VARIATION OF THE ELECTRONIC TRANSITION MOMENT AND FRANCK–CONDON FACTORS FOR THE GREEN VIBRATIONAL SYSTEM OF THE MgO MOLECULE

N. A. NAZIMOVA

For dipole radiation of diatomic molecules, the probability \(A_{v'v''}\) of the vibrational transition \(v' \to v''\), to within a constant factor, is determined by the expression

\[ A_{v'v''}=\operatorname{const}\,\nu_{v'v''}^{3}\left|\int \psi_{v'} R_e(r)\psi_{v''}\,dr\right|^{2}, \]

where \(\nu_{v'v''}\) is the frequency of the transition under consideration,
\(\psi_{v'}\) and \(\psi_{v''}\) are wave functions describing the \(v'\)- and \(v''\)-states,
\(R_e(r)\) is the electronic transition moment.

Thus, the quantity \(p_{v'v''}=\left|\int \psi_{v'}R_e(r)\psi_{v''}\,dr\right|^{2}\) is a measure of the transition probability.

In the first approximation, neglecting the dependence of the electronic transition moment \(R_e\) on the internuclear distance \(r\), one may assume that the probability of the vibrational transition is completely determined by the Franck–Condon factor

\[ q_{v'v''}=\left|\int \psi_{v'}\psi_{v''}\,dr\right|^{2}. \]

In the more general case, assuming a smooth variation of the electronic transition moment \(R_e\) with the internuclear distance \(r\), one may write

\[ p_{v'v''}=R_e^{2}(\bar r_{v'v''})\left|\int \psi_{v'}\psi_{v''}\,dr\right|^{2} \]

or

\[ p_{v'v''}=R_e^{2}(\bar r_{v'v''})q_{v'v''}, \]

where \(\bar r_{v'v''}\) is the \(r\)-centroid—the mean internuclear distance assigned in a definite way to each band

\[ v'\to v''. \]

The aim of the present work was to calculate the Franck–Condon factors and to find the functional dependence \(R_e=f(r)\) for the green vibrational system of the MgO molecule, corresponding to the electronic transition \(B^{1}\Sigma \to X^{1}\Sigma\).

The Franck—Condon factors for the selected band system were considered in Ortenberg’s work [1]. In the present work they were calculated using other values of the constants \(a_i\).

No data on the \(r\)-centroids and on the dependence \(R_e=f(r)\) for MgO could be found in the literature.

1. Calculation of the Franck—Condon factors

The Franck—Condon factors were found by the method of the “simplified Morse potential” [2].

The basic premise of the method is that the molecule is an anharmonic oscillator and is characterized by the Morse potential function

\[ U(r)=D[1-e^{-a(r-r_e)}]^2 . \]

The method of the “simplified potential” consists in replacing the constants \(a_1\) and \(a_2\), corresponding to the electronic states of the molecule, by the arithmetic mean

\[ a=\frac{a_1+a_2}{2}, \]

with the condition that

\[ \left|\frac{a_1-a_2}{a_1+a_2}\right|<5\%. \]

Then the overlap integrals can be found using the formulas:

\[ (0,0)=\left(\frac{1}{\rho_1}\right)^{\frac{\kappa_1-1}{2}} \left(\frac{1}{\rho_2}\right)^{\frac{\kappa_2-1}{2}} \frac{\Gamma(\kappa-1)} {\{\Gamma(\kappa_1-1)\Gamma(\kappa_2-1)\}^{\frac12}}, \]

\[ \frac{(v'v'')}{(0,0)} =(-1)^{v'+v''} \frac{N_1^{(v')}N_2^{(v'')}}{N_1^{(0)}N_2^{(0)}} \sum_{\sigma=0}^{v'+v''} (-1)^{\sigma} \frac{\Gamma(K-1-\sigma)}{\Gamma(K-1)} C(v'_1v'';\sigma), \]

where

\[ C(v'_1v'';\sigma)= \sum_{n=0}^{\sigma} C_{v'}^{n} C_{v''}^{\sigma-n} \rho_1^{n}\rho_2^{\sigma-n} \frac{\Gamma(K_1-v')\Gamma(K_2-v'')} {\Gamma(K_1-v'-n)\Gamma(K_2-v''-\sigma+n)}; \]

\[ K_1=\left(\frac{\omega_e}{\omega_e x_e}\right)_1 \left(\frac{a_1}{a}\right)^2; \qquad K_2=\left(\frac{\omega_e}{\omega_e x_e}\right)_2 \left(\frac{a_2}{a}\right)^2; \qquad K=\frac{K_1+K_2}{2}, \]

\[ \rho_1=\frac{\lambda}{\lambda_1}; \qquad \rho_2=\frac{\lambda}{\lambda_2}; \qquad \lambda_1=K_1 e^{ar_{e_1}}; \qquad \lambda_2=K_2 e^{ar_{e_2}}; \qquad \lambda=\frac{\lambda_1+\lambda_2}{2}. \]

