes. The probability distribution \(\Psi\Psi'\) for each \(\lambda\) does not depend on the angle \(\varphi\) [see equation (6)]; i.e., the molecule possesses perfect rotational symmetry with respect to the molecular axis.

We shall now consider the symmetry properties of the eigenfunctions with respect to reflection at the origin of coordinates. In the general case—when the nuclei are not identical—the eigenfunction remains unchanged in magnitude under simultaneous reflection of the nuclei and electrons in the origin of coordinates. Since the definition “even” or “odd,” by analogy with atoms, also applies in molecules to the reflection of only the electrons, for the symmetry property referring to the reflection of all particles the concept of positive and negative eigenfunctions is introduced. Positive eigenfunctions are those which, under such reflection, preserve their sign; negative ones are those which change their sign. With identical nuclei, the reflection of the nuclei and electrons separately must also leave unchanged the absolute value of the eigenfunctions. Depending on whether the sign is preserved or changed under this operation, one distinguishes even and odd functions (reflection of the electrons), and, with respect to the nuclei, symmetric or antisymmetric functions (reflection of the nuclei).

Fig. 8.

For illustration of the symmetry properties Fig. 8 serves. The following relations may be distinguished: 1) Reflection of the nuclei is equivalent to interchange of the nuclei; hence the names “symmetric” and “antisymmetric”: a function symmetric with respect to two variables is one which remains unchanged when they are interchanged (for example: \(a+b\)), and an antisymmetric one is a function which thereby changes its sign (for example: \(a-b\)). 2) Reflection of all particles is equivalent to changing all directions of rotation to the opposite ones (a clock in a mirror!). From 2 it is evident that, for example, the function \(\sin\varphi\) is negative, while the function \(\cos\varphi\) is positive. Indeed, under replacement

of the positive direction of rotation to the opposite one, \(\varphi\) is replaced by \(2\pi-\varphi\), but

\[ \sin \varphi=-\sin(2\pi-\varphi),\qquad \cos \varphi=\cos(2\pi-\varphi). \]

The property “even or odd” is completely independent of the property “positive or negative.” Thus, for example, the function \(\sin n\varphi\) is negative for every \(n\), but it is alternately even or odd, depending on whether \(n\) is an even or an odd integer. As can be seen from Fig. 7, the property of a function of being even or odd depends only on whether the sum \(n_i+\lambda=l\) is odd or even. The function \(\Psi_r\) plays no role here (just as the function \(\Psi_r\) in the atom plays no role), since in passing from any point to its mirror image one has to cross twice the nodal surfaces of this function. Since the nodal hyperboloids of the function \(\Psi_r\) are two-sheeted, when determining the symmetry property “even or odd” one must take into account not only the number \(\lambda\), but also whether or not there is, midway between the two nuclei, a degenerate hyperboloid \(r_1-r_2=0\) (Fig. 6, middle).

On the other hand, the property “symmetric or antisymmetric” with respect to the nuclei is determined unambiguously by the properties “even or odd” and “positive or negative,” for successive reflection of the nuclei and of the electrons must lead to the same result as their simultaneous reflection. For example, in order that an odd function be positive, it must also undergo a change of sign when the nuclei are interchanged, i.e. be antisymmetric, etc. Thus the following relations are obtained:

\[ \begin{aligned} \text{even and symmetric}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ . \\ \text{odd and antisymmetric}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ . \end{aligned} \right\}=\text{positive} \]

\[ \begin{aligned} \text{even and antisymmetric}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ . \\ \text{odd and symmetric}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ . \end{aligned} \right\}=\text{negative} \]

For example, the positive function \(\cos n\varphi\) for even \(n\) is even and symmetric, while for odd \(n\) it is odd and antisymmetric.

The property of being even or odd may be ascribed to any eigenfunction (for molecules with identical nuclei). The situation is different with the symmetry property—“positive or negative.” This distinction may be due, for example, to rotation of the entire molecule (cf. part IV). In a non-rotating state the direction of rotation can be determined only by the electronic structure. An electron that is a \(\sigma\)-electron (or any number of such electrons), in this sense, in no way ...

cannot be singled out, since in general no direction of rotation about the molecular axis corresponds to it. If there is a separate electron with \(\lambda \ne 0\) (\(\pi, \delta \ldots\)), then although the molecule possesses angular momentum, the two opposite directions of this angular momentum are energetically equal and therefore may be regarded as equally probable. The true term is, as it were, a superposition of both states with opposite angular momenta; it may also be described as a “degenerate” state, consisting of two energetically coincident terms—positive and negative. Only when a perturbation appears, caused, for example, by rotation, which affects the two coincident terms in different ways, does splitting into positive and negative terms occur. The special case in which such a perturbation occurs even without rotation is observed for two or more electrons with \(\lambda \ne 0\), whose angular momenta cancel one another, so that \(\lambda_1 + \lambda_2 + \ldots = 0\).

