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N2- and O2-broadening coefficients of C2H2 IR lines
IOURNAL
OF MOLECULAR
SPECTROSCOPY
ldtt,195-2
13 ( 1990)
N2- and 02-Broadening Coefficients of C2H2 IR Lines J. P. BOUANICH Laboratoire d’lnfrarouge, assock! au C.N.R.S., Universiti de Paris-&d, Bdtiment 350. F-91405 Orsay Cede.x, France AND
D. LAMBOT, G. BLANQUET, AND J. WALRAND Laboratoire de Spectroscopic Molhculaire. Facultb Universtaires N.D. de !a Pais. Rue de Bruxelies. 61, B-SOOO-Namur,Belgium Pressure-broadening parameters of six lines belonging to the vgband of C2Hz in collision with Nz have been measured with a tunable diode-laser spectrometer in order to complete up to J = 33 our earlier measurements (D. Lambot, G. Blanquet, and J. P. Bouanich, J. Mol. Spectrosc. 136, 86-92 ( 1989)) on the broadening of CzH2 by N2 and 0~ at 297 K. These N2- and 02broadening coefficients have been first calculated on the basis of the Anderson-Tsao-Curnutte theory; in this approach, we show that the short-range interactions which contribute significantly to the linewidths are not correctly treated. Next. we consider the improved semiclassical model proposed by Robert and Bonamy. The intermolecuIar potential consists in the addition of the atom-atom interaction model to the quadrupolar interactions. The limited radial spherical harmonics expansion of the atom-atom potential, from which expressions for the differential cross section were derived, appears to be quite insufficient at short intermolecular distances. Therefore, we use a more accurate representation of this potential, avoiding an inadequate truncation and keeping the analytic expressions obtained by Bonamy and Robert. In the calculations we take into account the contributions derived from the radial functions Um( r), U,,(r), and U,,,(r), as well as from Uew(r). A theoretical expression is obtained for the Il.,, contribution to the differential cross section. The results of the calculations arising from the exact radial expansion of the atom-atom potential appear to be significantly larger for high J lines than those arising from the truncated expansion. The latter results, which do not include adjustable atom-atom parameters, are in good agreement with experimental broadening coefficients for CzH2-O2 and in reasonable agreement (except at large J values) for CzHZ-N2. It is also shown that the contributions to the linewidths derived from 0’4~ are rather small for CzHz-N2 and more important for C2H2-02. Finally, by calculating the collisional linewidths of CZHZ-NI and C2H2-O2 at 200 K. we have predicted their temperature depcndences. 1~)1990 Academrc press. I”~. 1. INTRODUCTION
In a recent paper ( I), we reported NZ- and 02-broadening coefficients for several lines with J ranging from 2 to 29 ( NZ) and 3 to 33 (0,) in the v5 band of C2H2 (730 cm -’ ) . For high J values of ClH2 perturbed by N2, we measured only the collisional halfwidths of the R( 24) and P(29) lines. In order to complete this work, here we determine experimentally, using a tunable diode-laser (TDL) spectrometer, the N2 broadening coefficients of C2Hz for 6 lines with J ranging from 26 to 33 in the R branch of the v5 band. 195
0022-2852190 $3.00 Copyright 0
1990 by Academic Press, Lnc.
All righk of revcduction
in any form reserved.
196
BOUANICH
ET AL.
The experimental broadening coefficients for C2Hz-N2 and C2Hz-O2 are compared with theoretical results derived from the Anderson-Tsao-Curnutte (ATC) method (2. 3) and the more accurate semiclassical model developed by Robert and Bonamy (RB) (4). It may be noted that the contributions of vibrational effects should be small ( 1) and have been disregarded. Moreover, CzHz in the II, state (corresponding to the final level of the perpendicular u5 band) is not, strictly speaking, a linear molecule but is considered as linear since the two light H nuclei should rotate very fast around and close to the C-C line. For the intermolecular potential, we consider in addition to the electrostatic contribution (quadrupole-quadrupole interactions) an anisotropic dispersion contribution in the ATC calculations and an atom-atom (called also site-site) potential in the RB calculations. As was previously shown for C02-X systems (5)) the limited radial expansion of the atom-atom potential given in Ref. (4) is quite inaccurate at short distances. We have calculated exactly the main coefficients I!I.J~,~,,( r) of this expansion for CZH2-N2 as well as for CZH2-02, and shown that the radial function U&r) which was always neglected in the calculations has almost the same importance as U&r) or ~&a(r). Therefore, we have derived explicit formulas for the UdoOcontribution to the differential cross section. In