The overlap integrals \((v',v'')\) and the Franck—Condon factors \(q(v',v'')\) were found for the bands \((0,0)\), \((1,1)\), \((2,2)\), \((0,1)\), \((1,2)\), and \((2,3)\) of the green vibrational system of the MgO molecule; the corresponding wavelengths of the quanta of these bands are 5007.3; 4996.7; 4985.9; 5206.0; 5192.0; 5177.4 Å.

Calculation of the overlap integrals for other bands of the green vibrational system of the MgO molecule is not meaningful, since the method of the “simplified potential” is valid only for small values of \(v'\) and \(v''\).

For the band system under consideration,

\[ \left|\frac{a_1-a_2}{a_1+a_2}\right|=2.23\%<5\%; \]

therefore, the introduction of the arithmetic mean \(a\) is permissible. The values of the molecular constants \(\omega_e\), \(\omega_e x_e\) [3], as well as the values of \(a\), \(K\), \(\lambda\), and \(\rho\), are given in Table 1.

Table 1

Arithmetic mean values

\[ a = 1,6910\ \mathrm{\AA}; \qquad K = 161,95; \qquad \lambda = 3086,5. \]

The gamma functions entering into the expression for the overlap integral $(0,0)$ were found using the asymptotic expansion of the logarithm of the gamma function:

\[ \ln \Gamma(x) = \ln\left[\sqrt{2\pi}\,x^{x-\frac{1}{2}} e^{-x}\right] + \frac{1}{12x} + \cdots . \]

The ratio of the normalization constants was determined from the formula

\[ \frac{N_i^{(v)}}{N_i^{(v-1)}} = \left[ \frac{K_i - v}{v}\, \frac{K_i - 2v - 1}{K_i - 2v + 1} \right]^{\frac{1}{2}} . \]

The results of calculations of the overlap integrals $(v', v'')$ and the Franck–Condon factors $q(v', v'')$ are given in Table 2. Seven-place tables of logarithms were used in the calculations.

Table 2

2. Calculation of $r$-centroids

The mean internuclear distance $\bar r_{v'v''}$ associated with the vibrational transition $v' \to v''$ of a diatomic molecule, and called the $r$-centroid, is defined as

\[ \bar r_{v'v''} = \frac{\int \psi_{v'}\, r\, \psi_{v''}\, dr} {\int \psi_{v'}\, \psi_{v''}\, dr}. \]

It makes sense to speak of $r$-centroids when the following conditions are fulfilled [4]:

1) $\mu\omega_e \sim 10^4$, where $\mu$ is the reduced mass of the molecule under consideration in atomic-weight units;

2) $0,01\ \mathrm{\AA} < |r_{e_1} - r_{e_2}| < 0,25\ \mathrm{\AA}$;

3) $v' < 10;\quad v'' < 10$.

All the above-listed conditions are satisfied for the green vibrational system of the MgO molecule (electronic transition \(B^1\Sigma \to X^1\Sigma\)). The \(r\)-centroids were found for the following bands of this system: \((0,0)\), \((1,1)\), \((2,2)\), \((3,3)\), \((4,4)\), \((5,5)\), \((0,1)\), \((1,2)\), \((2,3)\); the corresponding wavelengths of the quanta of these bands are 5007.3; 4996.7; 4985.9; 4974.5; 4962.1; 4949.5; 5206.0; 5192.0; 5177.4 Å.

Several methods have been proposed for estimating \(r\)-centroids. In the present work the \(r\)-centroids were estimated by the method of the quadratic equation [4].

The MgO molecule is assumed to be an anharmonic oscillator in both the lower and upper electronic states; its potential function is described by the Morse potential, and the constants \(a_1\) and \(a_2\) of the lower and upper states are replaced by the arithmetic mean \(a\).

