Vibrational Energy and Luminescence of Complex Molecules

B. S. Neporent and B. I. Stepanov

Introduction

The most important distinctive feature of polyatomic molecules is the abundance of internal degrees of freedom. The majority of fluorescent polyatomic molecules are compounds of the aromatic series, from simple benzene derivatives to complex dye molecules of large molecular weight (of the order of several hundreds). The heat capacity of such molecules is correspondingly large, and already under normal conditions their store of vibrational energy amounts to several thousand reciprocal centimeters*), i.e., is close to the energy of electronic excitation.

An equally essential consequence of the large number of normal vibrations is the coupling between them, which in complex molecules leads to the blurring of vibrational levels and their transformation into broad bands.

In this article we shall consider certain well-known properties of the fluorescence of complex molecules from the point of view of the role of vibrational energy in the processes of absorption and emission of light. The work is divided into two parts. In the first part a review is given of experimental studies, established facts, results, and conclusions. In the second part a theoretical investigation is carried out of the questions considered in the first part**).

Within the limits of a single article it is impossible, even when considering the subject from a quite definite point of view, to cover all questions connected with the phenomena of luminescence. By the works of Soviet scientists, and first of all of S. I. Vavilov, the doctrine of luminescence

*) For convenience of comparison with experimental data, we shall express the energy of a molecule by using wave numbers, having in mind the known relations.

**) The first part was written by B. S. Neporent, the second by B. I. Stepanov.

has been transformed into a coherent, rigorously substantiated discipline. S. I. Vavilov[^1] gave a precise scientific definition of the phenomenon and a system of its principal characteristics. To him belongs the exhaustive elaboration of questions of the luminescence of solutions. The phenomena of fluorescence of complex molecules in connection with their structure and with the role of vibrational energy that interests us were comprehensively considered by A. N. Terenin[^2].

We shall not consider here many sections, such as the fundamental questions of the thermodynamics of luminescence developed by S. I. Vavilov[^3], questions of the role and nature of metastable states, the theory of which was given by A. N. Terenin[^4], questions of fluorescence polarization, questions of antistokes fluorescence of solutions and vapors, etc. We shall refer here to the recent complete reviews by P. P. Feofilov[^5] on fluorescence polarization and by E. I. Adirovich[^6] on the principles of the spectral transformation of light.

In the present article only a small circle of basic questions will be considered, in which the role of vibrational energy has been sufficiently clarified. These include: the distribution of energy within the molecule and its role in the formation of the spectra of complex molecules, the dependence of the yield and duration of fluorescence on the store of vibrational energy of the molecule, and also certain questions of intermolecular interactions, namely the quenching and enhancement of fluorescence by foreign substances, the stabilization of excited molecules, and the transfer of vibrational energy in collisions. The present article does not claim to be a complete review. Literature materials are cited only insofar as they are necessary for the discussion of the questions considered.

PART

1. THE ROLE OF THE AGGREGATE STATE*)

The greater part of the work on the study of the fluorescence of complex molecules, especially among quantitative investigations, concerns the study of solutions.

These investigations are made possible by the stability of the excited electronic states of aromatic and some other molecules with respect to collisions. Methodologically, investigations of solutions are simpler than investigations in vapors. In quantitative investigations in vapors there arises the problem of creating and maintaining a constant vapor pressure, requiring faultless thermostating, often at comparatively high temperatures. In addition, the brightness of fluorescence in vapors, as a rule,

*) In view of the further discussion, connected with averaging over all properties of the molecule, we shall not touch here upon the crystalline state, in which the interactions are regular and anisotropic.

less than in solutions, both because of difficulties connected with creating high concentrations and for other reasons. Overcoming these difficulties, however, proves fruitful, since studies in vapors open up new possibilities in comparison with solutions. It is essential that these new possibilities are especially important for the question that interests us, the role of vibrational energy in fluorescence phenomena.

Let us consider from this point of view the possibilities and features of the dissolved and gaseous states.

A. An excited molecule can be isolated from the action of other particles under the conditions of a rarefied gas. In solution this is impossible. The molecule continuously interacts with the solvent.

B. The average store of vibrational energy of an excited molecule \(Q^*\) under the conditions of a rarefied gas*) can easily be varied by exciting fluorescence with light of different wavelengths. Under the conditions of a solution this is impossible, since the molecules are in thermal equilibrium with the solvent.

C. The average store of vibrational energy of an excited molecule \(Q^*\) can also be regulated by changing the temperature. In the gaseous state \(Q^*\) cannot be made arbitrarily small, since the elasticity of vapors depends on temperature. In solutions \(Q^*\) can be made quite small**). Attaining large \(Q^*\) in this case is difficult because of the approach to the critical state.

D. In the gaseous state it is possible to create effects on an excited molecule that vary within wide limits in their character and intensity (various foreign gases at varying pressures). In solutions one can only change the nature of the solvent or act with an active quencher in a neutral solvent.

Let us also note that many fluorescent substances***) decompose without subliming, while some are insoluble.

It is clear from the foregoing that investigations of the fluorescence of vapors and solutions complement one another. In some respects of interest to us (the action of foreign gases on fluorescen-

*) The concept of a “rarefied gas” is not universal for all cases. The effective radii of molecules vary over wide limits depending on the nature of the interaction.

**) A change in the temperature of solutions may be associated with a change in the nature of the interaction between the molecules under study and the solvent.

***) In addition to the above, in solutions, owing to the viscosity of the solvent, it is sometimes possible to hinder the rotation of the molecules under study and of their parts during the excited state, which in some cases proves very significant. These questions lie outside the scope of the present article.

(to the pressure of the vapors) the fluorescence of solutions constitutes a limiting case. Already from the general considerations adduced, the very important role of studying the fluorescence of rarefied gases for elucidating the influence of vibrational energy on the optical properties of complex molecules is quite evident. Let us emphasize that the problem is not limited to optical properties. At the same time the chemical and photochemical properties of complex molecules are clarified, since molecular vibrations are an important source of chemical activation, and the stability of the energy of electronic excitation is essential for photochemical processes.

2. INTERACTION OF NORMAL VIBRATIONS AND CLASSIFICATION OF POLYATOMIC MOLECULES

In considering the experimental material we shall proceed from the classification of polyatomic molecules proposed by one of the authors of the present article[^1]. According to this classification, a polyatomic molecule is classified as complex if it is characterized by a large probability \(W\) of intramolecular redistribution of energy over degrees of freedom, substantially exceeding the reciprocal of the lifetime of the excited state and even approaching the frequencies of molecular vibrations. A simple polyatomic*) molecule we shall consider to be one in which the probability \(W\) is relatively small.

The probability of intramolecular redistribution of vibrational energy \(W\) is a measure of the degree of coupling between the normal vibrations of the molecule, generalized for all vibrations. The nature of this quantity is made more concrete in Section 10.

The duration of the molecule’s stay on an individual vibrational level is inversely proportional to the quantity \(W\), and, consequently, the width of the level is proportional to \(W\). Hence it is clear that, as \(W\) increases, the energy levels, and therefore the spectra of the molecule, will become diffuse and then continuous. Experience shows that the quantity \(W\) depends on many causes (see below) and increases with an increase in the store \(Q\) of vibrational energy of the molecule.

From the point of view of research, an important group is represented by molecules which at small \(Q\) are simple (small \(W\)), but with an increase in the store of vibrational energy \(Q\) become complex (large \(W\)). This is the group of “semicomplex” molecules. Such molecules will interest us as a transitional case from simple molecules to complex ones, for which \(W\) is large at all values of \(Q\).

*) In what follows, for brevity, we shall write “complex” or “simple” molecule.

In the following sections of the first part we shall consider the influence of vibrational energy on the fluorescence and absorption spectra, and also on the yield spectra and lifetimes of the excited state of complex molecules in solution and in the gas phase. We shall use only the most typical examples, which make it possible to establish general regularities. After this we shall consider questions connected with external actions on the excited molecule, and, finally, summarize all the data considered concerning the role of vibrational energy in these phenomena.

3. ABSORPTION AND FLUORESCENCE SPECTRA OF COMPLEX MOLECULES

The absorption spectra*) of vapors of polyatomic molecules have a typical character^8, schematically represented in Fig. 1. The spectrum consists of three regions—discrete, diffuse, and continuum. In simple molecules the discrete region is broad. In “semi-complex” molecules it is narrower or altogether absent. The spectra of complex

Discrete     Diffuse     Continuum

Fig. 1. Diagram of the absorption spectrum of a polyatomic molecule according to Terenin’s data.^8

molecules consist only of the continuum, representing a typical broad band.

For molecules of the first two types, complication of the structure, an increase in the elasticity of the vapors, an increase in temperature, and transition into the dissolved state, as well as advance along the spectrum into the short-wave region, lead to blurring of the structure and even to transformation of the spectrum into a continuum. The cause of this phenomenon is an increase in the probability of redistribution of vibrational energy.

The continuous band of the absorption spectra of solutions of complex dye molecules has, as S. I. Vavilov^9 showed, a universal form, connected with the statistical character of the interaction of generalized normal vibrations with electronic states (Fig. 2). V. L. Levshin^10 established, for molecules of this type in solutions, mirror symmetry of the absorption and fluorescence spectra in a number of cases. This rule has been generalized

*) Here and below we consider only the first, longest-wavelength absorption band.

author of this part of the article, who established general relations between the widths of the fluorescence and absorption spectra for various types of complex molecules in solutions and in the gas phase[^39].

The division of polyatomic molecules into simple and complex ones is supplemented here by consideration of two limiting cases for complex molecules. The first limit—complex molecules of the first group—corresponds to the absence or weak coupling of vibrational and electronic states at relatively small values of the probabilities \(W\). In this case the width of the spectra does not depend on the frequency of the electronic transition and the mirror-image rule is obeyed.

Fig. 2. Coincidence of the shape of the absorption band of various dyes according to Vavilov’s data[^9].

The second limit—complex molecules of the second group—corresponds to strong interaction between electronic and vibrational states,

Fig. 3. Typical spectra of polyatomic molecules[^31]. Simple or “semi-complex” molecule (9-aminoacridine; solution in alcohol).

associated with very large values of \(W\). In this case the width of the spectra depends on the frequency of the electronic transition and in the limiting case is apparently proportional to the square of the frequency.

Equal width of the spectra (“mirror symmetry”) occurs when they are plotted on a wavelength scale. This is illustrated by Figs. 3, 4, and 5, borrowed from ^31.

Membership in one or another group is determined not only by the structure of the molecule, but also depends on other factors determining the probability \(W\). The same molecule under different conditions may belong to different groups.

Fig. 4. Typical spectra of polyatomic molecules ^31. Complex molecule of the first group (3,6-diaminoacridine; solution in alcohol).

Fig. 5. Typical spectra of polyatomic molecules ^31. Complex molecule of the second group (3-aminoacridine; solution in alcohol). The small-scale curves are on the wavelength scale. For comparison, the spectra of a typical molecule of the first group—rhodamine 6—are shown by the dotted line.

A remarkable independence of the shape of the spectral curve of fluorescence of solutions from the wavelength of the exciting light, iz-

changing within wide limits[^3]. In the case of sufficiently complex molecules this property is also characteristic of vapors. Moreover, as N. A. Prilezhaeva[^11] has shown, even under anti-Stokes excitation the continuous fluorescence spectrum of superheated aniline vapor has practically the same form as under Stokes excitation.

