- Email: [email protected]
Study of the electronic transition moment for the CN red band system
1. Quonr. Specmsc.
Radial.
Transfer,
Vol. IS, pp. 571-573. PergamonPress 1975. Printed in Great Britain.
STUDY OF THE ELECTRONIC TRANSITION MOMENT FOR THE CN RED BAND SYSTEM D. C. JAIN York College of The City University of New York, Jamaica, NY 11432, U.S.A. (Received
30 August
1974)
Abstract-The Rydberg-Klein-Rees (RKR) potential energy curves of the X*Z’ state and the AZIIi state of CN have been calculated by using the recent data of POLEITO and RIGUTTI.‘~’ The RKR Franck-Condon factors and r-centroids for the CN red band system have been computed. These have been used, in conjunction with the relative band oscillator strengths reported by LAMBERT,‘“’to determine the variation of the electronic transition moment, R,(r), with the internuclear separation, r. The expression for R,(r) obtained has been found to be in good agreement with that obtained by using the data of DIXON and NICHOLLS”’ on the relative emission intensity distribution in the red band system of CN.
INTRODUCTlON
variation of the electronic transition moment, R,(r), with the internuclear separation, r, for the red band system of CN was determined byDIxoN and NICHOLLS”’ by measuring the relative intensities of bands in emission. They employed the Franck-Condon factors appropriate to the Morse potential function and applied the r-centroid method to obtain THE
R,(r)
=
constant(l.044 + 0*174r), 1.04A G r s 1*27A.
(1)
Equation (1) implies a very slight variation of R,(r) with r. JEUNEHOMME”’measured the radiative lifetimes, T,., for the vibrational levels of the A% state of CN and obtained R,(r)
= R,(?oo)[l +
(Y(r- ~oow)l,
(2)
where (Y= 1.9? 0.2 A-’ and rW is the r-centroid for the (0,O)band. He has observed that this value of a is much larger than that obtained by using the results of Dixon and Nicholls. LAMBERT’~’ has derived relative band oscillator strengths for 9 bands of the CN red system from equivalent-width measurements of CN lines in the solar spectrum. He has employed the Morse Franck-Condon factors as well, as the RKR Franck-Condon factors, for the interpretation of his experimental data. He has observed that the RKR Franck-Condon factors yield more consistent results. However, even with the RKR Franck-Condon factors and r-centroids, the value of (Yin equation (2) is found to be 1.6 ? 0.2 A-‘, which implies a much larger variation of R,(r) with r than that obtained by DIXONand NICHOLLS.“’In view of this divergence of results, a detailed study of the variation of R,(r) with r for this band system has been undertaken. THEORY
The relative band oscillator strength, fO,USZ, of the band characterized by the vibrational quantum numbers (a’, u”) is given by
f”,,s,= cv,,,,,((rCr,,lR,(r)l~",,))*,
(3)
where c is a constant and other symbols have their usual significance. Assuming that R,(r) = a’ + b’r,
equation (3) can be written as
(4)
