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Semiclassical line mixing analysis in the first overtone band of CO compressed by N2
f+wed Phys. Technol. Vol. 35. No. 7. pp. 897-903. 1994
13504495(94)EOO24-E
SEMICLASSICAL OVERTONE
Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 1350~4495194 $7.00 + 0.00
LINE MIXING ANALYSIS IN THE FIRST BAND OF CO COMPRESSED BY N,
N. N. FILIPPOV,’M. V. TONKOV’ and J.-P. BOUANICH’ ‘Institute of Physics, St Petersburg University, Peterhof, St Petersburg, 198904, Russia and 2Laboratoire de Physique Moleculaire et Applications, tU.P.R. 136 du C.N.R.S., Universitt de Paris&d,. Bat. 350, 91405 Orsay France (Received 7 March 1994) absorption band of CO compressed by N, at various densities Abstract-Infrared spectra of the 2 -0 up to 456 Amagat and at temperatures of 296, 188 and 122 K have been analyzed. To interpret these spectra we have applied a semiclassical model with adiabatic corrections which accounts for the line-mixing effects. This model provides a satisfactory agreement with experimental bandshapes for all temperatures, especially for the lower N, densities. The discrepancy arising at higher densities may be due to imperfections of the calculated relaxation matrix and/or to the non-validity of the impact and binary collision approximations.
INTRODUCTION
Collision-induced line mixing effects have given rise to some theoretical developments and these effects on IR absorption spectra have been mainly calculated for molecule-rare gas atom collisions.(‘m5) Experimental data on CO-N, system were previously obtained in the 2 CO band@) at such high densities that the line-mixing effects are very important and cannot be neglected. In a recent work”’ the contribution of the molecular van der Waals complex CO..N, to the 2 c 0 bandshape was estimated from the differences observed between the experimental profiles and those of the monomer which were calculated by using an empirical bandmodel. However, to account for the line mixing effect straightforwardly we have undertaken further investigations in the monomeric spectral profiles. To the best of our knowledge there is no ab initio calculation for the rotational relaxation matrix of the CO-N, system which is essential to generate the bandshapes. Moreover, it is difficult to use the empirical fitting laws@“) in matrix elements estimation as we have no definite means to determine the elements describing the interbranch line mixing. Therefore we have decided to use in a first step a semiclassical approach’5’ (SCA) which allows to treat some parameters as adjustable ones. This approach is based on the same collisional model as that considered by Gordon and McGinnis (‘I) to calculate the C&He bandshape but our calculation of the rotational relaxation matrix is notably simplified. Indeed, our method deals with the probability distribution functions instead of a model involving thousands of molecular trajectories.“” The experimental conditions of the recordings at room and low temperatures were discussed in detail previously.@‘,‘) In the present work, we consider a wider set of experimental data than in Ref. (7). We studied spectra at pressures up to 970 bar and at three temperatures: 296 k 1, 188 f 2 and 122 + 2 K. The densities of the gas mixtures were obtained from the state equation”*’ for nitrogen. tLaboratoire
associe
aux Universites
Paris-Sud
et Pierre et Marie Curie. 897
INF35,7--F
N. N. FILIPPOV et a/
898
Because absorption
the CO pressure was not accurately coefficients K(v) defined by
measured,
K(v) = !I (v)/
we have preferred
to study normalized
c1(v)dv,
(1)
s band where tl (v) is the measured
absorption
BANDSHAPE
The
absorption
coefficient
coefficient
IN
THE
at the wavenumber
IMPACT
V.
APPROXIMATION
CI(v) for a vibration-rotation
band
is given
by the well known
expression c(
By using the line space approximation by(13)
[I - exp(hcv/kT)]F(v).
