THEORY OF OPTICAL ROTATORY POWER¹

E. Condon, Princeton

1. Introduction

Optical rotatory power is the ability of a medium to rotate the plane of polarization of linearly polarized light passing through it. This phenomenon was first discovered by Arago in 1811. Arago found that quartz possesses the property of rotation in the direction of the optical axis, i.e., in the direction along which there is no ordinary double refraction.

The direction of rotation stands in a definite relation to the direction of propagation of light; thus, if light passes through one and the same medium in two mutually opposite directions—as if it were returning through an active medium upon reflection by a mirror—the rotation is equal to zero. A substance is said to possess positive rotatory power if the plane of polarization is turned clockwise from the point of view of an observer into whose eye the light ray falls. The rotation is proportional to the thickness of the layer of the active medium traversed, and the rotatory power is defined as the angle of rotation of the plane of polarization per unit length of path in the medium.

The phenomenon of the optical activity of liquids was discovered by Biot¹. Since here there is no preferential orientation of the molecules, the effect must be ascribed to structural features of the individual molecules. The modern theory, which connects this property of liquids with the structure of individual molecules, is the subject of the present article².

Conventionally, the rotatory power of a medium is expressed in degrees per decimeter. This quantity is denoted by the letter $\varphi$. In the CGS system it is measured in radians per centimeter; obviously, for conversion

¹ Reviews of Modern Physics, October 1937. Translation by M. V. Volkenshtein and N. I. Polyakova.

² Klaburs, “Optical Rotatory Power,” contains a very detailed survey of the experimental and empirical aspects of the question. In our article the basic principles are briefly set forth, and the chief attention is directed to the application of dispersion theory to the explanation of the phenomenon under consideration.

quantity expressed in degrees per decimeter into a quantity expressed in radians per centimeter, it must be multiplied by \(\frac{\pi}{1800}\).

Another measure of rotatory power is also in common use, called the specific rotation. This is the rotatory power divided by the density of the active medium in grams per cubic centimeter. It is denoted by \([\varphi]\). In the experimental literature the two quantities just mentioned are more commonly denoted respectively by \(\alpha\) and \([\alpha]\), but in the present article we abandon this notation in order to reserve the letter \(\alpha\) for the molecular polarizability, since this too is a well-established and generally used notation.

Finally, one more measure of rotatory power is used, called the molecular rotatory power. It is defined as the specific rotatory power \([\varphi]\), expressed in degrees/decimeter per \(g/M^3\), multiplied by one hundredth of the molecular weight, and is denoted by \([M]\). Thus

\[ [M]=\frac{[\varphi]}{100}, \qquad [\varphi]=\frac{\varphi}{\rho} \tag{1} \]

where \(\rho\) is the density, and \(M\) is the molecular weight.

In theoretical formulas one often encounters the combination \(\frac{\varphi M}{\rho}\), where \(\varphi\) is the rotatory power in radians per centimeter. It is convenient to remember the relation

\[ \frac{\varphi M}{\rho}=\frac{\pi}{18}[M]. \tag{2} \]

The principal feature of the propagation of light in an optically active medium, the feature that causes rotation of the plane of polarization, is circular double refraction. It was discovered by Fresnel\(^3\). A substance is called doubly refracting if, in a given direction, the phase velocity of propagation of light waves is different for two different states of polarization. In the case of an optically active substance, the velocity is different for waves polarized circularly to the right and to the left.

Although Fresnel’s work appeared much earlier than the electromagnetic theory of light, it is convenient for us to conduct our discussion at once in terms of the modern theory. In the electromagnetic theory there are two vectors associated with the wave and perpendicular to the direction of propagation of light. These are the electric induction vector \(\mathbf{D}\) and the magnetic induction vector \(\mathbf{B}\). Let the wave propagate in the direction of the unit vector \(\mathbf{k}\) with velocity \(\frac{c}{n}\), where \(n\) is the refractive index. If we introduce unit vectors \(\mathbf{i}\) and \(\mathbf{j}\), mutually orthogonal and orthogonal to the vector \(\mathbf{k}\), in such a way that the vectors \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) form a right-handed coordinate system,

then D and B may be written in the following form:

\[ \mathbf{D}=R\{ \mathbf{D}_0 e^{i\psi}\}; \quad \mathbf{B}=R\{ \mathbf{B}_0 e^{i\psi}\}; \]

\[ \psi=2\pi\nu\left(t-\frac{n}{c}\,\mathbf{k}\cdot\mathbf{r}\right); \tag{3} \]

where \(\mathbf{D}_0\) and \(\mathbf{B}_0\) are constant vectors expressed in terms of \(\mathbf{i}\) and \(\mathbf{j}\), and \(\psi\) is the phase of the wave at time \(t\) at the point \(\mathbf{r}\), it being assumed that \(\nu\) is the frequency of the wave. The symbol \(R\{\}\) means that the real part of the complex expression is to be taken.

For a wave polarized right-circularly, the constant amplitude must have the form of a constant multiplied by \((\mathbf{i}+\mathbf{i}\mathbf{j})\), for example \(D(\mathbf{i}+\mathbf{i}\mathbf{j})\); taking the real part of this expression, we obtain

\[ D(\mathbf{i}\cos\psi-\mathbf{j}\sin\psi). \]

When \(\psi=0\), the vector \(\mathbf{D}\) is parallel to \(\mathbf{i}\); with the passage of time \(\psi\) increases and the vector \(\mathbf{D}\) rotates clockwise from the point of view of an observer looking in the direction \(-\mathbf{k}\), i.e. in such a way that the light falls into his eye. Similarly, a wave polarized left-circularly is represented by a constant multiplied by \((\mathbf{i}-\mathbf{i}\mathbf{j})\).

The formulas obtained express the fact that light polarized circularly may be regarded as the superposition of two linearly polarized waves. The factor \(i\) in the expression \((\mathbf{i}+\mathbf{i}\mathbf{j})\) may be represented in the form \(e^{i\pi/2}\), whence it is clear that the phase of the linearly polarized component along \(\mathbf{j}\) is ahead by a quarter period of the linearly polarized component along \(\mathbf{i}\). Similarly, in a wave with left circular polarization the component along \(\mathbf{j}\) lags by a quarter period behind the component along \(\mathbf{i}\).

Linearly polarized light may also be regarded as the result of the superposition of two circularly polarized waves. Let us consider a wave obtained by superposing a right-circularly polarized wave with phase \(e^{i\delta}\), and a left-circularly polarized wave with phase \(e^{-i\delta}\). The expression for \(\mathbf{D}\) takes the form

\[ \mathbf{D}\simeq R\{(\mathbf{i}+i\mathbf{j})e^{i\delta}+(\mathbf{i}-i\mathbf{j})e^{-i\delta}\}= \]

\[ =2(\mathbf{i}\cos\delta-\mathbf{j}\sin\delta). \tag{4} \]

For \(\delta=0\) this expression represents a linearly polarized wave whose plane of polarization is determined by the vector \(\mathbf{i}\), and for \(\delta>0\) the plane of polarization is rotated clockwise through the angle \(\delta\) about the axis \(\mathbf{k}\).

Let us now suppose that the medium has different refractive indices for waves with right and left circular polarization; let us denote these refractive indices respectively by \(n_r\) and \(n_l\). Let the light enter the medium through the plane \(\mathbf{k}\cdot\mathbf{r}=0\) and leave

it through the plane, \(\mathbf{k}\cdot\mathbf{r}=d\), where \(d\) is the length of the path traversed by it. Suppose further that the incident light is linearly polarized in the direction \(\mathbf{i}\). On the plane through which the light emerges from the medium, the phases of the two components will be

\[ \psi_r=2\pi\nu\left(t-\frac{n_r d}{c}\right), \]

\[ \psi_l=2\pi\nu\left(t-\frac{n_l d}{c}\right), \]

so that we may write

\[ (\psi_r,\psi_l)=\psi\mp\delta, \]

where

\[ \psi=2\pi\nu\left(t-\frac{1}{2}\frac{(n_r+n_l)d}{c}\right) \]

is the phase corresponding to the mean refractive index, and the quantity

\[ \delta=\pi(n_l-n_r)\frac{d}{\lambda};\quad \left(\lambda=\frac{c}{\nu}\right) \tag{5} \]

arises as a consequence of the difference of the two refractive indices.

But, as we have seen, an advance in phase by \(\delta\) of the right component and a lag in phase by \(\delta\) of the left component gives, upon superposition, linearly polarized light whose plane of polarization is rotated through an angle \(\delta\). Therefore the rotation per unit length of path will be \(\dfrac{\delta}{d}\), and the rotatory power \(\varphi\) is expressed directly through the difference of the two refractive indices,

\[ \varphi=\frac{\pi}{\lambda}(n_l-n_r). \tag{6} \]

If in this equation \(\varphi\) is expressed in degrees/decimeter, then \(\lambda\)—the wavelength in vacuum—must be expressed in decimeters, and \(\pi=180\).

Since \(\lambda\) is very small in comparison with macroscopic values of \(d\), the quantity \(\dfrac{d}{\lambda}\) in formula (5) must be much greater than unity. Therefore we have a considerable rotation, despite the fact that the quantity \((n_l-n_r)\) is small in comparison with unity. It should be remembered that the sign of the rotation is determined by that one of the circularly polarized components which moves faster.

We have seen that optical activity is an indirect result of the difference in the velocities of propagation of waves polarized circularly to the right and to the left. Analogously, there exists

effect, called circular dichroism, arising owing to the difference in absorption of these two kinds of waves. This effect was discovered by Cotton^4 in 1896. It is well known that in the case of ordinary refraction the refractive power is closely connected with absorption bands; intense absorption bands give a greater refractive power. In the case of optically active liquids, the difference \(n_l - n_r\) is of the order of several millionths. The usual relation between refraction and absorption leads one to suppose that in the absorption coefficients there can be no difference of a larger order.

In view of the difficulty of measuring intensity, one cannot expect that so small a difference in absorption will be detected by a separate measurement of each of the two kinds of circularly polarized waves.

Instead, one should turn to the consideration of the differential effect that occurs as a consequence of the difference between the two absorption coefficients. For this it is necessary to study the propagation of a linearly polarized wave in an absorbing medium. Let \(\varepsilon_l\) and \(\varepsilon_r\) be the absorption coefficients for waves circularly polarized to the right and to the left; write

\[ (\varepsilon_l,\varepsilon_r)=\varepsilon\pm\varepsilon', \tag{7} \]

where \(\varepsilon\) is the mean absorption coefficient, and \(\varepsilon'\) is half the difference of \(\varepsilon_l\) and \(\varepsilon_r\). Then, since the intensity of light is proportional to \(D^2\), the amplitude \(D\) will decrease according to an exponential law because of the presence of the factors \(e^{-\varepsilon_r d/2}\) and \(e^{-\varepsilon_l d/2}\), respectively, for waves circularly polarized to the right and to the left after passing through a layer of medium of thickness \(d\). Consequently, both components, which at first combine to give a linearly polarized wave, after passing through the medium have unequal amplitudes. As a result they combine again and give elliptically polarized light, whose ellipticity is connected with the difference of the two absorption coefficients.

Each amplitude is diminished by the common factor \(e^{-\varepsilon d/2}\), corresponding to the mean absorption coefficient, but after passage through a thickness \(d\) the amplitude, in addition to this factor, is given by the formula

\[ e^{\varepsilon'd/2}(i+j)e^{i\delta}+e^{-\varepsilon'd/2}(i-j)e^{-i\delta}. \]

Denoting by \(i(\delta)\) and \(j(\delta)\) the unit vectors obtained by rotating the vectors \(i\) and \(j\) through an angle \(\delta\) clockwise, this expression can be written in the form

\[ i(\delta)\cosh\frac{\varepsilon'd}{2}+ij(\delta)\sinh\frac{\varepsilon'd}{2}. \tag{8} \]

For \(e>0\) this formula represents a wave elliptically polarized to the right, whose maximum amplitude is rotated by an angle \(\delta\) from the original direction of linear polarization. The ellipticity is measured by the angle \(\Psi\), whose tangent is equal to the ratio of the minimum amplitude to the maximum, i.e.

\[ \tg \Psi=\tgh \frac{\varepsilon' d}{2}, \tag{9} \]

if we agree to call the ellipticity positive for a wave polarized along a right-handed ellipse, and negative for a wave polarized to the left.

The observed phenomenon, known as circular dichroism, consists in the appearance of elliptical polarization when linearly polarized light is partially absorbed in passing through an active medium. In view of the fact that \(\varepsilon'\) is small in comparison with \(\varepsilon\), and that we must choose \(d\) in such a way that \(\varepsilon d\) is not too large in order to obtain a significant amount of transmitted light, one must always have \(\varepsilon' d \ll 1\), and therefore

\[ \Psi=\frac{1}{2}\varepsilon' d=\frac{1}{4}(\varepsilon_l-\varepsilon_n)d. \tag{10} \]

Thus the ellipticity is proportional to the thickness \(d\). Brouwer showed that the relative error in the measurement of \(\delta\) and \(\Psi\) will be minimal when such a thickness of the substance is chosen that the amplitude is diminished by the factor \(e^{-1}\), i.e. the energy is diminished by the factor \(e^{-2}\), that is, to \(13.5\%\) of its original value.

2. Electromagnetic theory

After the general introduction given in the preceding section, let us turn to the generalization of the usual elementary theory of light in order to give an explanation of circular double refraction.

The starting point is, of course, Maxwell’s equations

\[ \begin{aligned} \operatorname{div}\mathbf D&=0, & \operatorname{div}\mathbf B&=0,\\ \operatorname{rot}\mathbf E&=-\frac{1}{c}\dot{\mathbf B}, & \operatorname{rot}\mathbf H&=\frac{1}{c}\dot{\mathbf D},\\ \mathbf D&=\mathbf E+4\pi\mathbf P, & \mathbf B&=\mathbf H+4\pi\mathbf I . \end{aligned} \tag{11} \]

The properties of the medium are completely expressed in \(\mathbf P\)—the electric moment per unit volume—and in \(\mathbf I\)—the magnetic moment per unit volume. The theory of wave propagation, which underlies all dispersion effects, requires, in addition to equations (11), the establishment of a relation between \(\mathbf P\) and \(\mathbf I\), on the one hand, and \(\mathbf E\) and \(\mathbf H\), on the other. This relation may be obtained by the successive application of electrodynamics to a detailed model of the medium.

The simple theory of isotropic media gives us

\[ \mathbf{P}=\chi\mathbf{E},\qquad \mathbf{I}=\chi'\mathbf{H}, \]

where \(\chi\) and \(\chi'\) are scalars. Then

\[ \begin{aligned} \mathbf{D}&=(1+4\pi\chi)\mathbf{E}=\varepsilon\mathbf{E},\\ \mathbf{B}&=(1+4\pi\chi')\mathbf{H}=\mu\mathbf{H}, \end{aligned} \tag{12} \]

where \(\varepsilon\) and \(\mu\) are, respectively, the dielectric constant and the magnetic permeability. From this simple relation there follows, as is known, the propagation of waves with refractive index determined by the formula \(n^2=\varepsilon\mu\). As a rule, if strongly magnetic bodies are excluded, \(\chi'\simeq 10^{-4}\chi\), so that, generally speaking, one may write \(n^2=\varepsilon\).