512
D. C.
JAIN
or
Here q vluil is the Franck-Condon factor and T,,,Mis the r-centroid. It should be remarked that equation (5) does not involve any approximations. Thus, if the assumption regarding the linear dependence of R,(r) on r [equation (4)] is valid, then a plot of ~/CfU.V,./vV,V,,qD,U,,) vs f,,,” for the various bands should be a straight line within limits of experimental error. Further, a least square calculationof ~\lf,,,,,/v,,,,,q,.,.,against fUs,,,should yield the coefficients a and b [equation (S)]. The method described above can also be adopted for determining R,(r) for the band systems for which R,(r) can be represented by a quadratic expression in r. Thus, if R,(r)=a’+b’r+c’r*,
equation
(3) can be written as
Now a least-square adjustment of g(fU,Un/v .,,,,) against (&.I$“,.), (I,G~~IPII&)and ($,,lr2j$,u) will yield the coefficients a, b and c. The procedure adopted by Jain and Sahn? in the least-square adjustment of ~/(I,,,,~/v$~~~) against (I&\&), (&~lrJ&.$ and (+uslr21$us~) for the y-bands of NO can be applied in such cases. METHOD OF COMPUTATION
The RKR potential energy curves for the X’Z’ and A *IL states of CN have been calculated by the method suggested by WEISSMAN,VANDERSLICE and BATTINO.‘~’ The recent experimental data published by POLE~O and RIGUTTI@’ have been adopted for the calculation. The method described by JAIN and SAHNI’~’has been applied for calculating ($I&~), and (~,,~~r~~u~~). The transition moment, R,(r), for the red bands of CN has been determined by using the relative band oscillator strengths reported by LAMBERT. The fU,r,,value for the (6,2) band has for this band is very small. For comparison, been omitted because the value of vcf”,U,,/v,,,,, qU.Op,) the transition moment, R,(r), has also been obtained by adopting the relative intensity data reported by DIXONand NICHOLLS.‘~’by using the resealing procedure described by NICHOLLS.“’The data on the (0,l) band cannot be resealed and thus it has been omitted. The “smoothed” relative band strengths, p,,,,,, given by
and the relative band oscillator strengths fU,“,,have been computed by using both the expreseions for R,(r). The expression for R,(r) has also been computed by using the more extensive and recent experimental relative intensity data reported by LEBLANC.“’The data on the (13,5) and (25,13) bands cannot be resealed and thus it has not been used in the calculation. Even upon resealing of LeBlanc’s data, the plot of (I,,,,,/v;t,,,,),,,,,)“* vs ?“,,p,exhibits considerably larger scatter of points than similar plots made by employing the data of Dixon and Nicholls’4’ and of Lambert.“) RESULTS AND DISCUSSION
The RKR potential energy curves for the X’Z’ and A *n states of CN have been computed for vibrational levels up to 17 and 19 respectively.* POLETO and RIGUTTI@’ have reported the values of B, for the vibrational levels up to v” = 6 and v’ = 12 only. The expermental data for higher vibrational levels have been obtained by extrapolation for computing the RKR potential-energy curves. For the vibrational levels in question, the more recent data of LEBLANC”’are in good agreement with that of Poletto and Rigutto. The values of the constant wryr are comparatively small for both the electronic states. Thus the RKR curves have only slight deviation from the Morse potential function in the region of low vibrational levels. Consequently, the Franck*Tablesof theRKR potential energy curves, Franck-Condon factors and r-centroids can be obtained from the author
Study of the electronic transition moment
513
Condon factors and r-centroids involving low vibrational levels, are in fair agreement with those However, the small differences between the RKR and Morse reported by NICHOLLS.(~) Franck-Condon factors significantly affect the determination of R,(r) as has been also observed by LAMBERT.The expressions for R,(r) obtained by using Lambert’s, and Dixon and Nicholls’ data, respectively, are the following: R,(r) = constant(1 + 0_7318r), 1.08 A S r S I.21 A,
(6)
R,(r) = constant(1 + 0*5717r), 1.05 A s r s 1.27 A.
(7)
The expressions for R,(r) obtained by using the relative band oscillator strengths and the relative band intensities [equations (6) and (7)] are in agreement within limits of experimental errors. The following expression for R,(r) has been obtained by using the data of LeBlanc?’ R,(r) = constant( 1 + 3.46r), 0.97 A s r s 1.10 A.