(v) = g
formalism
we may
express
the spectral
density
F(r)
in the impact
where d is the dipole-moment operator of the system, m and m ’ represent the line numbers corresponding to the rotation-vibration transitions ( j,, z:,)+ ( j,, I’, ) and ( j ‘, , I’, ) + (.j ‘, . I‘, ). For parallel bands of linear molecules, it is convenient to consider M = -j, in the P-branch and m =j, + 1 in the R-branch. P, = P,.,P,, is the population of the absorbing molecules in the initial state, and d, is the reduced matrix element of the d-operator d,, = (t’~, Ildllr,~,) 1 in equation
(4) is the unit matrix,
and the L-matrix
elements
can be defined
Table 1. Matrix elements r,,,,, (in 10-l cm-‘. Amagatr’)
by
for CO-N: a: 296 K
m’
In
2 20 18 16 14 12 10 8 6 4 2
-2 -4 -6 -8 -10 - 12 - 14 - 16 -18 -20
4
79.68 -2
72
75.56 -3.19 -2.77 - 3.20
6
72.30 -4.51 -3.24 -2.62 -2.94 -2.67
8
69.63 -4.90 -4.1 I -2.84 -2.15 -2.35 -2.12 - I.71
10
67.19 -4.70 -4.17 -3.55 -2.22 - 1.56 ~ 1.68 - 1.52 - 1.25 -0.97
12
64.75 -4.25 -3.83 -3.23 ~ 2.46 - 1.56 - 1.03 ~ I.09
-0.99 - 0.85 -0.70 - 0.55
14
62.21 -3.72 -3.34 -2.86 -2.28 - I .65 - I .oo - 0.6 I
-0.64 -0.60 ~ 0.53 -0.47 -0.40 ~ 0.3.I
The elements presented in italics arise from the interbranch line-mixing effects.
16
59.60 -3.20 -2.83 - 2.42 - I .96 - I .48 - I.01 -0.58 ~ 0.3.7 - 0.z - 0.33 -0.31 -0.29 -0.27 - 0.25 ~ 0.21
IX 56.96 -2.14 -2.31 -2.01 ~ I .63 - I .24 -0.X8 -0.57 -0.3 I -0.17 - 0. I 7
-0.17 - 0. I 7 -0.17 -0.17 - 0. I 7 -0.16 ~ 0. I4
20 54.33 -2.33 ~ I .96 - 1.64 ~ I .32 ~ 1.01 -0.72 -0.4x -0.29
-0. I5 - 0. ox
-0.08 -0.0x - 0.09 -0.09 - 0. IO ~ 0. I I -- 0. I I -0.11 ~~0. IO
Semiclassical
line mixing
analysis
899
100
201 0
I 5
I 10
I 15
I 20
Iml Fig. I. The broadening coefficients for the 2 -0 band of CO perturbed by N,. The points are the experimental data of Ref. (18) at T = 298, 190 and 125 K. The curves represent the calculated results with adiabatic corrections [cf. equation (6)].
where v, is the line wavenumber, I is the relaxation matrix whose diagonal elements are the collisional line widths and shifts, and the off-diagonal elements are the line coupling parameters responsible for non-additive effects. By neglecting cross-relaxation terms, the function F(v) is given by
P,,d 5,Rer,, F,(v)=;~ (v - v, + ImI,,)* + (ReI,,,)”
(4)
“t
This expression represents the sum of lines with Lorentz profiles, where ReT,, = ny, are the line widths and - ImT,, = nA, are the line shifts, n is the perturber gas density. The actual bandshape differs from the sum of Lorentzian profiles mainly in the regions of overlapping lines due to line-mixing effects. Thus we should take into account a complete relaxation matrix to generate the smooth bandshapes observed at high gas pressures. To calculate this matrix, we use a semiclassical empirical approach”’ mvolving parameters which are determined by fitting experimental data. I--MATRIX
IN SEMICLASSICAL
APPROACH
In the SCA method”) the relaxation matrix elements I-,,, can be calculated by using the distribution function N (J, J’), which is the frequency of the collisions producing the change J + J’ in the classical angular momentum. Using the ovaloid-sphere model for the linear molecule-atom collisions the N-function has the simple analytical form”4’ N (J, J ‘) =
nzlaJ
(J’J’*)+?+
2x(1 -G2)ZkTexp
with
Z=l+G I-G
IJ'
-
JI*,
Z
(J’* -J*)* 22
,
(3
N. N. FILIPPOV et al.
900
where ~1is the mean relative velocity of the colliding particles, g is the collisional cross-section, I is the moment of inertia of the absorber, G is a parameter defining the type of collision. The values G z 1 correspond to weak collisions perturbing the rotational motion slightly, i.e. /J’ - Jl’<
(5): N,,,(J,
J’) = N(J, J’)/[l
+ (t,Q,
)‘I.