For terms for which the property “positive” or “negative” is indeterminate, the property “symmetric or antisymmetric” is, of course, equivalent to the property “even or odd.”

c) Correspondence between molecular and atomic electrons. The energy of an atomic electron is uniquely determined by its quantum numbers and by the charge of the nucleus. In the case of a molecular electron, the distance between the nuclei is added as a further parameter. Whereas the system of terms of an atom is represented graphically by a system of straight lines, in the graphical representation of the electronic terms of molecules one must also take into account the change of this term with the internuclear distance; in place of the usual term diagram there appear systems of curves, an example of which is Fig. 15. Here the abscissae are the internuclear distances (from zero practically to infinity), and the ordinates are the values of the terms. Each curve on the right merges with the term of the “separated atoms.”

In constructing term curves, certain difficulties arise. The curves must throughout represent one and the same term. However, it is not always easy to recognize that two segments of a potential curve belong to one term. Indeed, we know that quantum numbers have no strict meaning, but, when interactions are taken into account, are approximations. In passing from the case of nuclei close together to that of distant nuclei, it is expedient to change the approximation; the known quantum numbers lose their significance, and others appear in their place. This can be detected at once if one compares the two limiting cases of united and separated nuclei; in both cases the states

are described by the atomic quantum numbers $n$ and $l$, and, however, not every state which, for separated nuclei, has the quantum numbers $n$ and $l$ passes into a state with the same quantum numbers of the united atom; for sometimes one has to encounter here the prohibition imposed by the so-called Pauli rule. Let us imagine, for example, two helium atoms, each with two $1s$ electrons ($n = 1$, $l = 0$). If the nuclei are brought into coincidence, then all four electrons cannot remain $1s$ electrons; for, according to the Pauli rule, the group of $1s$ electrons (the so-called $K$ shell) in an atom can contain no more than two electrons. As the atoms approach one another, the initial quantum numbers $n$ and $l$ of the separated atoms lose their meaning; instead of them there appear molecular quantum numbers, so that upon further approach they give way to the quantum numbers of the united atom.

In order to find the correct correspondence of molecular electrons with atomic electrons, one may proceed by an experimental or a theoretical route. Experimentally, one can construct, by the method described in the second part of this article, the potential curves on the basis of a series of vibrational quanta. Theoretically, one can establish certain correspondence rules, which require the preservation of known properties despite changes in the bonding conditions. It should be borne in mind that theoretical and experimental correspondence do not always coincide. Theoretical correspondence refers to an “adiabatic” process, i.e. to an infinitely slow change of the internuclear distances with constant removal (or supply) of the released (or absorbed) energy. But dissociation as a consequence of vibrations is not such an adiabatic process. In any case, as long as the potential curves of different states pass far from one another, the correspondence is unambiguous, and the vibrations follow the adiabatic potential curve. The situation is different if two curves touch or even intersect. Then there exist several states which, at the same internuclear distance, possess equal or almost equal energy, and, according to the fundamental principles of wave mechanics, there is always a probability of mutual transformation of such states. Let us imagine the case of an “intersection” of two potential curves (Fig. 9); in Fig. 10 the region of intersection is shown once more on a larger scale. We shall approach the region of perturbations, for example, along the curve $HG$. With a sufficient approach to the curve $DC$ the molecule begins to oscillate between the two states. From the point of view of wave mechanics, such a periodic transition from one state to

another represents the result of beats between two wave processes, which are matched to the given two states. A wave-mechanical calculation shows that these beats lower the energy of the lower state \(HG\), and raise still further the energy of the upper state; the curves cease to intersect, but diverge, and as a result the branch \(HG\) passes into \(EF\), and the branch \(DC\) into the branch \(BA\), as is also shown by the dotted curves. But in order for this divergence to occur, the transition through the region of perturbations must take place sufficiently slowly, for beats can be formed only when the molecule remains in the region \(BCFG\) for a time that is large compared with the period of the beats. The dotted correspondence thus holds for a theoretical, “adiabatic” process, but not necessarily for the experimentally realized process of vibrational excitation. If, for example,

Fig. 9.

Fig. 10. Intersection of potential curves.

a molecule with vibrational energy \(T\) performs the vibration \(CADF\) (Fig. 9), then passage through point \(A\) may occur considerably faster than is necessary for beats to form and for the transition to the curve \(BAG\).

The greater the probability of mutual transformation of both molecular states in the region of perturbations, the smaller the beat period and the more strongly the adiabatic curves deviate from one another; at the same time, the greater is the probability that the vibrations follow the “adiabatic” path. In the limiting case of strong divergence, the theoretical and experimental correspondences coincide. In the intermediate region the adiabatic correspondence is still unambiguous, but the experimental one may be multivalued, since the beat period is of the order of the transit time, so that there is an approximately equal probability that, after passing through, the molecule will find itself either on one curve or on the other. In the limiting case of very slow beats, the divergence of the adiabatic curves is barely noticeable, and the established correspondence leads to an intersection of the potential curves.

