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Fitting of Line Intensities Using the Effective Operator Approach: The 4 μm Region of14N216O
JOURNAL OF MOLECULAR SPECTROSCOPY ARTICLE NO .
180, 72–74 (1996)
0224
Fitting of Line Intensities Using the Effective Operator Approach: The 4 mm Region of 14N2 16O O. M. Lyulin,* V. I. Perevalov,* and J.-L. Teffo† *Institute of Atmospheric Optics, Siberian Branch, Russian Academy of Sciences, 1, Akademicheskii av., 634055 Tomsk, Russia; and †Laboratoire de Physique Mole´culaire et Applications (laboratoire associe´ aux Universites Pierre et Marie Curie et Paris-Sud), C.N.R.S., Universite´ Pierre et Marie Curie, Boite 76, 4 Place Jussieu, 75252 Paris Cedex 05, France Received February 23, 1996; in revised form June 18, 1996
the linestrength of the transition N *J * e * R NJe between states labeled with the vibrational index N, the rotational quantum number J, and the parity e Å {1 is given by
In a series of recent papers (1–4), we have developed an effective Hamiltonian and an effective dipole moment approach for analyzing and predicting infrared spectra of CO2 and N2O molecules. In the first two papers of this series, we derived reduced effective Hamiltonians for a global treatment of vibrational–rotational energy levels of CO2 and N2O in their ground electronic states. The main feature of these effective Hamiltonians is their block-diagonal form, each block representing a polyad of interacting vibrational states in finite number. Thus less computational effort is required for solving the secular equation than using the well known DND technique (5). The parameters of these effective Hamiltonians have been obtained by fitting experimental spectroscopic constants Gv , Bv , and Dv . In the subsequent papers (3, 4), the eigenfunctions obtained from the above fits were successfully used for fitting experimental rotationless transition moments of CO2 and N2O to effective dipole moment parameters in various spectral regions. In this note we demonstrate the ability of the effective operators method to treat simultaneously line intensities of cold and corresponding hot bands of linear triatomic molecules, taking as an example line intensities of 14N2 16O bands in the 4 mm region, for which extensive and precise measurements have been performed recently (6–9). These measurements are used as input data. The line intensity, SbRa (T ), of a vibrational–rotational transition b R a is given in cm01 /moleculercm02 by the equation SbRa (T ) Å
8p 3 exp( 0hcEa /kT ) C sbRa 3hc Q(T )
eff eff 2 WN =J = e =RNJe Å 3 ∑ É» C eff N =J =M = e = É M Z ÉC NJMe …É MM *
Å3
∑É ∑ MM *
£1£2l 2£3
where the sum is taken over the magnetic quantum numbers M and M * of the lower and upper states, respectively. The J £1£2l2£3 C Ne are mixing coefficients determining the eigenfunction C eff NJMe Å
£1£2l2£3 É£1£2Él2É£3 JMe… ∑ JC Ne
of the effective Hamiltonian within the lower state polyad. In the same way J =C £N1==£e2==l 2=£ 3= stand for the mixing coefficients within the upper state polyad. The definition of the Wangtype basis function É£1£2Él2É£3 JMe… is given in Refs. (1–4). In the case of the N2O molecule, for which the approximate relations between harmonic frequencies are v3 á 2v1 á 4v2 ,
[1]
P Å 2£1 / £2 / 4£3 .
