Franck-Condon Factors and r-Centroids For Several Systems of CoF Molecule

. Franck–Condon factors factors and r-centroids have been calculated for the G 3 F 4 − X 3 F 4 , K 3 F 4 − X 3 F 4 , L 3 F 4 − X 3 F 4 , [20.6] 3 G 5 − X 3 F 4 and [34.7] 3 G 5 − X 3 F 4 bands of Cobalt monofluoride 59 Co 19 F from the best available molecular constants using Morse and Rydberg– Klein–Rees (RKR) methods. The reliability of the obtained values was verified by comparing the results of five different programs. Based on the calculated Franck–Condon factors values, the appearance of the bands in these systems are discussed and the relationship between the r-centroids and the band-origin wavenumbers and wavelengths are also determined along with accuracy factors.


INTRODUCTION
Franck-Condon factors (FCFs) are important parameters for every molecular band system, since they are involved in the calculation of the relative band intensities, the determination of the molecular structure, population of the levels in the upper electronic state involved in transition and the calculation of radiative lifetimes, vibrational temperature, kinetics of energy transfer, etc [1]. The FCFs are also useful in the studies of radiative transfer in the atmospheres of stellar and other astronomical objects, which contain molecular species [2]. On the other hand, the r-centroid [3][4][5] also has been found of much importance as it is used in the interpretation of intensity measurements of many band systems of astrophysically important molecules and, to determine the variation of the electronic transition moment with inter-nuclear separation, and thereby to derive band strengths, and other molecular transition probability parameters [6].
"The properties of transition metal compounds have fascinated physicists and chemists for a long time. These compounds have an enormous potential for future applications of solar energy conversion, information storage systems, chemical or biochemical sensors, low dimensional semiconductors, supramolecular systems, chemical synthesis, etc" [7]. In astrophysics it is the transition-metal-containing compounds are very important [8][9][10]. More over the presence of transition metal elements in cool stars has been known for some time [11].
In recent years, enough progress has been made in an understanding of the electronic structure of transition metal oxides and hydrides. Metal fluorides are also potential stellar and interstellar molecules, although Florin has relatively small cosmic abundance. Metal fluorides also have strong bonds, which favor their formation in stellar atmospheres [12]. In this paper, the FCFs and r-centroids for the bands of the G 3 F 4 − X 3 F 4 , K 3 F 4 − X 3 F 4 , L 3 F 4 − X 3 F 4 , [20.6] 3 G 5 − X 3 F 4 and [34.7] 3 G 5 − X 3 F 4 systems of the 59 Co 19 F molecule are reported. To the best of our knowledge, such information is not available in the literature. Therefore the reliable values of FCFs and r-centroids for these band systems have been determined using more accurate numerical integration procedure and suitable potentials. The required molecular constants are taken from the works of [8][9][10][11][12].

FRANCK CONDON FACTORS AND R-CENTROID
Within the framework of the Born-Oppenheimer approximation, the expression for band intensities in emission is given in as [13]: Where C is velocity of light, ′ is number of molecules in ′ th level, is the wave number of radiation of the ′ → ′′ vibronic transition; and ( ) is the average transition moment, which is commonly represented as a function of internuclear distance . ′ ( ) and ′′ ( ) are the wave functions which correspond to the rotationless (J = 0) vibrational levels in the upper and lower electronic states, respectively. They satisfy the time-independent Schrödinger radial wave equation: In which r is the inter-nuclear distance, ℏ is Planck's constant, μ is the reduced mass of the two atoms forming the molecule and V(r) is the potential energy.
For many band systems, ( ) can be well approximated by a constant or a linear function of r. Since the wave-mechanical formulation of the Franck-Condon principle rests on the assumption that this variation is slow. In such a situation, the r-centroid approximation can be employed. The quantity in the square bracket appearing in eq. (1) is the square of overlap integral known as Franck-Condon factor which is usually denoted by ′ ′′ and determines how the intensity is distributed among the vibrational bands of a band system. It is expressed as, (3) r-Centroid is the r-coordinate of the centroid of the area represented by the overlap integral and defined as: In the above equation is the r-centroid which is the weighted average with respect to ′ ′′ of the range of internuclear separation r values experienced by the molecule in both states of the ′ → ′′ transition [11] or may be looked upon as an effective value of r for the band in much the same way as the of eqn.(1) constitutes its effective wave number [14]. The r-centroid is the most practical method for calculation of the band strengths of the observed transitions in diatomic molecules .Since it is an approximation, there are limits to its use. According to [9] the approximation is good to the nth order if: and its accuracy factor that indicates the degree of its applicability given by Drake [15].
When ′ ′′ ( ) is unity the approximation implied in equations (4) and (6) is exact. Deviation from unity for (n  2) is a measure of the error in the approximation. The accuracy of the calculated FCFs and r-centroids depend on knowledge of the potential energy function of the lower and upper electronic states, which are used to solve the radial Schrödinger  [17][18][19].