*

It may be assumed with certainty that the curves intersect when, on theoretical grounds, radiationless transitions between the two states are “forbidden.” Such a prohibition always occurs when, with the transition, the properties of symmetry must change. Consequently, if, for example, an electron at a large separation of the nuclei possesses an “even” eigenfunction, then when the nuclei come closer together it cannot be transformed into an electron with an odd eigenfunction. Further, there must be obeyed the conservation of all the properties of the term characterized by definite quantum numbers. However, these quantum numbers must be such as have a sufficiently strict meaning at the corresponding internuclear distances. This applies above all to the angular momentum about the molecular axis $p_\varphi$, i.e. to the quantum number $\lambda$; from the graphical representations in Fig. 7 one can see that the existence and distribution of the nodal planes of the function $\Psi_\varphi$, which correspond to the quantum number $\lambda$ and to the angular momentum $p_\varphi$, do not depend on the internuclear distance. Thus we establish a rule according to which $\sigma$-electrons, by changing the nuclei, cannot be transformed into $\pi$- or $\delta$-electrons, etc. The matter is not so simple with the other quantum numbers $l$, $n_r$, and with the principal quantum number $n$ formed from them. On the basis of the same diagrams in Fig. 7 one can see that their nodal surfaces cannot remain unperturbed when the internuclear distance is changed; for example, at a large separation of the nuclei the place of nodal ellipsoids embracing both nuclei is taken by nodal spheres around each of the nuclei. The “molecular” quantum numbers $l$, $n_r$, and $n$ lose their meaning in the intermediate region of internuclear distances and are gradually replaced by “atomic” quantum numbers. If the approach of two potential curves occurs in a region where the molecular or atomic quantum numbers are still well defined, the requirement that they be conserved leads to crossing. However, until we have at our disposal a considerably larger amount of experimental material than we possess at present, it is rather difficult to sort out these relationships. Therefore we give in Fig. 11 a correlation constructed by Herzberg, on the assumption that only symmetry and the quantum number $\lambda$ give rise to crossing and that otherwise the curves diverge, i.e. that the lower states at close nuclei also correspond to lower states at distant nuclei, since symmetry and the quantum number $\lambda$ allow this. In Mulliken and Weizel one can find correlation tables in the construction of which the conservation of other ...

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quantum numbers and which therefore have a much larger number of intersections than Fig. 11. Apparently, however, for the theory of the chemical bond, for which we intend to use Fig. 11, the difference between the tables of Herzberg and Mulliken-Weitzel is not essential.

On the left in Fig. 11 are shown the electron shells in the “united” atom; the superscripts correspond to the possibilities of filling according to the Pauli rule. The next column gives the splitting of the shells for a small separation of the nuclei (an \(s\)-electron gives only \(s\sigma\), a \(p\)-electron gives \(p\sigma\) and \(p\pi\), etc.). The energy sequence is the same as that observed in the Stark effect: the energy is the greater, the stronger the precession about the axis of the molecule, i.e., the greater \(\lambda\). The fourth column, placed on the right, gives the energy sequence of the electron shells in the separated atoms; it is the same as in the first column, but the possibility of filling is twice as large as in the united atom. The third column indicates the splitting of these shells when the interaction between the atoms begins. And in this case, just as in passing from the first to the second column, \(s\)-electrons can form only \(\sigma\)-electrons, \(p\)-electrons \(\sigma\)- and \(\pi\)-electrons, etc.

1. United nuclei. 2. Near nuclei. 3. Distant nuclei.
2. Separated nuclei.

Fig. 11.

But besides this there is still another possibility of change: when the nuclei approach one another, a nodal plane in the middle between the two nuclei may or may not be formed. This can be seen by considering, with the aid of Fig. 7, the reverse process—dissociation. When two nuclei between which there is a nodal plane in the middle move apart, the nodal plane in the middle remains, and upon complete dissociation loses its significance, since nodal planes at infinity are not counted (for,

at infinite distance all eigenfunctions vanish). Thus the eigenfunctions 1 and 4, or 2 and 5, etc. (Fig. 7), upon dissociation must become identical. Conversely, from each eigenfunction of the separated nuclei there must arise one eigenfunction with a median nodal plane and one without this plane. The two eigenfunctions differ by one nodal plane. Consequently, if one of them is even, the other must be odd. Therefore each group of electrons in column 3 occurs twice: once with the index \(u\) and once with the index \(g\). It should be borne in mind that the emergence of an odd molecular function from an even atomic function (or conversely), for example the emergence of a \(2p\)-electron (\(l=1\), i.e. odd) from a \(1s\)-electron (\(l=0\), i.e. even), does not contradict the law of conservation of symmetry. In separated atoms the symmetry properties refer to reflection in the individual nuclei; in molecules—to reflection in the center of gravity of both nuclei; these properties have nothing in common with one another. But in the first, second, and third columns the origin of coordinates remains one and the same, and therefore within these three columns the symmetry must be preserved.