Crystal optics, insofar as it deals with ordinary double refraction, is obtained as a generalization in which \(\chi\) and \(\chi'\) are replaced by tensors, although usually \(\chi'\) is neglected in comparison with \(\chi\), and the refraction is considered to be caused only by electric polarization.

In order to construct a theory of optical activity, we need another series of generalizations of the material relations. It turns out that the essential point is that there is a part of \(\mathbf{P}\) proportional to \(\dot{\mathbf{B}}\), and a part of \(\mathbf{I}\) proportional to \(\dot{\mathbf{D}}\). Suppose for a moment that we have a molecular theory leading to relations of the form

\[ \mathbf{D}=\varepsilon\mathbf{E}-g\dot{\mathbf{H}}, \qquad \mathbf{E}=\frac{1}{\varepsilon}\mathbf{D}+\frac{1}{\varepsilon}g\dot{\mathbf{B}}, \]

or

\[ \mathbf{B}=\mathbf{H}+g\dot{\mathbf{E}}, \qquad \mathbf{H}=\mathbf{B}-\frac{1}{\varepsilon}g\dot{\mathbf{D}}, \tag{13} \]

and try to find solutions of Maxwell’s equations using these relations. Each vector may be regarded as a constant amplitude multiplied by \(e^{i\psi}\), where \(\psi\) is the phase, as in formula (3). The equations \(\operatorname{div}\mathbf{D}=0\) and \(\operatorname{div}\mathbf{B}=0\) give \(\mathbf{k}\cdot\mathbf{D}=0\) and \(\mathbf{k}\cdot\mathbf{B}=0\); consequently, \(\mathbf{D}\) and \(\mathbf{B}\) are perpendicular to the direction of propagation of the light.

Next, let us consider the curl equations. They give

\[ n\mathbf{k}\times\mathbf{E}=\mathbf{B}, \]

\[ n\mathbf{k}\times\mathbf{H}=-\mathbf{D}, \]

and the material relations (13) give

\[ \mathbf{E}=\frac{1}{\varepsilon}\mathbf{D}+i\gamma\mathbf{B}, \]

\[ \mathbf{H}=\mathbf{B}-i\gamma\mathbf{D}, \]

where

\[ \gamma=2\pi \nu g\,\frac{1}{\varepsilon}. \]

With the aid of these relations we can eliminate \(\mathbf E\) and \(\mathbf H\) from the equations for the curls and obtain

\[ n\left(\frac{1}{\varepsilon}\,\mathbf k\times \mathbf D+i\gamma\,\mathbf k\times \mathbf B\right)=\mathbf B, \]

\[ n\left(-i\gamma\,\mathbf k\times \mathbf D+\mathbf k\times \mathbf B\right)=-\mathbf D. \]

Expressing these equations in scalar form through the two components \(\mathbf B\) and \(\mathbf D\), we obtain four homogeneous equations for four unknown components. In order that they be compatible, the determinant of their coefficients must be equal to zero. This condition gives us the equation for \(n\), whose roots give the possible values of the refractive index. Writing

\[ \mathbf D=D_1\mathbf i+D_2\mathbf j, \]

\[ \mathbf B=B_1\mathbf i+B_2\mathbf j, \]

we obtain the equations

\[ \begin{array}{c|c} -n\left(\dfrac{1}{\varepsilon}D_2+i\gamma B_2\right)=B_1, & n(i\gamma D_2-B_2)=-D_1, \\[1.2em] n\left(\dfrac{1}{\varepsilon}D_1+i\gamma B_1\right)=B_2, & n(-i\gamma D_1+B_1)=-D_2. \end{array} \]

The admissible values of the refractive index are determined by the expression

\[ n^{-2}=\left(\varepsilon^{-\frac12}\pm\gamma\right)^2. \tag{14} \]

Both negative roots correspond to propagation in the direction \(-\mathbf k\) and are of no interest. The two positive roots give the indices for the propagation of waves in the direction \(+\mathbf k\). As is easy to see, the root with the positive sign of \(\gamma\) corresponds to the solution for \(\mathbf D\) and \(\mathbf B\) giving a wave polarized circularly to the right; the other root gives left circular polarization. Since \(\gamma\) is a quantity small in comparison with unity, it follows that

\[ \left. \begin{aligned} n_r&=\varepsilon^{\frac12}-2\pi\nu g,\\ n_l&=\varepsilon^{\frac12}+2\pi\nu g. \end{aligned} \right\} \tag{15} \]

This result, together with (6), gives the following relation between the rotatory power of the medium and the parameter \(g\) introduced in (13):

\[ \varphi=\left(\frac{2\pi}{\lambda}\right)^2 cg. \tag{16} \]

The next question is what must be the character of the reaction of individual molecules to the field created by light waves for the corresponding terms to appear in the macroscopic field equations of type (13). The effective electric field acting on a molecule is not only the vector \(\mathbf{E}\) from the macroscopic field theory, but also the mean field created by neighboring molecules. Lorentz\(^6\) showed that for a medium in which the molecules are distributed chaotically, we must take the effective field to be equal to

\[ \mathbf{E}'=\mathbf{E}+\frac{4\pi}{3}\mathbf{P}. \tag{17} \]

Analogous considerations are in principle valid for the magnetic field as well, but since the magnetization intensity of the medium is practically negligible in comparison with \(\mathbf{H}\), this may be disregarded.

Let us now suppose that the theory of the reaction of individual molecules to the external field gives the formulas

\[ \left. \begin{aligned} \mathbf{p}&=\alpha \mathbf{E}'-\frac{\beta}{c}\dot{\mathbf{H}},\\ \mathbf{m}&=+\frac{\beta}{c}\dot{\mathbf{E}}', \end{aligned} \right\} \tag{18} \]

where \(\mathbf{p}\) is the induced electric moment, and \(\mathbf{m}\) is the induced magnetic moment. The essential point here is the introduction of the term containing the parameter \(\beta\). Here \(\alpha\) is the usual polarizability term, giving an induced electric moment proportional to the applied electric field.

The total electric and magnetic moments of a unit volume will be

\[ \mathbf{P}=N_1\mathbf{p},\quad \mathbf{I}=N_1\mathbf{m}, \tag{19} \]

where \(N_1\) is the number of molecules per unit volume. In the case where the medium is a simple mixture of different kinds of molecules, there must be different coefficients \(\alpha_i\) and \(\beta_i\) for each kind. Then \(\mathbf{P}\) and \(\mathbf{I}\) are given by the sum of terms—one term for each kind of molecule, each \(\mathbf{P}_i\) and \(\mathbf{m}_i\) being multiplied by the corresponding \(N_i\), the number of molecules of the given kind per unit volume.

Combining (19), (18), and (17), it is easy to see that the assumed relations between \(\mathbf{D}\) and \(\mathbf{B}\), on the one hand, and between \(\mathbf{E}\) and \(\mathbf{H}\),

on the other hand, will be those given in (13). A new result is the relation between the individual molecular parameters \(\alpha\) and \(\beta\) and the molecular parameters \(\varepsilon\) and \(g\); this relation proves to be the following:

\[ \frac{4\pi N_1\alpha}{3}=\frac{\varepsilon-1}{\varepsilon+2}, \tag{20} \]

as is well known from the ordinary theory of dispersion; moreover, we have an analogous expression for the rotation parameter \(g\)

\[ \frac{4\pi N_1\beta}{3c}=\frac{g}{\varepsilon+2}. \tag{21} \]

Applying the known expression \(n^2=\varepsilon\), where \(n\) is the mean refractive index, and combining (21) with (16), we arrive at a formula giving a direct relation between the rotatory power and the molecular parameter \(\beta\)

\[ \varphi=\frac{16\pi^3 N_1\beta}{\lambda^2}\cdot\frac{n^2+2}{3}. \tag{22} \]

This is the principal result of the electromagnetic theory, relating the activity of the medium to the parameter \(\beta\). This parameter must find its explanation in terms of a more detailed molecular theory, as we shall see in the following sections.

From the definition of \(\beta\) by formulas (18) we see that molecules with nonvanishing \(\beta\) possess the property that an increasing electric field induces in them a magnetic moment, while an increasing magnetic field induces an electric moment. We shall try to explain this process visually.

The main effect observed when a molecule is placed in an electric field is the effect measured by the polarizability \(\alpha\); positive charges are displaced in the direction of \(\mathbf{E}\), and negative charges in the opposite direction, the displacement being proportional to the strength of the field. As a result, a dipole moment is induced in the molecule. If the electric field increases, the charges move in such a way that the dipole moment also increases. Let us now suppose that the molecular structure is such that these moving charges cannot pass directly from their initial to their final position, but must move along (to some extent) helical trajectories, so that there is a circular component of the motion around \(\mathbf{E}\), accompanying the general translational motion in the direction of \(\mathbf{E}\).

The currents associated with the circular component of the motion produce a magnetic moment proportional to the magnitude \(E\) and having the same direction as \(\dot{\mathbf{E}}\). Such is the visual description of the mechanism that creates the term containing \(\beta\) in the equation for the induced magnetic moment.

Accordingly, let us suppose that the molecule is in a variable magnetic field. The variable flux through the molecule causes induced currents in the molecule, i.e., induces a flow of charges around the axis \(\dot H\), in a direction determined by Lenz’s rule. The very same conditions which earlier caused the circular motion accompanying the general displacement will now cause the displacement of positive charges in one direction and of negative charges in the other, following the induced circular currents. Thus here we obtain a separation of positive and negative charges as a result of the action of induced currents. Such is the visual description of the origin of the electric moment in a variable magnetic field; the minus sign refers to another effect and is a simple consequence of Lenz’s rule for induced currents.

It should be emphasized here that a visual description can with equal success be constructed differently. In Maxwell’s equations, a variable electric field is invariably connected with the inhomogeneity of the magnetic field. Therefore it makes no sense to ask what “causes” the magnetic moment which we have formally and visually connected with the electric field—whether it is “caused” in reality by a variable electric field or by inhomogeneities of the magnetic field. An analogous remark is, of course, also valid for the electric moment which we have connected with a variable magnetic field.

In writing the present section it seemed expedient to us to place all references to the original works at the end. There is no point in giving a complete history of the question, since this would lead us to the phenomenological old theory of light, constructed on the theory of elasticity, a theory of light which at present has only antiquarian interest. Apparently, the first theory of the same type as that set forth in the present section belongs to Gibbs\(^7\). Among other investigators who tried to apply electromagnetic theory to the problem of optical activity, we shall mention Drude\(^8\), Lorentz\(^9\), and Livens\(^ {10}\).

3. The parameter \(\beta\) and rotatory dispersion

In the following section we shall set forth the quantum-mechanical theory of dispersion in the form which includes the theory of the parameter \(\beta\). But first it is of interest to consider the experimental data in connection with the theory.

It will be shown that \(\beta\) is given by the formula

\[ \beta_a=\frac{c}{3\pi h}\sum_b\frac{R_{ba}}{\nu_{ba}^{\,2}-\nu^2}. \tag{23} \]

Here \(\beta_a\) is the value of \(\beta\) belonging to molecules in the quantum state \(a\), \(\nu_{ba}\) is the frequency of the absorbed light in the transition \(a \to b\)

and, finally, \(R_{ba}\) is a constant characterizing the given individual absorption line. We shall call the quantity \(R_{ba}\) the rotational strength of the line \(\nu_{ba}\).

Let the molecules be distributed among the various states in thermal equilibrium corresponding to the temperature of the medium; the number of molecules in state \(a\) per unit volume is \(N_1(a)\); this number is given by the Boltzmann distribution formula. Then the effective value of \(\beta\), used in formula (22), is

\[ N_1\beta=\sum_a N_1(a)\beta_a . \tag{24} \]

Equation (23) for \(\beta_a\) is analogous to the better-known equation for the polarizability \(\alpha_a\) of molecules in the quantum state \(a\). This formula, given by Kramers and Heisenberg\(^{11}\), has the form

\[ \alpha_a=\frac{2}{3h}\sum_b \frac{\nu_{ba}S_{ba}}{\nu_{ba}^2-\nu^2}, \tag{25} \]

where \(S_{ba}\) is another characteristic of the line \(\nu_{ba}\), called its line strength\(^{12}\).

The line strength should not be confused with the oscillator strength, which is another measure of line intensity, important in the theory of dispersion. The oscillator strength is usually denoted by \(f_{ba}\). It is dimensionless and is defined by the equation

\[ f_{ba}=\frac{8\pi^2\mu}{3e^2h}\nu_{ba}S_{ba}, \tag{26} \]

where \(e\) and \(\mu\) denote the charge and mass of the electron.

Expressed through the oscillator strength, formula (25) for the polarizability takes the form

\[ \alpha_a=\frac{e^2}{4\pi^2\mu}\sum_b \frac{f_{ba}}{\nu_{ba}^2-\nu^2}. \tag{17} \]

The oscillator strength satisfies the important sum rule,

\[ \sum_b f_{ba}=n, \tag{28} \]

which was discovered independently by Thomas and Kuhn\(^{13}\). In (28) the letter \(n\) denotes the total number of electrons in the molecule. Equation (28) is valid for each state \(a\) of the molecule; the sum extends over all states.

Instead of the rotational strength of lines, Kuhn introduced another important quantity for a line, essential for optical rotatory power. He

is called the anisotropy factor of the line. In the notation adopted here, Kuhn’s anisotropy factor \(g_{ba}\) is

\[ g_{ba}=\frac{R_{ba}}{S_{ba}} . \tag{29} \]

As is easy to see, the anisotropy factor is a dimensionless number.

Let us now consider the experimental data in their relation to these equations. When one first begins the study of optical activity, one is usually struck by the enormous quantity of accumulated facts. But when attempting to connect them with theory, it turns out that only very few data are sufficiently complete and can be used. First, many measurements were made only for a single wavelength, usually for the sodium \(D\) lines, which is insufficient for determining the constants \(R_{ba}\) and \(\nu_{ba}\) in the dispersion formula for \(\beta\). This shortcoming was emphasized by Lowry\(^2\), to whose efforts we owe, to a considerable degree, the present tendency of experimenters to obtain more complete data on dispersion. Secondly, the refractive index is usually not measured, so that it is necessary to estimate the factor \(\dfrac{n^2+2}{3}\) entering equation (22).