The fluorescence spectrum of aniline vapor, studied in detail by A. T. Vartanyan[^12], who discovered “resonance” fluorescence here, represents a noteworthy transitional case from simple molecules to complex ones. With very low elasticity of the vapor and monochromatic excitation, at which

Fig. 6. Scheme of the fluorescence and absorption spectra of “semicomplex” molecules and a scheme of their origin.

the zero vibrational level of the upper electronic state is reached, the fluorescence spectrum has, on the short-wavelength side, a well-developed banded structure, which, on proceeding toward long wavelengths, becomes diffuse and passes into a continuum.

Thus, in vapors the absorption and fluorescence spectra of “semicomplex molecules” possess a peculiar “mirror symmetry.” They are oriented toward one another by several overlapping discrete regions with coincident bands.

As the distance from the axis of symmetry increases, the spectra become diffuse and, finally, continuous (Fig. 6). These relations become readily explicable if one takes into account that the probability of intramolecular redistribution, \(W\), which determines the degree of diffuseness of the levels, increases rapidly as the store of vibrational energy of the molecule increases, and that, to a first approximation, the absorption spectrum reflects the vibrational structure of the upper, while the fluorescence spectrum reflects the structure of the lower electronic—

states. Increasing the wavelength of the exciting light (the transition from arrow a to arrow b in Fig. 6) causes the discrete fluorescence spectrum of aniline vapor to become continuous. An increase in temperature acts in the same direction, as does an increase in the pressure of aniline vapor, which causes a broadening of the spectrum due to mutual perturbations of the electron shells of approaching aniline molecules. On passing into solution the spectrum becomes continuous. These facts are also connected with an increase in the probability \(W\).

Let us note that in some cases the action of the solvent may affect the redistribution probabilities \(W\) less than the increase in the reserve of vibrational energy \(Q\). A. N. Terenin\(^8\) established that decreasing the wavelength of the exciting light leads to the disappearance of diffuse bands in the fluorescence spectrum of anthracene. A. A. Shishlovskii\(^ {13}\), who investigated in detail the fluorescence of anthracene in various aggregate states, showed that broadening of the spectrum is characteristic only of the gas phase; diffuse bands are preserved in the spectra of solutions under any excitation conditions. This is a consequence of vibrational deactivation of the molecule.

A broad continuous band is formed in the spectra of complex molecules as a result of interaction among the molecule’s own vibrations. This is connected with the transformation of the system of vibrational levels into broad continuous zones of vibrational states. The distribution of molecules within these zones is statistical, and the properties of the various vibrational states (transition probabilities, etc.) become identical, in contrast to simple molecules, in which the properties of discrete states differ.

The universality of the form of the spectral curves of complex molecules, as well as the independence of fluorescence spectra from the wavelength of the exciting light, is characteristic of the molecules themselves and is not caused by the action of the solvent on them (see in more detail\(^7\)).

4. DURATION OF THE EXCITED STATE AND FLUORESCENCE YIELD

The works of S. I. Vavilov and his students\(^ {9,14,15}\) established that the quantum yield of fluorescence of solutions of complex molecules is independent of the wavelength of the exciting light over the entire region of the absorption spectrum. Only upon transition into the region of anti-Stokes fluorescence does the yield drop sharply. (See Fig. 7, borrowed from\(^ {14}\).)

In the case of vapor fluorescence, the spectrum of the quantum yield does not represent such a horizontal straight line. The author of the present article\(^ {7,17,18}\) established, using several aromatic

compounds that the quantum yield of fluorescence of vapors decreases as the quantum absorbed by the molecule increases, and it is shown that the phenomenon is connected with the concomitantly increasing probability of radiationless transitions of the excited molecule to the normal state. The foregoing is illustrated by the data of Fig. 8, borrowed from \(^{18*}\).

Fig. 7. Quantum yield of fluorescence of a fluorescein solution according to Vavilov’s data \(^{14}\).

The results cited explain why many aromatic compounds that fluoresce in the dissolved state do not fluoresce in the gaseous state. This is connected with large values of the probability of radiationless transitions already at small reserves of vibrational energy of the excited molecule. According to A. N. Terenin \(^{2,6}\), an electronic transition—absorption or emission of light—is, as a rule, accompanied by an increase in the molecule’s reserve of vibrational energy. A gaseous molecule after excitation is always rich in vibrational energy, as a result of which, in many compounds, radiationless transitions suppress fluorescence.

The decrease in the fluorescence yield of vapors in comparison with solutions, according to the classification of S. I. Vavilov \(^{19}\), belongs to the pro-

* In the figure the relative values of the quantum yield \(\gamma\) are given. The value \(\gamma = 0.84\) for \(\lambda_e = 3341\ \text{Å}\) at \(t = 156^\circ\text{C}\) was chosen for agreement with the results of lifetime measurements.

cesses of quenching of the second kind, since it is determined by radiationless transitions in the excited molecule or by “intramolecular collisions of the second kind.” In accordance with this, the lifetime of the excited state of vapors decreases when the wavelength of the exciting light is decreased. This was also established in^17 and ^18.

Fig. 8. Quantum yield of fluorescence of β-naphthylamine^18 in vapors (I), in vapors with the addition of 400 mm Hg of pentane (II), and in solution (III).

In Fig. 8 the curve for vapors is drawn through points representing the relative values of the quantum yield. The crosses on the same curve represent the relative values of the lifetime of the excited state \(\tau\), measured from the quenching of the fluorescence of β-naphthylamine vapors by oxygen. In absolute values the lifetime of the excited state changes in the case under consideration from \(1.67 \cdot 10^{-8}\) sec. to \(0.1 \cdot 10^{-8}\) sec. We shall give more detailed data below, after a detailed consideration of the question of the quenching of fluorescence of vapors of aromatic compounds by oxygen.

5. DETERMINATION OF THE LIFETIME OF THE EXCITED STATE, THE PROBABILITY OF EMISSION, AND THE PROBABILITY OF RADIATIONLESS TRANSITIONS FROM FLUORESCENCE QUENCHING

Quenching of fluorescence by foreign substances is connected with the loss of energy of electronic excitation when an excited molecule of the fluorescing substance collides with a molecule of the quencher. Here we shall consider these phenomena from the point of view of the possibility of determining the lifetimes of excited states. The nature of the interaction of the particles during quenching is of interest

VIBRATIONAL ENERGY AND LUMINESCENCE

interests us here only insofar as it determines the effectiveness \(\varepsilon\) of the act of quenching upon collision.

The number of collisions experienced by an excited molecule depends not only on the concentration of the quencher, but also on the properties of the solvent. The theory of quenching in solutions was developed by S. I. Vavilov, B. Ya. Sveshnikov, and I. M. Frank \(^{20,21}\). The expression relating the fluorescence yield to the concentration of the quencher in solution is complex and contains a number of constants. This makes it difficult to determine the lifetime of the excited state from fluorescence quenching in solutions. The situation is substantially improved in connection with the use of fluorometers \(^{22}\), by means of which it is possible to measure directly the lifetime of the excited state. Unfortunately, fluorometric measurements of the fluorescence lifetimes of vapors have not yet been carried out.

Determination of the lifetimes of the excited state of gas molecules from the quenching of fluorescence by oxygen is in many cases a quantitative method. The basis for this is the possibility of calculating the number of collisions from the equations of the kinetic theory of gases, as well as the constancy of the effectiveness of quenching collisions with oxygen. According to A. N. Terenin \(^{2,6}\), the effectiveness \(\varepsilon\) is connected with the paramagnetic properties of the oxygen molecule and should not depend on the store of vibrational energy of the excited molecule, the first stage of whose interaction with oxygen is associated with a change in the multiplicity of the electronic state. Evidence for the correctness of this proposition is the practical coincidence of the relative values of the fluorescence yield of \(\beta\)-naphthylamine vapors with the relative values of the lifetime of the excited state obtained from data on fluorescence quenching by oxygen (see Fig. 8). The results of a detailed experimental study of the mechanism of fluorescence quenching by oxygen are contained in the paper by A. V. Karyakin and A. N. Terenin \(^{23}\).

Let us note that the values of the effective radii of molecules relevant to quenching processes are, as a rule, unknown. The absolute values of the effectiveness \(\varepsilon\) of quenching collisions are also unknown. Usually gas-kinetic values of the radii are adopted and \(\varepsilon = 1\) is assumed. Therefore the lifetime \(\tau\) of the excited state is determined from quenching by oxygen with an accuracy up to a constant factor. From the work of A. V. Karyakin and M. D. Galanin \(^{24}\), however, it follows that the order of magnitude of \(\tau\) is correctly determined from quenching.

Let us turn to the results of measurements of the lifetime \(\tau\) of the excited state of complex molecules in the gas phase from quenching of fluorescence by oxygen. A systematic investigation was carried out by us \(^{18}\) for vapors of \(\beta\)-naphthylamine upon excitation of fluorescence...

Fig. 9. Quenching by oxygen of the fluorescence of β-naphthylamine vapor upon excitation by different wavelengths \[8\].

Fig. 10. Calculation of the data of Fig. 9 by formula (A).

rescence at various wavelengths and at different temperatures and vapor elasticities. Typical results of a series of quenching experiments are shown in Fig. 9. Along the abscissa is plotted the number of collisions per second \((z)\) of an excited β-naphthylamine molecule with oxygen molecules. Along the ordinate—the relative intensity of fluorescence \(\frac{F_z}{F_0}\).

As the quantum absorbed by the β-naphthylamine molecule is increased, or as the temperature is raised, the quenching action decreases, i.e., the lifetime \(\tau\) of the excited state decreases, as calculated from the well-known formula

\[ \frac{F_0}{F_z}=1+\varepsilon\tau z. \tag{A} \]

The observance of this law is illustrated by Fig. 10. We have shown \(^{17,18}\) that the decrease in the lifetime of the excited state is due to a rapid increase in the probability \(d\) of radiationless transitions, accompanying an increase in the store \(Q^*\) of vibrational energy of the excited molecule. From the relations \(^{17}\)

\[ \tau=\frac{1}{f+d}\quad \text{and}\quad \gamma=\frac{f}{f+d} \]

it is clear that the probability \(f\) of emission of a quantum changes little with changes in \(Q^*\), since \(\gamma\) changes in the same way as \(\tau\) (Fig. 8). For the probabilities \(d\) of radiationless transitions, an exponential dependence on the store of vibrational energy \(Q^*\) was found. This is illustrated by the data of Fig. 11. Along the ordinate is plotted the quantity \(f+d=\frac{1}{\tau}\), and along the abscissa—the energy of the absorbed quantum \(\nu=\nu_0+Q^*\) for a vapor temperature \(t=130^\circ\)*). For higher temperatures \(t=151, 172\), and \(193^\circ\), to the quantity \(\nu\) there have been added, respectively, \(500, 1000, 1500\ \mathrm{cm}^{-1}\). In Fig. 12 are given the values

Fig. 11. Sums of the probabilities of emission and radiationless transitions, as a function of the store of vibrational energy of the excited molecule \(^{18}\).

\[ \begin{array}{l} t=130^\circ\\ t=151^\circ\\ t=172^\circ\\ t=193^\circ \end{array} \]

*) There, along the abscissa, are plotted the values \(Q^*=\nu-\nu_0\). The frequency \(\nu_0\) of the level of zero vibrational energy of the excited state was determined \(^{18}\) from the temperature dependence of the anti-Stokes fluorescence.