(6)
where R,, = (J + _I’)/21 is the mean angular velocity of absorber molecule during the collision and T‘ is the effective duration of collisions. Since we have taken into account for the adiabatic correction the resonance function in equation (6) the N-function should be renormalized. The a-values presented hereafter correspond to the renormalized the N-function.
MODEL
CALCULATION
AND
DISCUSSION
We have used in this work the scheme of the SCA-calculation [described in detail in Ref. (S)] improved by the adiabatic correction (ACSCA). The parameters D and G of the semiclassical model were estimated for CO and CO, perturbed by noble gas atoms from ah initio potentials.“’ In the case of CO perturbed by Nz we have considered g, G and 1, = z’z, (r is the mean relative velocity) as adjustable parameters. The observed bandshapes as well as the experimental broadening coefficients data”*’ were used to determine these parameters
1.0
0
n = 456 Amagat
06.
v 0 t 2
0.6
Y
0.4
0.2
0.0 v (cm-‘) Fig. 2. Determination of the G-parameter from the normalized bandshape n = 456 Amagat and T = 296 K. Experimental bandshape: ---; calculated G = 0.6: 0. G = 0.7: +.
of CO compressed by N, at bandshapes with G = 0.5: x .
Semiclassical
line mixing
analysis
901
T=296K n = 54 Amagat
T=296K
v (cm-l) by Nz at Fig. 3. Comparison between normalized bandshapes for the 2 +O band of CO compressed T = 296 K and n = 54, 172 and 456 Amagat. Experimental bandshapes: -; calculated bandshapes from a sum of Lorentzian lines: 0 and from the ACSCA model: +.
For the relaxation matrix at room temperature we considered 33 lines in each branch. To shorten the presentation of this matrix, its elements are displayed (Table 1) only for even lines. Moreover, the matrix is given there in the following symmetrized form:
T‘,“,, =
(~,,Ip,)‘i2r,,,,,
(7)
such that T,,,,. = r‘,,, and moreover r’,,,,, = f _,,,_,. The spectroscopic constants for the 2 t-0 bands of 12C160, ‘%I60 and ‘*C”O required in our calculations were taken from Refs (19) and (20). The relative intensities of the isotopic bands were calculated from Ref. (21), by considering experimental Herman-Wallis factors.‘22’ The wavenumbers of the lines were corrected by the same line shift (-0.005 cm-‘/Amagat), derived from the data of Ref. (23). The parameters c and f< at room temperature were estimated by fitting broadening coefficients for each value of the parameters G (Fig. 1); next G was varied by steps of 0.1 and was adjusted to fit the calculated curve K(v) to the experimental bandshape at 456 Amagat, especially around the band center (Fig. 2). This spectrum was chosen since the line-mixing effects are the most important in the bandshape at the highest gas pressure, even if it is rather difficult to estimate the validity of the impact and binary collision approximations. Moreover, as shown in Ref. (24) for
902
N. N. FILIWOV PI nl.
such a high density
the collision
frequency
is probably
to the finite volume of the molecules. After a short iterative procedure the values
no more proportional
G = 0.6, c = 88 A’ and
to the density
owing
/, = 0.41 A have
been
adopted. It may be noted that the I, value has a reasonable order of magnitude.“” By comparing this value of G (0.6) with the results obtained in Ref. (5). it appears that the collisions CO-N? are stronger than the collisions CO-He, but they are weaker than those of CO-Ar. The possible reason of such a behavior
may be the significant
interaction to the rotational perturbation. We have postulated that the effective sufficient
to consider
length
of the multipole
/, is independent
(quadrupole-quadrupole)
of temperature.