09-09-96 21:55:26
[5]
Thus the polyads can be labeled with the integer P. The transitions in the 4 mm region we are dealing with in this work correspond to parallel bands with DP Å 4 for the difference between upper and lower polyad numbers. Summing in Eq. [2] over magnetic quantum numbers (10) one 72
6t10$$$221
[4]
each polyad is made up of vibrational basis states the quantum numbers of which fulfill the equation
0022-2852/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
/
[3]
£1£2l2£3
where T is the reference temperature, C is the isotopic abundance, sbRa is the wavenumber of the vibrational–rotational transition b R a, Ea is the energy of the lower state, k is the Boltzmann constant, Q(T ) is the partition function, and WbRa is the linestrength. Within the effective operators approach
JMS 7111
[2]
£1=£2=l2=£3=
2 1 »£*1 £*2 Él *2 É£*3 J *M * e *É M eff Z É£1£2Él2É£3 JMe…É ,
1 [1 0 exp( 0hc sbRa /kT )]WbRa ,
AID
£1£2l2£3 J = £ 1=£ 2=l 2=£ 3= C N=e= ∑ JC Ne
mspa
AP: Mol Spec
FITTING LINE INTENSITIES OF
N2 16O
14
73
TABLE 1 Parameters of Matrix Elements of the Effective Dipole Moment Operator
can obtain the following equation for the linestrengths of parallel bands (1, 3): WN =J = e =RNJe
∑
ÅÉ
∑
J
£1£2l2£3 J = £1/D£1£2/D£2l2£3/D£3 C Ne C N=e=
£1£2£3 D£1/D£2/D£3ÅDP
S H
q
l2Å0 l2Å0 1 MD LD f D£ DJ
1
D l2Å0 D£
( £, l2 )
[6]
1 / k 1D£ £1 / k 2D£ £2 / k 3D£ £3 / b JD£ m /
d JD£ [J(J / 1) 0 l 22 ], for Q branch
d JD£ [m2 0 l 22 ],
DZ
2
for P and R branches
.
l2Å0 is the Ho¨nl–London factor and m Å 0J, 0, J Here L D DJ / 1 for the P, Q, and R branches, respectively. The Herman– Wallis type rotational contributions in Eq. [6] are taken D l2 from Watson (11). The f D£ ( £, l2 ) functions are listed in Table 1 of Ref. (3) for small values of the quantum number differences D£. The MD£ , k iD£ (i Å 1, 2, 3), b JD£ , and d JD£ parameters of the effective dipole moment matrix elements describe the strengths of all lines of cold and hot bands belonging to a series of transitions characterized by a value of DP. Using a nonlinear least-squares adjustment, 652 line intensities belonging to P and R branches of the 10 parallel bands 0001 R 0000, 2000 R 0000, 1200 R 0000, 1310 R 0110, 2110 R 0110, 1400 R 0200, 3000 R 1000, 2200 R 1000, 2200 R 0200, and 3000 R 0200 have been fitted simultane-
Copyright q 1996 by Academic Press, Inc.
AID
JMS 7111
/
6t10$$$222
09-09-96 21:55:26
mspa
AP: Mol Spec
74
LYULIN, PEREVALOV, AND TEFFO
TABLE 2 Statistical Analysis of the Fit
ously to 12 effective dipole moment parameters. For this fit the mixing coefficients obtained from fit 8 of the spectroscopic constants in our previous work (2) have been used. Several fittings were performed using different sets of effective dipole moment parameters. The values of the effective dipole moment parameters and of the dimensionless standard deviation corresponding to the best residuals achieved are presented in Table 1. As the signs of all MD£ parameters (and of these parameters only) may be changed simultaneously, we have removed this ambiguity by constraining the leading parameter MD£3Å1 to a positive value. In all fittings we found that k iD£ (i Å 1, 2, 3) are not significant, except for the D£1 Å 2 matrix element, and the d JD£ parameters are very important in the two matrix elements D£1 Å 2, D£2 Å 0 and D£1 Å 1, D£2 Å 2. On the other hand, d JD£ in D£3 Å 1 and b JD£ in D£1 Å 2 matrix elements are not significant. It is to be noted that the corrections to linestrengths due to vibration–rotation interactions are accounted for partly by mixing coefficients and partly by parameters b JD£ and d JD£ . In the last column of Table 1 we report for comparison the purely vibrational parameters obtained from a previous fitting (4) of the band intensities measured by Toth (12). The agreement between the two sets of parameters is rather good except for high order ones.1 It is to be noted that the MD£ parameters for the matrix elements D£1 Å 1, D£2 Å 02, D£3 Å 1, on the one hand, and D£1 Å 3, D£2 Å 02, D£3 Å 0, on the other, have a rather large value compared with their theoretical orders of magnitude, which are third and fourth, respectively. A statistical analysis of our results is given in Table 2. As can be seen from the dimensionless standard deviation reported in Table 1, we have reached experimental accuracy. 1 The signs of all MD£ parameters reported in Ref. (4) have been changed simultaneously in order to make consistent the two sets of values compared in Table 1.