The introduction of the arithmetic mean \(a\) requires adjustment, to the new value of \(a\), of all quantities that depend on \(a\).

For the electronic transition \(B^1\Sigma \to X^1\Sigma\) of the MgO molecule under consideration, the dissociation energies \(D_e\), the anharmonicity constants \(x_e\), and the vibrational-level energies \(E_v\) for both electronic states \(B^1\Sigma\) and \(X^1\Sigma\) were adjusted to the new value of the parameter \(a\).

The values found for the \(r\)-centroids are given in Table 3.

Table 3

3. Determination of the dependence \(R_e = f(\bar r_{v'v''})\)

The intensity of a band corresponding to the vibrational transition \(v' \to v''\) can be written as follows:

\[ I_{v'v''}=A_{v'v''}N_{v'}E_{v'v''}, \]

where \(A_{v'v''}\) is the probability of the vibrational transition \(v' \to v''\),

\[ N_{v'} \text{ is the population of level } v', \]

\[ E_{v'v''} \text{ is the energy of the quantum emitted in the transition } v' \to v''. \]

Since

\[ A_{v'v''}=\mathrm{const}\,\nu_{v'v''}^{3}R_e^2(\bar r_{v'v''})q_{v'v''}, \]

the intensity of the band \((v',v'')\) is

\[ I_{v'v''}=\mathrm{const}\,N_{v'}E_{v'v''}^{4}R_e^2(\bar r_{v'v''})q_{v'v''}. \]

Hence

\[ \left(\frac{I_{v'v''}}{q_{v'v''}E_{v'v''}^{4}}\right)^{\frac12} = (\mathrm{const}\,N_{v'})^{\frac12}R_e(\bar r_{v'v''}). \]

Plotting the dependence

\[ \left(\frac{I_{v'v''}}{q_{v'v''}E_{v'v''}^{4}}\right)^{\frac12} \]

on \(\bar r_{v'v''}\), one can find the relative variation of \(R_e\) with \(r\) for each \(v''\)-progression (\(v'\) remains constant).

In the present work, the progressions \(v'=0\), bands \((0,0)\), \((0,1)\); \(v'=1\), bands \((1,1)\) and \((1,2)\); and \(v'=2\), bands \((2,2)\) and \((2,3)\), were considered.

For each of these bands, the \(r\)-centroids, Franck–Condon factors, and energies \(E_{v'v''}\) corresponding to the given transition \(v'\to v''\) were calculated. The data are given in Table 4.

Table 4

The intensities of the corresponding bands were determined experimentally. The spectrum of magnesium burning in air at atmospheric pressure was photographed on a KSA-1 spectrograph with glass optics. The plates were processed by the usual methods of photographic photometry.

For all three \(v''\)-progressions, plots were constructed of

\[ \left(\frac{I_{v'v''}}{q_{v'v''}E_{v'v''}^{4}}\right)^{1/2} \]

as a function of

\[ \bar r_{v'v''}. \]

The plots were then reduced to a single coordinate scale by the method described in [5].

The obtained dependence \(R_e(\bar r_{v'v''})=f(\bar r_{v'v''})\) for the entire green vibrational band system of the MgO molecule under consideration is shown in Fig. 1.

Fig. 1. 1—band \((2,2)\), 2—band \((1,1)\), 3—band \((0,0)\), 4—band \((2,3)\), 5—band \((0,1)\), 6—band \((1,2)\).

As a result of treating the data obtained by the least-squares method, the variation of the electronic transition moment \(R_e\) with the distance \(r\) was found to be

\[ R_e(r)=-0.6405+0.9778\,r. \]

Thus, one may conclude that, for the bands of the green vibrational system of the MgO molecule, the electronic transition moment \(R_e\) varies linearly with the change in internuclear distance \(r\).

References

1. F. S. Ortenberg, Opt. i spektr. 9, 151, 1960.
2. P. A. Fraser, W. R. Jarmain, Proc. Phys. Soc., a66, 1145, 1953.
3. A. Lagerqvist, A. Uhler, Ark. f. Fys., 1, 459, 1949.
4. R. W. Nicholls, W. R. Garmain, Proc. Phys. Soc., a69, 253, 1956.
5. R. G. Turner, R. W. Nicholls, Canad. J. Phys., 32, 475, 1954.