The energy sequence of the electron groups in the third column (for example, first \(\pi_u\), then \(\sigma_g\), \(\pi_g\) and, finally, \(\sigma_u\) for electrons arising from the \(2p\) group) was derived by Hund from theoretical considerations. However, for our purpose this sequence is not especially essential.

After the second and third columns have been established, it remains to correlate them, connecting the corresponding electrons with lines on the basis of the rules indicated above. Namely, the lowest state \(\sigma_g\) on the right corresponds to the lowest state \(\sigma_g\) on the left; the lowest state \(\sigma_u\) on the left—to the lowest state of the same type on the right, and so on. However, in Fig. 11 an attempt has also been made to indicate the approximate position of the intersections, i.e. it is indicated, on the basis of various considerations, whether the intersection occurs at large or at small internuclear distances.

From Fig. 11 it is seen that some connecting lines go from bottom to top (or at least horizontally), others—from top to bottom. By comparison with Fig. 7 one can see that those curves go upward which lead to terms with a nodal plane in the middle between the nuclei.

Herzberg and Hund suggested that those electrons which, upon bringing the nuclei together in Fig. 11, “rise upward” in fact increase their energy and therefore create repulsion (“loosening electrons”), whereas the remaining electrons (“binding”

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…electrons”) lower their energy when molecules are formed and therefore act in a bonding manner.* When two atoms approach one another, two types of interaction arise between them. First, the bond of the first electron in the field of two nuclei obviously becomes stronger than in the field of one nucleus; this gives a decrease of energy upon approach, i.e. attraction. On the other hand, like-charged nuclei repel one another; the true potential curve is the superposition of an attractive and a repulsive potential. When the attraction is strong, it gives a minimum at a certain distance between the nuclei and only at still closer approach does repulsion prevail; but when the attraction is weak, the atoms in practice repel each other at any distance. It is easy to see that the attraction will be weakest when the electron cannot enter the region midway between the two nuclei, where it gains most from interaction with the nuclei. But this will be the case when a nodal plane lies precisely midway between the nuclei. This argument shows that electrons with such a nodal plane situated in the middle apparently will be “loosening.”

One may try to find a simple relation between the number of valence bonds and the number of bonding and loosening electrons. It may be asserted, although probably as a rough approximation, that each bonding electron releases approximately the same amount of energy upon binding, while each loosening electron raises the energy of the molecule by approximately the same amount. The valence number of an atom in that case would be measured by the difference between the number of bonding and loosening electrons which this atom contributes to the compound. If, for example, two H atoms in the ground state combine, then each of them contains one \(1s\)-electron. Both of these electrons in the molecule may become \(1\sigma\)-electrons, i.e. act as bonding electrons. Therefore H atoms may be regarded as monovalent. Their bond is effected by one “bonding pair of electrons.” The theory set forth here is close to the conception of a bond by “pairs of electrons,” developed by Lewis, and also by Heitler and London. However, according to Fig. 11, there also exists the possibility that one or both \(1s\)-electrons become \(p\sigma\)-electrons; in that case repulsion results, and the molecule is not formed. Thus, alongside the potential curve which corresponds to the formation of a stable

* Similar ideas had been expressed still earlier by Mulliken, but they could not be carried out with sufficient success.

molecule H\(_2\), there also exists a repulsion curve. But exactly the same result had already been obtained earlier by Heitler and London on the basis of wave mechanics.*

Heitler and London established that the two possible combinations of H atoms are associated with different orientations of the electron spins; the bond corresponds to mutual saturation of both \(s\) vectors (\(s_1 + s_2 = 0\)), and repulsion to their parallel arrangement \(s_1 + s_2 = 1\). Generalizing this result, London put forward the assertion that the bonding of electron pairs is in general coupled with an antiparallel arrangement of their spin vectors. However, Heitler and London soon became convinced that such a representation can be carried through only in the case when at least one atom is in an \(S\) state. In the theory of bonding electrons no such obligatory saturation of spins is assumed, although, on the other hand, it turns out that there is often a parallelism between the two phenomena. In this way one explains, for example, the difficulty arising in the old theory in explaining the bond of the O\(_2\) molecule: spectroscopy shows that the ground state of the O\(_2\) molecule does not correspond to complete compensation of the spin vectors, but to a resultant spin of magnitude \(S = 1\) (the ground term of the O\(_2\) molecule is \({}^3\Sigma\), and not \({}^1\Sigma\), as might have been expected from the Heitler–London theory). According to the new theory, the lowest state of the molecule will be that to which the maximum possible number of bonding electrons corresponds. When there are several such terms, they are arranged in the same order as atomic terms are arranged for one and the same arrangement of electrons: lowest of all are the terms with the greatest multiplicity. But according to London the lowest state of the molecule should be the state with the least multiplicity. Agreement between the two theories is evidently possible only when the maximum number of bonding electrons is compatible only with the least multiplicity. This is the case when all bonding electrons are \(\sigma\)-electrons, or when there are four \(\pi\)- or \(\sigma\)-electrons each; such groups of electrons form a closed shell, which, by the Pauli principle, is possible only in \({}^1\Sigma\) states with compensated spin vectors. When, however, among the bonding—