Therefore
it is
coefficients “” in order to evaluate 0 and G at the lower temperatures. The calculated broadening coefficients are in reasonable agreement with experimental results (Fig. 1). The collision cross-section increases to 97 A’ at 190 K and to I I5 A’ at 122 K as the temperature decreases. The parameter G is practically independent on the temperature:
only experimental
contribution
it is taken
broadening
to be 0.5 for both
low temperatures.
These results
correspond
to those
obtained for the CO-He system.“’ Figure 3 displays the pressure dependence of the bandshapes at room temperature and demonstrates the insufficiency of a model based on a sum of Lorentzian lines as well as the substantial effect of line-mixing on the smooth bandshapes at high pressures. The discrepancies
n = 95 Amagat
n = 97 Amagat
T=122K n = 101 Amaaat
4200
4250 v (cm-l)
Fig. 4. Comparison between normalized bandshapes for the 7 -0 band of CO compressed by h’, at T = 296, 188 and 122 K and at densities of about 100 Amagat. Experimental bandshapes: - ~ ; calculated bandshapes from a sum of Lorentzian lines: l and from the ACSCA model: +
Semiclassical
between
calculated
increase
with increasing
and observed pressure.
spectra
line mixing
are almost
analysis
insignificant
For the low temperatures
903
at 54 Amagat,
and they slightly
(Fig. 4) the calculated
and observed
normalized spectra are also in satisfactory agreement although the calculated profiles are slightly shifted, especially in the R-branch, and are lowered too much at 122 K around the band center. Let us emphasize that the parameters used at the lower temperatures broadening coefficients. In conclusion the ACSCA model leads to a reasonable agreement
were derived between
only from
calculated
and
measured normalized bandprofiles for the overtone of CO in high density gas of N, at various temperatures. We have proved that this simple approach, developed initially for linear moleculeatom pairs, may be extended to more complicated systems which cannot be considered using ab initio calculations. The great effect of line-mixing on the bandshapes under consideration have clearly been shown by comparing the profiles derived from the sum of Lorentzian lines and those calculated
by using our model.
Nevertheless, there are small but systematic pressure-dependent deviations of the calculated spectra from the experimental ones. Firstly, these deviations can result from imperfections in the semiclassical model of the r-matrix. In particular, it may be a consequence of the finite duration of molecular collisions leading to the frequency dependence of the r-matrix, or the classical character of rotation which is the base of our model. The latter cause may produce a discrepancy near the band origin. Moreover, the binary collision approximation, as well as the neglect of the finite volume of the molecules may be incorrect at densities above 100-200 Amagat. Finally, since the discrepancies between calculated and observed spectra at high pressures are practically not T-dependent (Fig. 4) this study seems to indicate that the absorption due to van-der-Waals complexes does not contribute appreciably to the bandprofiles even at low temperature. However, we hope to clarify some of the points above mentioned, in particular the importance of the CO..N, complexes, from a new calculation involving the quantum description of molecular rotation. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. IO. Il. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
J. Boissoles. C. Boulet, D. Robert and S. Green, J. Chem. Phys. 87, 3436 (1987). J. Boissoles, C. Boulet, D. Robert and S. Green, J. Chem. Phys. 90, 5392 (1989). S. Green, J. Chem. Phys. 90, 3603 (1989). S. Green, J. Chem. Phys. 92, 4679 (1989). N. N. Fihppov and M. V. Tonkov, J. quanl. spectrosc. Radiut. Transfer 50, 111 (1993). J.-P. Bouanich, J. quunt. spectrosc. Radial. Transfer 27, 131 (1982). G. V. Tchlenova, A. A. Vigasin, J.-P. Bouanich and C. Boulet, Infrared Phys. 34, 289 (1993). L. L. Strow and B. Gentry, J. Chem. Phys. 84, 1149 (1986). B. Gentry and L. L. Strow, J. Chem. Phys. 86, 5722 (1987). T. Huet, N. Lacome and A. Levy, J. Molec. Spectrosc. 