The authors of Refs. (6–9) have given an absolute accuracy of 3% for their line intensity measurements. Of our calculated values, 99% lie within this accuracy. The list of calculated and experimental line intensities is available upon request to the authors. It is also on deposit in the editorial office of this journal. The set of matrix element parameters of the effective dipole moment operator reported in this study, together with the eigenfunctions obtained in our previous work (2), can be used for prediction of line intensities of hot bands in the 4 mm region and originating from higher excited vibrational states. ACKNOWLEDGMENTS Thanks are due to F. Rachet and A. Valentin for giving us their experimental data file.
REFERENCES 1. J.-L. Teffo, O. N. Sulakshina, and V. I. Perevalov, J. Mol. Spectrosc. 156, 48 (1992). 2. J.-L. Teffo, V. I. Perevalov, and O. M. Lyulin, J. Mol. Spectrosc. 168, 390 (1994). 3. V. I. Perevalov, E. I. Lobodenko, O. M. Lyulin, and J.-L. Teffo, J. Mol. Spectrosc. 171, 435 (1995). 4. O. M. Lyulin, V. I. Perevalov, and J.-L. Teffo, J. Mol. Spectrosc. 174, 566 (1995). 5. R. B. Wattson and L. S. Rothman, J. Quant. Spectrosc. Radiat. Transfer 48, 763 (1992). 6. M. El Azizi, F. Rachet, A. Henry, M. Margottin-Maclou, and A. Valentin, J. Mol. Spectrosc. 164, 180 (1994). 7. F. Rachet, M. Margottin-Maclou, M. El Azizi, A. Henry, and A. Valentin, J. Mol. Spectrosc. 164, 196 (1994). 8. F. Rachet, M. Margottin-Maclou, M. El Azizi, A. Henry, and A. Valentin, J. Mol. Spectrosc. 166, 79 (1994). 9. F. Rachet and A. Valentin, private communication. 10. F. Rasmussen and S. Brodersen, J. Mol. Spectrosc. 25, 166 (1968). 11. J. K. G. Watson, J. Mol. Spectrosc. 125, 428 (1987). 12. R. A. Toth, Appl. Opt. 32, 7326 (1993).
Copyright q 1996 by Academic Press, Inc.
AID
JMS 7111
/
6t10$$$222
09-09-96 21:55:26
mspa
AP: Mol Spec
180, 72–74 (1996)
0224
Fitting of Line Intensities Using the Effective Operator Approach: The 4 mm Region of 14N2 16O O. M. Lyulin,* V. I. Perevalov,* and J.-L. Teffo† *Institute of Atmospheric Optics, Siberian Branch, Russian Academy of Sciences, 1, Akademicheskii av., 634055 Tomsk, Russia; and †Laboratoire de Physique Mole´culaire et Applications (laboratoire associe´ aux Universites Pierre et Marie Curie et Paris-Sud), C.N.R.S., Universite´ Pierre et Marie Curie, Boite 76, 4 Place Jussieu, 75252 Paris Cedex 05, France Received February 23, 1996; in revised form June 18, 1996
the linestrength of the transition N *J * e * R NJe between states labeled with the vibrational index N, the rotational quantum number J, and the parity e Å {1 is given by
In a series of recent papers (1–4), we have developed an effective Hamiltonian and an effective dipole moment approach for analyzing and predicting infrared spectra of CO2 and N2O molecules. In the first two papers of this series, we derived reduced effective Hamiltonians for a global treatment of vibrational–rotational energy levels of CO2 and N2O in their ground electronic states. The main feature of these effective Hamiltonians is their block-diagonal form, each block representing a polyad of interacting vibrational states in finite number. Thus less computational effort is required for solving the secular equation than using the well known DND technique (5). The parameters of these effective Hamiltonians have been obtained by fitting experimental spectroscopic constants Gv , Bv , and Dv . In the subsequent papers (3, 4), the eigenfunctions obtained from the above fits were successfully used for fitting experimental rotationless transition moments of CO2 and N2O to effective dipole moment parameters in various spectral regions. In this note we demonstrate the ability of the effective operators method to treat simultaneously line intensities of cold and corresponding hot bands of linear triatomic molecules, taking as an example line intensities of 14N2 16O bands in the 4 mm region, for which extensive and precise measurements have been performed recently (6–9). These measurements are used as input data. The line intensity, SbRa (T ), of a vibrational–rotational transition b R a is given in cm01 /moleculercm02 by the equation SbRa (T ) Å
8p 3 exp( 0hcEa /kT ) C sbRa 3hc Q(T )
eff eff 2 WN =J = e =RNJe Å 3 ∑ É» C eff N =J =M = e = É M Z ÉC NJMe …É MM *
Å3
∑É ∑ MM *
£1£2l 2£3
where the sum is taken over the magnetic quantum numbers M and M * of the lower and upper states, respectively. The J £1£2l2£3 C Ne are mixing coefficients determining the eigenfunction C eff NJMe Å
£1£2l2£3 É£1£2Él2É£3 JMe… ∑ JC Ne
of the effective Hamiltonian within the lower state polyad. In the same way J =C £N1==£e2==l 2=£ 3= stand for the mixing coefficients within the upper state polyad. The definition of the Wangtype basis function É£1£2Él2É£3 JMe… is given in Refs. (1–4). In the case of the N2O molecule, for which the approximate relations between harmonic frequencies are v3 á 2v1 á 4v2 ,
[1]
P Å 2£1 / £2 / 4£3 .