* At first glance it may seem as though, on the basis of Fig. 11, three states, rather than two, should result by combining two H atoms, namely \(s\sigma^3\), \(s\sigma\), \(p\sigma\), and \(p\sigma^3\). As will be shown on p. 298, in fact there are even four terms, since the distributions \(s\sigma\) and \(p\sigma\) could correspond to two different states. But of these 4 terms only two can be assigned to the combination H \((1s)\) + H \((1s)\), whereas the other two give the dissociation products: H\(^+\) ion and H\(^-\) \((1s^2)\). Such an unsymmetrical dissociation obviously does not contradict Fig. 11.

BAND SPECTRA

If, of the remaining electrons, two are \(\pi\)- or \(\sigma\)-electrons (as also happens in the case of \(O_2\)), then the molecular shell formed from them is not yet saturated and therefore may possess spins directed in the same way. But the same direction of the spins gives a triplet term which, according to the rules mentioned above, lies lower than the singlet term corresponding to the same electron configuration.

Along with the elimination of difficulties in the case of \(O_2\) (and other molecules whose constituent parts are not in an \(S\) state), the conception of bonding and antibonding electrons also permits one to draw new conclusions in the case of unsymmetrical molecules. It is true that this conception was originally derived for identical nuclei, but it should also hold for nuclei that do not differ too greatly. According to this conception, the existence of a bonding pair of electrons is not an absolutely necessary condition for the occurrence of a bond; moreover, for this purpose even a single electron is sufficient. Thus there is a possibility for a bond between an “valence-deprived” atom and a monovalent atom, for example between Mg and F. According to London, in this case a true bond is impossible, since the atom has spin equal to zero—the ground term \(^{1}S\). According to Fig. 11, in this case there are 9 antibonding and 12 bonding electrons; consequently, there is the possibility of a fairly strong bond, which also agrees with spectroscopic experience, which proves the existence of a stable MgF molecule (the MgF molecule in the gaseous state is stable with respect to the free atoms Mg and F; this fact has no connection with the stability or instability of the chemical compound of this composition). It is true that a simple calculation indicates, for example, that in the combinations \(F + Ne\) or \(H + He\) there should also be an excess of bonding electrons. However, it is clear that the assumption of energetic equivalence of all bonding and antibonding electrons, even for identical nuclei, is only a crude approximation, and for different nuclei it has no place at all. The significance of these ideas consists only in the fact that they for the first time open the possibility of explaining the experimentally known bond between atoms deprived and not deprived of valence.

A “one-sided bond” may combine in a molecule with a “symmetrical bond” by pairs of electrons. Let us consider, for example, the molecule CO. The C atom can supply two bonding electrons, which determine a double bond with the corresponding electrons of the O atom by the formation of two pairs of electrons. However, there still remain two electrons of the O atom which, according to Fig. 11, can also act as bonding electrons. (The twelve electrons of the C atom together with

together with the twelve first electrons of the O atom fill the five lower groups in the third column of Fig. 11, including the group \(\pi_i^4\); the remaining two \(2P\)-electrons of the O atom can thus become bonding electrons \(\sigma_y\). At the same time, the bond in CO consists of one ordinary double bond and one additional one-sided bond, with the aid of two O electrons. This agrees well with the high bond energy of the CO molecule (about \(230\) kg-cal), which considerably exceeds the energy of simple double bonds in \(\mathrm{C}_2\) (about \(160\) kg-cal) and \(\mathrm{O}_2\) (\(117\) kg-cal).

4. Interaction between electrons; molecular terms

a) Vector model and symmetry properties with several electrons.

Up to now we have considered the ideal case in which all the electrons of a molecule can be separated; the state of each electron is described by its quantum numbers—\(n, l, \lambda, \omega\), and the state of the molecule by indicating the distribution of the electrons over the various shells. In reality, however, in a molecule, as in an atom, interactions occur between the individual electrons; their moment vectors and eigenfunctions, as a result of the coupling, create a common moment vector and an eigenfunction of the whole molecule.

The relative magnitude of the various interactions leads to the construction of the following vector model for a non-rotating molecule (when the molecular weight is not too high).