138, 141 (1989). R. G. Gordon and R. P. McGinnis, J. Chem. Phys. 55, 4898 (1971). R. T. Jacobsen and R. B. Stewart, J. Phys. Chem. Ref Data 2, 757 (1973). A. Ben-Reuven, Phys. Rea. 145, 7 (1966). N. N. Filippov, Khimicheskaju Fizikn (USSR) 6, 1085 (1987). A. E. De Pisto, S. D. Augustin, R. Ramaswamy and H. Rabitz, J. Chem. Phys. 71, 850 (1979). L. Bonamy, J. Bonamy, D. Robert. B. Lavorel, R. Saint-Loup, R. Chaux, J. Santos and H. Berger, J. Chem. Phys. 89, 5568 (1988). J. Boissoles, C. Boulet, L. Bonamy and D. Robert, J. quant. spectrosc. Radiar. Transfer 42, 509 (1989). J.-P. Bouanich, R. Farrenq and C. Brodbeck, Can. J. Phys. 61, 192 (1983). D.-W. Chen, K. N. Rao and R. S. McDowell, J. Molec. Specfrosc. 61, 71 (1976). C. R. Pollock, F. R. Petersen, D. A. Jennings, J. S. Wells and A. G. Maki, J. Molec. Spectrosc. 99, 357 (1983). L. S. Rothman, R. R. Gamache, A. Goldman, et al. Appl. Opf. 26, 4058 (1987). C. L. Korb, R. H. Hunt and E. K. Piyler, J. Chem. Phys. 48, 4252 (1968). J. P. Bouanich, Can. J. Phys. 61, 919 (1983). J. 0. Hirschfelder. C. F. Curtiss and R. B. Bird, Molecular Theory c~f Gases and Liquids. Wiley, New York (1967).
13504495(94)EOO24-E
SEMICLASSICAL OVERTONE
Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 1350~4495194 $7.00 + 0.00
LINE MIXING ANALYSIS IN THE FIRST BAND OF CO COMPRESSED BY N,
N. N. FILIPPOV,’M. V. TONKOV’ and J.-P. BOUANICH’ ‘Institute of Physics, St Petersburg University, Peterhof, St Petersburg, 198904, Russia and 2Laboratoire de Physique Moleculaire et Applications, tU.P.R. 136 du C.N.R.S., Universitt de Paris&d,. Bat. 350, 91405 Orsay France (Received 7 March 1994) absorption band of CO compressed by N, at various densities Abstract-Infrared spectra of the 2 -0 up to 456 Amagat and at temperatures of 296, 188 and 122 K have been analyzed. To interpret these spectra we have applied a semiclassical model with adiabatic corrections which accounts for the line-mixing effects. This model provides a satisfactory agreement with experimental bandshapes for all temperatures, especially for the lower N, densities. The discrepancy arising at higher densities may be due to imperfections of the calculated relaxation matrix and/or to the non-validity of the impact and binary collision approximations.
INTRODUCTION
Collision-induced line mixing effects have given rise to some theoretical developments and these effects on IR absorption spectra have been mainly calculated for molecule-rare gas atom collisions.(‘m5) Experimental data on CO-N, system were previously obtained in the 2 CO band@) at such high densities that the line-mixing effects are very important and cannot be neglected. In a recent work”’ the contribution of the molecular van der Waals complex CO..N, to the 2 c 0 bandshape was estimated from the differences observed between the experimental profiles and those of the monomer which were calculated by using an empirical bandmodel. However, to account for the line mixing effect straightforwardly we have undertaken further investigations in the monomeric spectral profiles. To the best of our knowledge there is no ab initio calculation for the rotational relaxation matrix of the CO-N, system which is essential to generate the bandshapes. Moreover, it is difficult to use the empirical fitting laws@“) in matrix elements estimation as we have no definite means to determine the elements describing the interbranch line mixing. Therefore we have decided to use in a first step a semiclassical approach’5’ (SCA) which allows to treat some parameters as adjustable ones. This approach is based on the same collisional model as that considered by Gordon and McGinnis (‘I) to calculate the C&He bandshape but our calculation of the rotational relaxation matrix is notably simplified. Indeed, our method deals with the probability distribution functions instead of a model involving thousands of molecular trajectories.“” The experimental conditions of the recordings at room and low temperatures were discussed in detail previously.@‘,‘) In the present work, we consider a wider set of experimental data than in Ref. (7). We studied spectra at pressures up to 970 bar and at three temperatures: 296 k 1, 188 f 2 and 122 + 2 K. The densities of the gas mixtures were obtained from the state equation”*’ for nitrogen. tLaboratoire
associe
aux Universites
Paris-Sud
et Pierre et Marie Curie. 897
INF35,7--F
N. N. FILIPPOV et a/
898
Because absorption
the CO pressure was not accurately coefficients K(v) defined by
measured,
K(v) = !I (v)/
we have preferred
to study normalized
c1(v)dv,
(1)
s band where tl (v) is the measured
absorption
BANDSHAPE
The
absorption
coefficient
coefficient
IN
THE
at the wavenumber
IMPACT
V.