order to use these expressions as well as those obtained by Bonamy and Robert (6) and to take into account the exact radial dependence of the atom-atom potential, we have fitted the atom-atom parameters in a given intermolecular range, following the method described by Rosenmann et al. (5). Thus, we have applied the RB model by including coherently the truncated and exact expansion of the atom-atom interaction potential and we have compared the theoretical results with experimental data. Finally, from these calculations performed for C2H2-N2 and C2H2-O2 at 297 K (experimental temperature) and 200 K, we have evaluated the coefficient n that defines the variation with temperature of the broadening coefficients through a simple analytic law. 2. EXPERIMENTAL
RESULTS
FOR Np-BROADENING
COEFFICIENTS
OF CzHz
The spectra of the lines R(26) and R(29) to R(33) in the v5 band of C2H2 were recorded on a TDL spectrometer interfaced to a minicomputer. The gas mixture was contained in a White-type cell with an optical path length of about 4 m. Nitrogen was supplied by 1’Air Liquide with a purity of 99.9% and acetylene by Air Products with a stated purity of 99.6%. The partial pressure of C2H2 ranged from 10e2 to 10-l mbar. Four pressures of N2 equal to about 50,65, 80, and 95 mbar were used. These pressures were measured using an MKS Baratron gauge. All the measurements were made at a room temperature of about 297 K. The relative wavelength calibration of the spectra was performed from a confocal etalon with a fringe spacing of 0.00977 11 cm-‘. Spectral purity of the laser modes was checked by recording a saturated line of CZHI. The procedure used in data reduction is similar to that described previously (I). The pressure-broadening coefficients y. (in cm-’ atm-‘) of the lines studied in this work as well as in Ref. (I) are listed in Table I with a relative error estimated to be about 5%. Within this accuracy, the halfwidths are only Im I -dependent [m = -J for P(J) lines and m = J + 1 for R(J) lines] and are represented as a function of (m 1 in Fig. 1. By using a least-squares procedure, we have obtained smoothed out data
197
BROADENING COEFFICIENTS IN C2H2 TABLE I
N,-Broadening Coefficients y0 ( IO-’ cm-’ atm-‘) in the v5 Band of CzH2 at 297 K, (the wavenumbers (cm-‘) of the unperturbed lines were measured by Hietanen a a/. ( 16))
T-
Wavenumbers
R
: :
5 !: 8 1’0 1:. 13 14 15 16 17 18 19 20 21 :: 24 Ei 27 28 ii 31 32 33 34
731.5084 733.8598 736.2130 738.5636 740.9146 743.2639 745.6135 747.9640 750.3 106 752.6580 755.0062 757.3511 759.6959 762.0396 764.3827 766.7241 769.0654 771.4048 713.7444 716.0822 778.4183 780.7531 783.0868 785.4191 787.7506 790.0803 792.4083 794.7345 797.0597 799.3835 801.7049 804.0250 806.3437 808.6601
R
P
724.4482 722.0948 719.7389 717.3861 715.0305 712.6757 710.3202 707.9657 705.6104 703.2556 700.9005 698.5436 696.1895 693.8339 691.4800 689.1254 686.7701 684.4164 682.0631 679.7094 677.3566 675.0040 672.6519 670.3000 667.9489 665.5980 663.2419 660.8979 658.5488 656.2013 653.8539 651.5072 649.1619
?(a P
109.0 102.0
104.2 91.6 91.7
86.2 83.6
84.1
82.5 80.3 80.0
76.8 75.0 71.7
72.8
62.1
smoolbed 116.6 109.1 103.1 98.4 94.6 91.7 89.4 87.6 86.2 85.0 83.9 82.9 81.9 80.9 79.7 78.5 77.1 75.6 73.9 72.1 70.3 68.3 66.3 64.2 62.2 60.1 58 56.1 54.2 52.4 50.7 49 1 47.6 46.1
1
58.5 52.7 53. I 51.3 49.8 46.8 46.1
(the curve of Fig. 1) which are more accurate at large ( m 1 values than those given in Ref. ( 1) . It may be seen that our results are in satisfactory agreement with the results of Devi el al. ( 7) obtained in the ( v4 + v5) band.
3. GENERAL FORMULATION
The collisional
halfwidth
OF
‘HE LINEWIDTH
of an isolated vibration-rotation
line f -
i may be written
as
where n2 is the number density of the colliding molecules; 03 is the partial optical cross section; pJ2 is the relative population of the perturber 1Jz, v2 = 0) state; F(v) is the distribution of the relative velocities v; S’i is the diffusion operator or the differential cross section representing the collisional efficiency; and b is the impact parameter. The integration over the velocity distribution is not generally performed and the calculations are made for 5, the mean collisional velocity. In the ATC theory, a straightpath trajectory described at constant velocity is assumed; S(b) is expanded through
198
BOUANICH ET AL.
0 01
”
“I’.
5 ‘.
10 ’
”
”
15
20
25
30
FIG. 1. Experimental N2-broadening coefficients y. for IR lines of CzHZ (for measurements in P(J) and R(J + 1) lines, only the average is represented): (0) our results (~5 band) (the smooth curve fits our experimental data); (0) results of Devi et Q/. ( 7) (~6 + u5 band); ( X ) results of Podolske et a/. (8) (“4 + vg band).