In all the remaining figures we shall give the values of \(Q^*\), calculated in the same way.

\(\lg d\) as a function of \(Q^*\). For values \(Q^*>6000\ \mathrm{cm}^{-1}\) the points fall on a common curve, i.e., the quantity \(d\) does not depend on the method of increasing \(Q^*\): either by increasing the absorbed quantum or by raising the temperature. The region of values \(Q^*>6000\ \mathrm{cm}^{-1}\) is the region of large values of the probability \(W\) for a β-naphthylamine molecule in vapor. In this region β-naphthylamine is a complex molecule. Let us note that in the absorption spectrum of β-naphthylamine vapor, near \(Q^*\sim 6000\ \mathrm{cm}^{-1}\) (which corresponds to \(\nu\sim 35\,000\ \mathrm{cm}^{-1}\)), diffuse maxima appear.

Fig. 12. Values of the logarithm of the probability of radiationless transitions as a function of the vibrational-energy reserve of the excited molecule\({}^{18}\).

For \(Q^*<6000\ \mathrm{cm}^{-1}\) the points do not fall on a common curve (Fig. 12). This is the region in which there is no equivalence of the thermal and optical methods of exciting vibrations in the molecule. Here the individual features of various molecular vibrations appear, and the β-naphthylamine molecule is closer to simple molecules than to complex ones*).

The value of \(f\) required for calculating \(d\) was taken to be \(f=0.6\cdot 10^8\ \mathrm{s}^{-1}\), as corresponding to the asymptote of the curve in Fig. 11. In the interval of \(Q^*\) values studied, the quantity \(d\) varies from

\[ d=6\times 10^6\ \mathrm{s}^{-1} \]

to

\[ d=9\cdot 10^8\ \mathrm{s}^{-1}. \]

It should also be noted that the values of \(d\) given for different conditions are in all cases averaged over a certain interval of \(Q^*\) values, determined by the distribution of the molecules absorbing monochromatic radiation according to their reserve of vibrational energy. This question is clarified in Sections 10 and 11 of the second part.

*) From the data presented, the vibrational heat capacity of β-naphthylamine vapor was determined:

\[ C_{\mathrm{vib}}=67\ \frac{\mathrm{cal}}{\mathrm{mol}\cdot\mathrm{deg}}. \]

Let us also consider the question of the nature of radiationless transitions. Here there are two main possibilities. The first is a photochemical transformation of the excited molecule. The second is an intramolecular transformation of electronic energy into vibrational energy. For processes of the first type—monomolecular reactions—the dependence of \(d\) on \(Q^*\) of the exponent type is predicted by the theory \(^{25}\). A. N. Terenin showed \(^{26}\) that the intramolecular transformation of energy is connected with the excitation, in the molecule, of strong vibrations of certain types. Thus, in mechanism, processes of the second type do not differ from processes of the first type. Their dependences on the store of vibrational energy should be similar.

Let us note in conclusion that, in studying the fluorescence of complex molecules in the gas phase, one should always use monochromatic light for excitation, since the properties of excited molecules depend substantially on the magnitude of the absorbed quantum.

5. ENHANCEMENT OF THE FLUORESCENCE OF VAPORS BY FOREIGN GASES AND STABILIZATION OF EXCITED MOLECULES UPON COLLISIONS

The phenomenon of enhancement of the fluorescence of vapors of complex aromatic compounds by foreign gases was discovered and studied by one of the authors of the present article \(^{7,17}\). Experimentally the phenomenon consists in a gradual increase in the brightness (yield) of fluorescence of the vapors as foreign gases that do not quench fluorescence are added to them (for example, \(\mathrm{H_2}\), \(\mathrm{N_2}\), and \(\mathrm{C_5H_{12}}\)). Typical curves of the dependence of the fluorescence brightness on the pressure of various foreign gases and on the wavelength of the exciting light are given for vapors of \(\beta\)-naphthylamine in Figs. 13 and 14. Along the abscissa axis in both figures are plotted the numbers of collisions per second \(z\) of an excited molecule with molecules of the foreign gas; along the ordinate axis—the values \(\frac{F_z}{F_0}\)—the brightness of fluorescence in the presence of the foreign gas, referred to that in its absence. (The data presented were obtained with allowance for changes in the absorption coefficient of \(\beta\)-naphthylamine vapors caused by the addition of the foreign gas \(^{27*}\).) The data of Fig. 14 are presented in Fig. 15 in the form of the dependence of the quantum yield of fluorescence of the vapors on the pressure of the foreign gas. Without considering here all the observed details of the phenomenon, we shall give only the general results and conclusions.

* This phenomenon, which has not yet received a definitive explanation, lies outside the range of questions considered in the present article.

Fig. 13. Enhancement of the fluorescence of β-naphthylamine vapor upon addition of hydrogen and upon excitation at different wavelengths.

Fig. 14. Enhancement of the fluorescence of β-naphthylamine vapor upon addition of pentane at different vapor pressures of β-naphthylamine and excitation at different wavelengths.

The enhancing action of foreign gases increases when the quantum absorbed by the β-naphthylamine molecule is increased, when the molecules of the foreign gas become more complex, and when its pressure is increased.

The phenomenon is associated with the loss, by excited β-naphthylamine molecules, of excess vibrational energy upon collisions with particles of the foreign gas and with the corresponding decrease in the probability of radiationless transitions. It was shown that this hypothesis explains all the features of the phenomenon, including a number of minor details not mentioned here. In addition, the hypothesis of vibrational deactivation—stabilization of excited molecules—is confirmed by independent data: in the fluorescence spectra of the vapors studied, small details characteristic only of long-wavelength excitation also appear upon excitation by shorter wavelengths if a large amount of a well-stabilizing polyatomic gas is added to the vapors under study, for example 300–500 mm Hg of pentane.

Fig. 15. Quantum yield of fluorescence of β-naphthylamine vapors as a function of the pressure of a foreign gas, upon excitation by various wavelengths.

In the process of stabilization, an excited molecule may, during its lifetime, undergo a whole series of collisions with molecules of the foreign gas. At each collision, part of the vibrational energy is lost, and the excited molecule passes through a series of states to which decreasing values of the probability \(d\) of radiationless transitions correspond.

As the pressure of the foreign gas is increased, the averaged (effective) value of \(d\) rapidly decreases. The probability of emission \(f\) changes little in this case, and the fluorescence yield increases. The transition to the dissolved state, in which maximum stabilization is achieved, is limiting in this sense. What has been said is illustrated by the data of Fig. 8, where the curve corresponding to the fluorescence of vapors with added pentane is located between the curves corresponding to vapors and to solution.

7. ANALYSIS OF THE PROCESS OF TRANSFER OF VIBRATIONAL ENERGY IN THE STABILIZATION OF EXCITED MOLECULES

The attempt \(^{17}\) to express in analytical form the regularities of the process considered in the preceding paragraph was not fruitful, since, in order to obtain an equation with a small number of constants, an extreme schematization of the phenomenon was required. Later we proposed \(^{7}\) a method for obtaining, from experimental data, information about the average amount of vibrational energy transferred by an excited molecule of the substance under study to a molecule of a foreign gas in a collision.

Fig. 16. Amount of vibrational energy lost in collision with a molecule of a foreign gas, as a function of the store of vibrational energy of an excited molecule of β-naphthylamine \(^{7}\).

For this, one chooses such a pressure \(P_{\tau}\) of the stabilizing gas at which the molecule undergoes, on average, one collision during the time of the excited state. Having determined from graphs of the type shown in Fig. 15 the increase in quantum yield corresponding to the pressure \(P_{\tau}\), one then finds from Fig. 8 the sought decrease in vibrational energy of the excited molecule. The data obtained in this way are given in Fig. 16, where the amount of vibrational energy \(\Delta Q\) lost by an excited molecule of β-naphthyl-

amine in a single collision, presented as a function of the store of vibrational energy \(Q^*\).

The region of the beginning of the rapid growth of the values \(\Delta Q\) near \(Q^* = 6000\ \mathrm{cm}^{-1}\) coincides with the boundary at which continuous maxima appear in the absorption spectrum of the vapors and with the boundary of equivalence of the thermal and optical methods of increasing the store of vibrational energy of the excited state. This is the boundary of the region of large probabilities of redistribution of vibrational energy. It is noteworthy that, in the range of values of \(Q^*\) under consideration, the amount of transferred energy \(\Delta Q\) increases rapidly and linearly with increasing \(Q^*\). Here the possibility of considering a complex molecule as a classical system with many degrees of freedom is manifested with particular clarity. The condition for this is large values of \(W\).

To characterize the process of transfer of vibrational energy, the concept of the accommodation coefficient \(\alpha\) has been introduced, characterizing the degree to which the system of colliding particles approaches thermal equilibrium. Using the known data on the heat capacity \(C'_v\) of \(\beta\)-naphthylamine vapors and the heat capacity \(C''_v\) of a foreign gas, it is easy to obtain:

\[ \Delta Q = \frac{\alpha C''_v}{C'_v+\alpha C''_v}\,Q^*, \]

i.e., the dependence of \(\Delta Q\) on \(Q^*\) should be linear if \(\alpha\) and \(C\) are constant. According to the values of \(\alpha\), the curves of Fig. 16 break up into two parts. The first of these is the region of small values of \(\alpha\) and \(W\); here the process is hindered by the impossibility of transferring arbitrary portions of energy, distributed in the form of quanta among several normal vibrations of the excited molecule. The second is the region of large \(\alpha\) and \(W\); here, owing to the interaction of vibrations, energy is transferred in arbitrary portions. We give a table of values of \(\alpha\), taken from \({}^{7}\).

Table I

Values of the accommodation coefficient \(\alpha\) in collisions of foreign-gas molecules with a \(\beta\)-naphthylamine molecule

Gas	First region: \(Q^* < 6000\ \mathrm{cm}^{-1}\)	Second region: \(Q^* > 6000\ \mathrm{cm}^{-1}\)
He	—	0.2
H\(_2\)	—	0.2
N\(_2\)	—	0.4
CO\(_2\)	—	1.4
NH\(_3\)	0.2	3.0
CHCl\(_3\)	0.2	2.5
C\(_5\)H\(_{12}\)	0.1	1.7
C\(_6\)H\(_6\)	—	2.3

Values of $\alpha > 1$ indicate that, for the process under consideration, the molecular radii exceed the gas-kinetic radii used for the calculation. Since the interaction involves electronically excited molecules, this agrees with the point of view of V. N. Kondrat’ev[^28]. In the cases of the action of helium, hydrogen, and, probably, nitrogen, the vibrational energy of the excited molecule is converted into translational and rotational energy of the foreign particle. For more complex molecules, processes of transfer of vibrational energy without conversion play an ever increasing role. The order in which the foreign gases are arranged according to the values of $\alpha$ indicates that the effective radius of interaction apparently has the greater value, the more capable the electronic system of the stabilizing molecule is of perturbing the electronic system of the $\beta$-naphthylamine molecule. This agrees with the hypothesis of Franck and Eucken[^29], confirmed by V. N. Kondrat’ev[^28].

8. CONCLUSIONS

From what has been set forth above there follows a general scheme of the behavior of complex molecules in the processes of absorption and emission of light, and the essential role played by vibrational energy in these processes becomes evident. We consider it rational to formulate the results in the form of a series of propositions. These conclusions are considered from the theoretical point of view in Part II.