Usually we may assume that all molecules are in the lowest electronic state, so that no averaging over all initial states, as in (24), is required; in that case \(\beta_a\) refers to the normal electronic state of the molecule. Combining (23) and (22), we obtain

\[ \frac{\dfrac{\varphi M}{\rho}}{\dfrac{1}{3}(n^2+2)} = \frac{16\pi^2 N}{3hc} \sum_b \frac{R_{ba}\nu^2}{\nu_{ba}^{\,2}-\nu^2}, \tag{30} \]

where \(M\) is the molecular weight, \(\rho\) the density, and \(N\) Avogadro’s number. In this equation \(\varphi\) denotes the rotation in radians/centimeter. Using formula (2), we can write (30) as an equation for the molecular rotatory power \([M]\) in the corresponding units of § 1. As a result we obtain

\[ \frac{[M]}{\dfrac{1}{3}(n^2+2)} = \frac{96\pi N}{hc} \sum_b \frac{R_{ba}\nu^2}{\nu_{ba}^{\,2}-\nu^2}. \tag{31} \]

For an ordinary transparent liquid the absorption frequencies \(\nu_{ab}\) lie in the ultraviolet; at the same time, experimental data concerning the magnitude \([M]\) are usually limited to the visible part of the spectrum. Therefore, if in one and the same part of the ultraviolet there are many different absorption bands, this cannot be detected from experimental data relating only to the visible spectrum. Instead, all these frequencies are combined together

and form one effective term, whose effective frequency \(\nu_{ba}\) represents a certain average of the individual \(\nu_{ba}\) actual lines. The effective rotational strength \(R_{ba}\) represents the sum of the individual rotational strengths. This picture is well known from consideration of the dispersion formula for ordinary refraction.

A good example of the application of formula (31) and of the analogous formula just discussed is given by consideration of Günther’s data\(^{15}\) for \(d\)-secondary octyl alcohol. These data have the rare advantage that they include measurements of the refractive index. The data are given in Table 1; they may be represented by the following empirical formulas:

\[ n^2 = 1.6913 + \frac{0.313\,\lambda^2}{\lambda^2 - 0.0283}, \]

\[ [\varphi] = \frac{3.14}{\lambda^2 - 0.0283}, \]

where \(\lambda\) is given in microns.

TABLE 1

Refractive index and specific rotation of \(d\)-secondary octyl alcohol\(^{15}\)

\(\lambda\)	\(n\)	\([\varphi]\)
6438	1.4238	8.12
5896	1.4256	9.86
5461	1.4273	11.65
5086	1.4292	13.58
4800	1.4311	15.46
4678	1.4320	16.42
4358	1.4349	19.49
4251		20.6
3969		24.2
3790		27.3
3650		29.9

According to (31), formulas of this type, as used in practice, must be satisfied not by \([\varphi]\) and \([M]\), but by these quantities divided by

\[ \frac{n^2 + 2}{3}. \]

But in the example given this factor changes from the beginning to the end of the table from 1.34 to 1.36, so that, without making a large error, we may regard it as constant and equal to 1.35. The molecular weight is \(M = 130\), and therefore, applying Günther’s empirical formula, we obtain

\[ \frac{[M]}{\frac{1}{3}(n^2 + 2)} = \frac{107\,\sigma^2}{35.6 - \sigma^2}, \]

where \(\sigma\) is the wave number in inverse microns (\(10^4\ \mathrm{cm}^{-1}\)). Since \(\sigma\) is proportional to \(\nu\) and formula (31) is homogeneous with respect to frequency, it follows that

\[ \frac{96\pi N R_{ba}}{hc} = 107, \]

where \(R_{ba}\) is the effective rotational strength of the group of ultraviolet bands whose effective wave number is \(58\,800\ \mathrm{cm}^{-1}\),

i.e., the effective wavelength is equal to \(1700\ \text{\AA}\). The combination of universal constants entering here is

\[ \frac{96\pi N}{hc}=0.943\cdot 10^{42}, \]

and therefore the effective rotational strength obtained from these data is

\[ R_{ba}=1.13\cdot 10^{-40}. \]

Very few cases are known in which the data on rotatory dispersion are so complete as to allow calculations of this kind to be carried out. And even when this is possible, the results obtained are simply the result of combining all active absorption bands into one effective mean frequency in the ultraviolet.

The quantum-mechanical theory set forth in the following paragraph gives a formula for the rotational strength \(R_{ba}\) of an individual line, expressed in terms of the matrix components of the unperturbed molecule. Thus it is shown that

\[ R_{ba}=\operatorname{Im}\{(a|\mathbf p|b)\cdot(b|\mathbf m|a)\}, \tag{32} \]

where \((a|\mathbf p|b)\) and \((b|\mathbf m|a)\) denote, respectively, the matrix components of the electric and magnetic moments of the molecule connecting the states \(b\) and \(a\). The symbol \(\operatorname{Im}\{\}\) denotes “the imaginary part of” in the sense

\[ \operatorname{Im}\{u+i v\}=v, \]

if \(u\) and \(v\) are real. The electric moment of the molecule is

\[ \mathbf p=\sum e\mathbf r_i, \]

where \(\mathbf r_i\) is the position vector of the \(i\)-th electron. Then one may expect that the nonvanishing matrix components of \(\mathbf p\) will be of the order of the electron charge multiplied by the radius of the first Bohr orbit, \(a_H\), i.e., of the order

\[ ea_H=2.53\cdot 10^{-18}\ \mathrm{CGS}. \]

Similarly, the nonvanishing matrix components of the magnetic moment should be of the order of the Bohr magneton, or approximately

\[ \frac{e\hbar}{2\mu c}=0.92\cdot 10^{-20}\ \mathrm{CGS}. \]

Thus, from an approximate estimate one should expect that the rotational strength \(R_{ba}\) is a quantity of the order

\[ \frac{ea_H\cdot e\hbar}{2\mu c}=2.32\cdot 10^{-38}\ \mathrm{CGS}, \]

The value found for octyl alcohol is approximately one two-hundredth of this roughly estimated value. This discrepancy is quite understandable if one takes into account that our estimate was too high for a number of reasons: a) the scalar product of two matrix components is smaller than the product of their magnitudes, since they are usually not parallel; b) usually the product of two matrix components is not purely imaginary, so that, when we take its imaginary part, some fraction is lost; c) the empirical value is a sum of values referring to different absorption bands in the molecule, some of these values being positive and others negative, so that the sum of several such terms tends to be smaller than the average term.

It is of interest to continue the consideration of the above example and to analyze the data on the refractive index in the same way as we did for rotatory dispersion. Combining (25) and (20), we obtain the formula

\[ \frac{(n^2-1)\dfrac{M}{\rho}}{\dfrac{1}{3}(n^2+2)} = \frac{8\pi N}{3hc} \sum_b \frac{\sigma_{ba}S_{ba}}{\sigma_{ba}^2-\sigma^2}, \tag{33} \]

where \(\sigma_{ba}\) and \(\sigma\) are the wave numbers equivalent to the corresponding frequencies \(\nu_{ab}\) and \(\nu\), and \(\rho\) is the density. The data on refraction for octyl alcohol, already given above, then give

\[ S_{ba}=83.7\cdot 10^{-36}\,CGS \]

for the effective strength of the ultraviolet absorption bands.

In the standard theory of dispersion the formula for \(S_{ba}\), analogous to (32), is

\[ S_{ba}= |(a|\mathbf p|b)|^2 . \tag{34} \]

Thus the expected order of magnitude of the term \(S_{ba}\) will be

\[ (ea_{\mathrm H})^2 = 6.40\cdot 10^{-36}\,CGS. \]

The experimental value is 13.1 times larger than this natural atomic value, which is quite understandable, since none of the reasons given above for explaining the smallness of \(R_{ba}\) is applicable here.

We shall conclude the consideration of our example by calculating the effective \(f_{ba}\) (oscillator strength) and \(g_{ba}\) (anisotropy factor) for the ultraviolet bands joined at \(1700\,\text{\AA}\). From (26) we have, generally speaking,

\[ f_{ba}=4.79\cdot 10^{29}\sigma_{ma}S_{ba}, \]

and, substituting the quantities \(\sigma_{ba}\) and \(S_{ba}\), we obtain

\[ f_{ba}=2.36. \]

This shows that transitions of more than one electron are included in the set of bands that participate in the dispersion according to the empirical formula given by Gunter. The Kuhn anisotropy factor for these bands is easily computed from (29); it turns out to be equal to

\[ g_{ba}=1.35\cdot 10^{-6}. \]

This value is typical for values of \(g_{ba}\) in the case of strong absorption bands, for which \(f_{ba}\) is of the order of unity.

In considering our example, we referred to the fact that the computed value \(R_{ba}\) is an effective value representing the total participation of a whole group of bands. For greater accuracy let us suppose that the group of bands is represented by one term in the formula for rotational dispersion. Then

\[ \frac{R_1}{\nu_1^2-\nu^2} = \sum_b \frac{R_{ba}}{\nu_{ba}^2-\nu^2}, \tag{35} \]

where the equality holds for small values of \(\nu\). In order to obtain agreement of both sides in the first two terms of the power series in the expansion in \(\nu\), we must have

\[ \left. \begin{aligned} \nu_1^{-2} &= \sum \frac{R_{ba}}{\nu_{ba}^4}\div \sum \frac{R_{ba}}{\nu_{ba}^2},\\ R_1 &= \left(\sum \frac{R_{ba}}{\nu_{ba}^2}\right)^2 \div \sum \frac{R_{ba}}{\nu_{ba}^4}. \end{aligned} \right\} \tag{36} \]

Similarly, if we wish to represent the entire group of ultraviolet bands by the usual dispersion formula with one generalized term, then, as is easy to see, the analogues of (35) and (36) will be

\[ \frac{\nu_1 S_1}{\nu_1^2-\nu^2} \approx \sum \frac{\nu_{ba}S_{ba}}{\nu_{ba}^2-\nu^2}, \tag{37} \]

where, in order to obtain agreement in the first two terms in the power series for the frequencies, we must have

\[ \left. \begin{aligned} \nu_1^{-2} &= \sum \frac{S_{ba}}{\nu_{ba}^3}\div \sum \frac{S_{ba}}{\nu_{ba}},\\ S_1^2 &= \left(\sum \frac{S_{ba}}{\nu_{ba}}\right)^3 \div \sum \frac{S_{ba}}{\nu_{ba}^3}. \end{aligned} \right\} \tag{38} \]

Since formulas (36) and (38) are completely different, there is no reason to expect that the effective frequency \(\nu_1\) in the ordinary dispersion formula will be the same as in the formula for rotational disper-

values, even if the individual frequencies \(\nu_{ba}\) are identical. An interesting illustration of this fact is provided by the experimental data obtained by Folkmann\(^{16}\). He found that for limonene the critical wavelength in the formula with one term for refraction is \(998\ \text{Å}\), whereas in the corresponding one-term formula for the rotatory power the same wavelength was equal to \(1878\ \text{Å}\), i.e. was almost twice as large. This occurs because the \(R_{ba}\) are not all of one sign, and thus the participation of different bands in the far ultraviolet leads to mutual cancellation of \(R_{ba}\). At the same time the \(S_{ba}\) are all positive; thus, with respect to refraction, the bands in the far ultraviolet are fully effective.

The case of ethyl tartrate is of interest because here two terms are needed in the formula for rotatory dispersion. The experimental data were obtained by Lowry and Cutter\(^{17}\) and can be represented by the following empirical formula:

\[ [\varphi]=\frac{25.005}{\lambda^2-0.03}-\frac{20.678}{\lambda^2-0.056}, \]

corresponding to absorption bands with opposite rotatory power at \(1730\ \text{Å}\) and \(2360\ \text{Å}\).

As a result of the combined action of these two different groups of absorption bands, the rotatory power of ethyl tartrate passes through zero with a change of sign in the neighborhood of \(4520\ \text{Å}\), although at this point of the spectrum there is no characteristic frequency of the molecule. Data concerning the refractive index are not given, and for want of anything better we simply take the quantity

\[ \frac{n^2+2}{3}, \]

equal to \(1.30\). Carrying out the calculation in exactly the same way as in the preceding case, we obtain the following results:

Ethyl tartrate

\[ \lambda_1=1730\ \text{Å}, \qquad \lambda_2=2360\ \text{Å}, \]

\[ R_1=12.1\cdot 10^{-40}, \qquad R_2=5.42\cdot 10^{-40}. \]

All the critical frequencies entering the empirically found formula for rotatory dispersion lie in the ultraviolet or in the visible part of the spectrum, if the substance absorbs in the visible part. Such frequencies are associated with electronic transitions in the molecule. All these compounds possess infrared absorption spectra corresponding to changes in the state of vibration of the nuclei, and therefore the question arises: does absorption at infrared frequencies actually participate in the rotatory power? A great many different investigations\(^{18}\) have been carried out on this subject; the rotatory power of various substances was measured for wavelengths up to \(2.14\ \mu\). In this, no ...

no irregularity in the rotation that could be associated with infrared absorption. This is not surprising, since we know from the ordinary theory of dispersion that the contribution of infrared bands to refraction is, by about 2000 times in order of magnitude (the order of the mass ratio), smaller than the contribution of electronic bands. This is true even for fundamental vibrations, in which the quantum number changes by unity. It is all the more true for studies that concerned only the infrared region of the harmonics, where the vibrational quantum number changes by 2, which makes the action of these bands exceedingly weak.

4. Quantum mechanics of rotational dispersion

The modern development of the problem of obtaining the parameter \(\beta\) from a molecular model began with the independent discovery by Born, Oseen, and Gray\(^{19}\), in 1915, of the fact that the calculation of \(\beta\) depends mainly on taking into account the finite ratio of the molecular diameter to the wavelength of light. In other words, the fundamental fact is that the phase of the light wave is different for different parts of the molecule.

In doing this they used a molecular model that represented a spatial distribution of bound oscillators. This corresponds to the natural extension of the form of electronic theory that was current at that time. More or less independently of this work, the same point of view was developed by Thomson, de Mallemann, and Boys\(^{20}\). Kuhn\(^{21}\) also contributed substantially to the solution of this problem by a detailed consideration of the simplest special case of a model of bound oscillators possessing optical activity. His work on this subject played a large role in the modern development of the question.

The calculation of \(\beta\) on the basis of quantum mechanics was first carried out by Rosenfeld\(^{22}\). His calculations lead to formulas (23) and (32), discussed in the preceding sections. This work undoubtedly gives an approximation much better than the former model of bound oscillators, so that one may hope that further work will be built on quantum-mechanical theory, and not on further study of the model of bound oscillators.

We shall now proceed to set out the quantum-mechanical theory of \(\beta\), following mainly Rosenfeld’s papers, departing from them only in some details of the calculations.

Readers who are not interested in the details of the quantum-mechanical calculations will gladly skip the remainder of the present section, since everything that will be done is merely a development of equations (23) and (32).

The fields of a light wave propagating in the direction of the unit vector \(\mathbf{k}\) can be obtained from the vector potential \(\mathbf{A}\).

\[ \mathbf{A}=R\left\{\mathbf{A}e^{\frac{i\left(t-\frac{\mathbf{k}\mathbf{r}}{c}\right)E}{\hbar}}\right\}. \tag{39} \]

Here \(E=h\nu\) represents the quantum of energy associated with the wave. The electric and magnetic vectors of the wave are given by the formulas

\[ \left. \begin{aligned} \mathbf{E}&=-\frac{\dot{\mathbf{A}}}{c} =-\frac{E}{\hbar c}R\left\{i\mathbf{A}e^{\dagger}\right\},\\ \mathbf{H}&=\operatorname{rot}\mathbf{A} =-\frac{E}{\hbar c}R\left\{i\mathbf{k}\times\mathbf{A}e^{\dagger}\right\}, \end{aligned} \right\} \tag{40} \]

where the powers \(e\) are the same as in (39).