a second-order perturbation treatment which requires avoid divergence of S for small values of b) so that
a cutoff procedure
(in order to
i)ri a/i = 7rbi +
f b0
2irb&( b)db,
(2)
where b. is evaluated for each Jz from &( bo) = 1. If b. is smaller than the distance of closest approach rO, b. is replaced by r. in Eq. (2). In the RB model, the diffusion operator is expressed in a closed form through the linked cluster theorem, which allows a partial resummation of all V-orders, such as S(b)
= 1 - expl-(&
+ 5’2~2 +
&,&I,
(3)
where &,a, &,fz, and Sz,/ziz are respectively equal to Sq>ter, Sqy, and Syiddle, the second order terms of the perturbation treatment of S(b) in the ATC theory. The trajectory model includes the influence of the isotropic potential vi,, in energy conservation and of the isotropic force F, (F, = -grad I&) in the equation of motion around the distance of closest approach rc. The actual trajectory is replaced near Y, by a parabolic trajectory described with an apparent velocity vi given by
BROADENING
COEFFICIENTS
199
IN &HZ
where v),is the relative velocity at the closest approach and m is the reduced mass of the colliding partners. In our calculations, the equivalent straight-path trajectory (with velocity v’,) defined in Ref. ( 9) is used instead of the parabolic trajectory. This model greatly simplifies the expressions of the resonance functions given in Ref. (6) and should lead to nearly identical results of yo. 4. CALCULATED
RESULTS
FROM
A SIMPLE
INTERMOLECULAR
POTENTIAL
In the ATC approach, only electrostatic interactions are accounted for in the intermolecular potential, Here, we consider in addition an anisotropic dispersion contribution according to (10) f hPz(cos 0,),
V= Vo,gz - 4ty, 0
(5)
where Q1 and Q2 are the quadrupole moments of molecule I (absorber) and molecule 2 (perturber); yl denotes here the polarizability anisotropy of C2H2 defined by ( ali - aI)/( a// + 2aL); Pz is the second Legendre polynomial; and 8, is the angle between the axis of molecule 1 and the intermolecular axis. The Lennard-Jones (LJ) parameters t and c for the molecular pairs CZH*-N2 and CZHZ-O2 are determined from the usual combination rules, t = (E~c~)~” and u = ((T] + u2)/2. The quadrupole moment of C2H2 is evaluated from self-broadening linewidths measured in the 5~~ ( 11) and ( v, + q) ( 12) bands. Using the ATC and the RB formulations with I/defined by Eq. ( 5 ), we obtain approximately 4.3 and 3.5 D * A, respectively. We have adopted Q, = 4 D * A, a value notably smaller than the result ( 5.42 D - A) derived recently from collision-induced spectra of C2HZ-Ar ( 13). For the quadrupole moment of N2, we have considered the value - 1.4 D - A obtained experimentally ( 14), and we have used the O2 quadrupole moment recommended by Stogryn and Stogryn ( 15), Q2 = -0.39 D - A. The molecular parameters introduced in the computations are summarized in Table II. The coefficients y. at 297 K were computed by including the contributions from perturbing molecules having J? values up to 42 for N2 and 5 1 for 02, weighted by the TABLE 11 Molecular
Parameters
Used in the Calculations
wb M (g.mol-1)
26.04
B, (cm’)
1.1766406 a
B; @ml)
1.176432 1
N2
9
28.0134
31.9988
1.989622 b 5.763 x lo-6 b
D, (cm-‘)
1.6221 x 10-6 a
Q@,A)
4.0
-1.4d
199.2 f
95.1 f
E (K) 0 (A)
4.523 f
n
0.18Og
3.75 f
a Reference (16)
dRef-
b Reference (17)
e Reference (15)
c Reference (18)
fRefmnce(19)
(14)
g Calculated from polarizabilities values given in Ref. (20 p. 581).
1.437654 C 4.837 x 10-6 C -0.39 = 116f 3.57 f
200
BOUANICH
ET AL.
Boltzmann factor and the nuclear spin factor (( - 1)J2 + 3)/2 for N2 and ( 1 - ( -l)J2)/2 for Oz. As shown in Fig. 2, the results for C2H2-N2 derived from the ATC model are in reasonable agreement, especially at large [ml values, with the experimental data. The agreement is notably poorer for C2H2-02 in the range 7 < )m 1 < 30 (Fig. 3 ) . It may be shown that at low 1m 1, the predominant contribution arises from the dispersion terms of the potential and, at large 1m 1, the calculated yo- values derive mainly from the cutoff procedure S, ( b < ro) = 1. For C2HZ-02, this procedure leads to broadening coefficients nearly constant and independent of the lines with 1m ( > 14. It may be also noted that the consideration of a repulsive contribution in addition to the dispersion term [see Eq. (6) of Ref. (21)] or the inclusion of higher order electrostatic interactions involving the hexadecapole moments of C2Hz and the penurber would not notably change the results of yo. By using the RB model, we have computed the N2 - and 02-broadening coefficients from exactly the same interaction potential, and the molecular parameters given in Table II. As may be seen in Fig. 2, the results obtained for CZH2-N2 are close to the experimental data up to 1m I = 10 but decrease notably faster at higher I m (. The discrepancy is much more pronounced for CzH2-O2 and the theoretical results decrease dramatically with increasing (m ( (Fig. 3). In the latter case, the electrostatic contribution is weak owing to the smallness of the quadrupole moment of 02 and the interaction potential [ Eq. (5 )] appears to be quite insufficient: the short-range forces which are dominant in the broadening of the lines at large 1m) are greatly underes-
120
-0
5
10
20
15
25
30
35
Iml FIG. 2. N2 -broadening coefficients y. for IR lines of C2 H, . Experimental values ( l, 0, X ) . Results from ATC formalism applied with V defined by Eq. (5): curve (a); results from the RB model applied with the same potential: curve (b) .
BROADENING
o’.-..a....s 0
5
10
COEFFKIENTS
“‘*.““““”
15
20
201
IN C2Hz
25
“I..”
30
35
RG. 3. 02-broadening coeficients y0 for IR lines of C2H2. Experimental values given in Ref. (I): ( l ). Results from ATC formalism applied with I/ defined by Eq. (5): curve (a); results from the RB model applied with the same potential: curve (b).
ATC theory leads artificially to improved results at high Im I values only through the cutoff procedure used for b0 < ro. timated.
The
5. ATOM-ATOM
INTERACTION
MODEL
The interaction potential used in the RB model consists in adding to the electrostatic interactions V. the atom-atom interactions V,. V, is the sum of elementary atomatom contributions described by Lennard-Jones expressions:
Here, the indices 1i and 2 j refer to the ith atom of molecule I and the jth atom of molecule 2; rli,zj is the distance between these atoms and dij and eU are the atomic pair parameters. To our knowledge these parameters are not available for acetylene. For C-C interactions, we have considered the parameters derived from COz , and we have adopted the rather low values of e and d given by Murthy et al. (22). The H-H parameters are evaluated from eji = 4ta6 and dj, = 4~‘~ with t = 8.6 K and u = 2.8 13 A, LJ parameters derived from CH4 (Ref. (20, p. 112)). The parameters used for NN and O-O are the mean values determined by Oobatake and Ooi (23) from virial coefficients of NZ and 02. For the interaction between different atoms i and j, eii and dl, are calculated from the combination rules eiI = (eiiej,) “2, dg = ei,cT$,and alj = (oil
202
BOUANICH ET AL.
