On the Method for Deciphering the Vibrational Structure of Electronic Transitions in Diatomic Molecules

G. V. Kovalenok

(Presented by Prof. V. A. Sokolov)

In connection with the development of methods of molecular spectroscopy, one of the important tasks is the deciphering of spectrograms—the identification of the lines and bands present in a spectrum with a definite state of atoms or molecules.

The well-known book by R. Pearse and A. Gaydon [1] contains the necessary information on most known spectra of diatomic molecules: wavelengths of band heads, intensities of heads, the appearance of bands and the conditions of their occurrence, an indication of the electronic transition, and references to the corresponding original papers. However, for a number of systems a considerable incompleteness of essential data has been found; some systems have still not been identified.

The only work in which attention is paid to the question of deciphering spectra is the book by G. Herzberg [2], which gives an exhaustive picture of the level of knowledge of the diatomic molecule, but the question of deciphering is presented in the form of an already completed sum of results and their agreement with theoretical conclusions.

In [3] a deciphering was proposed of an extremely complex vibrational structure observed when calcium salts are introduced into a flame or arc, in an arc between electrodes of metallic calcium.

The region 5900–6500 Å, containing 71 bands grouped into seven sequences, was analyzed. The results are in good agreement with the experimental data of other authors [1].

The assumption that this system belongs to the CaO molecule was not confirmed. In a further study of the system it was possible to prove convincingly that the vibrational structure belongs to the Ca₂ molecule. In the next article [4] of this series we shall dwell in detail on the method that made it possible to identify the deciphered spectrum.

In the present article we shall examine in detail the questions that facilitate the process of deciphering a vibrational structure.

It is known that the spectra of diatomic molecules consist of a system of bands with more or less pronounced heads. In many cases several systems of bands are observed for molecules. Sometimes different band systems of one molecule lie in different regions, but often they overlap. The wave numbers of the band heads of each system are arranged—

are given in the form of the so-called Deslandres table (or quantum scheme). Equation (1) gives the positions in the spectrum of all quanta:

\[ \nu=\nu_{00}+(a'v'-b'v'^2)-(a''v''-b''v''^2), \tag{1} \]

where

\[ \begin{aligned} \nu &\text{— the wave number of the quanta of any band,}\\ \nu_{00} &\text{— the wave number of the }0\text{--}0\text{ band,}\\ a' \text{ and } a'' &\text{— the distances (in wave-number units) between the bands}\\ &\quad \text{of the } v' \text{ and } v'' \text{ (progression and sequence) series in the case}\\ &\quad \text{of bands equally spaced from one another,}\\ b' \text{ and } b'' &\text{— give the observed gradual decrease of the spacing}\\ &\quad \text{between bands }(b \ll a),\\ v' \text{ and } v'' &\text{— integers }(0,1,2,\ldots)\text{ determining the band number.} \end{aligned} \]

Thus, if the five quantities \(\nu_{00}\), \(a'\), \(a''\), \(b'\), and \(b''\) are known, the entire system of band quanta can be obtained by combining different \(v'\) and \(v''\) in (1).

The value \(\nu_{00}\) is helped by the special feature of the \(0\)--\(0\) band and its quantum—it may be stated with confidence that these are always the most intense band and quantum among all observed bands and quanta. The wavelength of the most intense quantum determines its wave number.

The next question is the interval of the sequence and progression series, i.e., the distance between the quanta of the \(0\)--\(0\) and \(0\)--\(1\), or \(0\)--\(0\) and \(1\)--\(0\), bands, respectively.

Comparing the Deslandres table (Table 1) with the diagrams in Fig. 1 (taken from [2]), we note that the constant distance between neighboring lines for the first two sequence series of Deslandres (\(v'=0\) and \(v'=1\)) gives the first vibrational quantum \(\Delta G'_{1/2}\) of the upper state, while the constant distance between neighboring lines for the first two progression series (\(v'=0\), \(v''=1\)) gives the first vibrational quantum \(\Delta G''_{1/2}\) of the lower state.

Fig. 1. Vibrational transitions between two electronic states. \(a\)—series \(v''\); \(b\)—series \(v'\). Individual series are indicated by braces.

The experimental equation (1) coincides with the theoretical equation (2) for band systems,

\[ \nu=\nu_{00}+\omega'_0 v'-\omega'_0 x'_0 v'^2+\omega'_0 y'_0 v'^3+\ldots \]

\[ -\left(\omega''_0 v''-\omega''_0 x''_0 v''^2+\omega''_0 y''_0 v''^3+\ldots\right), \tag{2} \]

if the very small values \(\omega_0 y_0\) (equal to \(\omega_e y_e\)) are neglected. According to (1) and (2), the empirical constants \(a'\) and \(a''\) are identical with the vibrational cha-

states \(\omega'_0\) and \(\omega''_0\), while the empirical constants \(b'\) and \(b''\) are the anharmonicities \(\omega'_0 x'_0\) and \(\omega''_0 x''_0\); from the values of \(\omega_0\) and \(\omega_0 x_0\), by means of equation (3), one can pass to \(\omega_e\) and \(\omega_e x_e\).