We may neglect the direct interaction of the atomic nuclei in the molecule with the light wave, owing to their comparatively large mass. Thus, the interaction gives the perturbing term

\[ \mathbf{H}=-\frac{e}{mc}\sum_i\left[\mathbf{p}_i\cdot\mathbf{A}_i+\mathbf{S}_i\cdot(\operatorname{rot}\mathbf{A})_i\right]. \tag{41} \]

Here \(\frac{e}{mc}\) refers to the charge and mass of the electron, and the index \(i\) on \(\mathbf{A}\) and \(\operatorname{rot}\mathbf{A}\) denotes that they are taken at the position of the \(i\)-th electron.

We must determine the influence of the perturbing action of the light wave on the molecule in a particular state characterized by the quantum numbers \(a\). For the perturbed wave function we may write

\[ \Psi=\psi(a)e^{-\frac{iW_a t}{\hbar}}+\psi_1(a), \tag{42} \]

where \(\psi_1(a)\) must be determined from the dynamical equation of quantum mechanics

\[ i\hbar\frac{\partial\Psi}{dt}=(H_0+H)\Psi, \tag{43} \]

where \(H_0\) is the Hamiltonian for the unperturbed molecule. This gives the following equations for \(\psi_1(a)\):

\[ \left(H_0-i\hbar\frac{\partial}{\partial t}\right)\psi_1(a) =-H\psi(a)e^{-\frac{iW_a t}{\hbar}}. \tag{44} \]

This equation is solved in the usual way. The right-hand side is decomposed

is divided into terms containing unperturbed wave functions:

\[ - H\psi(a)e^{-\frac{iW_a t}{\hbar}} = \frac{1}{2}\sum_b \psi(b)\left[ (b|H_+|a)e^{\frac{i(E-W_a)t}{\hbar}} + (b|H_-|a)e^{-\frac{i(E+W_a)t}{\hbar}} \right], \tag{45} \]

where the coefficients are

\[ (b|H_{\pm}|a) = \frac{e}{mc} \left\{ \psi(b)\left(\sum_i \mathbf{p}_i e^{\mp \frac{i\mathbf{k}\cdot\mathbf{r}_i E}{\hbar c}}\right)\psi(a)\cdot \mathbf{A} \mp \frac{iE}{\hbar c}\psi(b)\left(\sum_i \mathbf{S}_i\right)\psi(a)\cdot(\mathbf{k}\times \mathbf{A}) \right\}. \tag{46} \]

Here it is necessary to remember that \(\overline{\mathbf{A}}\) is to be put in place of \(\mathbf{A}\) when the lower sign is taken, and, in addition, we neglect the retardation factor in the small spin term in the second line. In (46) integration of the wave functions over the configuration space of the molecule is also understood.

If we now expand the retardation factor and retain only the first two terms, then after some simple transformations we obtain

\[ (b|H_{\pm}|a) = \frac{i}{\hbar c}W_{ba}(b|\mathbf{p}|a)\cdot\mathbf{A} \pm \]

\[ \pm \frac{EW_{ba}}{2\hbar^2c^2}(b|\mathbf{N}|a)\cdot\mathbf{A} \mp \frac{iE}{\hbar c}(b|\mathbf{m}|a)\cdot(\mathbf{k}\times\mathbf{A}), \tag{47} \]

where, as above, \(\overline{\mathbf{A}}\) should be written instead of \(\mathbf{A}\) when the lower sign is taken. In (47) the following abbreviations have been introduced:

\[ \left. \begin{aligned} W_{ba}&=W_b-W_a; \qquad \mathbf{p}=e\sum_i \mathbf{r}_i,\\ \mathbf{N}&=e\sum_i \mathbf{r}_i\mathbf{r}_i; \qquad \mathbf{m}=\frac{e}{2mc}\sum_i(\mathbf{r}_i\times\mathbf{p}_i+2\mathbf{S}_i). \end{aligned} \right\} \tag{48} \]

Thus \(\mathbf{p}\), \(\mathbf{N}\), and \(\mathbf{m}\) are, respectively, the electronic parts of the electric dipole moment, the electric quadrupole moment, and the magnetic moment of the molecule.

The terms of the electric quadrupole moment do not introduce a new type of propagation of light in a medium. They give only a small correction, of the order of several millionths, to the usual relation between the mean refractive index and the electro-

... electric dipole moment and, thus, in what follows we shall neglect them. Therefore, in subsequent calculations using equation (47), the terms with \(\dot{\mathbf N}\) may simply be discarded.

Following the usual procedure, one then expands \(\psi(a)\) in the unperturbed wave functions and determines the coefficients in the expansion by equating the coefficients of both sides in (44). The final formula for \(\psi_1(a)\) has the form

\[ \psi_1(a) = \frac{1}{2}\sum_b \psi(b) \left[ \frac{(b|H_{+}|a)}{W_{ba}+E}\, e^{\frac{i(E-W_a)t}{\hbar}} + \frac{(b|H_{-}|a)}{W_{ba}-E}\, e^{-\frac{i(E+W_a)t}{\hbar}} \right]. \tag{49} \]

The first-order correction to the diagonal matrix elements of any observable \(F\), pertaining to atoms in the given state \(a\), will then be

\[ 2R\left\{ \bar{\psi}(a)F\psi_1(a) e^{\frac{iW_a t}{\hbar}} \right\} = \]

\[ = R\cdot \sum_b \left\{ \frac{(a|F|b)(b|H_{+}|a)}{W_{ba}+E}\, e^{\frac{iEt}{\hbar}} + \frac{(a|F|b)(b|H_{-}|a)}{W_{ba}-E}\, e^{-\frac{iEt}{\hbar}} \right\}. \tag{50} \]

For us, the particular cases of (50) are important when the values (47) are applied to \((b|H_{\pm}|a)\), and \(F\) is identified with \(\mathbf p\) and \(\mathbf m\). If we denote the induced values of \(\mathbf p\) and \(\mathbf m\) by \(\mathbf p_1\) and \(\mathbf m_1\), then we obtain

\[ \mathbf p_1 = \sum_b R \left\{ \frac{(a|\mathbf p|b)(b|H_{+}|a)}{W_{ba}+E}\, e^{\frac{iEt}{\hbar}} + \frac{(a|\mathbf p|b)(b|H_{-}|a)}{W_{ba}-E}\, e^{-\frac{iEt}{\hbar}} \right\} \tag{51} \]

and an analogous expression for \(\mathbf m_1\). Substituting expressions (47) and neglecting the quadrupole terms, as has already been indicated, we may write

\[ \mathbf p_1 = 2\sum_b R \left\{ \frac{W_{ba}}{W_{ba}^2-E^2} (a|\mathbf p|b)(b|\mathbf p|a)\cdot \mathbf E + \right. \]

\[ \left. + i\, \frac{\frac{\hbar W_{ba}^2}{E^2}}{W_{ba}^2-E^2} (a|\mathbf p|b)(b|\mathbf p|a)\cdot \dot{\mathbf E} + \frac{W_{ba}}{W_{ba}^2-E^2} (a|\mathbf p|b)(b|\mathbf m|a)\cdot \mathbf H + i\, \frac{\hbar}{W_{ba}^2-E^2} (a|\mathbf p|b)(b|\mathbf m|a)\cdot \dot{\mathbf H} \right\}. \tag{52} \]

and the corresponding expression for \(\mathbf{m}_1\) takes the form

\[ \mathbf{m}_1 = 2 \sum_b R \left\{ -\frac{W_{ba}}{W_{ba}^2-E^2}(a|\mathbf{m}|b)(b|\mathbf{p}|a)\mathbf{E} + i\,\frac{\dfrac{W_{ba}}{E^2}}{W_{ba}^2-E^2}\,(a|\mathbf{m}|b)(b|\mathbf{p}|a)\cdot \dot{\mathbf{E}} \right\}. \tag{53} \]

Each of these expressions contains a term including

\[ \frac{\dfrac{W_{ba}}{E^2}}{W_{ba}^2-E^2} = E^{-2}+\frac{1}{W_{ba}-E^2}. \]

The part of \(\mathbf{p}_1\), expressed in terms of \(E^{-2}\), can be written in the form

\[ R\left\{ \frac{2i\hbar}{E^2}\sum_b (a|\dot{\mathbf{p}}|b)(b|\mathbf{p}|a)\cdot \dot{\mathbf{E}} \right\} = \frac{2\hbar}{E^2}R\left\{ i(a|\mathbf{pp}|a)\cdot \dot{\mathbf{E}}\right\} \]

by the law of matrix multiplication. The expression \((a|\mathbf{pp}|a)\) is a diagonal matrix element of a real observable; thus it is real. Consequently, the expression whose real part we have to take is purely imaginary, and its real part is equal to zero.

An analogous reduction can be carried out for the corresponding term in (53). Since the real part of \(iX\) is \(-\operatorname{Im}\{X\}\), where \(\operatorname{Im}\{X\}\) denotes the imaginary part of \(X\), the expressions for the induced moments may finally be written in the form

\[ \left. \begin{aligned} \mathbf{p}_1 &= 2\sum_b \frac{W_{ba}}{W_{ba}^2-E^2} R\left\{(a|\mathbf{p}|b)(b|\mathbf{p}|a)\right\}\cdot \mathbf{E} \\ &\quad -\frac{\hbar}{W_{ba}^2-E^2} \operatorname{Im}\left\{(a|\mathbf{p}|b)(b|\mathbf{p}|a)\right\}\cdot \dot{\mathbf{E}} \\ &\quad +\frac{W_{ba}}{W_{ba}^2-E^2} R\left\{(a|\mathbf{p}|b)(b|\mathbf{m}|a)\right\}\cdot \dot{\mathbf{H}} \\ &\quad -\frac{\hbar}{W_{ba}^2-E^2} \operatorname{Im}\left\{(a|\mathbf{p}|b)(b|\mathbf{m}|a)\right\}\dot{\mathbf{H}}; \\[4pt] \mathbf{m}_1 &= 2\sum_b \frac{W_{ba}}{W_{ba}^2-E^2} R\left\{(a|\mathbf{m}|b)(b|\mathbf{p}|a)\right\}\cdot \mathbf{E} \\ &\quad -\frac{\hbar}{W_{ba}^2-E^2} \operatorname{Im}\left\{(a|\mathbf{m}|b)(b|\mathbf{p}|a)\right\}\cdot \dot{\mathbf{E}} \end{aligned} \right\} \tag{54} \]

These expressions give the coherent induced electric and magnetic moments that are needed for calculating the relation in the medium between \(\mathbf{D}\) and \(\mathbf{B}\), on the one hand, and \(\mathbf{E}\) and \(\mathbf{H}\), on the other. Further, they

can be simplified by averaging over all orientations of the molecules in space. Usually this means that all orientations are equivalent. If they are not equivalent and there exists some preferred direction of the molecules, caused, for example, by an external field, then special effects may arise, such as, for example, the effect discovered by Kuhn and Böckok \(^{23}\),—the influence of an external electric field on the rotatory power of an active liquid.

The averaging over all directions is carried out as follows:

We must average expressions of the type \(\mathbf{p}\mathbf{m}\cdot\mathbf{H}\) over all orientations of \(\mathbf{p}\) and \(\mathbf{m}\), regarding the absolute magnitudes \(\mathbf{p}\) and \(\mathbf{m}\) and the angle between them as fixed; \(\mathbf{p}\mathbf{m}\cdot\mathbf{H}\) is a vector in the direction of \(\mathbf{p}\). When \(\mathbf{p}\) takes all possible directions consistent with a fixed direction \(\mathbf{m}\), the mean value of \(\mathbf{p}\mathbf{m}\cdot\mathbf{H}\) will be its component along \(\mathbf{m}\), i.e. \(\mathbf{p}\cdot\mathbf{m}_0\,\mathbf{m}\cdot\mathbf{H}\,\mathbf{m}_0\), where \(\mathbf{m}_0\) is the unit vector in the direction \(\mathbf{m}\). Next, we average over all directions of \(\mathbf{m}\), considering first of all the directions that make a fixed angle \(\theta\) with \(\mathbf{H}\). As a result we obtain a vector directed along \(\mathbf{H}\), whose magnitude is \(\mathbf{p}\cdot\mathbf{m}\cos^2\theta\,\mathbf{H}\). In conclusion we average over all directions \(\theta\), and the final result is

\[ \mathbf{p}\cdot\mathbf{m}\,(\cos^2\theta)_{\mathrm{av}}\,\mathbf{H} = \frac{1}{3}\,\mathbf{p}\cdot\mathbf{m}\,\mathbf{H}. \]

Thus, after averaging over all orientations of the molecules, the expressions for \(\mathbf{p}_1\) and \(\mathbf{m}_1\) simplify as follows:

\[ \left. \begin{aligned} \mathbf{p}_1 &= \alpha_a \mathbf{E}+\gamma_a \mathbf{H} -\frac{1}{c}\beta_a \dot{\mathbf{H}},\\ \mathbf{m}_1 &= \gamma_a \mathbf{E}+\frac{1}{c}\beta_a \dot{\mathbf{E}}. \end{aligned} \right\} \tag{55} \]

These equations are almost completely similar to equations (18), whose existence was assumed by us in the phenomenological discussion of § 2. Here, however, we have additional results relating \(\alpha\), \(\beta\), and \(\gamma\) to the quantum-mechanical description of the molecular model; they are determined by the following equations:

\[ \alpha_a=\frac{2}{3h}\sum_b \frac{\nu_{ba}\, |(a|\mathbf{p}|b)|^2}{\nu_{ba}^{\,2}-\nu^2}, \tag{56a} \]

\[ \frac{2\pi\nu\beta_a}{c} = \frac{2}{3h}\sum_b \frac{\nu\,\mathrm{Im}\left\{(a|\mathbf{p}|b)\cdot(b|\mathbf{m}|a)\right\}} {\nu_{ba}^{\,2}-\nu^2}, \tag{56β} \]

\[ \gamma_a= \frac{2}{3h}\sum_b \frac{\nu_{ba}\,R\left\{(a|\mathbf{p}|b)\cdot(b|\mathbf{m}|a)\right\}} {\nu_{ba}^{\,2}-\nu^2}. \tag{56γ} \]

Here \(\nu\) denotes the frequency of the light wave, and \(\nu_{ba}=\dfrac{W_{ba}}{h}\). The index \(a\) on the coefficients \(a,\ \beta,\ \gamma\) denotes that they characterize molecules in the state \(a\).

The final result of the calculations is contained in (55) and (56). These equations without the terms containing \(\beta\) and \(\gamma\) correspond to the usual theory of the refractive index in an isotropic medium.