+ Uij)/2. The intramolecular distances Tii for CzH2 are taken from Ref. (24). The parameters used for the atom-atom potential are listed in Table III. In Ref. (4), rii,zj is expanded in power series of the intermolecular distance r to include the fourth-order contribution in r-‘o-r-‘6 (for even I, and Z2), and V, is expressed in terms of the intramolecular distances rli, rzj and the spherical harmonics Y, tied to each molecule, as
where 8,) 4, and &, 42 are the polar angles for the two linear colliding molecules (Fig. 4). Here, the total order of the spherical harmonics 1, + Z2is limited to 4 in accordance with the electrostatic potential limited to quadrupolar interactions. Rosenmann et al. (5) have shown that the expressions given in Ref. (4) for the coefficients U,,& r) are generally not valid for small values of r. We have calculated the exact radial functions UI,,,,( r) for C2H2-N2 and C2H2-O2 in two different ways. The first is based on the properties of spherical harmonics (see Ref. (20, p, 450)). If w and w’ denote the polar angles 04, 19’4’of the two linear molecules, these radial functions are given by
Ul,i,dr) =
V(r, w, 0’)
J-s
4?r
Y,:,(w)Y;-,(w')dwdw'
with
U!,I~, may be computed as a function of r by expressing rii,zj in terms of r, rli, rzj, 0,) 19,,c#P,,ti2, and perfornring a triple integral over ~8,) d&, and d( 4, - $2). The second
method is based on the expressions obtained by Downs et al. (25) for the harmonic expansion of the site-site potential. In the coordinate axes of Fig. 4 where the two 2 axes coincide, the spherical harmonic expansion of V, may be expressed as V, = 2
21+ 1 “2
2 ~~(1~1~1,W(lJ2l;
m
-
m
O)Y/,,YI,-,
l,l~hn ij
( i --yg-
,
TABLE III Atom-Atom
Parameters, Intramolecular Distances, and LJ (6-n) Parameters Fitting U,(r) for CZHz-N2 and C, Hz-02 Cij (m-l’+A~) C.&N= 0.154 C2Hz
Nz
E(K)
o(A)
n
1, (A)
61.44
3.886
16
3.687
3.685
16.5
3.471
dc_R = 0.133
CH_N= 0.0862 dw, = 0.0764 ec_o = 0.122
CzH2 - 02
rli,r2i
d(w7c;Aq
r,.H = 1.6614
d%xJ= 0.0682
.ZHXJ = 0.0682 dw, = 0.0394
75.63
(A)
r,_c = 0.604 n-2-N = 0.550 r*xJ = 0.605
(8)
BROADENING
FIG. 4. Orientational
COEFFICIENTS
and radial coordinates
203
IN C2H2
for two interacting
linear molecules.
where C( lIIIl; mlm2m) is a Clebsch-Gordan coefficient and 1 varies from 11, - lz 1 to 1, + 12.(Only terms with even { 1,121} are nonzero for the symmetric linear molecules considered here.) By comparing Eqs. ( 7) and ( 8), we obtain Ul,i,&r) = (4~)~~~~ C C uij(l1121,r)C(I,lJ; I
m - m O))tGi
ij
with
where the coefficientsS,( l,lJ, Y) have been given in Eq. (4) of Ref. (25). This determination of the Ul,,,,,,(r) functions is more accurate and needs much less computing time than the first method. As shown in Figs. 5 and 6, the exact and truncated radial functions U,,,,,(Y), calculated from the atom-atom parameters of C2HZ-N2 and C2H2-02 given in Table III, are very different for r < 4.5 A. It appears also that the Uz2, and U2z2contributions may be neglected in the linewidth calculations but the exact Udoocontribution has almost the same importance as the I&O and Uz20contributions. This result is probably caused by the relatively long tetraatomic molecule C2H2. Therefore, starting from the truncated expression of U400( r) ,
204
BOUANICH ET AL.
1400
c
tf
-u, 1200 1000
-._
u tr
-_-__.
”
--
u,tr
tr
600
Go -_- Go ______ UZ
23 c@O 3 400
-_
200
.
“u,
0 -200 3.00
4.50
5.00
5.50
r CA> FIG.5. Exact and truncated radial functions U,,& r) of the spherical harmonic expansion of the atomatom potential for C2H2-N2.
(9) we have obtained (see the Appendix) the 4305’zcontribution to the differential cross section. The isotropic potential U,,(r) used in the trajectory model cannot be well represented by a 6- 12 LJ potential. We have fitted the truncated and exact U,( r) functions by a 6-n LJ potential, where n > 12, such as
By considering
the conservation
of angular
momentum
and energy, we obtain
from
Eq. (4)
The parameters E, 6, n for the 6-n potential, as well as ro( V), the minimum distance of approach corresponding to b = 0 at 297 K, are given in Tables III, IV, and V. In order to take into account the exact U2oo, &o, and U4m contributions within the RB formalism, we have used the approach described in Ref. (5 ): New parameters denoted by dg@‘, ego’, ds2’, eF”, d$w, ezm are obtained in four intermolecular ranges
205
BROADENING COEFFICIENTS IN CzH2
.