APPROXIMATION
CI(v) for a vibration-rotation
band
is given
by the well known
expression c(
By using the line space approximation by(13)
[I - exp(hcv/kT)]F(v).
(v) = g
formalism
we may
express
the spectral
density
F(r)
in the impact
where d is the dipole-moment operator of the system, m and m ’ represent the line numbers corresponding to the rotation-vibration transitions ( j,, z:,)+ ( j,, I’, ) and ( j ‘, , I’, ) + (.j ‘, . I‘, ). For parallel bands of linear molecules, it is convenient to consider M = -j, in the P-branch and m =j, + 1 in the R-branch. P, = P,.,P,, is the population of the absorbing molecules in the initial state, and d, is the reduced matrix element of the d-operator d,, = (t’~, Ildllr,~,) 1 in equation
(4) is the unit matrix,
and the L-matrix
elements
can be defined
Table 1. Matrix elements r,,,,, (in 10-l cm-‘. Amagatr’)
by
for CO-N: a: 296 K
m’
In
2 20 18 16 14 12 10 8 6 4 2
-2 -4 -6 -8 -10 - 12 - 14 - 16 -18 -20
4
79.68 -2
72
75.56 -3.19 -2.77 - 3.20
6
72.30 -4.51 -3.24 -2.62 -2.94 -2.67
8
69.63 -4.90 -4.1 I -2.84 -2.15 -2.35 -2.12 - I.71
10
67.19 -4.70 -4.17 -3.55 -2.22 - 1.56 ~ 1.68 - 1.52 - 1.25 -0.97
12
64.75 -4.25 -3.83 -3.23 ~ 2.46 - 1.56 - 1.03 ~ I.09
-0.99 - 0.85 -0.70 - 0.55
14
62.21 -3.72 -3.34 -2.86 -2.28 - I .65 - I .oo - 0.6 I
-0.64 -0.60 ~ 0.53 -0.47 -0.40 ~ 0.3.I
The elements presented in italics arise from the interbranch line-mixing effects.
16
59.60 -3.20 -2.83 - 2.42 - I .96 - I .48 - I.01 -0.58 ~ 0.3.7 - 0.z - 0.33 -0.31 -0.29 -0.27 - 0.25 ~ 0.21
IX 56.96 -2.14 -2.31 -2.01 ~ I .63 - I .24 -0.X8 -0.57 -0.3 I -0.17 - 0. I 7
-0.17 - 0. I 7 -0.17 -0.17 - 0. I 7 -0.16 ~ 0. I4
20 54.33 -2.33 ~ I .96 - 1.64 ~ I .32 ~ 1.01 -0.72 -0.4x -0.29
-0. I5 - 0. ox
-0.08 -0.0x - 0.09 -0.09 - 0. IO ~ 0. I I -- 0. I I -0.11 ~~0. IO
Semiclassical
line mixing
analysis
899
100
201 0
I 5
I 10
I 15
I 20
Iml Fig. I. The broadening coefficients for the 2 -0 band of CO perturbed by N,. The points are the experimental data of Ref. (18) at T = 298, 190 and 125 K. The curves represent the calculated results with adiabatic corrections [cf. equation (6)].