09-09-96 21:55:26
[5]
Thus the polyads can be labeled with the integer P. The transitions in the 4 mm region we are dealing with in this work correspond to parallel bands with DP Å 4 for the difference between upper and lower polyad numbers. Summing in Eq. [2] over magnetic quantum numbers (10) one 72
6t10$$$221
[4]
each polyad is made up of vibrational basis states the quantum numbers of which fulfill the equation
0022-2852/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
/
[3]
£1£2l2£3
where T is the reference temperature, C is the isotopic abundance, sbRa is the wavenumber of the vibrational–rotational transition b R a, Ea is the energy of the lower state, k is the Boltzmann constant, Q(T ) is the partition function, and WbRa is the linestrength. Within the effective operators approach
JMS 7111
[2]
£1=£2=l2=£3=
2 1 »£*1 £*2 Él *2 É£*3 J *M * e *É M eff Z É£1£2Él2É£3 JMe…É ,
1 [1 0 exp( 0hc sbRa /kT )]WbRa ,
AID
£1£2l2£3 J = £ 1=£ 2=l 2=£ 3= C N=e= ∑ JC Ne
mspa
AP: Mol Spec
FITTING LINE INTENSITIES OF
N2 16O
14
73
TABLE 1 Parameters of Matrix Elements of the Effective Dipole Moment Operator
can obtain the following equation for the linestrengths of parallel bands (1, 3): WN =J = e =RNJe
∑
ÅÉ
∑
J
£1£2l2£3 J = £1/D£1£2/D£2l2£3/D£3 C Ne C N=e=
£1£2£3 D£1/D£2/D£3ÅDP
S H
q
l2Å0 l2Å0 1 MD LD f D£ DJ
1
D l2Å0 D£
( £, l2 )
[6]
1 / k 1D£ £1 / k 2D£ £2 / k 3D£ £3 / b JD£ m /
d JD£ [J(J / 1) 0 l 22 ], for Q branch
d JD£ [m2 0 l 22 ],
DZ
2
for P and R branches
.
l2Å0 is the Ho¨nl–London factor and m Å 0J, 0, J Here L D DJ / 1 for the P, Q, and R branches, respectively. The Herman– Wallis type rotational contributions in Eq. [6] are taken D l2 from Watson (11). The f D£ ( £, l2 ) functions are listed in Table 1 of Ref. (3) for small values of the quantum number differences D£. The MD£ , k iD£ (i Å 1, 2, 3), b JD£ , and d JD£ parameters of the effective dipole moment matrix elements describe the strengths of all lines of cold and hot bands belonging to a series of transitions characterized by a value of DP. Using a nonlinear least-squares adjustment, 652 line intensities belonging to P and R branches of the 10 parallel bands 0001 R 0000, 2000 R 0000, 1200 R 0000, 1310 R 0110, 2110 R 0110, 1400 R 0200, 3000 R 1000, 2200 R 1000, 2200 R 0200, and 3000 R 0200 have been fitted simultane-
Copyright q 1996 by Academic Press, Inc.