1400
tr
-U
1200
000
--I
”
tr
..11.1
”
tr
1000
tr
--u, 800 3 -600 s 3
-
u&
--_
u
20 _-----u zl -_ U.&
400
#
200 0 -200 3.50
3.00
4.00
FIG. 6. Exact and truncated radial functions Z/,,& atom potential for C2HZ-Oz.
4.50 r (A) r)
5.50
5.00
in the spherical harmonic expansion of the atom-
so that the truncated radial functions are fitted through a least-squares procedure to the exact radial functions such as U?W(&~, $0, r) r, Ugo(dij, eij, r), U&(dyO, ey, Y) % Uzo(dij, eij. r) and U&(d~oo, e$w, r) x U&(dij, eij, r). In this fit, we have limited U&(r) to the contributions in Y-’ and r-14. For C2Hz-XZ (X = N or TABLE IV Atom-Atom Parameters ei’2m(IO-” erg ti”) and dzhrn (IO-’ erg A’*), and LJ (6-n) Parameters for CZH2-N2 Deduced from the Fit to the Exact Radial Functions V,,(r), UzzO(r), U,& r), and C&(r)
rg
1.053
2.587
1.881
4.500
rg 5 r s 4.35
3.319
3.259
5.927
5.667
4.35 < I 5 4.85
0.324
1.161
0.578
2.019
4.3s < r 5 4.85
0.847
1SO4
1.513
2.615
4.85 < r S 6.55
0.151
0.509
0.269
0.884
4.85 < r < 6.35
0.198
0.606
0.353
1.054
r > 6.55
0.0862
0.0764
0.154
0.133
I > 6.35
0.0862
0.0764
0.154
0.133
~ntennolecular range
rgsrI;4.35
(A) 1.842
2.146
3.290
4.35 < I 5 4.85
0.549
1.078
0.980
1.876
4.85 < r z 5.75
0.174
0.469
0.310
0.816
r > 5.75
0.0862
0.0764
0.154
0.133
3.732
E(K)
och
30.92
4.370
n
ro (A)
21.5
4.071
206
BOLJANICH ET AL. TABLE V
Atom-Atom Parameters eihrn (lo-” erg A6) and df:“” ( IO-’ erg A I*), and LJ (6-n) Parameters for Cz Hz-02 Deduced from the Fit to the Exact Radial Functions l&(r) , U,,,(r) , U,,(r) , and u,(r) lntemolsularrange (.k)
e$j
dg
e
dzw co
lnterm~lecularran~c (A)
e$$'
d220 Ho
220
%a
d220 co 3.856
r#J5 I 5 4.35
0.799
1.777
1.430
3.074
q 5 r s 4.35
2.533
2.229
4.532
4.35 < I5 4.91
0.224
0.677
0.401
1.171
4.35 < r i 4.92
0.515
0.836
0.922
1.446
4.91
0.104
0.198
0.187
0.342
4.92 < r i 6.55
0.138
0.272
0.247
0.471
0.0682
0.0394
0.122
0.0682
I > 6.55
0.0682
0.0394
0.122
0.0682
7.05
r > 7.05
29.9 1.705
1.692
3.050
4.35 < r 5 4.92
rgsrs4.35
0.391
0.676
0.699
1.169
4.92 < r 5 6.95
0.120
0.217
0.214
0.376
r > 6.95
0.0682
0.0394
0.122
0.0682
4.256
23
3.981
2.928
0), the four energy parameters eu-x, dH-x, Q_~, de-x, are considered as adjustable variables. Since the truncated expressions of the u’s are given by the difference between two monomials in do and e,, it may be shown that only two variables are independent. Therefore the fit to the exact U(Y) functions is rather poor even if four regions have been considered. It should be noted that this approach involves another approximation because the integrations over z = vkt/rc (leading to the resonance functions in the RB formalism) imply that the atom-atom parameters are not r-dependent. To obtain the atom-atom parameters given in Tables IV and V, we have arbitrarily set
At long distance, these fitted parameters of V, is then convergent.
6. CALCULATED
are taken as do and e,, since the radial expansion
RESULTS FROM THE RB MODEL
The RB formalism has been applied by considering, in addition to the electrostatic (quadrupole-quadrupole ) contributions, the atom-atom contributions derived from either truncated or exact radial functions UOoo(r), lJ2~ ( r), and U2zo( r). All the terms ‘1bS2[r,(b)] with even I, and I2 of the atom-atom and cross contributions given in Ref. (6) have been introduced in the computation. When truncated radial functions are considered, we have also taken into account all the terms arising from the r-I0 and r-l6 contributions in the potential according to Ref. (26). In both cases, we will show the influence of the U,, contribution on the broadening coefficients. Calculations were generally made for the mean relative velocity 3 = ( 8kBT/mn) ‘I*. However, for a few lines of CZ H2 perturbed by NZ , we have performed the average over the MaxwellBoltzmann distribution of velocities [see Eq. ( 1)] by using the method of integration
BROADENING
COEFFIClENTS
IN C1H2
207
described previously (IO), combined with that developed by Looney and Herman (27). Slightly larger results of y. ( 1 to 4%) were obtained. So the very costly consideration of the velocity distribution does not affect the calculated linewidths appreciably and was not generalized. 6.1. C2H2-N2 The collisional linewidths y. at 297 K have been computed with the spectroscopic constants given in Table II, the atom-atom parameters given in Tables III and IV. QGH~= 4 D-A, and QN, = -1.4 D *A. The broadening of the lines resulting only from quadrupole-quadrupole interactions depends strongly on the isotropic potential used in the trajectory model (Fig. 7 ): the exact VoW(r) function described by a 62 1.5 potential leads to a rather small electrostatic contribution. Using the truncated expansion of the atom-atom potential, we obtain calculated linewidths that are smaller than the experimental results at low ) m 1 values and significantly larger at great )m 1 values. For ) m 1 > 22, the difference is outside the range of the estimated experimental uncertainty. It appears also (Fig. 7) that the contribution arising from 4,0S2, for which analytic expressions are presented in the Appendix, is rather weak for all the lines. The consideration of the exact radial functions through the fitted atom-atom parameters yields somewhat larger results at intermediate or high ) m I values than the first
IL”
lml FIG. 7. N2-broadening coefficients yo for IR lines of C2Hz. Experimental values ( l, 0, X). results from the RB model: V = I/ala2 with U’&(r): curve (a); V = V,,, with Ue&,( r): V = Vo,a + V, with U&(r). U&&r). U&(r): curve (b); same interactions plus t&,(r): V= C’Q,Q~+ r\ with (i&(r). U%(r). and U&(r): curve(c); same interactions plus C&(r):
Calculated curve (a’); curve (b’): curve (c’).