The rotational angular momenta \(l\) and \(s\) of the individual electrons retain, in a first approximation, their value. The coupling between the angular momenta \(l\) and \(s\) of a separate electron, as already mentioned above, is disturbed by the forces which orient \(l\) relative to the molecular axis and lead to the formation of \(\lambda\). The vectors \(s\) are coupled only weakly with the axis; their mutual interaction is stronger, leading to the appearance of a quantized resultant \(S\), i.e. the total moment of the molecule. At the same time the individual quantum number \(\omega\) loses its meaning. The separate values of \(\lambda\) add up to the total orbital moment \(\Lambda\) relative to the axis. Further, the magnetic interaction of \(S\) and \(\Lambda\) comes into play; since \(\Lambda\) is connected with the molecular axis, this interaction leads to the formation of a quantized spin component in the direction of the axis \(\Sigma\). \(\Lambda\) and \(\Sigma\) together form the total moment of the molecule relative to the axis \(\Omega\). An example of such a vector scheme is given in Fig. 12.

The state (term) of a non-rotating diatomic molecule is thus characterized by the quantum numbers \(n, l\), and \(\lambda\) of the individual electrons and by the quantum numbers \(S, \Lambda\), and \(\Omega\)

of the totality of all electrons. In this, just as in the case of atoms, the so-called closed electron shells, in constructing the vector model and in determining terms, may be left out of consideration, since within these shells all vectors mutually compensate one another. Consequently, in determining the multiplicity of terms in optical spectra, only those electrons play a role which lie outside the last closed shell (valence electrons).

States which correspond to the various possible values of $\Omega$ for a given distribution of electrons over the shells $(n, l, \lambda)$ and a given quantum number $\Lambda$ are called components of a multiplet term. The number of these components is $(2S+1)$, for it is equal to the number of admissible projections $\Sigma$ of the vector $S$ on the molecular axis. If $\Lambda = 0$, then magnetic coupling with $S$ is impossible, and the formation of a quantized spin component $\Sigma$ can be effected only by a very weak, energetically negligible direct action of the molecular axis on the spin. Therefore all $2S+1$ components coincide. The quantity $B = 2S+1$ is called, as in the case of atoms, the multiplicity of the term; it is written in the form of an index at the upper left of the term symbol, whereas the number $\Omega$ is written as an index at the lower right. The quantum numbers $\Lambda$ are replaced by capital Greek letters $\Sigma, \Pi, \Delta \ldots$ $(\Lambda = 0, 1, 2 \ldots)$. In addition, indices are also appended to denote the symmetry properties of the term— even or odd, positive or negative. In this way there arise term symbols having, for example, the following form: ${}^3\Pi_{1}^{+}$, “triplet—pi—one—odd—positive,” etc. The components of the multiplet in light molecules lie very close together and diverge as the molecular weight increases, just as occurs in atoms (the reason is the increasing interaction energy of $L$ and $S$). Of course, to

\[ \begin{gathered} l_1=2,\quad l_2=3,\quad l_3=1,\\ \lambda_1=0,\quad \lambda_2=2,\quad \lambda_3=1,\quad \lambda=3,\\ S=s_1+s_2+s_3=1\tfrac{1}{2},\quad \Sigma=\tfrac{1}{2},\\ \Omega=3\tfrac{1}{2}. \end{gathered} \]

Fig. 12. Example of a vector model of a molecule with 3 electrons.

to the term symbol one may also attach indications of the quantum numbers of the individual electrons, for example:

\[ 1s^2\,2s^2\,2p\sigma^2\,2p\pi^4\,3s\sigma^2\,3p\sigma^2,\;{}^1\Sigma_{0g}^{+}. \]

This formula represents the ground state of the molecule \(N_2\). However, when using such formulas one must bear in mind the circumstances already indicated earlier: characterization by the quantum numbers \(n, l\) of the combined atom is a more or less expedient approximation only for the outer electrons directly taking part in the bond. In the case of \(N_2\), apparently only six electrons, \(2p\pi^4\,3s\sigma^2\), belong to this number; they arise from the \(2p\)-electrons of the separated atoms (cf. Fig. 11). The remaining eight electrons are certainly in a state which can most closely be characterized as belonging to the \(1s\) and \(2s\) groups of the separated atoms. Lennard-Jones therefore proposed dividing the electrons in a molecule into two groups: “atomic electrons,” which can be assigned to one of the two atoms, and “molecular electrons,” belonging to the molecule as a whole. However, this division is to a considerable degree arbitrary; the corresponding symbols, such as, for example,

\[ (1s^2+1s^2)\,(2s^2+2s^2)\,2p\pi^4\,3s\sigma^2\cdot{}^1\Sigma \]

(for the ground state of \(N_2\)) are not very convenient.

Just as the eigenfunctions of individual electrons, so also the eigenfunction of the whole term may possess the property and symmetry “even or odd” or “symmetric or antisymmetric” with respect to the nuclei (for identical nuclei), or “positive or negative” (for different nuclei). It is clear that a term can be odd only when it contains an odd number of odd electrons; this can be verified if one successively performs reflection of the electrons at the origin of coordinates.