Using the method given in § 2, it is easy to show that the term containing \(\gamma\) gives only a second-order effect for the mean refractive index of light polarized circularly to the right and to the left.¹ Therefore the indicated effect may be neglected,

¹ In accordance with the introduction of \(g\) from (13) and (21), the quantity \(\gamma\) leads to

\[ f=4\pi N\gamma\left(1-\frac{4}{3}\pi Na\right)^{-1}, \]

where \(N\) is the number of molecules per \(\mathrm{cm}^3\), and thus, instead of (11), we have

\[ \mathbf{D}=\varepsilon\mathbf{E}+f\mathbf{H}-g\dot{\mathbf{H}}, \]

\[ \mathbf{B}=\mathbf{H}+f\mathbf{E}+g\dot{\mathbf{E}}. \]

Applying Maxwell’s equations, as in Section 4, and assuming

\[ \mathbf{D}=d_1(1-i\mathbf{m})+d_2(1+i\mathbf{m}), \]

\[ \mathbf{B}=b_1(1-i\mathbf{m})+b_2(1+i\mathbf{m}), \]

we find that the curl equations give

\[ (in^{-1}d_1-b_1+Fd_1)(1-i\mathbf{m})+(-in^{-1}d_2-b_2+\bar Fd_2)(1+i\mathbf{m})=0, \]

\[ (in^{-1}b_1+\varepsilon^{-1}d_1-Fb_1)(1-i\mathbf{m})+(-in^{-1}b_2+\varepsilon^{-1}d_2-\bar Fb_2)(1+i\mathbf{m})=0. \]

We may separately equate to zero the coefficients of \((1-i\mathbf{m})\) and \((1+i\mathbf{m})\) in these equations, in order to find the refractive index for light polarized circularly to the left and to the right. Here

\[ F=\varepsilon^{-1}(f-2\pi i\nu g), \]

and \(\bar F\) is the complex conjugate. The resulting equations give

\[ \varepsilon^{\frac12}n_l^{-1}=\left(1-\frac{f^2}{\varepsilon}\right)^{\frac12}-2\pi\nu g\varepsilon^{-1}, \]

\[ \varepsilon^{\frac12}n_r^{-1}=\left(1-\frac{f^2}{\varepsilon}\right)^{\frac12}+2\pi\nu g\varepsilon^{-1}. \]

This shows that \(f\) enters only into the term containing \(f^2\), and does not affect the difference \((n_l-n_r)\). Therefore we have

\[ n_l=\varepsilon^{\frac12}+2\pi\nu g \quad\text{and}\quad n_r=\varepsilon^{\frac12}-2\pi\nu g, \]

as in (15).

and (55) passes into (18). We see that the calculation of the rotatory power of a given substance includes the calculation of the numerators

\[ R_{ba}=\operatorname{Im}\{(a|\mathbf{p}|b)\cdot(b|\mathbf{m}|a)\}, \tag{32} \]

which enter formula (56) for \(\rho\). This part of the problem is discussed in the following sections.

We see that \(\nu_{ba}\) enters the numerator of the formula for \(\alpha\), just as \(\nu\) enters the numerator of the formula for \(\beta\). The appearance of \(\nu_{ba}\) gives rise to the phenomenon of negative dispersion in substances with a large number of atoms in excited electronic states.\(^{24}\) The question then arises: does an analogous negative rotatory dispersion exist? This question is rather of purely theoretical interest, since it is unlikely that a sufficient concentration of excited optically active molecules could be obtained for an experimental study of the question. The answer must be positive, and it reduces to the theorem

\[ R_{ba}=-R_{ab}. \]

Since \(R_{ba}\) is the strength associated with resonance with a virtual transition from state \(a\) to state \(b\) in molecules that are in state \(a\), it follows that \(R_{ab}\) is the strength associated with resonance with a virtual transition from state \(b\) to state \(a\) for molecules that are in state \(a\). The equation given above is obtained from (32), since upon interchanging \(a\) and \(b\) the matrix components are replaced by their complex conjugates, as a result of which the sign of the imaginary part changes.

5. General properties of the rotatory strength

Equation (31) makes it possible to express completely the rotatory power of a medium in terms of the strengths of rotation of the absorption lines, which in turn are expressed by means of equation (32) through the components of the electric- and magnetic-dipole matrices. We shall now consider some properties of \(R_{ba}\), which can be obtained without specifying the molecular model.

First of all, we have a sum rule analogous to (28),

\[ \sum_b R_{ba}=0, \tag{57} \]

valid for all states \(a\), the sum extending over all states \(b\). This rule was established by Kuhn in connection with the model of coupled oscillators. It is easy to give a general quantum-mechanical proof of this rule, occupying only one line:

\[ \sum_b R_{ba} = \operatorname{Im}\left\{ \sum_b (a|\mathbf{p}|b)(b|\mathbf{m}|a) \right\} = \operatorname{Im}\{(a|\mathbf{pm}|a)\} =0. \]

Equality to zero follows from the fact that any diagonal matrix element of the matrix will be real, and therefore its imaginary part is equal to zero.

On the basis of the sum rule, the optical activity of all substances must disappear in the limits where \(\nu \gg\) all \(\nu_{ba}\), as is easily seen from equations (23) and (31). From (31) it is also obvious that the rotatory power disappears as \(\nu \to 0\), owing to the factor \(\nu^2\) in the numerator. Consequently, the rotatory power is a property that tends to zero at both ends of the spectrum.

Next, we shall consider the symmetry properties of optical activity. This can be done independently of any special theory of the phenomenon, and in fact these considerations, expressed by Pasteur, Van’t Hoff, and Le Bel, underlie modern stereochemistry. The main result is the following: the quantity \(\beta\) is a pseudoscalar, i.e., it changes sign upon passing from a right coordinate system to a left one. This means that two molecules which are mirror images of one another have equal and opposite rotatory powers. This is in complete agreement with equation (32), since the electric dipole moment is a polar vector, whereas the magnetic dipole moment is an axial vector, and thus their scalar product is a pseudoscalar and not a true invariant1.

A point \(P\) with coordinates \(x, y, z\) in the right system will have coordinates \(-x, -y, -z\) in the left system. Coordinates of a point change sign under a change of system: a vector whose components possess this property is called a polar vector.

For the vector product we had, in the right system,
\[ \mathbf{i}\times\mathbf{j}=\mathbf{k}, \ldots, \]
whereas in the left system, applying the indicated transformations, we obtain
\[ \mathbf{i}'\times\mathbf{j}'=-\mathbf{k}'. \]
It is therefore obvious that the vector product of two polar vectors has components that do not change sign upon passing from the right system to the left. Such a vector is called axial.

It is easy to see that the differential operator
\[ \nabla=\mathbf{i}\frac{\partial}{\partial x}+\mathbf{j}\frac{\partial}{\partial y}+\mathbf{k}\frac{\partial}{\partial z} =\mathbf{i}'\frac{\partial}{\partial x_1}+\cdots+\cdots \]
is a polar vector, and that the curl of the field of a polar vector gives the field of an axial vector, whereas the curl of the field of an axial vector gives the field of a polar vector.

We see that two molecules which are mirror images of one another will have equal and opposite rotatory powers, but at the same time we have not the slightest indication as to which of the structural features of the molecule causes its activity. Any pseudoscalar associated with the structure of the molecule can cause its activity, and a symmetry consideration is incapable of leading to the selection of one of them. Attempts to relate optical activity to the structure of molecules without an in-depth theory have reached the indicated limit, but have not gone beyond it.

Thus, Crum Brown \(^{25}\) proposed a formula for a molecule in which one asymmetric carbon atom is bonded to four different atoms or radicals \(A, B, C, D\). Let \(k\) be some scalar attribute of each of the four radicals; then the quantity

\[ K=(k_A-k_B)(k_A-k_C)(k_A-k_D)\times \]

\[ \times (k_B-k_C)(k_B-k_D)(k_C-k_D) \]

is, evidently, a pseudoscalar, since we can pass to the mirror image by interchanging any two groups, and this changes the sign of \(K\). Of course, \(K\) vanishes if any two \(k\)'s are equal to one another. Crum Brown identified the attribute \(k\) with the mass of the attached group, and he was partly successful in relating this to experimental data, but at present this representation has been discredited. For example, Walden showed that propyl-isopropyl-cyano-acetic acid

\[ \begin{array}{cc} n\!-\!\mathrm{C}_3\mathrm{H}_7 & \mathrm{CN}\\ & \diagdown \quad \diagup\\ & \mathrm{C}\\ & \diagup \quad \diagdown\\ iso\!-\!\mathrm{C}_3\mathrm{H}_7 & \mathrm{COOH} \end{array} \]

is optically active, despite the equal masses of the propyl and iso-

Consequently, if the equations of the rotor of the electromagnetic field are to be invariant under the transition from a right-handed system to a left-handed one, then \(B\) and \(E\) must have opposite character, and, on the other hand, \(D\) and \(H\) must also have opposite character. It has been agreed to assume that \(D\) and \(E\) are polar vectors, and \(B\) and \(H\) axial ones, since the theory requires only that both pairs have opposite character. The choice is determined by the assumption that charge density is a true scalar, and density is a polar vector. If we have a simple relation in which a vector of one type is equated to a scalar multiplied by a vector of the other type, then it is clear that this scalar cannot be a true scalar, but must change sign under the transition from a left-handed coordinate system to a right-handed one. Such a scalar is called a pseudoscalar. Since \(D\) and \(H\) are vectors of opposite type, it follows from this that \(g\) in (18) must be a pseudoscalar, and consequently \(\beta\) in (18) is also a pseudoscalar.

groups. Later other researchers\(^{26}\) tried to apply other group attributes in formulas of this type, but without great success.

6. The coupled-oscillator model

We now turn to the problem of calculating rotational strengths for definite molecular models. They may be of two kinds: the coupled-oscillator model of Born, Oseen\(^{19}\), and Kuhn\(^{21}\), and the ordinary oscillator model, recently proposed by Condon, Altar, and Eyring\(^{27}\). The present section will be devoted to a consideration of the main results obtained with the coupled-oscillator model, while we shall examine the one-oscillator model in the following section.

Fig. 1.              Fig. 2.

At first it is convenient to become acquainted with an exceptionally simple variety of the coupled-oscillator model, which was proposed by Kuhn, since it displays all the basic features of models of this type\(^{28}\). Let us have two particles, whose coordinates are \(x_1, 0_1, -\dfrac{d}{2}\) and \(0, y_2, +\dfrac{d}{2}\), where \(d\) is a constant. Their arrangement in space is shown in Fig. 1. Let the charges and masses be, respectively, \(e_1, e_2\) and \(m_1, m_2\), and let us suppose that each particle is elastically bound to its equilibrium position. If we specify a quadratic interaction term, then the potential energy of the system is expressed by the formula

\[ U=\frac{1}{2}k_1x_1^2+k_{12}x_1y_1+\frac{1}{2}k_2y_2^2, \tag{58} \]

and the kinetic energy by the formula

\[ T=\frac{1}{2}m_1\dot{x}_1^{\,2}+\frac{1}{2}m_1\dot{y}_2^{\,2}. \tag{59} \]

The motion is expressed through the normal coordinates \(\xi_1\) and \(\xi_2\) as follows:

as follows:

\[ \left. \begin{aligned} x_1 (m_1)^{\frac12} &= \xi_1 \cos \alpha + \xi_2 \sin \alpha,\\ y_2 (m_2)^{\frac12} &= -\xi_1 \sin \alpha + \xi_2 \cos \alpha, \end{aligned} \right\} \tag{60} \]

where the parameter \(\alpha\) must be chosen in such a way that the potential energy (58) is transformed into a sum of squares of \(\xi_1\) and \(\xi_2\). Then, as is easily seen, the value of \(\alpha\) is given by the formula

\[ \left(\frac{k_1}{m_1}-\frac{k_2}{m_2}\right)\sin 2\alpha + \frac{k_{12}}{(m_1m_2)^{\frac12}}\cos 2\alpha =0. \tag{61} \]

In this case the expression for the kinetic and potential energy is as follows:

\[ T=\frac12(\dot{\xi}_1^{\,2}+\dot{\xi}_2^{\,2}),\qquad U=2\pi^2(\nu_1^2\xi_1^2+\nu_2^2\xi_2^2), \]

where

\[ \left. \begin{aligned} (2\pi\nu_1)^2 &= \frac{k_1}{m_1}\cos^2\alpha - 2\,\frac{k_{12}}{(m_1m_2)^{\frac12}}\sin\alpha\cos\alpha + \frac{k_2}{m_2}\sin^2\alpha,\\[6pt] (2\pi\nu_2)^2 &= \frac{k_1}{m_1}\sin^2\alpha + 2\,\frac{k_{12}}{(m_1m_2)^{\frac12}}\sin\alpha\cos\alpha + \frac{k_2}{m_2}\cos^2\alpha. \end{aligned} \right\} \tag{62} \]

Consequently, the general motion will be a superposition of a simple harmonic motion \(\xi_1\) with frequency \(\nu_1\) and a simple harmonic motion \(\xi_2\) with frequency \(\nu_2\). It is interesting to consider some qualitative properties of the motion. Suppose that

\[ \frac{k_{12}}{(m_1m_2)^{\frac12}} \ll \left(\frac{k_1}{m_1}-\frac{k_2}{m_2}\right), \]

then (61) shows that \(\alpha\) is a small angle. Consequently, in (60) the motion \(\xi_1\) will mainly be the motion \(x_1\) with a much smaller amplitude for \(y_2\) (assuming that the masses are quantities of the same order). Similarly, in the case of \(\xi_2\) the motion \(y_2\) is large in comparison with the motion \(x_1\). Qualitatively, both types of motion have the character of the oscillations shown in Fig. 2. In the diagrams the arrows show the opposite character of the screw motion for both types \(\xi_1\) and \(\xi_2\). Here the screw motion is determined by the direction of rotation about the \(z\)-axis and is associated with a displacement in the \(+z\) direction, necessary in order that the direction

the displacement of the 1st particle coincided with the direction of displacement of the 2nd particle.

The next step in the classical treatment of the model will be the determination of the forced oscillations caused by waves polarized circularly, respectively to the right and to the left. These forced oscillations cause coherent scattering, which determines the refractive index. This has been set out in detail by Kuhn and Freudenberg \(^{28}\), and therefore we shall not repeat it. Instead, we shall derive the formula for the rotatory power of this model, applying quantum-mechanical theory to it. This will give us an internal variant of its interpretation and at the same time proof that there is no difference between the classical and the quantum-mechanical treatment of this model.