208
BOUANICH ET AL.
0
5
10
15
20
25
30
35
IN
FIG. 8. 02-broadening coefficients y. for IR lines of C2H2. Experimental values (1): (0). Calculated results from the RB model: V = Vplo2 with U&(r): curve (a); V = Vala2 with U&(r): curve (a’); V = VQ,~ + V. with U&(r), C.&,(r), ugo(r): curve (b); same interactions plus U&,(r): curve (b’); V = ValQ + V, with Q&(r), UT&.,(r)and @&(r): curve (c); same interactions plus U&(r): curve (c’).
calculations. The theoretical results remain satisfactory for the lines in the range ( m 1 = 5 to 19 but the agreement with our experimental data is poorer at large (m ( values.
The linewidths of &Hz broadened by O2 are a good test for theoretical models because they are very sensitive to short-range interactions owing to the very small electrostatic contribution (Fig. 8). The calculation performed with the atom-atom parameters given in Table III involving the truncated UO,( r), Uzoo(r), U,,,(r), and U,,( r) functions provides satisfactory O2 -broadening coefficients. The consideration of the exact U(r) functions through the fitted parameters e$12” and df,l”“’ given in Table V leads to significantly larger results for 1m 1 > 10. We can explain this result as follows: the contributions to the differential cross section derived from the truncated radial functions Uzoo(r), U2,,( r), and Um( Y) are notably smaller for close collisions (dominating the widths at large (m I values) than those derived from these exact functions. The difference between these contributions is not entirely compensated by the respective influences, through the trajectory model, of the truncated and exact isotropic potentials Uoo~(r). From Fig. 8, it appears also that the 4,0S~contribution is important, especially for ) m I < 15, and cannot be neglected for CzH2-02.
BROADENING
6.3.
COEFFICIENTS
209
IN Cz Hz
Temperature Dependence of Linewidths
To our knowledge, no work has been yet devoted to the determination of the temperature dependence of y. for C2H2-Nz or CZH2-Oz. This temperature dependence can be accurately represented by using the simple relation TO(T) = YO(To)( Tl To)-“, where TO = 297 K (our reference temperature) and T has been arbitrarily taken as 200 K. By comparing for CZH2-NZ the calculated values of y. at 297 K and 200 K arising from the RB model including either the truncated or the exact radial expansion of the atom-atom potential (with or without the U4oocontribution ), we have derived (Figs. 9 and 10) for the n exponent values which are strongly 1m )-dependent. The decrease of n as ( m ( increases has been obtained theoretically and experimentally for other radiative molecules such as CO1 on collision with N2 or O2 (28). It may be seen (Figs. 9 and 10) that the influence of 4*oS2is rather small, but the exact atom-atom model provides values of n which are significantly smaller, for I m ( > 8, than those derived from the truncated model. Only accurate experimental values of broadening coefficients at low temperature would allow us to reach a conclusion about these theoretical results.
03L.““““’ 0
“’ 5
10
“““‘c”.‘n’.” 20
15
25
30
5
14 FIG. 9. variationof the temperatureexponentn with )m 1 for C2H2-N2. Calculated results from the RB model: V = VQla + V, with U&,(r), U&&r), U&,(r): curve (a): some interactions plus U&,(r):CUIYS (a’): I,’ = I’Q,~, + V, with U&(r), Uym( r), and @&(r): curve(b); same interactions plus U&(r): curve (b’).
210
BOUANICH ET AL. 0.9,
lml FIG. 10. Variation of the temperature exponent n with I m / for C,H,-0, Calculated results from the RB model: V = Vola2 + V, with U&+(r), uF&r), V&,(r): curve (a); same interactions plus u&,(r): curve (a’); I’= Vala2 + V. with U&(r), @&(r) and u;“,,(r): curve (b); same interactions plus Cl&(r): curve (b’).