Analogously to the way in which, according to Fig. 11, each molecular electron can be correlated with a definite atomic electron, each molecular term must arise from a combination of two definite atomic terms (with the same reservation concerning the one-to-one nature of the correlation). Of course, the two correlations—of the individual electrons and of the complete term—cannot contradict one another, but must mutually complement one another.

For the establishment of molecular terms, the following general rules hold, established chiefly by Wigner and Witmer.

1) Spin \(S\): if one atom has total spin \(S_1\),

and the second total spin \(S_2\), then when these atoms approach, a coupling arises between these spins, as a result of which the resultant total spin of the molecule is obtained, for which all quantized values between \(S_1+S_2\) and \(S_1-S_2\) are possible. Upon further approach the coupling conditions change so that the decomposition of \(S\) into \(S_1\) and \(S_2\) is no longer possible; however, a law holds according to which the total spin \(S\) is conserved, even if this should require a crossing of the potential curves.

2) Orbital angular momentum relative to the molecular axis. Each atom, upon approach of the other atom, forms a quantized component \(\Lambda_1\) or \(\Lambda_2\) of the total orbital angular momentum \(L_1\) or \(L_2\) in the direction of the axis connecting the nuclei. Upon further approach, the quantum numbers \(L_1\) and \(L_2\), owing to the increasing coupling of the individual electrons with the molecular axis, lose their significance; however the quantum number \(\Lambda=\Lambda_1+\Lambda_2\) is conserved.

1. Molecule.  2. Distant nuclei.  3. Separated nuclei.

Fig. 13.

On the basis of both rules one may, for example, predict that two atoms in the state \({}^3P(L_1=L_2=1;\ S_1=S_2=1)\) can give terms \(\Sigma,\Pi,\Delta\) in the system of singlets, triplets, and quintets.

3) Symmetry properties. All symmetry properties in the formation of a molecule must be preserved and therefore may give occasion for crossings of potential curves. The quantum numbers \(\Lambda\) and \(S\), together with the symmetry properties, are called the race of the term. We assume, consequently, that potential curves, insofar as possible, do not cross, and that a crossing is necessary only for the preservation of the race of the term.

Let us consider (Fig. 13) a very simple example, namely the case of two nuclei and two \(1s\)-electrons. With separated nuclei two states \(1s+1s\) may arise

(for example, two neutral H atoms) and \(1s^2+\) nucleus (for example, \(H^-+H^+\)). The first distribution gives a combination of terms \({}^2S+{}^2S\), the second—a combination of terms \({}^1S+{}^1S\). According to rules 1 and 2, the first combination can give molecular terms \({}^1\Sigma\) and \({}^3\Sigma\), while the second gives only one term \({}^1\Sigma\). On the other hand, according to p. 278, from each combination there can arise one even and one odd term. In this way, from column 3 of Fig. 13, column 2 is obtained. According to Fig. 11, from \(1s\)-electrons in the molecule there arise \(1s\sigma\)- or \(2p\sigma\)-electrons. In column 1 are written all the terms of the electronic distributions \(1s\sigma^2\), \(1s\sigma\), \(2p\sigma\), \(2p\sigma^2\) that are possible according to the Pauli principle. The energetic order of the terms is obtained from the general rule according to which, all other conditions being equal, the terms with the greatest multiplicity lie lower. From the conservation laws there are then obtained groupings of terms corresponding to definite lines. In this case, as is seen, no crossings arise at all. If, however, one takes into account that in reality between the states \(H+H\) and \(H^+ + H^-\) in column 3 there also lie all the states \(H+H^*\) (normal + excited atom), which in turn give a considerable number of molecular terms—though for the most part lying above the molecular terms given in column 1—then one can see that in reality the correlation curves (with the exception of the very lowest) must give a considerable number of crossings.

5. System of electronic terms.

The theoretical results—especially the Wigner and Witmer rule—make it possible to predict a very large number of possible molecular terms even for the simplest combinations of atoms. It must, however, unfortunately be stated that, despite the enormous amount of work expended on the study of band spectra, there is still little experimental material that would be suitable for verifying and extending these results. Whereas for the majority of free atoms dozens, and sometimes hundreds, of electronic terms are known, for the majority of investigated molecules only two or three, quite rarely more than ten, terms are known (see Table 1). In turn, only for an insignificant number of these terms is a series of vibrational terms also known, so that it has been possible empirically to construct potential curves and to know with certainty into which atomic terms the corresponding molecular term dissociates. Thus, a very large number of theoretically possible terms corresponds to a considerably smaller number of empirically known ...

are not present. Under such circumstances the assignment of empirical terms to a theoretical scheme is always possible, but it is rarely unambiguous. The reason for this unfavorable situation is that a large number of theoretically possible terms do not correspond to stable molecular states, but to repulsion curves. Transitions to such curves do indeed appear in the spectrum through the occurrence of continuous regions of absorption and emission; however, since such continuous regions have no features by which one could judge the quantum numbers of the corresponding states, their theoretical interpretation has hitherto been possible only in rare cases. Only for H₂ and the haloids is something known regarding repulsion curves. In particular, from Witmer and Stueckelberg it is known that the continuous spectrum of hydrogen corresponds to a transition from a higher-lying stable molecular curve to the repulsion curve \(1s\sigma\,2p\sigma\,{}^{3}\Sigma\), also shown in Fig. 13.