We need an expression for the variable part of the electric dipole moment. As is easy to see, it has the form

\[ \mathbf{p}=e_1 x_1 \mathbf{i}+e_2 y_2 \mathbf{j} = \left( \frac{e_1}{m_1^{1/2}}\cos\alpha\,\mathbf{i} - \frac{e_2}{m_2^{1/2}}\sin\alpha\,\mathbf{j} \right)\xi_1 + \left( \frac{e_1}{m_1^{1/2}}\sin\alpha\,\mathbf{i} + \frac{e_2}{m_2^{1/2}}\cos\alpha\,\mathbf{j} \right)\xi_2 . \tag{63} \]

Similarly, the magnetic moment of the orbital motion of two charges is given by the formula

\[ \mathbf{m}=\frac{1}{2c}\left[e_1\mathbf{r}_1\times\mathbf{v}_1+e_2\mathbf{r}_2\times\mathbf{v}_2\right], \]

where \(\mathbf{r}_1\) and \(\mathbf{r}_2\) are radius vectors, and \(\mathbf{v}_1\) and \(\mathbf{v}_2\) are the velocities of the particles. When we apply the relations

\[ \mathbf{r}_1=-\frac{d}{2}\mathbf{k}+x_1\mathbf{i},\qquad \mathbf{r}_2=+\frac{d}{2}\mathbf{k}+y_2\mathbf{j}, \]

\[ \mathbf{v}_1=\dot{x}_1\mathbf{i},\qquad \mathbf{v}_2=\dot{y}_2\mathbf{j}, \]

then one may write

\[ \mathbf{m} = -\frac{d}{2c}\left(e_1\dot{x}_1\mathbf{j}+e_2\dot{y}_2\mathbf{i}\right) = -\frac{d}{2c} \left( \frac{e_1}{m_1^{1/2}}\cos\alpha\,\mathbf{j} - \frac{e_2}{m_2^{1/2}}\sin\alpha\,\mathbf{i} \right)\dot{\xi}_1 - \frac{d}{2c} \left( \frac{e_1}{m_1^{1/2}}\sin\alpha\,\mathbf{j} + \frac{e_2}{m_2^{1/2}}\cos\alpha\,\mathbf{i} \right)\dot{\xi}_2 . \tag{64} \]

Now we must compute the matrix components \(p\) and \(m\), in order to obtain the rotational strength given by formula (32). The energy levels are characterized by two quantum numbers of harmonic oscillators \(n_1\) and \(n_2\), corresponding to the normal coordinates \(\xi_1\) and \(\xi_2\). The energy is given by the formula

\[ W(n_1,n_2)=\left(n_1+\frac{1}{2}\right)h\nu_1+\left(n_2+\frac{1}{2}\right)h\nu_2, \tag{65} \]

and the corresponding wave functions will be

\[ \psi(n_1,n_2)=\varphi_{n_1}\left(\frac{\xi_1}{a_1}\right)\varphi_{n_2}\left(\frac{\xi_2}{a_2}\right), \tag{66} \]

where

\[ \varphi_n(z)=\frac{1}{\left[2^n n!(\pi)\right]^{\frac{1}{2}}}\,H_n(z)e^{-\frac{z^2}{2}}; \tag{67} \]

here \(H_n(z)\) denotes the \(n\)-th Hermite polynomial, and

\[ a_i=\frac{1}{2\pi}\left(\frac{h}{\nu_i}\right)^{\frac{1}{2}}\quad (i=1,2). \tag{68} \]

The matrix components needed for \(\xi_1\) and \(\xi_2\), \(\dot{\xi}_1\) and \(\dot{\xi}_2\), appear in formulas (63) and (64). They are well known from the quantum mechanics of the harmonic oscillator:

\[ (n_1 n_2|\xi_1|n_1' n_2')= a_1\left(\frac{\bar n_1}{2}\right)^{\frac{1}{2}}\delta(n_2,n_2'). \tag{69} \]

Here \(n_1'=n_1\pm1\), and \(\bar n_1\) denotes the larger of the numbers \(n_1\) and \(n_1'\). The effective mass of the oscillator is equal to unity, and therefore \(\dot{\xi}_1\) is the same as the momentum \(P_{\xi_1}\); thus the matrix components \(\dot{\xi}_1\) will be

\[ (n_1'n_2'|\dot{\xi}_1|n_1n_2)=\mp i\,\frac{h}{a_1}\left(\frac{\bar n_1}{2}\right)^{\frac{1}{2}}\delta(n_2,n_2'), \tag{70} \]

where \(\bar n_1\) has the same value as before, and the sign \(\mp\) has the same meaning as in \(n_1'=n_1\pm1\). The formulas for the matrix components \(\xi_2\) and \(\dot{\xi}_2\) are obtained from (69) and (70) by simple transformations.

When we compute the parameter \(\beta\) for molecules in the state \((n_1,n_2)\), we obtain four other states with nonvanishing values of the rotational strength, namely:

\[ (n_1+1,n_2)\quad \text{and}\quad (n_1-1,n_2), \]

also

\[ (n_1,n_2+1)\quad \text{and}\quad (n_1,n_2-1). \]

The pair in the first row has the same resonance denominator as in formula (23). This is also true for the pair in the second row; thus, although here we have four quantum transitions, in the dispersion formula we obtain only two critical frequencies.

If we write \(R_1\) and \(R_2\) for the two rotational strengths, then it is easy to calculate, by combining (63), (64), (69), and (70), that

\[ R_1=-R_2=\frac{\hbar}{d}\, \frac{e_1 e_2}{4c\,(m_1m_2)^{1/2}}\, \sin\alpha\cos\alpha . \tag{71} \]

This result is in agreement with the sum rule of § 5, since it gives \(R_1+R_2=0\). It should be noted that the upper virtual transition gives for \(R_1\) a value proportional to \((n_1+1)\), whereas the lower virtual transition gives a value proportional to \((-n_1)\). Thus, as a result of both jumps, we obtain a quantity independent of the quantum number \(n_1\).

As we remember, this is completely analogous to the quantum-mechanical theory of the ordinary dispersion of a harmonic oscillator. There the positive dispersion caused by the upper virtual transition increases together with the quantum number, and the same is true for the negative dispersion caused by the virtual transition downward; moreover, analogously to the preceding case, the difference does not depend on the quantum number.

If we substitute these values into (23) and use the value of \(\beta\) from (22), then finally, for the rotatory power of a medium containing \(N_1\) such models per unit volume, oriented chaotically, we obtain

\[ \varphi=\frac{2\pi N_1}{3}\lambda^{-2}\cdot\frac{1}{3}(n^2+2)\cdot d\sin\alpha\cos\alpha\times \]

\[ \times \frac{e_1e_2}{(m_1m_2)^{1/2}} \left[ \frac{1}{\nu_1^{2}-\nu^{2}} - \frac{1}{\nu_2^{2}-\nu^{2}} \right]. \tag{72} \]

This agrees perfectly with the results obtained by Kuhn and Freudenberg,\(^{29}\) except for the factor \(1/3\) in (72), which appears because (72) is the final formula, applicable for a chaotic orientation of the molecules, whereas equations (46) and (46a) of Kuhn and Freudenberg must still be averaged (cf. reference \(^{28}\) on pp. 72–76).

It is interesting to note how (72) expresses the basic properties of the model: 1) the oscillators must be coupled, since if \(k_{12}=0\), then according to (61) \(2\alpha=0\), and the rotation disappears; and 2) the oscillators must be separated by a distance \(d\), which is evident from the proportionality of \(\varphi\) to \(d\), explicitly appearing in (72).

Since the preceding calculations illustrate all the basic properties of the model of coupled oscillators, we consider it unnecessary to give detailed calculations for the general case. They may be found in the original papers of Born and Oseen[^19], as well as in the already cited article by Kuhn and Freudenberg[^28] (pp. 69–72).

The results of these calculations may be formulated as follows. Let the model consist of \(s\) particles; let the charge and mass of the \(k\)-th particle be denoted by \(e_k\) and \(m_k\), and let \(x_k, y_k,\) and \(z_k\) be the coordinates of the equilibrium position of the \(k\)-th particle relative to the coordinate system fixed in the molecule.

We assume that the potential energy is a general quadratic form with respect to the displacements of the particles from the equilibrium position, and denote the displacements of the \(k\)-th particle by \(u_k, v_k,\) and \(w_k\). This means that, generally speaking, there exist \(3s\) different natural frequencies and \(3s\) normal vibrations, described by \(3s\) normal coordinates \((a = 1, 2, \ldots, 3s)\). Analogously to (60), we may express the relation between the displacements and the normal coordinates by the formula

\[ \xi_a=\sum_{k=1}^{s}(m_k)^{\frac12} \left(\alpha_{ka}u_k+\beta_{ka}v_k+\gamma_{ka}w_k\right), \tag{73} \]

where \(a = 1, 2, \ldots, 3s\). Here the coefficients \(\alpha_{ka}, \beta_{ka}, \gamma_{ka}\) determine an orthogonal transformation which, as usual, is determined by the requirement that the potential energy be expressed as a sum of squares of the coordinates \(\xi_a\). Therefore the coefficients depend on the coupling term in the expression for the potential energy.

The final result of the calculations shows that the rotatory power is given by the formula

\[ \varphi=\frac{2\pi N_1}{3}\lambda^{-2}\cdot \frac13\left(n^2+2\right) \sum_{a=1}^{3s} \frac{\mathbf{L}_a\mathbf{M}_a}{\nu_a^2-\nu^2}. \tag{74} \]

Here \(\nu_a\) is the frequency associated with the normal coordinate \(\xi_a\), and the vectors \(\mathbf{L}_a\) and \(\mathbf{M}_a\) are determined by

\[ \left. \begin{aligned} \mathbf{L}_a &= \sum_{k=1}^{s} \frac{e_k}{(m_k)^{\frac12}} \left(\alpha_{ka}\mathbf{i}+\beta_{ka}\mathbf{j}+\gamma_{ka}\mathbf{k}\right), \\[4pt] \mathbf{M}_a &= \sum_{k=1}^{s} \frac{e_k}{(m_k)^{\frac12}} \left(\alpha_{ka}\mathbf{i}+\beta_{ka}\mathbf{j}+\gamma_{ka}\mathbf{k}\right) \times \left(x_k\mathbf{i}+y_k\mathbf{j}+z_k\mathbf{k}\right). \end{aligned} \right\} \tag{75} \]

It is easy to see that the vector \(\mathbf{L}_a\) is connected with the electric dipole moment corresponding to the vibrational state \(\xi_a\), and that \(\mathbf{M}_a\) is connected with the corresponding magnetic dipole moment. Consequently, this classical formula is in the same corre-

in agreement with the formula of quantum mechanics, as in the case which we encountered in our detailed calculations for Kuhn’s model.

7. Model with One Oscillator

Drude^30 proposed for optical activity a model in which one electron must move along a spiral while at the same time being bound to its equilibrium position on a curve. This model was accepted for many years, until Kuhn^31 in 1933 found an error in the calculations. Kuhn showed that, when treated correctly, this model does not exhibit rotatory power. This result was exceptionally interesting in that it presented a case in which the rotatory power vanished although the required kind of dissymmetry was present. On the basis of this result Kuhn was inclined to conclude that the coupled oscillators discussed in the preceding section determine the optical activity of a substance. The assertion that rotatory power requires the existence of coupled oscillators for its explanation has appeared many times in the literature of recent years.

At present this view must be abandoned, since the work of Condon, Altar, and Eyring^27 showed that it is possible to construct a model of a single charged particle moving in a dissymmetric field and possessing rotatory power. In the present article we give a brief account of this work; for a more detailed study of the question one should consult the original.

A large part of our modern ideas about the electronic structure of the molecule is expressed by means of the Hartree approximation. Each electron is considered, in the first approximation, as moving in a static potential field caused by the averaged distribution of the charges of the nuclei and of the remaining electrons of the molecule. As has been shown, this is the correct path for calculating the greater part of the interaction between electrons. The dynamic coupling of electrons appears only in subsequent approximations. The model of coupled oscillators of Born, Oseen, Kuhn, and others takes this dynamic coupling of electrons into account. But here the question arises: if we follow Hartree, will optical activity appear at the outset, when each electron moves in the mean static asymmetric field of the rest of the molecule, or does it vanish in this approximation and appear for the first time only when the dynamic coupling of electrons is considered?

Kuhn’s elimination of Drude’s spiral model was intended to show that the dynamic coupling of more than one electron was a necessary condition, until the single-oscillator model was developed.

The model with one oscillator assumes that the electron moves—

is in a dissymmetric potential field, in which the potential energy as a function of the Cartesian coordinates \(x_1, x_2, x_3\) is given by the formula

\[ V=\frac{1}{2}k_1x_1^2+\frac{1}{2}k_2x_2^2+\frac{1}{2}k_3x_3^2+Ax_1x_2x_3 . \tag{76} \]

The term containing \(A\) gives the necessary dissymmetry. It is easy to see that the potential surface (76) qualitatively represents the same surface as we obtain if we take an ellipsoid with three unequal axes and apply to it a twisting force. The equipotential surface has elliptical cross-sections if it is cut by one of the planes parallel to a principal plane of the coordinate system. For definiteness, let us consider the section of the surface \(V=\mathrm{const}\) by the plane \(x_3=\mathrm{const}\). The section by the plane \(x_3=0\) is an ellipse whose principal axes are the axes \(x_1\) and \(x_2\). For positive \(x_3\) the section is an ellipse whose principal axes are rotated in a right-handed screw system, if \(k_1>k_2\) and \(A>0\). The same direction of the screw occurs for \(x_3<0\). Consequently, under the indicated conditions, a right-handed motion is associated with \(x_3\).

The character of the screw associated with the other coordinate axes is determined in the same way. In each case there is a definite kind of screw associated with each axis, two of them being the same and the third different. Thus, in the special case

\[ k_1>k_2>k_3 \quad \text{and} \quad A>0 \]

it turns out that for the first and third axes the screw is right-handed, and for the second it is left-handed.

In the present calculations the term containing \(Ax_1x_2x_3\) is determined by perturbation theory based on the solution of the problem of the anisotropic oscillator represented by the quadratic terms in the potential. Thus the stationary states are characterized by the quantum numbers of harmonic oscillators \((n_1 n_2 n_3)\), corresponding to the energy levels

\[ W(n_1n_2n_3)=\left(n_1+\frac{1}{2}\right)h\nu_1+ \left(n_2+\frac{1}{2}\right)h\nu_2+ \left(n_3+\frac{1}{2}\right)h\nu_3, \tag{77} \]

which are not affected by the energy of the perturbations, at least in first approximation. Here \(\nu_1,\nu_2,\nu_3\) denote the frequencies determined by the constants \(k_1,k_2,k_3\) in the usual way; \(\nu_i=\frac{1}{2\pi}\left(\frac{k_i}{\mu}\right)^{1/2}\). With the aid of standard perturbation theory one can find the wave functions of the first order, and from them the matrix components of the electric and magnetic dipole moments in the first-order approximation.

If we take into consideration that the molecules represented by

model, are in the lowest quantum state \((0\ 0\ 0)\), then in the zero approximation only three upper states can be reached by ordinary electric dipole absorption of light, namely \((1\ 0\ 0)\), \((0\ 1\ 0)\), and \((0\ 0\ 1)\). Similarly, in the zero approximation the magnetic dipole moment has nonvanishing matrix components connecting \((0\ 0\ 0)\) only with \((0\ 1\ 1)\), \((1\ 0\ 1)\), and \((1\ 1\ 0)\). Since these selection rules are mutually exclusive, it follows that there is no transition from the normal state that would have nonvanishing matrix components simultaneously both for \(\mathbf p\) and for \(\mathbf m\). Consequently, the rotational strength vanishes.