RESULTS AND DISCUSSION
Using above mentioned approaches, FCFs and r-centroids arrays have been computed for the following Systems: G-X, K-X, L-X, [ . ] G -X and [ . ] G -X of cobalt monofluoride. Tables 2, 3,4,5 and 6, show the results obtained using the approach based on Espy RKR/FCF computer programmes along with the wave number, wavelengths and accuracy factor of rcentroides for n = 3. The Comparison of the FCFs and r-centroides calculated using the five approaches for G-X system is presented in Table 7 and 8. In these tables the values are calculated for many bands which were not experimentally observed.
The ∆ = ′ − ′′ for the band systems G-X, K-X, L-X, [ . ] G -X and [ . ] G -X are: 0.088, 0.027, 0.026, 0.045 and 0.016 respectively. These small values indicate that the minima of the potential energy curves involved in K-X, L-X ,and [ . ] G -X transitions lie one above the other ( Fig.1) and their Condon parabola is very narrow, appear to lie along the main diagonal ′ = ′′ in the Deslandres table and has its maximum value at the (0,0) band. In a three dimensional representation, the FCFs surface looks like a diagonal ridge along ′ = ′′ . The situation is somewhat different for [ . ] G -X band system in which ∆ is a little larger ( 0.05 Å), the ridge widens into a very narrow parabolic ridge with an axial valley. For G-X band system ∆ is relatively larger and the Condon parabola widens and its vertex moves right the prime diagonal away from (0,0) in accordance with the observations of Nicholls [27]. Condon parabola is a locus of intense bands in ′′ progressions which often pass through (0,0) and are not completely symmetric about ′ = ′′ unless the ′ and ′′ are very close together [28]. FCFs of such bands are shown in bold figures in corresponding tables.

ILCPA Volume 58
The percentage error (deviation) between the experimental and computed values of the rotational constant  for X, G, K, L, [ . ] G and [ . ] G systems are: 46 %, 15 %, 4 %, 19 %, 15 % and 8 % respectively .This suggests that the lower part of the potential energy curves of the excited states may be described by a simple Morse function but it cannot describe the ground state properly. The r-centroid of the observed bands vary systematically from band to band of a system. In the case of very small overlap integrals, r-centroids have irregular behavior, and such values are not used in our consideration. The sequence difference ∆ = ′ + , ′′ + − ′ ′′ for the observed bands are found generally to be constant for the strong sequences. In G-X, L-X, [ . ] G -X and [ . ] G -X systems the sequence difference is found to be constant amounting to about 0.01(Å). But in the case of K-X system, the sequence difference is not a constant and is varying from 0.004-0.01 (Å). For all these band systems, it is found that the r-centroids increase systematically with increasing wavelength of the corresponding ′ , ′′ bands, which is expected for transitions between states with ′ > ′′ and the bands are red degraded. For example the wavelength and wavenumbers versus rcentroids of few bands of ∆ = 3 sequence of the G-X system are displayed in Table 9 and wavelength versus r-centroids are plotted in Fig.3 which show a systematic relationship. Nicholls et al. [11] have shown that , is usually greater than ( ′ + ′′ )/ for systems whose potentials are not very anharmonic. In the present case, , is found to be slightly greater than ( ′ + ′′ )/ by 0.004, 0.005, 0.005, 0.004 and 0.005 respectively, suggesting that the potentials are almost harmonic. The vibrational sum rule is satisfied for lower and upper states progressions up to (2, ′′ ), ( ′ ,0) progressions for G-X, [ . ] G -X and up to (2, ′′ ), ( ′ ,1) progressions for K-X, L-X and to (2, ′′ ), ( ′ , 2) progressions for [ . ] G -X, indicating that further experimental work will not be helpful to observe more bands in these progressions. To assess the reliability of r-centroids values we have calculated the factor ′ ′′ ( ) for n = 3. In terms of this factor 1% agreement means that this quantity, which should be exactly 1, is smaller than 1.01 or | − ′ ′′ ( ) | < 0.01. According to McCallum [30] the obtained values may be considered reliable for which | − ′ ′′ ( ) |< 0.02. International Letters of Chemistry, Physics and Astronomy Vol. 58       The result order is respectively for Le Roy, Tele, Espy, Jarmain and Felenbok The result order is respectively for Le Roy, Tele, Espy, Jarmain and Felenbok

Figures and Tables
International Letters of Chemistry, Physics and Astronomy Vol. 58

CONCLUSION
FCFs and r-centroids for the G-X, K-X, L-X,[20.6] 3 G 5 -X and [34.7] 3 G 5 -X systems of cobalt monofluorid have been calculated for the first time using the Morse and RKR potentials. The generated values using Morse potential show's deviation with those generated by RKR potentials even for low υ-level transitions, however, both sets of the results follow the same pattern. This deviation does not depend on Morse program that we used but on nature of Morse model. We also checked the Morse result using the standard program LEVEL8 which gives fairly the same result presented here. The calculated values of FCFs may help in the search and experimental identification of new bands in these systems. On the other hand, the r-centroid values will be helpfull in studies of variations in electronic transition moment with internuclear distance as well as in the search for other molecular properties as a function of r.