Another reason for the small number of known electronic distances is that each separate electronic jump in a molecule corresponds to an entire system of bands with an innumerable number of individual rotational lines. The energy that can be expended for excitation or radiation is always limited; but in order for the individual lines to be visible throughout the entire system of bands as a whole, a very large amount of energy must be expended. For this reason, only those systems of bands appear in the spectrum which correspond to the most intense (i.e., most probable) electronic jumps. Thus, with respect to molecular spectra, we find ourselves in the same position as we would with respect to atomic spectra if in them we could detect only the strongest lines—for example, only a single resonance line.

The situation is somewhat more favorable in the case of the lightest molecules. Owing to the magnitude of the rotational quanta, each band consists of a small number of lines. Correspondingly, the number of systems of bands in this case is considerably larger. In Figs. 14 and 15 a system of H₂ bands is presented. In Fig. 14 there are given (according to Richardson) only the magnitudes of the electronic terms for the equilibrium positions of the nuclei and, in addition, the principal quantum numbers of the optical electron (the second electron remains throughout in the ground state \(1s\sigma\)) and the multiplicities (on the left, the system of singlets; on the right, the system of triplets). The designations of the terms by capital letters are arbitrary. In Fig. 15 the potential curves and their theoretical interpretation according to Hund are given (for part of the terms of Fig. 14). The plotted terms are dis-

associate into two normal H atoms or into one normal and one excited atom in an \(n\)-quantum state (\(n = 2, 3\), and 4).

As is known, in the case of atoms the terms form series, which correspond to an increase of the principal quantum number while all the other quantum numbers are preserved; moreover, the point of convergence of these terms corresponds to the work of complete removal of the optical electron—the ionization energy. From

Fig. 14. Electronic terms of the \(H_2\) molecule.

Fig. 14 it is seen that in the case of \(H_2\) there are also indications of such series, which already by approximate extrapolation make it possible to determine the ionization potential. Such series are known, however, almost only in the cases of \(H_2\) and \(He_2\). Therefore we still, unfortunately, know very little about the ionization potentials of the majority of molecules (values obtained by the method of electron impacts are relatively few and, moreover, precisely in the case of molecules can easily be excessively high).

How are electronic terms and their quantum numbers determined? From the calculated or observed frequency of “zero-

...of the place” or the zero band in the system of bands one obtains a “pure electronic jump,” i.e. the difference of two electronic terms. If the ionization potential or the position of the ground term is known, then the values of the terms can be referred to one of these values. True, this cannot always be done; for example, sometimes there are band systems of different multiplicity, and it is impossible to indicate the relative height of their terms, since there are no transitions from one system to another or to a common zero level. If the potential curves and the products of dissociation are known, then this value often makes it possible to establish the absolute height of the terms (cf., for example, Fig. 15).

Fig. 15. Potential curves for part of the terms of Fig. 14.

Possible multiplicities are obtained from the number of valence electrons. In the case of light molecules, the multiplet splitting can be established only with the aid of the most refined spectroscopic methods (in this way, for example, the classification of the system of terms of H₂ into singlets and triplets was recently directly confirmed). In the case of molecules of medium mass the multiplicity can be readily detected; finally, in the case of heavy molecules the multiplet components diverge so far that establishing their connection often presents difficulties. The value of the quantum number \(\Lambda\) and the property of symmetry are often most easily established by investigating the rotational structure (see the following Part IV). Likewise, the establishment of the potential curve and of the dissociation products can often serve to derive the possible values of \(\Lambda\) and the symmetry properties.

Literature

Since the appearance of the first part of this survey, the following general accounts devoted to the field under consideration have been published.

1. R. de L. Kronig, Band Spectra and Molecules and Structure, Cambridge University Press, 1930.

2. A. E. Ruark, R. C. Urey, Atoms, Molecules and Quanta, 1930.

3. F. W. Kohlrausch, Ramaneneffekt; der Sammlung Struktur der Materie; Berlin 1930.

4. G. Herzberg, Die Prädissoziation und verwandte Erscheinung, in Bd. X der “Ergebnisse der exakten Naturwissenschaften,” S. 207 bis 284; Berlin 1931.

5. “Molekülstruktur,” Leipziger Vorträge, 1931.

6. W. Weizel*, Bandenspekter., Bd. I, des Ergänzungswerkes zum Handbuch der Experimentalphysi, herausgegeben von M. Wien und C. Joos, Leipzig 1931.

* The book contains a complete bibliographic index in which all newly appearing works in the field of band spectra are taken into account.