1. Analysis of the experimental data confirms the important significance of the probability $W$ of intramolecular redistribution of vibrational energy. This probability, expressing the degree of interaction of normal vibrations and determining to a large extent the properties of complex molecules, is one of the basic characteristics of a polyatomic molecule.

2. The probability $W$ of intramolecular redistribution of energy increases rapidly as the degree of symmetry of the molecule decreases, and, for a given molecule, depends on its store of vibrational energy $Q$ and on the degree of interaction with the surrounding medium. As the store of vibrational energy $Q$ increases, the value of the probability $W$ increases rapidly, just as it does when external actions on the molecule are intensified at constant $Q$.

3. At large values of the probability $W$, the concept of normal vibrations becomes inapplicable, and the vibrational levels merge, forming continuous bands. The properties of the molecule that depend on vibrational states are averaged, while the properties connected with the store of vibrational energy become monotonic functions of $Q$. (The latter also applies to $W$.)

4. Complex molecules include those for which the probability $W$ has a large value. The degree of complexity is determined

therefore not only by the structure, but also by the store of vibrational energy of the molecule, as well as by the character and degree of influence of the surrounding medium. Molecules of one and the same substance under different conditions may be classed both as complex and as simple, characterized by small values of \(W\).

1. A complex molecule, characterized by large values of \(W\), should be regarded as a classical system possessing many degrees of freedom. In this case the relations of classical statistics are applicable. In a certain sense, complex polyatomic molecules are simpler than simple polyatomic molecules.

In simple molecules the electronic state is associated with a large number of vibrational levels; moreover, molecules situated on nearby vibrational levels may possess sharply differing properties.

1. The degree of diffuseness of the absorption and emission spectra of polyatomic molecules is determined by the value of \(W\). Simple molecules are characterized by spectra with a discrete portion, while complex ones by continuous spectra having a typical form. The width and mutual arrangement of these spectra obey definite relations connected with the degree of interaction of the electronic and vibrational degrees of freedom, which depends on the value of \(W\).

2. The principal characteristics of a complex molecule are the probability of emission \(f\) and the probability of radiationless transitions \(d\). The first of these depends little on the store of vibrational energy \(Q\), whereas the second increases rapidly with increasing \(Q\). The relation between \(f\) and \(d\) determines the fluorescence yield \(\gamma\) and the lifetime of the excited state \(\tau\), which decrease according to known relations as the probability of radiationless transitions \(d\) increases.

What was said in item 3 about averaging at large values of \(W\) applies fully to the probabilities \(f\) and \(d\).

The conclusions formulated in items 1–7 determine the character of the phenomena occurring in a complex molecule during the absorption and emission of light. From these conclusions there follows a number of consequences. Some of these consequences, mainly concerning the material considered in the article, are listed in items 8–10.

1. In solutions, owing to the continuous action of the solvent on the molecule of the fluorescent substance, the probability of redistribution \(W\) is somewhat increased in comparison with the gas phase. On the other hand, since a high temperature is required for the evaporation of the substances under investigation, the probability \(W\) may be greater in the gas phase than in solutions. Therefore the spectra of solutions are in some cases more structured (in the sense of preserving individual maxima), and in other cases less structured, than the spectra of vapors. With sufficiently

large values of \(W\) the general form of the spectra is independent of the surrounding medium.

1. With respect to the spectra of the yield \(\gamma\) and the duration \(\tau\), the dissolved state leads to simpler relations (constancy of \(\gamma\) and \(\tau\)) than do the conditions of rarefied gases. This is due to vibrational deactivation of excited molecules, which rapidly come into thermal equilibrium with the solvent.

2. The preservation, in rarefied gases, of a store of vibrational energy by an excited molecule—which determines the difference between the properties of its fluorescence and those in solution—opens up new possibilities in experiment. By using monochromatic excitation, it becomes possible to dose the store of vibrational energy of the excited molecules (with an accuracy determined by the sharpness of the thermal distribution). By acting with foreign gases, it becomes possible to regulate this store. This determines ways of studying the influence of vibrational energy on the properties of the excited molecule, as well as of investigating the processes of transfer of vibrational energy in collisions. Such investigations substantially supplement the study of fluorescence in solutions. Unfortunately, to date there have been only isolated quantitative studies of the fluorescence of complex molecules in the gas phase.

PART II

INTRODUCTION

In Part I the principal experimental facts were considered and a general scheme was given for the phenomena relating to the influence of vibrational energy on the processes of absorption and fluorescence of complex molecules. As we have seen, its influence is extremely great.

In the present part an attempt is made at a theoretical investigation of these same phenomena. The starting points for this part of the article were the conclusions on the nature of the phenomena under consideration, formulated in §§ 7, 17, and 18 on the basis of an analysis of the experimental data. These conclusions were summarized in Section 8. Section 9 deals with the possible levels of vibrational energy of a complex molecule and with the localization of its vibrational energy in particular degrees of freedom. In the following sections, questions are considered concerning the redistribution of vibrational energy within a molecule and between molecules and the medium, and the law of fluorescence decay in solutions and vapors. Next, the dependences of the quantum yield of fluorescence on the store of vibrational energy of the excited molecule are considered, as well as certain questions of temperature quenching of fluorescence.

In this part of the article the conclusions of Section 8 are confirmed and developed, and relations are obtained which will undoubtedly be useful for explaining certain other properties of fluorescence.

9. ENERGY LEVELS AND LOCALIZATION OF VIBRATIONAL ENERGY IN COMPLEX MOLECULES

Let us briefly consider the question of the possible values of the vibrational energy of complex molecules. All conclusions will be valid both for unexcited and for excited electronic states. In doing so we shall assume that the variables of electronic motion and the variables of nuclear vibration are independent of one another, and that one may speak independently of the electronic and the vibrational energies. Of course, this is true only approximately.

The value of the vibrational energy of a given electronic state is expressed by the formula:

\[ \left. \begin{aligned} Q={}&\sum_{i=1}^{L}\nu_i\left(v_i+\frac{1}{2}\right) +\sum_{i,j}\alpha_{ij}\left(v_i+\frac{1}{2}\right)\left(v_j+\frac{1}{2}\right)+{}\\ &+\sum \beta_{ijk}\left(v_i+\frac{1}{2}\right)\left(v_j+\frac{1}{2}\right)\left(v_k+\frac{1}{2}\right)+\ldots \end{aligned} \right\} \tag{1} \]

Here \(L\) is the total number of vibrational degrees of freedom, \(v_i, v_j,\ldots\) are vibrational quantum numbers, and \(\nu_i\) is the frequency of the \(i\)-th normal vibration. The first sum gives the energy in the harmonic approximation; all the others are corrections for anharmonicity*). For complex molecules the anharmonicity is very large, since the total store of vibrational energy at ordinary temperatures is very high and the vibrations cannot be regarded as small. One could limit oneself to the first term only at very low temperatures.

It is not difficult to show that the vibrational energy of complex molecules passes through a continuous series of values, and that discreteness of the energy is retained only for very small \(Q\). To this end let us make the simplest calculation, using only the first sum of expression (1). Suppose we have a system consisting

*) In the concept of anharmonicity we include the interaction between normal vibrations.

only of two degrees of freedom with frequencies \(\nu_1 = 101\ \mathrm{cm}^{-1}\) and \(\nu_2 = 130\ \mathrm{cm}^{-1}\).

It follows from (1) that in the interval from \(0\) to \(300\ \mathrm{cm}^{-1}\) there are in all only six different levels*:

\[ E(v_1 = 0,\ v_2 = 0)=0; \]

\[ E(v_1 = 1,\ v_2 = 0)=101\ \mathrm{cm}^{-1}; \quad E(v_1 = 0,\ v_2 = 1)=130\ \mathrm{cm}^{-1}; \]

\[ E(v_1 = 2,\ v_2 = 0)=202\ \mathrm{cm}^{-1}; \quad E(v_1 = 1,\ v_2 = 1)=231\ \mathrm{cm}^{-1}; \]

\[ E(v_1 = 0,\ v_2 = 2)=260\ \mathrm{cm}^{-1}. \]

In the energy interval \(1000\)—\(1300\ \mathrm{cm}^{-1}\), equal in magnitude but lying somewhat higher, there are already not six, but 30 energy levels: \(1010\), \(1011\), \(1025\), \(1039\), \(1040\), \(1054\), \(1068\), \(1083\), \(1097\), \(1111\), \(1112\), \(1126\), \(1140\), \(1141\), \(1155\), \(1169\), \(1170\), \(1184\), \(1198\), \(1212\), \(1227\), \(1241\), \(1242\), \(1256\), \(1270\), \(1271\), \(1285\), \(1299\), and \(1300\ \mathrm{cm}^{-1}\).

The average distance between energy levels decreases from \(50\ \mathrm{cm}^{-1}\) to \(10\ \mathrm{cm}^{-1}\). If one carries out an analogous calculation for the energy interval \(5000\)—\(5300\ \mathrm{cm}^{-1}\), it will contain 120 levels, with an average spacing of only \(2.5\ \mathrm{cm}^{-1}\). The result obtained is not difficult to generalize also to complex molecules with a large number of degrees of freedom. In this case the number of different \(\nu_i\) is very large. In calculations by (1) we obtain a sharp increase in the number of possible energy levels, growing as the molecule becomes more complex. In all cases the density of the levels will rapidly increase as \(Q\) increases. It follows from this that in a complex molecule all energy values are admissible, each of which is multiply expressed. Discrete values can be observed only for values of \(Q\) very little different from the values of zero energy, and only for “semi-complex” molecules. At first glance it may seem that this continuity is only apparent, and that different states of the molecule possessing one and the same, or almost one and the same, value of the energy may be completely different. Let us explain this by an example. For our system of two degrees of freedom, an energy value close to \(1110\ \mathrm{cm}^{-1}\) can be realized in two ways. It may be that all the energy is concentrated in the first degree of freedom and its 11th vibrational quantum is excited \((Q = 1111\ \mathrm{cm}^{-1})\). On the other hand, there may be realized a state in which two quanta of the first degree of freedom and seven quanta of the second degree of freedom are excited \((Q = 1112\ \mathrm{cm}^{-1})\). At first glance both these states are physically different, since they correspond to an entirely different localization of the vibrational energy. In reality this is not so. Such a conclusion would be valid if the anharmonicity of the vibrations were negligible and if the separate degrees of freedom were completely independent—

* The value of the zero energy is taken as zero.

tems from one another. In the presence of anharmonicity the concept of a normal vibration loses its meaning. In the potential function there appear terms characterizing the interaction of normal coordinates, as a result of which localization of the energy in a given degree of freedom becomes impossible. Under the influence of anharmonicity the molecule continuously passes from one state into another (for example, from the state \(v_1=11,\ v_2=0\) to the state \(v_1=2,\ v_2=7\)), i.e., the localization of vibrational energy changes continuously, and the vibrational energy is continuously displaced over the molecule. If at the initial moment of time an additional reserve of vibrational energy has been released in one or several degrees of freedom, then a redistribution of energy over all degrees of freedom takes place, as a result of which a certain equilibrium distribution is established. Subsequently only certain comparatively small fluctuations may arise.

The time required for the establishment of equilibrium depends on the magnitude of the interaction of the degrees of freedom and decreases as the molecule becomes more complex or as the reserve of vibrational energy increases (i.e., in particular, as the temperature increases). Usually it is considerably shorter than the lifetime of the molecule in the excited electronic state, although, of course, exceptions may occur.