(8)
This equation implies a much stronger dependence of R,(r) on r than equations (6) and (7). This result is spurious and has been obtained most probably due to the scatter of points in the plot of (I”‘““/Y4v’v”q”.“.~)“2 vs ?“~“~~. Equations (6) and (7) can be put in the form of equation (2) to yield the values of (Yequal to 0.39 and 0.34, respectively. The “smoothed” relative band strengths, P~,“~,and relative band oscillator have been obtained by using these expressions for R,(r). It is seen that the two strengths, fVr,rS, sets of values of p,,,,, are in good agreement as are the two sets of values of f”.,,,. This result is obtained because the transition moment represented by both the expressions [equations (6) and (7)] has a variation of about 8% within the range 1.05 w G r G 1.27 A. Thus, the experimental data of Dixon and Nicholls and of Lambert yield essentially the same variation of R,(r) with r. The variation of R,(r) represented by equations (6) and (7) is quite small. This result also has been recently observed by Arnold and Nicholls”” and by Carbon.‘“’ The present approach of determining R,(r) by using the relative band oscillator strengths is free from any uncertainties which are involved in the resealing that has to be done in most cases if the relative band intensities are employed. Acknowledgements-The author is indebted to the City College of the City University of New York for the use of their IBM 360/50computer. The Franck-Condon factors and r-centroids were computed on the IMBM 360/95computer of the Institute for Space Studies, 2880 Broadway, New York, N.Y. REFERENCES I. R. N. DIXONand R. W. NICHOLLS,Can. J. Phys. 36, 127 (1958). 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
M. JEUNEHOMME, J. Chem. Phys. 42, 4086 (1965). D. L. LAMBERT,JQSRT 8, 1265 (1968). D. C. JAIN and R. C. SAHNI, Trans. Faraday Sot. 64, 3169 (1968). S. WEISSMAN, J. T. VANDERSLICE and R. J. BAITINO,J. Chem. Phys. 39, 2226 (1%3). G. POLE~O and M. RIGUTTI,Nuouo Cimmto 39, 519 (1965). R. W. NICHOLLS,Proc. Phys. Sot. A69, 741 (1956). F. J. LEBLANC, .J. Chem. Phys. 48, 1980 (1968). R. W. NICHOLLS,J. Res. Nat. Bur. Stand. 68A, 75 (1964). J. 0. ARNOLDand R. W. NICHOLLS,JQSRT 12, 1435 (1972). D. F. CARBON,Astrophys. J. 183, 903 (1973).
Radial.
Transfer,
Vol. IS, pp. 571-573. PergamonPress 1975. Printed in Great Britain.
STUDY OF THE ELECTRONIC TRANSITION MOMENT FOR THE CN RED BAND SYSTEM D. C. JAIN York College of The City University of New York, Jamaica, NY 11432, U.S.A. (Received
30 August
1974)
Abstract-The Rydberg-Klein-Rees (RKR) potential energy curves of the X*Z’ state and the AZIIi state of CN have been calculated by using the recent data of POLEITO and RIGUTTI.‘~’ The RKR Franck-Condon factors and r-centroids for the CN red band system have been computed. These have been used, in conjunction with the relative band oscillator strengths reported by LAMBERT,‘“’to determine the variation of the electronic transition moment, R,(r), with the internuclear separation, r. The expression for R,(r) obtained has been found to be in good agreement with that obtained by using the data of DIXON and NICHOLLS”’ on the relative emission intensity distribution in the red band system of CN.
INTRODUCTlON
variation of the electronic transition moment, R,(r), with the internuclear separation, r, for the red band system of CN was determined byDIxoN and NICHOLLS”’ by measuring the relative intensities of bands in emission. They employed the Franck-Condon factors appropriate to the Morse potential function and applied the r-centroid method to obtain THE
R,(r)
=
constant(l.044 + 0*174r), 1.04A G r s 1*27A.