6.4. Concluding Remurks By using the RB model, we have performed different calculations for the Nz- and 02-broadening coefficients of C2Hz. Contrary to our expectations, the truncated radial expansion of the atom-atom potential provides generally more satisfactory results than the exact expansion. We do not believe that this result is due to the way we have taken into account the exact radial spherical harmonics development of I’, by still using the RB formalism, although this approach may be criticized. We estimate that the discrepancy with experimental data may be caused by the atom-atom potential itself, which accounts approximately for short-range interactions but has some deficiencies related to the assumed pair-wise additivity (see Ref. (20, p. 34)). It should be noted that no parameters of the interaction potential (except the quadrupole moment of CZH2) were fitted to obtain the theoretical results; in particular the parameters dv and eii used for C2H2 arise from CO* (C) and CH4 (H). As for Qc2uZ, the value adopted, 4 D +A, is much lower than the other recent results (see Ref. (20, p. 581)) but represents a high limit for the RB calculations of self-broadening coefficients. Another explanation of this discrepancy may originate in the semiclassical approximation where the relative translational motion is assumed to be independent of the rational motion, which should limit the accuracy of the results, especially for the high J states. Moreover, as shown by Margottin-Maclou et al. (29)) a small modification in the trajectory model (which is derived from UO,( r) only) may induce large changes in the calculated values of yo. These authors have obtained a much better agreement
BROADENING
COEFFICIENTS
211
IN C>Hz
between experimental and calculated broadening coefficients of CO*-Nz and C02-O2 by slightly modifying the apparent velocity V: (F, in Eq. (4) was arbitrarily lowered by 15%). Finally, from our results, we estimate that the 4, 0, 0 contribution of the atomatom potential may be important for perturbers having a small quadrupole moment such as Oz. APPENDIX: THE 4, 0, 0 ATOM-ATOM CONTRIBUTION TO THE DIFFERENTIAL CROSS-SECTION
In the coordinate axes represented in Fig. Al of Ref. (27) (see also Fig, 2 of Ref. (3)) for the collision pair molecule 1 (radiator)-molecule 2 (perturber). the II = 4, fz = 0 truncated contribution to the atom-atom potential may be expressed as (25) 4’oVa = 4lr C 2 rt ij
m
~-~)YZ(n-il,1)~4m(I)l;o(2). (
with m varying from -4 to 4. Using the notations of Bonamy and Robert (6), we obtain the following expressions for the 4.0Sz[v,( b)] function:
+
c
cy “,“f$ k)
+
04)
-
z
(C
ii
di
elj)(
C
hi
x (C c)(4)“.“f$k Ji
) +
c
c:;“’
4.0f$k)
Gidij>
+
i,j
+
04)
+
2;(y;;;;3
Ji
with
“,““f ;:( k) = 13 ;9ik825 ( 14k12 + 252k” + 2466k” + 16 980k9 + 90
405k8+ 388 980k7+ 1 380 690k6 + 4 062 420k5 + 9 810 450k4 + 18918900k3+27584550k’+27
187650k+
13593825)
e-2k
4,0f;;(k)
=
47 760 763 050 + 1 156 437k”
( 14k15 + 504k14 + 9432k13 + 120 234k” + 8 826 894k”
+ 54 977 031k9 + 283 885 560k8
+1225445760k7+4430855520k6+13349311920k5 +33007768920k4+64972076400k3+96167055600k2 +95521526lOOk+47760763050)
BOUANICH
212
ET AL.
e-2k
4q-;&k) =
(70k’* + 3780k17 + 104 760k16 + 1968
857266364354400
120k15
+ 27 953 424k14 + 317 538 144k13 -t 2 981 059 200k12 + 23597619
192k”+
159422475996k”+925
186646160k9
+4621044715560k8+19817845462800k7 +72462716144280k6+222971619356640k5 + 564969
197816
100k4+
1
137433809862350k3
+1711738724070375k2+1714532728708800k + 857 266 364 354 400). In these expressions,
C,!?’ are squared
Clebsch-Gordan
coefficients
according
to
c!4’ I’ = [ C(j4j’. >000) (2 and 04 =
(-1)““/2[(2j,
where Wis a Racah coefficient. ofj and m in Ref. (30).
+ 1)(2jf+
1)c:;4’c::4)]“*u/(jii/jri~;
The coefficients
14),
Ci?’ and D4 have been given in terms
ACKNOWLEDGMENTS
Support by the “Accords Culturels” between France and Belgium is gratefully acknowledged. We also thank Professor C. Boulet for many helpful discussions. REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9. IO. II. 12. 13. 14. 15. 16.
D. LAMBOT, G. BLANQUET, AND J. P. BOUANICH, J. Mol. Spectrosc. 136, 86-92 ( 1989). P. W. ANDERSON, Phys. Rev. 76,647-661 ( 1949). C. J. TSAO AND B. CURNUTTE, J. Quant. Spectrosc. Radiat. Transfer 2, 41-91 ( 1962). D. ROBERT AND J. BONAMY, J. Phys. 40,923-943 ( 1979). L. ROSENMANN, J. M. HARTMANN, M. Y. PERRIN, AND J. TAINE, J. Chem. Phys. 88, 2999-3006 (1988). J. B~NAMY AND D. ROBERT, unpublished. V. MALATHY DEVI, D. C. BENNER,C. P. RINSLAND,M. A. SMITH,AND B. D. SIDNEY,J. Mol. Spectrosc. 114,49-53 (1985). J. R. PODOLSKE,M. LOEWENSTEIN,AND P. VARANASI, J. Mol. Spectrosc. 107,241-249 ( 1984). J. BONAMY, L. BONAMY, AND D. ROBERT, J. Chem. Phys. 67,4441-4453 ( 1977). I. P. BOUANICH,C. CAMPERS,G. BLANQUET,AND J. WALRAND, J. Quant. Spectrosc. Radiat. Transfer 39,353-365 (1988). J. S. WONG, J. Mol. Spectrosc. 82,449-45 1 (1980). P. VARANASI AND B. R. P. BANGARU, J. Quant. Spectrosc. Radiat. Transfer. 15, 267-273 ( 1975). I. R. DAGG, A. ANDERSON,W. SMITH, M. MISSIO,C. G. JOSLIN,AND L. A. A. READ, Canad. J. Phys. 66,453-459 (1988). W. H. FLYGARE AND R. C. BENSON,Mol. Phys. 20,225-250 ( 1971). D. E. ST~CRYN ANDA. P. STOGRYN, Mol. Phys. 11, 371-393 (1966). J. HIETANEN AND J. KAUPPINEN, Mol. Phys. 42,41 I-423 ( 1981).