Thus, the energy levels of a complex molecule form a continuous spectrum. They correspond to a continuous totality of infinitely close states of the molecule. To each value of the energy there corresponds a very large number of ways of localizing the energy among the various degrees of freedom. It follows from this that any discreteness of the properties of the vibrational energy is completely lost, and a complex molecule with a large reserve of vibrational energy may be regarded as a classical system obeying the laws of classical mechanics and classical statistics. In particular, in the presence of an equilibrium distribution of the vibrational energy of a molecule among the various degrees of freedom, one may speak of the intrinsic temperature of a complex molecule. The latter is valid, of course, only for the limiting case of very complex molecules.

The considerations set forth are in complete agreement with the conclusions drawn in Sections 2, 3, and 7.

10. REDISTRIBUTION OF VIBRATIONAL ENERGY WITHIN A MOLECULE

We have already established that within a molecule there occurs a continuous redistribution of vibrational energy among the various degrees of freedom. To each possible value of the energy there corresponds a very large number of ways of localizing this energy—

a large number of degenerate states. Let us consider the process of establishing an equilibrium distribution under arbitrary initial conditions. The initial state is nonequilibrium and is connected with an act of absorption or emission of light, or with some other external action on the system. We shall consider the process of redistribution of vibrational energy in the upper electronic state. In view of this it is necessary simultaneously to consider the course in time of two other processes—the processes of transition to the lower electronic state with radiation and without radiation. The physical characteristic of nonradiative transitions was given above (see Section 5). It is more convenient to carry out the mathematical solution of the problem for two degenerate states and then to make the corresponding generalization. In Fig. 17 two degenerate vibrational levels of the molecule are shown, corresponding to the energy \(Q^*\). Let us denote by the letters \(f_2\) and \(f_3\), \(d_2\) and \(d_3\) the probabilities of transition (per 1 second) to the lower electronic state with radiation and without radiation. Let us denote by the letter \(r\) the probability of transition of the molecule from the excited level 2 to the excited level 3, and by the letter \(s\) the probability of the reverse transition.

By the letters \(n_2(t)\) and \(n_3(t)\) we shall denote the number of molecules on the corresponding levels. The change in the number of particles \(n_2\) and \(n_3\) with time is determined by the obvious equations \((p_2=f_2+d_2,\ p_3=f_3+d_3)\):

\[ \begin{aligned} dn_2&=-p_2 n_2\,dt-rn_2\,dt+sn_3\,dt,\\ dn_3&=-p_3 n_3\,dt-sn_3\,dt+rn_2\,dt. \end{aligned} \tag{2} \]

The solution of this system for given initial conditions has the form

\[ \begin{aligned} n_2={}&\left\{\left(\frac{p_2-p_3+r-s+2M}{4M}\right)n_2(0)-\frac{2s}{4M}n_3(0)\right\}e^{-P_1t}+\\ &+\left\{\left(\frac{p_3-p_2+s-r+2M}{4M}\right)n_2(0)+\frac{2s}{4M}n_3(0)\right\}e^{-P_2t},\\[6pt] n_3={}&\left\{\left(\frac{p_2-p_3+s-r+2M}{4M}\right)n_3(0)-\frac{2r}{4M}n_2(0)\right\}e^{-P_1t}+\\ &+\left\{\left(\frac{p_2-p_3+r-s+2M}{4M}\right)n_3(0)+\frac{2r}{4M}n_2(0)\right\}e^{-P_2t}, \end{aligned} \tag{3} \]

where

\[ P_{1,2}=\frac{1}{2}\left(p_2+p_3+r+s\pm 2M\right), \]

\[ M=\frac{1}{2}\sqrt{(p_2+p_3+r+s)^2-4(p_2s+p_3r+p_3p_2)}. \]

From formula (3) it follows that, in the presence of interaction of the excited levels, the numbers of molecules \(n_2\) and \(n_3\) decrease not according to an exponential law. The same applies to the sum \(n_2+n_3\), i.e., to the total number of molecules in the excited electronic state. This result is very essential, since for a nondegenerate state (for a single excited level) the decrease in the number of particles during radiation and in the presence of radiationless transitions is always strictly exponential.

Fig. 17. Scheme of various processes in a complex molecule with two pronounced levels of the upper electronic state.

Let us now consider the most important particular case of formulas (3). Let \(r\) and \(s\) be much larger than \(p_2\) and \(p_3\), i.e., let the probabilities of redistribution of vibrational energy be much larger than the probabilities of fluorescence and of radiationless transitions. In this approximation \([n_2(0)+n_3(0)=n(0)]\)

\[ \left. \begin{aligned} n_2 &= \left[\frac{r}{r+s}\,n(0)-n_3(0)\right] e^{-(r+s)t} + \frac{s}{r+s}\,n(0)\, e^{-\left(\frac{s}{r+s}p_2+\frac{r}{r+s}p_3\right)t}, \\ n_3 &= \left[n_3(0)-\frac{r}{r+s}\,n(0)\right] e^{-(r+s)t} + \frac{r}{r+s}\,n(0)\, e^{-\left(\frac{s}{r+s}p_2+\frac{r}{r+s}p_3\right)t}. \end{aligned} \right\} \tag{4} \]

The expressions for \(n_2\) and \(n_3\) are the sum of two terms, which determine an entirely different dependence of \(n_2\) and \(n_3\) on time. Let, for example, the order of magnitude of \(p_2\) and \(p_3\) be \(10^8\ \mathrm{sec}^{-1}\), while \(r\) and \(s\) are \(10^{13}\ \mathrm{sec}^{-1}\). Then the first term has an appreciable value only during the time interval from \(t=0\) to \(t=10^{-13}\)—\(10^{-12}\ \mathrm{sec}\). At larger values of \(t\) the first term becomes vanishingly small and, consequently, the subsequent change of \(n_2(t)+n_3(t)\) is determined only by the second term, i.e., it is strictly exponential. Thus, if we investigate the dependences \(n_2(t)\) and \(n_3(t)\) experimentally, but cannot control time intervals significantly smaller than \(10^{-9}\ \mathrm{sec}\), then we shall detect an exponential decrease of \(n_2(t)\) and \(n_3(t)\), as well as of \(n_2(t)+n_3(t)\).

The result obtained is of very great importance. If the probabilities of redistribution of vibrational energy within the molecule are considerably greater than the probabilities of fluorescence and radiationless transitions, then the decrease in the number of molecules with a prescribed value of the energy \(Q^*\) must be exponential. This conclusion is also valid for molecules with a large number of pronounced vibrational levels of the excited electronic state. If their number is equal to \(Z\), then expressions of type (3) for \(n_1, n_2,\ldots,n_Z\) will contain \(Z\) terms, and in \((Z-1)\) terms the values of \(p_i\) will be, in order of magnitude, equal to the probabilities of redistribution of vibrational energy and, consequently, will have appreciable magnitude only in the first instants of time, which usually elude observation.

Let us consider the dependences \(n_2(t)\) and \(n_3(t)\) in more detail. Neglecting the first term in (4), we obtain:

\[ \left. \begin{aligned} n_2(t)&=\frac{s}{r+s}\,n(0)\,e^{-\left(\frac{s}{r+s}p_2+\frac{r}{r+s}p_3\right)t},\\[6pt] n_3(t)&=\frac{r}{r+s}\,n(0)\,e^{-\left(\frac{s}{r+s}p_2+\frac{r}{r+s}p_3\right)t}. \end{aligned} \right\} \tag{5} \]

The coefficients \(\dfrac{s}{r+s}\) and \(\dfrac{r}{r+s}\), standing before the exponential, have a quite definite meaning. Namely, \(\dfrac{s}{r+s}\) gives the fraction of molecules located on level 2, and \(\dfrac{r}{r+s}\) on level 3.

Indeed,

\[ \frac{s}{r+s}=\frac{n_2}{n_2+n_3}=g_2,\qquad \frac{r}{r+s}=\frac{n_3}{n_2+n_3}=g_3. \tag{6} \]

It is very important that the values \(g_2\) and \(g_3\) do not depend on time. Thus, in those intervals of time when formula (5) is valid \(\left(t\gg \dfrac{1}{r+s}\right)\), the distribution of molecules over the different states (localization of vibrational energy over the different degrees of freedom) is stationary. It is not stationary for the preceding instants of time. In the initial instants of time the relative fraction of molecules on levels 2 and 3 changes continuously \(\left(g_2 \text{ from } \dfrac{n_2(0)}{n(0)} \text{ to } \dfrac{s}{r+s},\; g_3 \text{ from } \dfrac{n_3(0)}{n(0)} \text{ to } \dfrac{r}{r+s}\right)\), and only thereafter is the equilibrium distribution established. It should be noted that the equilibrium distribution is completely

does not depend on the initial conditions or on the method of excitation and is determined by the internal properties of the molecule (the probabilities of distribution \(r\) and \(s\)).

Let us now consider the exponents in expression (5). They are the same for \(n_2(t)\) and \(n_3(t)\) and are equal to

\[ P = g_2 p_2 + g_3 p_3 . \tag{7} \]

The value \(P\) determines the true probability of transition from any sublevel of the excited state to the lower electronic state. For \(r=s\), \(P=\frac{1}{2}(p_1+p_2)\), i.e., the mean value of the initial transition probabilities \((2\to 1)\) and \((3\to 1)\). For \(s \ne r\), the value \(P\) is the mean of \(p_2\) and \(p_3\), taken with the weights \(g_2\) and \(g_3\), respectively.

In the general case of a large number of expressed states strongly interacting with one another, instead of (5) we have:

\[ n_i = n(0) g_i e^{-Pt}, \tag{8} \]

where

\[ g_i = \frac{n_i}{\sum_i n_i}, \tag{9} \]

\[ P = \sum_i g_i p_i . \tag{10} \]

Here \(g_i\) is the distribution function over the various excited states after the attainment of an equilibrium distribution, and \(P\) is the total probability of transition from the excited electronic state to the lower state. The value \(P\) is the mean of all \(P_i\) and is calculated in the usual way. The transition probabilities with and without radiation undergo analogous averaging. Indeed,

\[ P = \sum_i g_i(f_i+d_i)=\sum_i g_i f_i+\sum_i g_i d_i=f+d. \]

It follows from this that the true probability of transition with radiation from the excited electronic state to the unexcited state is

\[ f=\sum_i g_i f_i . \tag{11} \]

Similarly we have:

\[ d=\sum_i g_i d_i . \tag{12} \]

The change in the number of excited molecules with time is determined by the simple formula:

\[ n=\sum_i n_i=\sum_i g_i n(0)e^{-(f+d)t}=Ne^{-(f+d)t}. \tag{13} \]

The lifetime of a molecule in the excited state is equal to

\[ \tau=\frac{1}{f+d}=\frac{1}{\sum_i g_i(f_i+d_i)} . \tag{14} \]

Thus, owing to the redistribution of the vibrational energy of the molecule over the various possible states, the latter is characterized by averaged values of all transition probabilities and by an averaged lifetime in the excited state. The probabilities \(r\) and \(s\), generalized for the case of a complex molecule, represent the probability \(W\), introduced in Section 2, of redistribution of vibrational energy within the molecule.