However, in the first approximation we can obtain a nonvanishing rotational strength associated with transitions from the normal state to all six excited states just listed. It is useful to consider in detail the pair \((0\ 0\ 0)\to(1\ 0\ 0)\) and \((0\ 0\ 0)\to(0\ 1\ 1)\), since the other two pairs behave in a completely analogous way, and the results for them can be obtained by a cyclic permutation of the indices. As regards \((0\ 0\ 0)\to(1\ 0\ 0)\), here we have a zero-order component of the matrix \(\mathbf p\) and a first-order component of the matrix \(\mathbf m\) under a dissymmetric perturbation. As a result, the indicated line is strong in ordinary absorption and takes part in the rotational strength. On the other hand, \((0\ 0\ 0)\to(0\ 1\ 1)\) has a vanishing zero-order component of \(\mathbf p\), so that its ordinary absorptive power is weak, since the component \(\mathbf p\) arises exclusively owing to the dissymmetric perturbation. Conversely, the rotational strength of this transition is comparable with the strength of the other absorption line; it is equal to it and opposite to it.

This important qualitative difference between the two lines exactly corresponds to the empirical generalization made by Kuhn.^32 He emphasizes that empirically strong bands \((f\simeq 1)\) have very small anisotropy factors \((g\simeq 10^{-5})\), whereas weak absorption bands \((f\simeq 10^{-3})\) possess much larger anisotropy factors \((g\simeq 10^{-2})\), so that, in order of magnitude, the rotational strengths are approximately the same \((fg\simeq 10^{-5})\).

This result receives a very simple interpretation in the one-oscillator model, independently of the special form, assumed in (76), for the effective force field. The dissymmetry of the effective field in which the individual electron moves is caused by the action of atoms other than the one or two atoms to which the given electron belongs. Thus this dissymmetry is very weak, since the other atoms are far away and because the dissymmetry is a residual effect of higher order, caused by the combined action of many neighbors. Consequently, the electron will tend to obey the selection rules that are strictly satisfied in the absence of dissymmetry. In this case the selection rules are mutually exclusive: the component of the matrix \(\mathbf p\) vanishes if \(\mathbf m\) does not vanish, and conversely.

If the selection rules are removed by a dissymmetric field, then it will produce two classes of active bands: those which arise from a large factor \(p\) and a small \(m\), and which will have a large value of \(f\), and those which arise from a large \(m\) in combination with a small \(p\), and have a small value of \(f\).

Continuing the consideration of the model of a separate unit oscillator which makes use of field (76), we find the magnitude \(\beta\) for a particle with charge \(e\) and mass \(\mu\), if it is in the normal state.

\[ \begin{aligned} \beta_{000}={}& \frac{A\hbar e^2}{12(2\pi)^5\mu^3} \Bigg\{ \left(\frac{1}{\nu_2}-\frac{1}{\nu_3}\right) \frac{1}{(\nu_2+\nu_3)^2-\nu_1^2} \left[ \frac{1}{\nu_2^2-\nu^2} + \frac{1}{(\nu_2-\nu_3)^2-\nu^2} \right] \\ &+ \left(\frac{1}{\nu_3}-\frac{1}{\nu_1}\right) \frac{1}{(\nu_3+\nu_1)^2-\nu_2^2} \left[ -\frac{1}{\nu_2^2-\nu^2} + \frac{1}{(\nu_3+\nu_1)^2-\nu^2} \right] \\ &+ \left(\frac{1}{\nu_1}-\frac{1}{\nu_2}\right) \frac{1}{(\nu_2+\nu_1)^2-\nu_3^2} \left[ -\frac{1}{\nu_3^2-\nu^2} + \frac{1}{(\nu_1+\nu_2)^2-\nu^2} \right] \Bigg\}. \tag{78} \end{aligned} \]

A similar formula is obtained for molecules in an arbitrary state \((n_1, n_2, n_3)\). The general formula was given by Condon, Altar, and Eyring \(^{27}\). It is of interest to mention it here only in connection with the fact that Planck’s constant cancels in the expression for the rotatory power; whence it follows that, according to the correspondence principle, this model would show optical activity even in the treatment of classical mechanics. In other words, the nonvanishing rotatory power of the model is not a specifically quantum-mechanical effect, but is a property of the model itself from both the classical and the quantum-mechanical points of view.

Returning to the question of the possibility of applying this model to real molecules, we see that above all it is necessary to make definite assumptions as to what part of the molecule gives rise to the absorption band. This part is usually called the chromophoric group, and there is a fairly extensive body of experimental material on the absorption spectra of polyatomic molecules which makes it possible to make the required choice. The strongest part of the field in which the chromophoric electron moves is created by the atom to which it belongs, or by the two atoms which it binds. This part of the field is determined by methods developed in another connection for the consideration of molecular orbits. Superposed on this field is the dissymmetry-producing field of the other atoms. This field can be approximately calculated by assuming that it is created by point charges situated at the center of each atom, the magnitudes of these charges being chosen in such a way as to represent the observed values of the static dipole moments caused by each bond. Further information on this question may be obtained from the literature

according to dipole moments \(^{33}\). However, there is a lack of information as to how much partially hindered free rotation is retarded, and on other similar questions. The relation of optical rotatory power to these questions may mean that, in the future, an expanded study of rotatory power will be able to shed light on these and related problems of molecular structure.

After the best possible assumption has been made concerning the system of effective charges obtained in this way, it is necessary to calculate the action of the field on an individual electron by expanding the field in powers of the displacement of this electron from its mean position. Thus one obtains quadratic and cubic terms of the type introduced in (76), which manifests itself in a nonvanishing rotational strength for transitions of this electron. Such a method of treating the field created by neighboring atoms is, of course, to a large extent identical with the method of Van Vleck and others \(^{34}\), successfully applied by them in considering the influence of the crystal field on magnetic permeability.

More detailed calculations of this type may be found in the paper by Condon, Altar, and Eyring \(^{27}\).

7a. Rotatory Power and Polarizability of Groups

Immediately after the completion of the present review, an important paper by Kirkwood \(^{35}\) appeared, which showed how to relate the quantum-mechanical theory of rotatory power to the polarizability of groups and their mutual coupling. In the present section a brief review and commentary on the indicated work is given; but since this section was written as an addendum, after the completion of the article, the equations here are renumbered in a separate sequence, namely (1a), (2a), etc. The notations occurring in Kirkwood’s paper have been changed where this was necessary for agreement with the system of notation adopted in the present article.

We shall suppose that the electrons in the molecule can be assigned unambiguously to \(N\) different groups connected with a central group. Then for the total electric dipole moment we may write

\[ \mathbf{p}=\sum_{i=1}^{N+1}\mathbf{p}^{(i)}, \tag{1a} \]

where \(\mathbf{p}^{(i)}\) is the electric dipole moment of the \(i\)-th group, defined as

\[ \mathbf{p}^{(i)}=\sum_s e\mathbf{r}_s . \]

where \(\mathbf r_s\) is the radius vector of the electron of the \(i\)-th group, referred to the center of mass of this group, and not to the center of mass of the molecule, as before. Let \(\mathbf R_k\) be the radius vector of the center of mass of the \(k\)-th group relative to the center of mass of the whole molecule; then for the magnetic moment we obtain

\[ \mathbf m=\frac{e}{2mc}\sum_k \mathbf R_k \times \mathbf P_k+\sum_k \mathbf m^{(k)}, \tag{2a} \]

where \(\mathbf P_k\) denotes the total electronic momentum of the electrons of the \(k\)-th group, and \(\mathbf m^{(k)}\) denotes the magnetic dipole moment of the \(k\)-th group, calculated with the radius vectors of the electrons measured from the center of mass of the given group. Let us now consider the electric and magnetic moments, written as sums of terms from the various groups, as in (1a) and (2a); then the rotational strength \(R_{ba}\), associated with some transition \(a \to b\), may be written, according to (32), in the form

\[ \begin{aligned} R_{ba} &=\operatorname{Im}\left\{ \left(a\left|\sum_i \mathbf p^{(i)}\right|b\right)\cdot \left(b\left|\frac{e}{2mc}\sum_k \mathbf R_k\times \mathbf P_k+\sum_k \mathbf m^{(k)}\right|a\right) \right\} \\ &=\operatorname{Im}\left\{ \sum_i (a|\mathbf p^{(i)}|b)\cdot(b|\mathbf m^{(i)}|a)+\right. \\ &\qquad\left. +\sum_{i\ne k}(a|\mathbf p^{(i)}|b)\cdot \mathbf R_k\times (b|\mathbf P_k|a)\,\frac{e}{2mc} +\right. \\ &\qquad\left. +\sum_{i\ne k}(a|\mathbf p^{(i)}|b)\cdot(b|\mathbf m^{(k)}|a) \right\}. \end{aligned} \tag{3a} \]

This equation corresponds to equation (24) in Kirkwood’s paper. In the first line we have a sum characterizing the contributions of separate groups. They have a nonzero value owing to the action of neighboring groups, which produce a dissymmetric field, as was shown in the preceding section. The second line corresponds to the most important part of the theory of coupled oscillators and is given here in a form which by itself calls for discussion and constitutes the main point of Kirkwood’s work. The third line was discarded by Kirkwood as inessential, but without a detailed investigation of its magnitude; this point deserves more thorough study.

Let us now consider the terms of the second line. From the commutation rules we have, for any electron,

\[ H\mathbf r_i-\mathbf r_i H=\frac{\hbar \mathbf p_i}{im}, \]

and, consequently, summing over all the electrons of some, for example the \(k\)-th, group, we have

\[ H\mathbf p^{(k)}-\mathbf p^{(k)}H=\frac{e\hbar}{im}\mathbf P^{(k)}, \tag{4a} \]

\[ * \]

which allows us to express the components of the matrix of the electronic moment in terms of the matrix components of the dipole moments of the groups

\[ -\frac{e}{2mc}\,(b|P^{(k)}|a) = -\frac{\pi i\nu_{ba}}{c}\,(b|p^{(k)}|a), \]

and consequently the contribution of the corresponding term \(R_{ba}\) in the second line of (3a), which we shall denote by \(R^\circ_{ba}\), is, since \(\operatorname{Im}\{iz\}=R\{z\}\),

\[ R^\circ_{ba} = \frac{\pi\nu_{ba}}{c}\, R\left\{ \sum_{i\ne k} (a|p^{(i)}|b)\cdot R_k \times (b|p^{(k)}|a) \right\}. \]

Using the standard properties of the mixed triple product and the relation \(R\{\bar z\}=R\{z\}\), this may be written as

\[ R^\circ_{ba} = \frac{\pi\nu_{ba}}{2c}\, R\left\{ \sum_{i\ne k} (R_k-R_i)\cdot (a|p^{(k)}|b)\times (b|p^{(i)}|a) \right\}. \tag{5a} \]

In the sum each pair \((i,k)\) is repeated twice—once as \((i,k)\) and once as \((k,i)\). The result obtained clearly reveals the properties emphasized by us in § 6 in a more classical treatment of the model of coupled oscillators: \(R^\circ_{ba}\) has the same dimensions as the ordinary strength \(S_{ba}\), namely the square of a dipole moment, but \(R^\circ_{ba}\) is small for two reasons: first, \(R_{ba}\) is small, since the vector distance between the groups \(i\) and \(k\) is divided by \(\dfrac{c}{\nu_{ba}}\), i.e. by the wavelength corresponding to the frequency of the quantum transition of the active band in question. We know that this wavelength is large in comparison with the dimensions of the molecule. Second, \(R^\circ_{ba}\) is small because it owes its existence entirely to the weak dynamical coupling between the separate groups.

To see this, let us consider the case in which the dynamical coupling between the groups may be neglected. This means that the Hamiltonian can be adequately represented as a sum of separate Hamiltonians, one for each group, and that the set of quantum numbers, symbolically represented by the letters \(a\) and \(b\), splits into separate series, each of which corresponds to only one group. Thus, for example, \(a\) is split into \(a_1, a_2,\ldots,a_{n+1}\), where \(a_k\) refers only to the \(k\)-th group. If this were in fact so, then the only nonvanishing matrix components \((a|p^{(i)}|b)\) would be those components in which \(b\) is identical with \(a\) with respect to all quantum numbers not belonging to the \(i\)-th group; the same would hold for \((a|p^{(k)}|b)\). Therefore, when the dynamical coupling is absent, the selection rules for \(p^{(i)}\) and \(p^{(k)}\) mutually exclude one another, and \(R^\circ_{ba}\) vanishes. Therefore

To obtain optical rotatory power it is very important to consider the connection between the electronic groups in the molecule.

Various assumptions may be made about the nature of this connection, but if we assume the usual interaction according to Coulomb’s law between the electrons in the groups, then the first term in the expansion in inverse powers of the distance is the interaction energy of a dipole with a dipole, which may be written in the form

\[ V=\sum_{l>j=1}^{N}\mathbf{p}^{(l)}\cdot \mathbf{T}_{lj}\cdot \mathbf{p}^{(j)}, \tag{6a} \]

where

\[ \mathbf{T}_{lj}=R_{jl}^{-3}\left[1-3\mathbf{R}_{jl}\mathbf{R}_{jl}\frac{1}{R_{jl}^{2}}\right]. \]

The influence of this term on the wave functions and, consequently, on the matrix components can be taken into account by the usual perturbation theory. A detailed discussion of this question may be found in Kirkwood’s work; the final result is a formula for the rotatory power, which depends on the polarizability of the interacting groups.

8. Influence of the Solvent and the Correction for the Effective Field

According to what was set out in § 2 and the equation (22) obtained there, the parameter \(\beta\) is a property of individual active molecules. If the rotatory power \(\varphi\) has been measured and the molecular density \(N_1\) and refractive index \(n\) are known, then, using (22), one can calculate the empirical value of \(\beta\). This will be an effective value of \(\beta\) for the active molecules under the average conditions in which they exist in the particular medium for which the measurements were made.

The question arises: is the parameter \(\beta\) a constant property of the molecule, completely independent of the molecular environment? In the present section we shall consider facts showing that the answer to this question must be negative. It turns out that \(\beta\), generally speaking, is very sensitive to the molecular environment. This is the so-called solvent effect. In this respect \(\beta\) is quite different from the ordinary polarizability \(\alpha\), which, generally speaking, has a constant value and gives the known additivity laws for the ordinary molecular refraction.

Let us first consider what happens in the transition from the liquid state to the gaseous state. In the gaseous state the molecules are at a large distance from one another; in the liquid state they are closely packed.

The simplest assumption is that, despite the change in the environment, the value of \(\beta\) remains the same in both states. If this condition is in fact fulfilled, then it is easy to see from (22) that

\[ \frac{[\varphi]}{\frac{1}{3}(n^{2}+2)} \]

must be continuous under changes of state, since \(N_1\) is proportional to the density \(\rho\). For all vapors the refractive index is so close to unity that one may put \(\frac{1}{3}(n^{2}+2)=1\). For liquids, however, this factor usually lies between 1.30 and 1.50. Consequently, if \(\beta\) does not change, then the specific rotating power \([\varphi]\) must exhibit the same discontinuity under a change of state as the factor \(\frac{1}{3}(n^{2}+2)\), namely, it decreases by 30–50% in passing from the liquid state to the gaseous state.