(1)
Equation (1) implies a very slight variation of R,(r) with r. JEUNEHOMME”’measured the radiative lifetimes, T,., for the vibrational levels of the A% state of CN and obtained R,(r)
= R,(?oo)[l +
(Y(r- ~oow)l,
(2)
where (Y= 1.9? 0.2 A-’ and rW is the r-centroid for the (0,O)band. He has observed that this value of a is much larger than that obtained by using the results of Dixon and Nicholls. LAMBERT’~’ has derived relative band oscillator strengths for 9 bands of the CN red system from equivalent-width measurements of CN lines in the solar spectrum. He has employed the Morse Franck-Condon factors as well, as the RKR Franck-Condon factors, for the interpretation of his experimental data. He has observed that the RKR Franck-Condon factors yield more consistent results. However, even with the RKR Franck-Condon factors and r-centroids, the value of (Yin equation (2) is found to be 1.6 ? 0.2 A-‘, which implies a much larger variation of R,(r) with r than that obtained by DIXONand NICHOLLS.“’In view of this divergence of results, a detailed study of the variation of R,(r) with r for this band system has been undertaken. THEORY
The relative band oscillator strength, fO,USZ, of the band characterized by the vibrational quantum numbers (a’, u”) is given by
f”,,s,= cv,,,,,((rCr,,lR,(r)l~",,))*,
(3)
where c is a constant and other symbols have their usual significance. Assuming that R,(r) = a’ + b’r,
equation (3) can be written as
(4)
512
D. C.
JAIN
or
Here q vluil is the Franck-Condon factor and T,,,Mis the r-centroid. It should be remarked that equation (5) does not involve any approximations. Thus, if the assumption regarding the linear dependence of R,(r) on r [equation (4)] is valid, then a plot of ~/CfU.V,./vV,V,,qD,U,,) vs f,,,” for the various bands should be a straight line within limits of experimental error. Further, a least square calculationof ~\lf,,,,,/v,,,,,q,.,.,against fUs,,,should yield the coefficients a and b [equation (S)]. The method described above can also be adopted for determining R,(r) for the band systems for which R,(r) can be represented by a quadratic expression in r. Thus, if R,(r)=a’+b’r+c’r*,
equation
(3) can be written as
Now a least-square adjustment of g(fU,Un/v .,,,,) against (&.I$“,.), (I,G~~IPII&)and ($,,lr2j$,u) will yield the coefficients a, b and c. The procedure adopted by Jain and Sahn? in the least-square adjustment of ~/(I,,,,~/v$~~~) against (I&\&), (&~lrJ&.$ and (+uslr21$us~) for the y-bands of NO can be applied in such cases. METHOD OF COMPUTATION
The RKR potential energy curves for the X’Z’ and A *IL states of CN have been calculated by the method suggested by WEISSMAN,VANDERSLICE and BATTINO.‘~’ The recent experimental data published by POLE~O and RIGUTTI@’ have been adopted for the calculation. The method described by JAIN and SAHNI’~’has been applied for calculating ($I&~), and (~,,~~r~~u~~). The transition moment, R,(r), for the red bands of CN has been determined by using the relative band oscillator strengths reported by LAMBERT. The fU,r,,value for the (6,2) band has for this band is very small. For comparison, been omitted because the value of vcf”,U,,/v,,,,, qU.Op,) the transition moment, R,(r), has also been obtained by adopting the relative intensity data reported by DIXONand NICHOLLS.‘~’by using the resealing procedure described by NICHOLLS.“’The data on the (0,l) band cannot be resealed and thus it has been omitted. The “smoothed” relative band strengths, p,,,,,, given by
and the relative band oscillator strengths fU,“,,have been computed by using both the expreseions for R,(r). The expression for R,(r) has also been computed by using the more extensive and recent experimental relative intensity data reported by LEBLANC.“’The data on the (13,5) and (25,13) bands cannot be resealed and thus it has not been used in the calculation. Even upon resealing of LeBlanc’s data, the plot of (I,,,,,/v;t,,,,),,,,,)“* vs ?“,,p,exhibits considerably larger scatter of points than similar plots made by employing the data of Dixon and Nicholls’4’ and of Lambert.“) RESULTS AND DISCUSSION
The RKR potential energy curves for the X’Z’ and A *n states of CN have been computed for vibrational levels up to 17 and 19 respectively.* POLETO and RIGUTTI@’ have reported the values of B, for the vibrational levels up to v” = 6 and v’ = 12 only. The expermental data for higher vibrational levels have been obtained by extrapolation for computing the RKR potential-energy curves. For the vibrational levels in question, the more recent data of LEBLANC”’are in good agreement with that of Poletto and Rigutto. The values of the constant wryr are comparatively small for both the electronic states. Thus the RKR curves have only slight deviation from the Morse potential function in the region of low vibrational levels. Consequently, the Franck*Tablesof theRKR potential energy curves, Franck-Condon factors and r-centroids can be obtained from the author
Study of the electronic transition moment
513
Condon factors and r-centroids involving low vibrational levels, are in fair agreement with those However, the small differences between the RKR and Morse reported by NICHOLLS.(~) Franck-Condon factors significantly affect the determination of R,(r) as has been also observed by LAMBERT.The expressions for R,(r) obtained by using Lambert’s, and Dixon and Nicholls’ data, respectively, are the following: R,(r) = constant(1 + 0_7318r), 1.08 A S r S I.21 A,
(6)
R,(r) = constant(1 + 0*5717r), 1.05 A s r s 1.27 A.