All the conclusions obtained are valid for an excited molecule with a quite definite store of vibrational energy \(Q^*\). For another value of \(Q^*\), analogous results are valid. However, another value of \(Q^*\) will correspond to its own value of the averaged transition probabilities with and without radiation. Thus, the probabilities \(f\) and \(d\) are functions of the store of vibrational energy of the molecule. If the molecule under investigation is sufficiently complex, and its store of vibrational energy is large and, consequently, it may be regarded as a classical system, then \(f\) and \(d\) must be monotonic functions. The dependences of \(f\) and \(d\) on \(Q^*\) for each molecule may be different and are determined by its structure and, in particular, by the interaction of the motion of the electrons and nuclei. At present it is impossible to obtain the form of the functions \(f(Q^*)\) and \(d(Q^*)\) from any theoretical considerations, and they are subject to experimental determination. Let us recall that from the experimental data\(^{18}\) (see also Section 2) it follows that \(f\)—the probability of a transition with radiation—depends very little on \(Q^*\), and in approximate calculations it may be regarded as constant. On the other hand, experiment shows that the probabilities of transitions without radiation rapidly increase with increasing store of vibrational energy of the molecule (see Fig. 11). This result is valid only for large \(Q^*\). It should be emphasized that the quantities \(f(Q^*)\) and \(d(Q^*)\) are the most important characteristics of the excited electronic state.

11. DISTRIBUTION OF EXCITED MOLECULES OVER VIBRATIONAL ENERGY LEVELS

In any experimental investigation of the phenomena of fluorescence and absorption, we are dealing with an enormous ensemble of molecules. Since the individual properties of molecules depend on the store of vibrational energy, an essential characteristic of the entire ensemble of molecules is their distribution with respect to the store of vibrational energy in both the lower and upper electronic states. As we shall see below, all the basic properties of the fluorescence of complex molecules, both in vapors and in solutions, depend on the form of the distribution function. Let us first consider the form of the distribution function of molecules over the vibrational levels of the lower electronic state prior to the act of excitation. It corresponds to the state of equilibrium established through collisions of molecules with one another and interactions with the external medium, and is determined by the temperature of the medium. Since each molecule is a complex classical system and is part of the entire gas or solution with which it is in a state of statistical equilibrium, the probability of finding a molecule with an energy value in the interval from \(Q\) to \(Q + dQ\) is determined by the usual Gibbs function

\[ dW = AZ(Q)e^{-\frac{Q}{kT}}\,dQ = \rho(Q)\,dQ, \tag{15} \]

where \(Z(Q)\) is the total number of possible expressed states of the molecule in the given energy interval, i.e., the number of ways of distributing the vibrational energy differently within the molecule. The form of the function \(Z(Q)\) is unknown. From the considerations of Section 9 it is clear only that \(Z\) increases very rapidly with \(Q\), and, for a given \(Q\), increases rapidly as the molecule becomes more complex*). The function \(\rho(Q)\) has a sharp maximum close to \(E = LkT\), where \(L\) is the total number of effective vibrational degrees of freedom. We use the concept of effective degrees of freedom because some vibrational degrees of freedom may not participate in the exchange of energy, and therefore \(L\) is smaller than the total number of degrees of freedom. Only at very high temperatures can \(L\) reach the total number of degrees of freedom of the molecule. Let us now determine the form of the distribution function of molecules over the vibrational levels

*) In the literature one usually uses the expression \(Z(Q)=Q^{\frac{L}{2}-1}\), where \(L\) is the number of degrees of freedom of the molecule. It should be kept in mind that this expression is inaccurate, since it was obtained under the assumption of the independence of the individual degrees of freedom. It can serve only for qualitative estimates.

energy of the excited electronic state. It is different for gases and solutions, and therefore a special study of each case is necessary.

Let us begin with vapors. We shall consider the case of monochromatic excitation in the Stokes region of the spectrum, i.e., for \(\nu>\nu_{\mathrm{el}}\). Each molecule that has absorbed will receive one and the same additional store of vibrational energy \(\Delta Q^* = h\nu - h\nu_{\mathrm{el}}\). If it is assumed that the probability of absorption of a quantum does not depend on the initial store of vibrational energy*), then the form of the distribution function over vibrational levels will be the same in the upper and lower electronic states, but shifted along the vibrational-energy scale by the amount \(\Delta Q^*\) (Fig. 18). The probability of finding an excited molecule with a store of vibrational energy smaller than \(\Delta Q^*\) is equal to zero. What is most important for what follows is that the mean vibrational energy of all excited molecules is greater by \(\Delta Q^*\) than the mean vibrational energy of the unexcited molecules. It should also be emphasized that, under continuous illumination, the form of the distribution function will remain unchanged. This is true, of course, only for vapors, since in this case there is no exchange of vibrational energy and the molecule retains its entire store of vibrational energy until the act of fluorescence or some nonradiative transition takes place.

After the excitation is stopped, the form of the distribution function will change, since molecules with a large store of energy leave the excited state considerably earlier (\(d(Q^*)\) increases with \(Q^*\)). Figure 19 gives the change in the distribution function that should be expected in this case.

The form of the distribution function considered is valid only under monochromatic excitation. Under excitation by several wavelengths, the result of the action of each of them may be considered independently, and the distribution function will have several maxima. Under excitation by a continuous spectrum, the integral result must be considered. In any case it is obvious that, under excitation by a continuous spectrum, the distribution function will not be sharp, and the possible values of the store of vibrational energy will be highly varied.

Let us now turn to the form of the distribution function of excited molecules located in solution. In this case, owing to the strong interaction of the fluorescing molecules with mole-

*) This assumption, of course, is not strictly fulfilled. However, we have already seen above that, from the experimental data, it follows that the probability of emission \(f(Q^*)\) depends only weakly on \(Q^*\). This, of course, is also true for the probability of absorption. The dependence of the absorption probability on \(Q\) will somewhat change the form of the distribution function, which, however, will not affect the results of our arguments.

Fig. 18. Distribution of vapor molecules by their store of vibrational energy in the upper and lower electronic states.

Labels in the figure: \(\Delta Q^*\), \(\nu_{\text{el}}\).

Fig. 19. Change in the form of the distribution function of vapor molecules in the excited electronic state after the cessation of excitation.

Labels in the figure: \(\rho(Q^*, t)\), \(Q^*\);
I — \(t = 0\),
II — \(t_1\),
III — \(t_2 > t_1\).

of solvent molecules, there occurs a continuous and rapid exchange of vibrational energy. As in the case of vapors, after the act of absorption of light with frequency \(\nu > \nu_{\mathrm{el}}\), all absorbing molecules receive an additional amount of vibrational energy \(\Delta E\). Possessing an excess of energy, they pass into a nonequilibrium state and must give up their excess energy to the molecules of the medium. If the interaction with the medium is sufficiently great, then in a time considerably shorter than the lifetime in the excited electronic state, all excited molecules will come into thermal equilibrium with the medium. After equilibrium is established, the form of the distribution function in the excited electronic state will be the same as in the lower electronic state (see formula (15)); in Fig. 20 this is shown graphically. All that has been said is valid, of course, if the probability of exchange of vibrational energy is very large. Such an assumption is entirely justified and is supported by the whole set of experimental facts considered above in Part I. If the lifetime of molecules in the excited electronic state is of the order of \(10^{-8}\) sec, then the establishment of thermal equilibrium usually occurs in a time two or three orders of magnitude shorter.

Fig. 20. Distribution of solute molecules according to the store of vibrational energy in the upper and lower electronic states.

In contrast to vapors, this form of the distribution function over the vibrational levels of the excited state is established not only under monochromatic excitation, but also under excitation by a continuous spectrum.

Moreover, after the excitation ceases, the form of the function also remains unchanged, since any disturbance of it is very rapidly eliminated by the continuing exchange of vibrational energy with the medium.

12. LAWS OF FLUORESCENCE DECAY OF VAPORS AND SOLUTIONS

Let us first consider the law of fluorescence decay of vapors. The form of the distribution function of vapor molecules over the vibrational levels of the excited state is shown in Fig. 18 (under monochromatic excitation). The number of molecules with energy from \(Q^*\) to \(Q^* + dQ^*\) is equal to \(N \rho(Q^*)\,dQ^*\), where \(N\) is the total number of excit-

excited molecules, which remains constant under continuous illumination.

The number of molecules passing into the unexcited state with emission per unit time, under conditions of a stationary regime, is equal to

\[ \int_{Q^*=0}^{\infty} f(Q^*)\,N\rho(Q^*)\,dQ^* . \tag{16} \]

It also remains constant. After the excitation is stopped, the total number of excited molecules rapidly decreases. Correspondingly, the fluorescence intensity falls. The change in the number of excited molecules with an energy reserve \(Q^*\) is given by expression (13). The change in the number of excited molecules with energy in the interval \(dQ^*\) is equal to

\[ dN(Q^*,t)=dN(0)e^{-[f(Q^*)+d(Q^*)]t}= \]

\[ =N\rho(Q^*)e^{-[f(Q^*)+d(Q^*)]t}\,dQ^* . \tag{17} \]

Since in vapors the molecules retain their reserve of vibrational energy during the entire time they remain in the excited state, \(f\) and \(d\) do not depend on time. The number of molecules passing, in 1 sec., into the unexcited state with emission of light quanta, at the time \(t\) after the cessation of excitation, is equal to

\[ \int_{Q^*=0}^{\infty} f(Q^*)\,dN(t) = N\int_{Q^*=0}^{\infty} f(Q^*)\rho(Q^*)e^{-[f(Q^*)+d(Q^*)]t}\,dQ^* . \tag{18} \]

The fluorescence intensity is equal to

\[ I(t)=N\cdot h\nu_{\mathrm{cp}} \int_{Q^*=0}^{\infty} f(Q^*)\rho(Q^*)e^{-[f(Q^*)+d(Q^*)]t}\,dQ^*, \tag{19} \]

where \(\nu_{\mathrm{cp}}\) is the mean frequency of fluorescence. Since a sum of exponentials is not an exponential, \(I(t)\) in the general case does not change according to an exponential law. An exponential law should be realized approximately, but only under monochromatic excitation. Indeed, under monochromatic excitation the distribution function is sufficiently sharp. Therefore, by the theorem on the mean,

\[ I(t)\cong Nh\nu_{\mathrm{cp}}\overline{f(Q^*)}\, e^{-[\overline{f(Q^*)}+\overline{d(Q^*)}]t} = I_0e^{-[\overline{f(Q^*)}+\overline{d(Q^*)}]t}. \tag{20} \]

The sharper the distribution function and the weaker the dependence of \(f\) and \(d\) on \(Q^*\), the more accurately the exponential law of fluorescence decay is obeyed. Since, as the molecule becomes more complicated, both the absolute width (absolute fluctuations) of the distribution function and the value

\[ \frac{d[d(Q^*)]}{dQ^*} \]

increase, the deviation from the exponential should be more clearly expressed precisely for the most complex molecules. Further, it is not difficult to show that the deviation from the exponential is most substantial in the first moments of time after the excitation is discontinued. Indeed, in Fig. 19 we saw that after the excitation is discontinued the form of the distribution function changes in the direction of its narrowing. Therefore, as \(t\) increases, the approximate formula (20) will become ever more accurate*).

It must be emphasized that the duration of the excited electronic state of molecules in the gas phase must depend on the wavelength of the exciting light. From formula (20) it follows that the value of \(\tau\) is equal to

\[ \tau=\frac{1}{[f(Q^*)+d(Q^*)]}. \tag{21} \]

If the distribution function is sufficiently sharp, then one may write approximately:

\[ \tau=\frac{1}{f(\overline{Q^*})+d(\overline{Q^*})}. \tag{21a} \]

With an increase in the frequency of the exciting light, the reserve of vibrational energy of each excited molecule increases \((\Delta Q^*=h\nu-h\nu_{\text{el}}\) increases). As a result, their average energy also increases. Since \(d\) rapidly increases with \(Q^*\), as the wavelength of the exciting light decreases, a very rapid decrease of \(\tau\) should occur.