This is in contradiction with the well-known assertion \(^{35}\) that \([\varphi]\) itself is continuous under a change of state. But the data at our disposal are very scanty. The most detailed investigation was carried out by Gouy and Amaral \(^{36}\). However, their results are not sufficiently accurate to permit a final conclusion to be drawn. A summary of the results obtained by them \(^{38}\) is pre-

TABLE 2

Rotating power in the vapor and liquid states \(^{38}\)

Substance	\([\varphi]\) (vapor)	\([\varphi]\) (liquid)
Valeraldehyde	7.1 to 6.4	14.6
Amyl acetate	2.6 3.2	2.8
Methyl valerate	14.3 14.5	16.4
Amyl chloracetate	1.9 1.6	3.1
Diamyl	10.7 10.9	11.1
Amylamine	2.1 2.2	2.8
Amyl bromide	1.9	2.8
Amyl iodide	3.9 to 4.1	5.6
Amyl nitrite	5.8 6.5	5.1
Valeric acid	10.7 10.9	13.5

entered in Table 2. From Table 2 we see that \([\varphi]\) does not exhibit any large discontinuity upon a change of state; that in most cases \([\varphi]\) for liquids is greater than for vapor, as is required by the constancy of \(\beta\), but this is not always true, and in no case do we have sufficiently good data to draw a definitive conclusion. Therefore, apparently, we can say only that \(\beta\) changes little in passing from one state to another, or does not change at all.

The most recent investigations of rotatory dispersion in the vapor phase were carried out by Lowry and Gore[^37]. They studied camphor vapor at \(180^\circ\) and its solution in cyclohexane at \(20^\circ\mathrm{C}\). These data cannot be compared with one another because of the difference in temperature.

Similarly, in the case of an inactive solvent, where there is only a very small perturbation of the molecules of the dissolved substance by the solvent molecules, we should expect the same value of \(\beta\), applying (22) to the observed rotations, independently of the choice of solvent. This question was first studied from this point of view by Wolf and Volkmann[^38], and recently it has been the subject of a whole series of experimental investigations carried out by Rule[^39] and his collaborators. The results of these investigations show that for a nonpolar active substance in a nonpolar solvent the quantity

\[ \frac{[\varphi]}{\frac{1}{3}\left(n^2+2\right)} \]

is more constant than \([\varphi]\). But in the case of polar solvents and polar active substances there is a large difference, which indicates large changes in the effective value of \(\beta\) caused by dissolution. Thus Pickard and Kenyon[^40] found that the sign of the rotatory power is different for \(\beta\)-hexyl stearate in two different solvents:

\[ [\varphi] = +20.21 \quad (\text{alcohol}) \]

\[ [\varphi] = -8.93 \quad (\text{carbon disulfide}) \]

Much information on this question may be found in Lowry’s book.

It is very important, although this is often neglected, to study rotatory dispersion in connection with the influence of the solvent. One may expect that the rotatory power of various chromophoric groups will change in different ways as a result of association or the formation of unstable compounds with the solvent. The data must be sufficiently detailed to show how the individual rotatory strengths \(R_{ba}\) of the various bands \(\nu_{ba}\) change upon dissolution.

A phenomenological theory of the action of the solvent was recently given by Beckmann and Cohen[^41]. In the main, they relate the action of the solvent to deformation of the molecule.

9. Circular Dichroism

As was indicated in the introduction, circular dichroism consists in the difference in absorption by a medium of light polarized circularly to the right and to the left. Circular dichroism is observed when determining the ellipticity of polarized light obtained as a result of the passage through an active medium of a beam which was initially linearly polarized. This property stands in the same relation to optical rotatory power as ordinary dispersion does to ordinary absorption.

The experimental technique for observing circular dichroism is described in the books of Lowry and Bruhat already cited by us, and also in Eger’s lectures. \(^{42}\)

In the old electron theory of dispersion, the connection between refraction and dispersion is obtained by the formal introduction of a damping term into the equation of motion of the electron. The damping term in the expression for the force is proportional to the velocity and is opposite to it in direction, so that, regardless of how the electron moves, energy must always do work against this force. As is known, this destroys the sharpness of the resonance and gives appreciable absorption of energy at frequencies close to the resonant frequency, where the forced oscillations are comparatively large. Various attempts have been made to give a physical interpretation of the damping term. The two most important of these are Planck’s “damping due to radiation” \(^{43}\) and Lorentz’s “damping due to collisions.” \(^{44}\) In the picture of damping due to radiation, the loss of energy of the initial beam is explained by the scattering of radiation in all directions by radiation caused by the forced oscillations of the electron. Damping due to collisions is an expression of the mean loss of energy associated with the interruption of forced oscillations as a consequence of collisions of other molecules with the resonator. \(^{45}\)

When considering the absorption of light, a whole series of different measurements of the absorbing capacity is used. For theoretical treatment the most convenient method of description is given by the complex refractive index, which is usually represented in the form \(n = n(1 - i\chi)\). Introducing the refractive index in this form, we obtain from equation (3) for the propagation of the electric induction

\[ \mathbf{D}=R\{D_{0}e^{i\psi}\} = e^{-\frac{2\pi \nu \chi\, \mathbf{k}\cdot \mathbf{r}}{c}} \times R\left\{ D_{0}e^{2\pi i\nu\left(t-\frac{n\mathbf{k}\cdot\mathbf{r}}{c}\right)} \right\}, \tag{79} \]

so that the amplitude \(D\) decreases with an exponential factor which depends on the imaginary part of the index. Since the intensity of a light wave is proportional to the mean value of \(D^{2}\), the intensity falls according to an exponential law, in accordance with the formula

\[ I=I_{0}e^{-\frac{4\pi \nu \chi z}{c}}, \tag{80} \]

where \(z\) is the distance traversed in the medium. The coefficient of \(z\) in the exponent is here called the “attenuation coefficient” of the medium and is usually denoted (as in § 1) by the letter \(\varepsilon\), so that

\[ \varepsilon=\frac{4\pi n\varkappa}{c}=\frac{4\pi n\varkappa}{\lambda}, \tag{81} \]

where, as everywhere in the present article, \(\lambda\) denotes the wavelength in vacuum (sometimes \(\lambda\) denotes the wavelength in the medium, i.e. our \(\frac{\lambda}{n}\), so that \(n\) should not explicitly enter the numerator).

The reader will easily show that (6) and (10) can be combined with the aid of the complex refractive index. We shall write \(\varphi'\) for \(\frac{\Psi}{d}\), which denotes the ellipticity per unit length, and consider \(\varphi\) and \(\varphi'\), combined into one complex quantity \((\varphi+i\varphi')\), which we shall call the “complex rotatory power.”

Then (6) and (10) can be combined into one equation

\[ (\varphi-i\varphi')=\frac{\pi}{\lambda}(n_l-n_r), \tag{82} \]

where \(n_l\) and \(n_r\) are the complex refractive indices, respectively, for light circularly polarized to the right and to the left. In other words, the complex rotatory power is related to the complex refractive index in the same way as the ordinary rotatory power is to the real refractive index. This result is very important for establishing the connection between rotatory power and circular dichroism.

Before returning to the further study of the theory of circular dichroism, it is useful to define another measure of absorptive power, often encountered in the experimental literature. This is the molecular or molar absorption, usually denoted by the letter \(x\), although in the present article we shall denote it by \(x'\), so as not to confuse it with the imaginary part of the complex refractive index. \(x\) is defined by the equation

\[ I=I_0\,10^{-x'Cz}, \tag{83} \]

where \(z\) is expressed in cm, and \(C\) is the concentration of the absorbing substance in moles per liter. The relation between \(x\) and \(\varepsilon\) will obviously be

\[ \varepsilon=2{,}303\,x'C. \tag{84} \]

One may also introduce a quite intuitive quantity, closely connected with the coefficient of molecular absorption, namely the effective cross section of a molecule for absorption of a light quantum of the type under consideration. Let \(A\) be the effective cross section; then, for \(N_1\) absorbing molecules per cubic centimeter, the probability that a light quantum will pass through

through a thickness \(z\), without being absorbed, will be \(e^{-N_1 A z}\). On the other hand, if the concentration is equal to \(C\) mole/l, then \(N_1=NC/1000\), where \(N\) is Avogadro’s number, and therefore

\[ A\left(\frac{2.303}{N}\right)x' = x'\,3.81\cdot 10^{-21}\ \mathrm{cm}^2 . \tag{85} \]

Since in ordinary absorption bands \(x'\) is of the order from 10 to \(10^3\), we see that the effective area of the molecular cross sections is, generally speaking, small in comparison with the actual area of the molecular cross sections \((10^{-16}\ \mathrm{cm}^2)\).

In the classical electron theory, absorption is investigated by introducing a damping term into the equation of motion of the electronic oscillator. If the electron is elastically bound to the origin of coordinates and its natural frequency is \(\nu_0\), then its equation of motion must have the form

\[ m\ddot r+2\pi m\Gamma_0\dot r+(2\pi\nu_0)^2mr^2=eF, \tag{86} \]

where \(\Gamma_0\) measures the strength of the damping term, and in what follows we shall assume \(\Gamma_0\ll \nu_0\). Here \(F\) denotes the action of the effective field on the electron and is equal to \(E'\) in (17). If \(F=F_0 e^{2\pi i\nu t}\), so that the frequency of the light producing the forced oscillations is \(\nu\), then for the stationary state the solution of (86), as is well known, takes the form

\[ er=er_0e^{2\pi i\nu t}= \frac{\dfrac{e^2}{m}}{4\pi^2\left[(\nu_0^2-\nu^2)+i\nu\Gamma_0\right]}F . \tag{87} \]

This gives us the dipole moment caused by coherent forced oscillations, and thus we obtain the classical analogue of the dispersion formula (56a) of the quantum theory. In establishing the correspondence, we multiply the coefficient of \(F\) in (87) by the oscillator strength \(f_{ba}\), defined by (26), and set it equal to the coefficient of \(F\) in (87).

If we do this, then for the polarizability we obtain the modified form of (27), in which the effect of damping is included in the denominator,

\[ \alpha_a=\frac{2}{3h}\sum_b \frac{\nu_{ba}S_{ba}}{(\nu_{ba}^2-\nu^2)+i\nu\Gamma_{ba}} . \tag{88} \]

This result, obtained by a formal change of the denominators, agrees with quantum theory, but the complete proof of this fact is rather complicated and leads to the theory of the natural line width of Wigner–Weisskopf\(^{48}\). The corresponding generalization of the theory of rotatory power, including damping, has not yet been developed, but from general considerations it is clear that the final result must be a similar change of (56β) or (23)

with the same change of the resonance denominator. Thus it is quite possible that (23) will take the form

\[ \beta_a=\frac{c}{3\pi h}\sum_b \frac{R_{ba}}{\nu_{ba}^{\,2}-\nu^2+2\pi i\nu\Gamma_{ba}}, \tag{89} \]

where the damping constants \(\Gamma_{ba}\) are the same as in (88). With this complex value for \(\beta_a\) we obtain, for the complex rotatory power, the obvious generalization of (22)

\[ (\varphi-i\varphi')=\frac{16\pi^3N_1}{\lambda^2}\cdot \frac{n^2+2}{3}\cdot\beta . \tag{90} \]

Separating the real and imaginary parts, we obtain the formulas for the rotation per unit length and, respectively, the ellipticity per unit length

\[ \left. \begin{aligned} \varphi&=\frac{16\pi^2N_1}{3hc}\sum_b \frac{\nu^2(\nu_{ba}^{\,2}-\nu^2)R_{ba}} {(\nu_{ba}^{\,2}-\nu^2)^2+\nu^2\Gamma_{ba}^{\,2}},\\[6pt] \varphi'&=\frac{16\pi^2N_1}{3hc}\sum_b \frac{\nu^3\Gamma_{ba}R_{ba}} {(\nu_{ba}^{\,2}-\nu^2)^2+\nu^2\Gamma_{ba}^{\,2}} . \end{aligned} \right\} \tag{91} \]

These formulas have the same form as the formulas obtained by Drude\(^{47}\), except that they have been brought into correspondence with the rotatory powers \(R_{ba}\), defined from the point of view of quantum mechanics.

In applying these formulas to experimental data, one should remember that the radiation damping coefficient is very small. Consequently, when the experiment gives a broad band many hundreds of angstroms wide, one cannot choose \(\Gamma_a\) so large that the absorption band is in fact so broad according to (91) (it is easy to see that \(\Gamma_{ba}\) denotes the full width from the absorption half-maximum on the low-frequency side of the band to the absorption half-maximum on the high-frequency side of the band). To do this would require inconceivably large values of \(\Gamma_{ab}\), and this would give an incorrect form of the absorption band; weak absorption would take place over too broad a region of frequencies around the maximum.

The actual absorption bands obtained from molecules in the gas phase or, especially, in solution must be regarded as formed by an enormous number of very sharp lines corresponding to a large number of possible rotational and vibrational transitions. One may even think of the broadening of individual lines in collisions. A broadened line may be regarded as the resulting effect of a large number of sharp lines directly adjacent to one another, arising from different individual molecules which have been subjected, to varying degrees, to the perturbing action of the surrounding molecules. Equations of the type (91) were also obtained by Natanson\(^{48}\) in a serious

classical work on circular dichroism. In this work a generalization was stated, known as Nathanson’s rule: “The most strongly absorbed wave propagates more slowly for frequencies lower than the absorption frequency.” The equations were studied from the experimental point of view by Bruhat,^49 who found that they reproduce very accurately his observations on tartrates of metallic ions with absorption in the visible region, such as chromium and copper.

For the modern development of the question of circular dichroism we are indebted to a considerable extent to Kuhn and his collaborators.^50 Bjelicki and Andrée^52 showed that in many absorption bands the absorption coefficient for complex molecular bands obeys the law

\[ \varepsilon(\nu)\sim e^{-\frac{(\nu-\nu_0)^2}{\theta^2}} . \]

This simply means that if the logarithm of the absorption coefficient is plotted against frequency, the resulting curve will be a parabola with its vertex upward. Kuhn and his collaborators adopted an analogous representation for the spectral distribution of the rotatory power in an absorption band, for the empirical representation of the data they obtained. This procedure proved much better than allowing for the width of the band by simply choosing a very large value of \(\Gamma\) for this band.

In the work of Condon, Altar, and Eyring^28 detailed calculations based on quantum mechanics are given, showing that the probability of transitions with absorption of light is different for waves polarized right and left circularly, provided, of course, that the calculations are carried out with the same degree of approximation as is necessary to obtain the rotatory power (see § 4).

LITERATURE

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1. Breit, Rev. Mod. Phys., 5, 91, 1933.
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1. It may be that these remarks should be made in more detail, since in the usual exposition of electromagnetic theory very little attention is paid to this question. Let \(\mathbf{i}, \mathbf{j}\), and \(\mathbf{k}\) be the unit vectors of the right coordinate system, and \(\mathbf{i}', \mathbf{j}', \mathbf{k}'\) the vectors of the left system, related to the right vectors by the relations
   \[ \mathbf{i}'=-\mathbf{i}, \qquad \mathbf{j}'=-\mathbf{j}, \qquad \mathbf{k}'=-\mathbf{k}. \] ↩