where v, is the line wavenumber, I is the relaxation matrix whose diagonal elements are the collisional line widths and shifts, and the off-diagonal elements are the line coupling parameters responsible for non-additive effects. By neglecting cross-relaxation terms, the function F(v) is given by
P,,d 5,Rer,, F,(v)=;~ (v - v, + ImI,,)* + (ReI,,,)”
(4)
“t
This expression represents the sum of lines with Lorentz profiles, where ReT,, = ny, are the line widths and - ImT,, = nA, are the line shifts, n is the perturber gas density. The actual bandshape differs from the sum of Lorentzian profiles mainly in the regions of overlapping lines due to line-mixing effects. Thus we should take into account a complete relaxation matrix to generate the smooth bandshapes observed at high gas pressures. To calculate this matrix, we use a semiclassical empirical approach”’ mvolving parameters which are determined by fitting experimental data. I--MATRIX
IN SEMICLASSICAL
APPROACH
In the SCA method”) the relaxation matrix elements I-,,, can be calculated by using the distribution function N (J, J’), which is the frequency of the collisions producing the change J + J’ in the classical angular momentum. Using the ovaloid-sphere model for the linear molecule-atom collisions the N-function has the simple analytical form”4’ N (J, J ‘) =
nzlaJ
(J’J’*)+?+
2x(1 -G2)ZkTexp
with
Z=l+G I-G
IJ'
-
JI*,
Z
(J’* -J*)* 22
,
(3
N. N. FILIPPOV et al.
900
where ~1is the mean relative velocity of the colliding particles, g is the collisional cross-section, I is the moment of inertia of the absorber, G is a parameter defining the type of collision. The values G z 1 correspond to weak collisions perturbing the rotational motion slightly, i.e. /J’ - Jl’<
(5): N,,,(J,
J’) = N(J, J’)/[l
+ (t,Q,
)‘I.
(6)
where R,, = (J + _I’)/21 is the mean angular velocity of absorber molecule during the collision and T‘ is the effective duration of collisions. Since we have taken into account for the adiabatic correction the resonance function in equation (6) the N-function should be renormalized. The a-values presented hereafter correspond to the renormalized the N-function.
MODEL
CALCULATION
AND
DISCUSSION
We have used in this work the scheme of the SCA-calculation [described in detail in Ref. (S)] improved by the adiabatic correction (ACSCA). The parameters D and G of the semiclassical model were estimated for CO and CO, perturbed by noble gas atoms from ah initio potentials.“’ In the case of CO perturbed by Nz we have considered g, G and 1, = z’z, (r is the mean relative velocity) as adjustable parameters. The observed bandshapes as well as the experimental broadening coefficients data”*’ were used to determine these parameters
1.0
0
n = 456 Amagat
06.
v 0 t 2
0.6
Y
0.4
0.2
0.0 v (cm-‘) Fig. 2. Determination of the G-parameter from the normalized bandshape n = 456 Amagat and T = 296 K. Experimental bandshape: ---; calculated G = 0.6: 0. G = 0.7: +.
of CO compressed by N, at bandshapes with G = 0.5: x .
Semiclassical
line mixing
analysis
901
T=296K n = 54 Amagat
T=296K
v (cm-l) by Nz at Fig. 3. Comparison between normalized bandshapes for the 2 +O band of CO compressed T = 296 K and n = 54, 172 and 456 Amagat. Experimental bandshapes: -; calculated bandshapes from a sum of Lorentzian lines: 0 and from the ACSCA model: +.
For the relaxation matrix at room temperature we considered 33 lines in each branch. To shorten the presentation of this matrix, its elements are displayed (Table 1) only for even lines. Moreover, the matrix is given there in the following symmetrized form:
T‘,“,, =
(~,,Ip,)‘i2r,,,,,
(7)
such that T,,,,. = r‘,,, and moreover r’,,,,, = f _,,,_,. The spectroscopic constants for the 2 t-0 bands of 12C160, ‘%I60 and ‘*C”O required in our calculations were taken from Refs (19) and (20). The relative intensities of the isotopic bands were calculated from Ref. (21), by considering experimental Herman-Wallis factors.‘22’ The wavenumbers of the lines were corrected by the same line shift (-0.005 cm-‘/Amagat), derived from the data of Ref. (23). The parameters c and f< at room temperature were estimated by fitting broadening coefficients for each value of the parameters G (Fig. 1); next G was varied by steps of 0.1 and was adjusted to fit the calculated curve K(v) to the experimental bandshape at 456 Amagat, especially around the band center (Fig. 2). This spectrum was chosen since the line-mixing effects are the most important in the bandshape at the highest gas pressure, even if it is rather difficult to estimate the validity of the impact and binary collision approximations. Moreover, as shown in Ref. (24) for