Everything said here agrees with the explanation given in \(^{18}\), where the phenomenon of a decrease of \(\tau\) with increasing \(Q^*\) for vapors of aromatic compounds was discovered.

Under nonmonochromatic excitation the law of fluorescence decay is not exponential. Indeed, if the data of Fig. 10 had been obtained not for each wavelength of the exciting light independently, but under simultaneous ex-

*) We made an approximate calculation by integrating (19) under the assumption that \(d=Ae^{\alpha Q^*}\), and that \(\rho(Q^*)\) has a simple rectangular form. The corresponding values of \(A\) and \(\alpha\) were taken from the work on the fluorescence of \(\beta\)-naphthylamine. It turned out that for \(t>0.8\tau\) the fluorescence decay is almost strictly exponential. Deviations from the exponential for \(t<0.8\tau\) are also small.

VIBRATIONAL ENERGY AND LUMINESCENCE

excitation by all wavelengths, then the dependence of \(\dfrac{F_0}{F_z}\) on \(z\) would not be rectilinear.

Let us turn to the decay of fluorescence in solutions. Strange as it may seem at first glance, the laws of decay, as well as other properties of the fluorescence of solutions, sometimes prove to be simpler (see also section 3). The reason for this is the very strong interaction between the molecules of the fluorescing substance and the molecules of the solvent. We have already seen that, as a result of such interaction, the form of the distribution function does not depend on the wavelength of the exciting light and is determined only by the temperature of the medium. As a result of this interaction it also turns out that the law of fluorescence decay is strictly exponential even under nonmonochromatic excitation. To prove this fundamental proposition one may use formulas (3)—(5) of section 10. Let us consider two vibrational energy levels of an excited molecule (Fig. 21): \(Q_2^*\) and \(Q_3^*\). The probabilities of transitions to the lower electronic state with emission are \(f(Q_2^*)\) and \(f(Q_3^*)\), the probabilities of transitions without emission are \(d(Q_2^*)\) and \(d(Q_3^*)\). Let us denote by the letter \(s\) the probability of transfer of vibrational energy \(Q_3^* - Q_2^*\) from the molecule to the medium, i.e. the probability of the transition of the molecule from the state with energy \(Q_3^*\) to the state with energy \(Q_2^*\), and by the letter \(r\) the probability of receiving from the medium the energy \(Q_3^* - Q_2^*\), i.e. the probability of the transition of the molecule from the state \(Q_2^*\) to the state \(Q_3^*\). In the course of the last two transitions the molecule retains its excited electronic state. The probabilities \(r\) and \(s\) are not equal to one another. If \(Q_3^*\) is large compared with the mean value of the molecular energy at a given temperature, then \(s\) is considerably greater than \(r\) (the probability of approaching equilibrium is considerably greater than the probability of fluctuation). Suppose further that at the moment of time \(t=0\), i.e. at the moment of excitation of the molecules, the numbers of molecules on levels (2) and (3) are equal to \(N_2(0)\) and \(N_3(0)\). Then the number of molecules on these levels at any moment of time will be determined by formulas (3). If \(r\) and \(s\), or at least only \(s\), are considerably greater, \(p(Q_2^*)=\)

Fig. 21. Diagram of various processes occurring in a molecule in the presence of exchange of vibrational energy with solvent molecules.

\[ =f(Q_2^*)+d(Q_2^*) \quad \text{and} \quad p(Q_3^*)=f(Q_3^*)+d(Q_3^*), \]
i.e., if the probability of redistribution of vibrational energy between the fluorescing molecules and the medium is significantly greater than the probability of transitions to the lower electronic state, then formula (4) is applicable, with the first term needing to be taken into account only during a very short interval of time preceding the establishment of equilibrium. A detailed analysis of these formulas was given in Section 10 and is fully applicable to the present case as well (one need only remember that the physical content of the probabilities \(p(Q_2^*)\), \(p(Q_3^*)\) and \(r\) and \(s\) is now different). From formulas (5) and (6) it follows that the number of molecules with energies \(Q_2^*\) and \(Q_3^*\) is equal to \((N=N_2(0)+N_3(0))\):

\[ \left. \begin{aligned} N_2&=\frac{s}{r+s}Ne^{-\left[\frac{s}{r+s}p(Q_2^*)+\frac{r}{r+s}p(Q_3^*)\right]t} \\ &=\rho_2Ne^{-\left[\rho_2p(Q_2^*)+\rho_3p(Q_3^*)\right]t}, \\[6pt] N_3&=\frac{r}{r+s}Ne^{-\left[\frac{s}{r+s}p(Q_2^*)+\frac{r}{r+s}p(Q_3^*)\right]t} \\ &=\rho_3Ne^{-\left[\rho_2p(Q_2^*)+\rho_3p(Q_3^*)\right]t}, \end{aligned} \right\} \tag{22} \]

where \(\rho_2=\dfrac{N_2}{N_2+N_3}\) and \(\rho_3=\dfrac{N_3}{N_2+N_3}\) are the fractions of molecules with energies \(Q_2^*\) and \(Q_3^*\), respectively. It is seen from (22) that these fractions do not depend on time, i.e., the distribution of molecules over the vibrational energy levels remains unchanged throughout the entire lifetime of the molecules in the excited electronic state.

The result obtained is not difficult to generalize to the case of all possible values of the vibrational energies of the excited molecules. In this case the number of molecules in the energy interval from \(Q^*\) to \(Q^*+dQ^*\), after equilibrium with the medium has been established, is equal to

\[ dN(Q^*,t)=N(t)\rho(Q^*)\,dQ^* = \]

\[ =N(0)\rho(Q^*)e^{-\left[\int_{Q^*=0}^{\infty}\{f(Q^*)+d(Q^*)\}\rho(Q^*)\,dQ^*\right]t}\,dQ^*, \tag{22a} \]

where \(\rho(Q^*)\) is the unchanged distribution function of the excited molecules over the vibrational energy levels of the solution (see Fig. 20), and \(N(t)\) is the total number of excited molecules at time \(t\). It follows from (22) that the decrease in the number of excited molecules as a result of fluorescence and radiationless transitions does not depend on \(Q^*\) and obeys one and the same exponential law. Thus, although \(p(Q^*)=f(Q^*)+d(Q^*)\) depends strongly on the store of vibrational energy \(Q^*\), the lifetime of the molecule at

of all vibrational levels is identical. Any violation of the equilibrium distribution that might arise owing to the nonuniformity of transitions to the lower electronic state from different vibrational levels is immediately compensated by the continuous redistribution of the vibrational energy of the excited molecules and the medium. It should be mentioned once again that the form of the distribution function does not depend on the method of excitation and, in particular, on the wavelength of the exciting light. For monochromatic and nonmonochromatic excitation the value of \(\rho(E)\) is the same. This is obvious, since expression (22) does not depend on the initial conditions.

Let us now write the expression for the law of decay of the fluorescence intensity after the cessation of excitation. The energy of radiation of all molecules passing into the unexcited electronic state in 1 sec. is equal to

\[ \begin{aligned} J(t) &= h\nu_{\mathrm{cp}}\int_{Q^*=0}^{\infty} f(Q^*)\,dN(Q^*,t) \\ &= h\nu_{\mathrm{cp}}\cdot N(0)\int_{Q^*=0}^{\infty} f(Q^*)\rho(Q^*)\,dQ^* \,e^{-\left[\int_{Q^*=0}^{\infty}\{f(Q^*)+d(Q^*)\}\rho(Q^*)\,dQ^*\right]t} \\ &= h\nu_{\mathrm{cp}}\cdot N(0)\overline{f(Q^*)}\,e^{-(\overline f+\overline d)t} = I_0 e^{-(\overline f+\overline d)t}. \end{aligned} \tag{23} \]

Here

\[ \overline{f(Q^*)}=\int_{Q^*=0}^{\infty} f(Q^*)\rho(Q^*)\,dQ^* \tag{24} \]

and

\[ \overline{d(Q^*)}=\int_{Q^*=0}^{\infty} d(Q^*)\rho(Q^*)\,dQ^* \tag{25} \]

are the averaged values of \(f(Q^*)\) and \(d(Q^*)\) over all excited molecules. The redistribution of vibrational energy between fluorescing molecules and the medium leads to the loss of the individual properties of each separate molecule. They are characterized not by their own probabilities of transition to the lower electronic state, but by one and the same probability common to all excited molecules.

Expression (23) shows that the decay of the fluorescence of solutions is strictly exponential. The lifetime of all molecules in the excited state is equal to

\[ \tau=\frac{1}{\overline{f(Q^*)}+\overline{d(Q^*)}}. \tag{26} \]

Since \(\rho(Q^*)\) does not depend on \(\lambda\), it follows from formulas (23) and (26) that the fluorescence decay law and the lifetime of molecules in the excited electronic state do not depend on the wavelength of the exciting light. It is essential to note that the duration of the excited state of solutions must be greater than the duration of the excited state of vapor molecules. Indeed, upon excitation in the Stokes region of the spectrum, the mean vibrational energy of the excited molecules increases. In vapors the excess energy is conserved; in solutions it is transferred to the medium. The mean vibrational energy of solutions is lower than in vapors. Since \(\overline{d(Q^*)}\sim d(\overline{Q^*})\), it follows hence that the value of \(d(\overline{Q^*})\) in solutions is lower than in vapors, and consequently, \(\tau_{\text{soln}}>\tau_{\text{vap}}\).

All these results are in complete agreement with experiment. (See above, Sections 4 and 8 of the first part.)

In conclusion to the present section, let us consider the specific features of fluorescence processes before equilibrium is established. All our conclusions were valid under the assumption that the probabilities of redistribution of vibrational energy within a molecule and between fluorescing molecules and the medium are significantly greater than the probabilities of transitions with and without radiation.

If this condition is not satisfied, then formula (21) and its more general form (22) are invalid. The interval of time during which the number of particles will undergo an exponential decrease and, consequently, the fluorescence intensity will undergo an exponential decrease will be preceded by a period of establishment of equilibrium, characterized by deviation from the exponential. The magnitude of this period will be the greater, the smaller the probability of redistribution of vibrational energy. Since the probabilities of redistribution are the smaller, the smaller the reserve of vibrational energy, the period of deviation from the exponential should increase as the temperature decreases. Indications of the presence of such a phenomenon are found in S. I. Vavilov\({}^{5}\).

13. QUANTUM YIELD OF FLUORESCENCE OF SOLUTIONS AND VAPORS

The quantum yield of fluorescence is usually understood as the ratio of the total number of emitted quanta to the total number of absorbed quanta. If one considers the interval of time after the cessation of excitation, then the quantum yield should be understood as the ratio of the total number of emitted quanta to the total number of excited molecules at the moment \(t=0\). It is not difficult to show that calculations carried out by both methods give identical results.