In the lowest vibrational state there is vibrational energy, called the zero-point energy. If the energy levels are referred to this lowest state, then we obtain

\[ G_0(v)=\omega_0 v-\omega_0 x_0 v^2+\ldots, \]

where

\[ \omega_0=\omega_e-\omega_e x_e, \]

\[ \omega_0 x_0=\omega_e x_e. \tag{3} \]

The first vibrational quantum, according to (3), is equal to

\[ \Delta G_{1/2}=\omega_0-\omega_0 x_0=\omega_e-2\omega_e x_e \tag{4} \]

or, in the notation of equation (1),

\[ \Delta G_{1/2}=a-b=\omega_e-2b, \tag{5} \]

whence

\[ \omega_e=a+b. \tag{6} \]

The distance between the 0—0 and 0—1 bands is equal to the difference \(a\) and \(b\), so that, knowing the value of \(\omega_e\), we find the position of the 0—1 bands \(2b\) farther than the actual one. In practice, the quantity \(a\) is greater than the quantity \(b\) by 2–3 orders of magnitude, and one may take \(\omega_e \approx a\), or, more roughly, \(\omega_e \approx a-b\). For approximate searches for the positions of band edges such an assumption is justified. The exact value of the wave number of an edge is found from the spectrum.

Thus, it is desirable to know the vibrational constant \(\omega_e\). It seems possible, in order to find \(\omega''_e\) of the normal lower state of the molecule, to use the interesting dependence of \(\omega''_e\) on the mass of the molecules \(M\) (the sum of the masses of the atoms forming the molecule). We shall use the values of \(\omega''_e\) taken from Table No. 36 of Herzberg’s book [2].

Table 1

Delandre table of PN bands

Each group of elements of the Mendeleev table has its own profile of curves in the coordinates \(\omega''_e\) and \(M\), strictly constant for all elements of this group. In Fig. 2 the curves are presented for elements of group II: Be, Mg, Cd, and Hg (the curves for Ca, Sr, and Ba are omitted so as not to complicate the figure). Each curve is constructed on the basis of the values of \(\omega''_e\) and \(M\) of molecules composed in a definite way: an atom of an element of group II

groups of compounds with hydrogen and with atoms of the halogen series. The curves are valuable both from the standpoint of checking already known but doubtful \(\omega''_e\), and for finding as yet unknown vibrational constants. In particular, the value \(\omega''_e = 312\ \text{cm}^{-1}\) was confirmed for the MgJ molecule; values of \(\omega_e\) were found for some molecules; the data are given in Table 2.

Fig. 2. Vibrational constants of molecules as functions of their molecular weight (curves of type I)

Table 2

Vibrational constants \(\omega''_e\) obtained by means of curves of type 1 for some diatomic molecules

It is impossible to find the value of \(\omega'_e\) in this way, but, as we shall see below, this will not be needed.

As regards determining the quantity \(b''\), the following cases are possible:

1. The bands are clearly visible and do not overlap (or overlap only partially); then we may assume the sequence of bands \(\Delta V = 0\) and \(\Delta V = 1\), set up, for several quanta, a system of equations of the form (1), and, solving it, find the values of all the coefficients: \(a'\), \(a''\), \(b'\), \(b''\).

2. In our system the sequence \(\Delta V = -1\) was very weak, while the sequence \(\Delta V = -2\) (which we assumed) was intense; then a system of equations was set up for some quanta of the sequences \(\Delta V = 0\) and \(\Delta V = -2\).

3. The spectrum is seen weakly, the bands overlap, and it is impossible to assume the bands of the sequences \(\Delta V = 0\) and \(\Delta V = -1\); then we propose using curves of the mutual relation of the vibrational constants \(\omega''_e\) and \(\omega''_e x''_e\). As an example, in Fig. 3 we give such a dependence for molecules with a copper atom. The curves are constructed, like the preceding curves, for definitely selected molecules; for a known \(\omega''_e\) they give the value \(\omega''_e x''_e = b''\).

Of course, in case 3 one can count on only a small accuracy in the quantity \(b'' = \omega''_e x''_e\), since the points do not always lie as fortunately as in Fig. 3.

Curves of types I and II made it possible to confirm the values of \(\omega''_e\) and to find the values of \(\omega''_e x''_e\) for some molecules with a Li atom; the data are entered in Table 3. It must be emphasized that this is an approximate estimate of the quantities \(a''\) and \(b''\); they are determined more accurately from a system of equations of type (1), but even this degree of approximation helps one to make sense of comp—

... arrangement of bands. However, the experimentally found coefficients \(a\) and \(b\) will not give an exact scheme of quanta, owing to the fact that in equation (1) terms containing \(v'\) and \(v''\) in higher powers are omitted.

Fig. 3. Curves of mutual correlation of vibrational constants
(type-II curves)

The systematic deviation of the differences between the observed and calculated values of the wave numbers of quanta can be corrected by means of insignificant changes of the constants.

Table 3

Vibrational constants \(\omega_e'' x_e''\), the values of which were determined with the aid of type-II curves

The question arises of a criterion that would make it possible to determine the correctness of the deciphering of the vibrational structure. The very fact that all the bands investigated fit into the Delandre table is already a reliable guarantee of correctness. One may use the conclusion drawn by G. Herzberg for the case of a system in the ultraviolet region, where the necessity is noted for equality of the first vibrational quanta of the lower state for bands in the ultraviolet and infrared regions.

LITERATURE

1. R. Pearse and A. Gaydon. The Identification of Molecular Spectra. IL, 1949.
2. G. Herzberg. Spectra and Structure of Diatomic Molecules. IL, 1949.
3. G. V. Kovalenok, V. A. Sokolov. Vibrational quantum analysis of the orange bands of the CaO molecule. Proceedings of the Symposium on Spectroscopy. Izvestiya SO AN SSSR, chemical series, issue 4, 1966.
4. G. V. Kovalenok. Identification of the deciphered vibrational structure of diatomic molecules (in this collection).