AID
JMS 7111
/
6t10$$$222
09-09-96 21:55:26
mspa
AP: Mol Spec
74
LYULIN, PEREVALOV, AND TEFFO
TABLE 2 Statistical Analysis of the Fit
ously to 12 effective dipole moment parameters. For this fit the mixing coefficients obtained from fit 8 of the spectroscopic constants in our previous work (2) have been used. Several fittings were performed using different sets of effective dipole moment parameters. The values of the effective dipole moment parameters and of the dimensionless standard deviation corresponding to the best residuals achieved are presented in Table 1. As the signs of all MD£ parameters (and of these parameters only) may be changed simultaneously, we have removed this ambiguity by constraining the leading parameter MD£3Å1 to a positive value. In all fittings we found that k iD£ (i Å 1, 2, 3) are not significant, except for the D£1 Å 2 matrix element, and the d JD£ parameters are very important in the two matrix elements D£1 Å 2, D£2 Å 0 and D£1 Å 1, D£2 Å 2. On the other hand, d JD£ in D£3 Å 1 and b JD£ in D£1 Å 2 matrix elements are not significant. It is to be noted that the corrections to linestrengths due to vibration–rotation interactions are accounted for partly by mixing coefficients and partly by parameters b JD£ and d JD£ . In the last column of Table 1 we report for comparison the purely vibrational parameters obtained from a previous fitting (4) of the band intensities measured by Toth (12). The agreement between the two sets of parameters is rather good except for high order ones.1 It is to be noted that the MD£ parameters for the matrix elements D£1 Å 1, D£2 Å 02, D£3 Å 1, on the one hand, and D£1 Å 3, D£2 Å 02, D£3 Å 0, on the other, have a rather large value compared with their theoretical orders of magnitude, which are third and fourth, respectively. A statistical analysis of our results is given in Table 2. As can be seen from the dimensionless standard deviation reported in Table 1, we have reached experimental accuracy. 1 The signs of all MD£ parameters reported in Ref. (4) have been changed simultaneously in order to make consistent the two sets of values compared in Table 1.
The authors of Refs. (6–9) have given an absolute accuracy of 3% for their line intensity measurements. Of our calculated values, 99% lie within this accuracy. The list of calculated and experimental line intensities is available upon request to the authors. It is also on deposit in the editorial office of this journal. The set of matrix element parameters of the effective dipole moment operator reported in this study, together with the eigenfunctions obtained in our previous work (2), can be used for prediction of line intensities of hot bands in the 4 mm region and originating from higher excited vibrational states. ACKNOWLEDGMENTS Thanks are due to F. Rachet and A. Valentin for giving us their experimental data file.
REFERENCES 1. J.-L. Teffo, O. N. Sulakshina, and V. I. Perevalov, J. Mol. Spectrosc. 156, 48 (1992). 2. J.-L. Teffo, V. I. Perevalov, and O. M. Lyulin, J. Mol. Spectrosc. 168, 390 (1994). 3. V. I. Perevalov, E. I. Lobodenko, O. M. Lyulin, and J.-L. Teffo, J. Mol. Spectrosc. 171, 435 (1995). 4. O. M. Lyulin, V. I. Perevalov, and J.-L. Teffo, J. Mol. Spectrosc. 174, 566 (1995). 5. R. B. Wattson and L. S. Rothman, J. Quant. Spectrosc. Radiat. Transfer 48, 763 (1992). 6. M. El Azizi, F. Rachet, A. Henry, M. Margottin-Maclou, and A. Valentin, J. Mol. Spectrosc. 164, 180 (1994). 7. F. Rachet, M. Margottin-Maclou, M. El Azizi, A. Henry, and A. Valentin, J. Mol. Spectrosc. 164, 196 (1994). 8. F. Rachet, M. Margottin-Maclou, M. El Azizi, A. Henry, and A. Valentin, J. Mol. Spectrosc. 166, 79 (1994). 9. F. Rachet and A. Valentin, private communication. 10. F. Rasmussen and S. Brodersen, J. Mol. Spectrosc. 25, 166 (1968). 11. J. K. G. Watson, J. Mol. Spectrosc. 125, 428 (1987). 12. R. A. Toth, Appl. Opt. 32, 7326 (1993).
Copyright q 1996 by Academic Press, Inc.
AID
JMS 7111
/
6t10$$$222
09-09-96 21:55:26
mspa
AP: Mol Spec