Let us first calculate the quantum yield of fluorescence of solutions. The total number of transitions with radiation during the time \(dt\) of all molecules

in the energy interval from \(Q^*\) to \(Q^*+dQ^*\) is equal to

\[ dN(Q^*,t)\,f(Q^*)\,dt, \]

where \(dN(Q^*,t)\) is given by formula (22). The total number of all transitions from the excited to the unexcited electronic state during the time from \(t=0\) (the moment when excitation ceases) to \(t=\infty\) is equal to

\[ \int_{Q^*=0}^{\infty}\int_{t=0}^{\infty} dN(Q^*,t)f(Q^*)\,dt = \]

\[ = N(0)\int_{Q^*=0}^{\infty}\int_{t=0}^{\infty}\rho(Q^*)f(Q^*)e^{-[\bar f+\bar d]t}\,dQ^*\,dt . \]

According to the definition, the quantum yield is equal to the ratio of this quantity to \(N(0)\), i.e.,

\[ \gamma = \int_{Q^*=0}^{\infty}\rho(Q^*)f(Q^*)\,dQ^* \int_{t=0}^{\infty} e^{-[\bar f+\bar d]t}\,dt = \]

\[ =\overline{f(Q^*)}\cdot \frac{1}{\overline{f(Q^*)}+\overline{d(Q^*)}} . \tag{27} \]

Here \(\overline{f(Q^*)}\) and \(\overline{d(Q^*)}\) are the mean values of the transition probabilities for all excited molecules. They are given by formulas (24) and (25). It should be emphasized that this basic formula is valid only in the case where the probabilities of redistribution of vibrational energy are much greater than the probabilities of transitions with and without radiation.

From formula (27) one immediately obtains the law of independence of the quantum yield of fluorescence from the wavelength of the exciting light. Indeed, in solutions the form of the distribution function \(\rho(Q^*)\) does not depend on \(\lambda\). The same applies, according to (24) and (25), to \(\overline{f(Q^*)}\) and \(\overline{d(Q^*)}\), and consequently to \(\gamma\).

It should be emphasized at once that experiments have also revealed a deviation from this law. If fluorescence is excited by light in the so-called anti-Stokes region of the spectrum, then the quantum yield of fluorescence falls extremely rapidly. A molecular theory of this phenomenon is lacking.* However, from all the preceding considerations it is clear that such a violation of one of the basic laws of fluorescence of solutions can occur only if, upon excitation in the anti-Stokes region of the spectrum, for one reason or another (as yet unknown), equilibrium is not established between the excited molecules

* S. I. Vavilov\(^1\) elegantly explained the inevitability of the decrease in the quantum yield in the anti-Stokes region from purely thermodynamic considerations. A review by Adirovich\(^6\) is also devoted to this phenomenon.

and by the medium, and the form of the distribution function is not given by the usual statistical expression (15).

The second most important consequence of formula (27) is that the quantum yield of fluorescence can be equal to unity only in the case when there are no radiationless transitions whatsoever \((d(Q^*)=0\) for all \(Q^*)\). As a rule, this is not the case. Hence it follows that usually \(\gamma<1\). It can be equal to unity only at very low temperatures, i.e., for very small stores of vibrational energy in the excited molecules.

Let us consider the relation between the quantum yield of fluorescence of solutions and the duration of the excited state. From (27) and (26) it follows that:

\[ \frac{\gamma}{\tau} = \int_{Q^*=0}^{\infty} f(Q^*)\rho(Q^*)\,dQ^* = \overline{f(Q^*)}. \tag{28} \]

Since, for a given kind of molecule, the distribution function depends only on temperature, the quantity \(\dfrac{\gamma}{\tau}\) can also depend only on temperature. Experiments on the temperature quenching of fluorescence show that \(\dfrac{\gamma}{\tau}\), as a rule, does not depend on \(T\). This fact is an additional confirmation of the already mentioned circumstance that \(f(Q^*)\) depends only very weakly on \(Q^*\). Indeed, when the temperature changes, \(\overline{f(Q^*)}\) can remain unchanged only in the absence of a strong dependence of \(f\) on \(Q^*\). It is important to note, further, that the right-hand side of (28) contains no probabilities of radiationless transitions. When \(d\) is changed, for example by introducing foreign quenching substances into the solution, the ratio \(\dfrac{\gamma}{\tau}\) must remain unchanged. This result agrees well with experiment. Moreover, on the basis of this fact S. I. Vavilov \(^{19}\) proposed his criterion for distinguishing different types of fluorescence quenching from one another.

Let us now find an expression for the quantum yield of fluorescence of vapors. Since the total number of excited molecules in the energy interval from \(Q^*\) to \(Q^*+dQ^*\) is given by formula (17), the total number of transitions to the lower state during the time from \(t=0\) to \(t=\infty\) is equal to

\[ \int_{Q^*=0}^{\infty}\int_{t=0}^{\infty} dN(Q^*,t)\,f(Q^*)\,dt = \]

\[ = N(0)\int_{Q^*=0}^{\infty}\int_{t=0}^{\infty} f(Q^*)\rho(Q^*)e^{-[f(Q^*)+d(Q^*)]t}\,dQ^*\,dt. \]

The value of the quantum yield is equal to

\[ \gamma = \int_{Q^*=0}^{\infty}\int_{t=0}^{\infty} f(Q^*)\rho(Q^*) e^{-[f(Q)+d(Q)]t}\, dQ^*\, dt = \]

\[ = \int_{Q^*=0}^{\infty} \frac{f(Q^*)}{f(Q^*)+d(Q^*)}\rho(Q^*)\,dQ^* . \tag{29} \]

Expression (29) differs substantially from expression (27) for solutions. First of all, it is obvious that in vapors the quantum yield (in the Stokes region) depends on the wavelength of the exciting light. This is connected with the dependence on \(\lambda\) of the form of the distribution function \(\rho(Q^*)\). The greater \(\Delta Q^* = h\nu - h\nu_{\mathrm{el}}\), the greater the displacement of the maximum of the distribution function, and the greater the average vibrational energy of the excited molecules*). Applying the mean-value theorem, we have:

\[ \gamma = \overline{ \frac{f(Q^*)}{f(Q^*)+d(Q^*)} }. \tag{30} \]

As \(\lambda\) decreases, the mean value \(\overline{\frac{f}{f+d}}\) will be taken for ever larger values of \(Q^*\). Since \(f\) remains (approximately) constant, while \(d\) increases rapidly, \(\gamma\) will rapidly decrease. Expressions (27), (28), and (29), as well as expression (22), are in complete agreement with experiment. They are similar to the formulas obtained by B. S. Neporentov\(^{17,18}\), but in our expressions the character of the averaging carried out in the cited works is specified.

14. TEMPERATURE QUENCHING OF FLUORESCENCE

Let us now consider the influence of temperature on the lifetime of the excited state and on the quantum yield of fluorescence. In the case of solutions all the reasoning is especially simple. According to (27) and (26) we have:

\[ \tau = \frac{1}{\overline{f(Q^*)}+\overline{d(Q^*)}}, \qquad \gamma = \frac{\overline{f(Q^*)}}{\overline{f(Q^*)}+\overline{d(Q^*)}} . \]

Since the average vibrational energy of molecules in the excited electronic state increases with increasing temperature (\(\overline{Q^*}=LkT\)), and since \(\overline{d(Q^*)}\) may approximately be set equal to \(d(\overline{Q^*})\) (the distribution function is sharp), \(\overline{d(Q^*)}\) increases with increasing temperature. It follows from this that both \(\tau\) and \(\gamma\) must decrease with increasing \(T\). This phenomenon is well known experimentally for

*) These arguments apply only to Stokes excitation.

under the name of temperature quenching of fluorescence (see 30 and 23)*). It should, however, be borne in mind that this result is valid only in the case when the probability of nonradiative transitions increases with increasing \(Q^*\). It may turn out that, in some interval of values of \(Q^*\), the value of \(d(Q^*)\) changes comparatively little. This may occur, for example, at small \(Q^*\) (see Fig. 11). If the maximum of the distribution function falls, in some temperature interval, into this region of values of \(Q^*\), then \(d(Q^*)\) will change negligibly. In these cases temperature quenching of fluorescence will be absent, especially if the absolute value of \(f(Q^*)\) is sufficiently large in comparison with \(d(Q^*)\). A similar case occurs in fluorescence, where no appreciable temperature quenching is found.

Analogous results are valid also for the gas phase. The difference consists in the fact that in solutions the mean vibrational energy of the excited molecules depends only on temperature, whereas in vapors it depends, in addition, also on the wavelength of the exciting light. Upon excitation in the Stokes region of the spectrum, the store of vibrational energy is higher than in solutions. Therefore we are always in a region where \(d\) depends more strongly on \(Q^*\), and the phenomenon of temperature quenching will be sharper. As we saw in Part I, increasing the store of vibrational energy of vapors by raising the temperature and by decreasing the wavelength of the exciting light is equivalent in its results.

We believe that further investigation of the role of vibrational energy in the processes of light absorption and fluorescence should be very fruitful.

CITED LITERATURE

1. S. I. Vavilov, Izv. AN SSSR, ser. fiz. 9, 283 (1945); Jubilee collection of the Academy of Sciences of the USSR, 1947, 2, 1, p. 377.
2. A. N. Terenin, Photochemistry of Dyes, Publishing House of the Academy of Sciences of the USSR, L., 1947.
3. S. I. Vavilov, Izv. AN SSSR, ser. fiz. 7, 3 (1943); Journ. Phys. USSR 9, 68 (1945).
4. A. N. Terenin, Zhurn. fiz. khim. 18, 1 (1943); Acta Physicochimica USSR 18, 210 (1943).
5. P. P. Feofilov, UFN 36, 417 (1948).
6. E. M. Al’perin, UFN 40, 341 (1950).
7. B. S. Neporent, Zhurn. fiz. khim. 24, 1219 (1950).
8. A. N. Terenin, Acta Physica Polonica 5, 229 (1936).
9. S. I. Vavilov, Phil. Mag. 43, 307 (1922).
10. V. L. Levshin, Zhurn. fiz. khim. 9, 1 (1937).
11. N. A. Prilezhaeva, Acta Physicochimica USSR 1, 785 (1935).
12. A. T. Vartanyan, Izv. AN SSSR, OMEN, 341 (1938).
13. A. A. Shishlovskii, ZhETF 7, 1252 (1937).

*) It is possible that, besides the indicated cause of temperature quenching of fluorescence, there are also other, as yet unknown, causes.

1. S. I. Vavilov, Zeits. f. Physik 42, 311 (1927).
2. S. S. Solomon, DAN 31, 741 (1941).
3. A. N. Terenin, A. T. Vartanyan, B. S. Neporent, Trans. Farad. Soc. 35, 39 (1939).
4. B. S. Neporent, Zhurn. fiz. khim. 13, 965 (1939).
5. B. S. Neporent, Zhurn. fiz. khim. 21, 1111 (1947).
6. S. I. Vavilov, DAN 3, 271 (1936).
7. I. M. Frank, S. I. Vavilov, Zeits. f. Physik 69, 100 (1931).
8. B. Ya. Sveshnikov, Acta Physicochimica USSR 4, 453 (1936).
9. M. D. Galanin, Trudy FIAN 5, 339 (1950).
10. A. V. Karyakin and A. N. Terenin, Izv. AN SSSR, ser. fiz. 13, 9 (1949).
11. A. V. Karyakin, M. D. Galanin, DAN 64, 37 (1949).
12. L. S. Kassel, Kinetics of Homogeneous Gas Reactions, ONTI, L., 1937.
13. A. N. Terenin, Izv. AN SSSR, ser. fiz. 9, 305 (1945).
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