902
N. N. FILIWOV PI nl.
such a high density
the collision
frequency
is probably
to the finite volume of the molecules. After a short iterative procedure the values
no more proportional
G = 0.6, c = 88 A’ and
to the density
owing
/, = 0.41 A have
been
adopted. It may be noted that the I, value has a reasonable order of magnitude.“” By comparing this value of G (0.6) with the results obtained in Ref. (5). it appears that the collisions CO-N? are stronger than the collisions CO-He, but they are weaker than those of CO-Ar. The possible reason of such a behavior
may be the significant
interaction to the rotational perturbation. We have postulated that the effective sufficient
to consider
length
of the multipole
/, is independent
(quadrupole-quadrupole)
of temperature.
Therefore
it is
coefficients “” in order to evaluate 0 and G at the lower temperatures. The calculated broadening coefficients are in reasonable agreement with experimental results (Fig. 1). The collision cross-section increases to 97 A’ at 190 K and to I I5 A’ at 122 K as the temperature decreases. The parameter G is practically independent on the temperature:
only experimental
contribution
it is taken
broadening
to be 0.5 for both
low temperatures.
These results
correspond
to those
obtained for the CO-He system.“’ Figure 3 displays the pressure dependence of the bandshapes at room temperature and demonstrates the insufficiency of a model based on a sum of Lorentzian lines as well as the substantial effect of line-mixing on the smooth bandshapes at high pressures. The discrepancies
n = 95 Amagat
n = 97 Amagat
T=122K n = 101 Amaaat
4200
4250 v (cm-l)
Fig. 4. Comparison between normalized bandshapes for the 7 -0 band of CO compressed by h’, at T = 296, 188 and 122 K and at densities of about 100 Amagat. Experimental bandshapes: - ~ ; calculated bandshapes from a sum of Lorentzian lines: l and from the ACSCA model: +
Semiclassical
between
calculated
increase
with increasing
and observed pressure.
spectra
line mixing
are almost
analysis
insignificant
For the low temperatures
903
at 54 Amagat,
and they slightly
(Fig. 4) the calculated
and observed
normalized spectra are also in satisfactory agreement although the calculated profiles are slightly shifted, especially in the R-branch, and are lowered too much at 122 K around the band center. Let us emphasize that the parameters used at the lower temperatures broadening coefficients. In conclusion the ACSCA model leads to a reasonable agreement
were derived between
only from
calculated
and
measured normalized bandprofiles for the overtone of CO in high density gas of N, at various temperatures. We have proved that this simple approach, developed initially for linear moleculeatom pairs, may be extended to more complicated systems which cannot be considered using ab initio calculations. The great effect of line-mixing on the bandshapes under consideration have clearly been shown by comparing the profiles derived from the sum of Lorentzian lines and those calculated
by using our model.
Nevertheless, there are small but systematic pressure-dependent deviations of the calculated spectra from the experimental ones. Firstly, these deviations can result from imperfections in the semiclassical model of the r-matrix. In particular, it may be a consequence of the finite duration of molecular collisions leading to the frequency dependence of the r-matrix, or the classical character of rotation which is the base of our model. The latter cause may produce a discrepancy near the band origin. Moreover, the binary collision approximation, as well as the neglect of the finite volume of the molecules may be incorrect at densities above 100-200 Amagat. Finally, since the discrepancies between calculated and observed spectra at high pressures are practically not T-dependent (Fig. 4) this study seems to indicate that the absorption due to van-der-Waals complexes does not contribute appreciably to the bandprofiles even at low temperature. However, we hope to clarify some of the points above mentioned, in particular the importance of the CO..N, complexes, from a new calculation involving the quantum description of molecular rotation. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. IO. Il. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
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