(7)
The expressions for R,(r) obtained by using the relative band oscillator strengths and the relative band intensities [equations (6) and (7)] are in agreement within limits of experimental errors. The following expression for R,(r) has been obtained by using the data of LeBlanc?’ R,(r) = constant( 1 + 3.46r), 0.97 A s r s 1.10 A.
(8)
This equation implies a much stronger dependence of R,(r) on r than equations (6) and (7). This result is spurious and has been obtained most probably due to the scatter of points in the plot of (I”‘““/Y4v’v”q”.“.~)“2 vs ?“~“~~. Equations (6) and (7) can be put in the form of equation (2) to yield the values of (Yequal to 0.39 and 0.34, respectively. The “smoothed” relative band strengths, P~,“~,and relative band oscillator have been obtained by using these expressions for R,(r). It is seen that the two strengths, fVr,rS, sets of values of p,,,,, are in good agreement as are the two sets of values of f”.,,,. This result is obtained because the transition moment represented by both the expressions [equations (6) and (7)] has a variation of about 8% within the range 1.05 w G r G 1.27 A. Thus, the experimental data of Dixon and Nicholls and of Lambert yield essentially the same variation of R,(r) with r. The variation of R,(r) represented by equations (6) and (7) is quite small. This result also has been recently observed by Arnold and Nicholls”” and by Carbon.‘“’ The present approach of determining R,(r) by using the relative band oscillator strengths is free from any uncertainties which are involved in the resealing that has to be done in most cases if the relative band intensities are employed. Acknowledgements-The author is indebted to the City College of the City University of New York for the use of their IBM 360/50computer. The Franck-Condon factors and r-centroids were computed on the IMBM 360/95computer of the Institute for Space Studies, 2880 Broadway, New York, N.Y. REFERENCES I. R. N. DIXONand R. W. NICHOLLS,Can. J. Phys. 36, 127 (1958). 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
M. JEUNEHOMME, J. Chem. Phys. 42, 4086 (1965). D. L. LAMBERT,JQSRT 8, 1265 (1968). D. C. JAIN and R. C. SAHNI, Trans. Faraday Sot. 64, 3169 (1968). S. WEISSMAN, J. T. VANDERSLICE and R. J. BAITINO,J. Chem. Phys. 39, 2226 (1%3). G. POLE~O and M. RIGUTTI,Nuouo Cimmto 39, 519 (1965). R. W. NICHOLLS,Proc. Phys. Sot. A69, 741 (1956). F. J. LEBLANC, .J. Chem. Phys. 48, 1980 (1968). R. W. NICHOLLS,J. Res. Nat. Bur. Stand. 68A, 75 (1964). J. 0. ARNOLDand R. W. NICHOLLS,JQSRT 12, 1435 (1972). D. F. CARBON,Astrophys. J. 183, 903 (1973).