BROADENING
COEFFICIENTS
IN C2H2
213
17. D. REUTER. D. E. JENNINGS,AND J. W. BRAULT, J. Mol. Spectrosc. 115,294-304 ( 1986). 18. T. AMANO AND E. HIROTA, J. Mol. Spectrosc. 53, 346-363 ( 1974). 19. J. 0. HIRSCHFELDER,C. F. CURTISS. AND R. B. BIRD, “Molecular Theory of Gases and Liquids,” Wiley, New York, 1967. 20. C. G. GRAY ANDK. E. GUBBINS, “Theory of Molecular Fluids,” Vol. 1, “Fundamentals,” Oxford Univ. Press, New York, 1984. 21. J. P. BOUANICHAND G. BLANQUET, J. Quant. Spectrosc. Radiat. Transfer. 40, 205-220 ( 1988 ). 22. C. S. MURTHY, K. SINGER, AND I. R. MCDONALD, Mol. Phys. 44, 135-143 (1981). 23. M. O~BATAKE AND T. 001, Prog. Theor. Phys. 48,2 (32-2 143 ( 1972). 24. M. HERMAN AND R. COLIN, BuII. Sot. Chim Belg. 89, 335-342 ( 1980). 25. J. DOWNS. K. E. GUBBINS,S. MURAD, AND C. G. GRAY, WII. Phys. 37, 129-140. 26. J. BONAMY, D. ROBERT,AND C. BOULET. J. Quant. Spectrosc. Radiat. Transfer. 31, 23-34 ( 1984). 27. J. P. L~ONEY AND R. M. HERMAN, J. Quant. Spectrosc. Radiat. Transfer. 31, 547-557 ( 1987 ). 2X. L. ROSENMANN, J. M. HARTMANN, M. Y. PERRIN. AND J. TAINE, Appl. Opt. 27, 3902-3907 (1988). 29. M. MARGOTTIN-MACLOU.P. DAHOO, A. HENRY, A. VALENTIN.AND L. HENRY, J. Mol. Spectrosc. 131,21-35 (1988). 30. J. P. BOUANICHAND C. BRODBECK,J. Quant. Spectrosc. Radiat. Tramftir. 14, 141-151 ( 1974).
OF MOLECULAR
SPECTROSCOPY
ldtt,195-2
13 ( 1990)
N2- and 02-Broadening Coefficients of C2H2 IR Lines J. P. BOUANICH Laboratoire d’lnfrarouge, assock! au C.N.R.S., Universiti de Paris-&d, Bdtiment 350. F-91405 Orsay Cede.x, France AND
D. LAMBOT, G. BLANQUET, AND J. WALRAND Laboratoire de Spectroscopic Molhculaire. Facultb Universtaires N.D. de !a Pais. Rue de Bruxelies. 61, B-SOOO-Namur,Belgium Pressure-broadening parameters of six lines belonging to the vgband of C2Hz in collision with Nz have been measured with a tunable diode-laser spectrometer in order to complete up to J = 33 our earlier measurements (D. Lambot, G. Blanquet, and J. P. Bouanich, J. Mol. Spectrosc. 136, 86-92 ( 1989)) on the broadening of CzH2 by N2 and 0~ at 297 K. These N2- and 02broadening coefficients have been first calculated on the basis of the Anderson-Tsao-Curnutte theory; in this approach, we show that the short-range interactions which contribute significantly to the linewidths are not correctly treated. Next. we consider the improved semiclassical model proposed by Robert and Bonamy. The intermolecuIar potential consists in the addition of the atom-atom interaction model to the quadrupolar interactions. The limited radial spherical harmonics expansion of the atom-atom potential, from which expressions for the differential cross section were derived, appears to be quite insufficient at short intermolecular distances. Therefore, we use a more accurate representation of this potential, avoiding an inadequate truncation and keeping the analytic expressions obtained by Bonamy and Robert. In the calculations we take into account the contributions derived from the radial functions Um( r), U,,(r), and U,,,(r), as well as from Uew(r). A theoretical expression is obtained for the Il.,, contribution to the differential cross section. The results of the calculations arising from the exact radial expansion of the atom-atom potential appear to be significantly larger for high J lines than those arising from the truncated expansion. The latter results, which do not include adjustable atom-atom parameters, are in good agreement with experimental broadening coefficients for CzH2-O2 and in reasonable agreement (except at large J values) for CzHZ-N2. It is also shown that the contributions to the linewidths derived from 0’4~ are rather small for CzHz-N2 and more important for C2H2-02. Finally, by calculating the collisional linewidths of CZHZ-NI and C2H2-O2 at 200 K. we have predicted their temperature depcndences. 1~)1990 Academrc press. I”~. 1. INTRODUCTION
In a recent paper ( I), we reported NZ- and 02-broadening coefficients for several lines with J ranging from 2 to 29 ( NZ) and 3 to 33 (0,) in the v5 band of C2H2 (730 cm -’ ) . For high J values of ClH2 perturbed by N2, we measured only the collisional halfwidths of the R( 24) and P(29) lines. In order to complete this work, here we determine experimentally, using a tunable diode-laser (TDL) spectrometer, the N2 broadening coefficients of C2Hz for 6 lines with J ranging from 26 to 33 in the R branch of the v5 band. 195
0022-2852190 $3.00 Copyright 0
1990 by Academic Press, Lnc.
All righk of revcduction
in any form reserved.
196
BOUANICH
ET AL.
The experimental broadening coefficients for C2Hz-N2 and C2Hz-O2 are compared with theoretical results derived from the Anderson-Tsao-Curnutte (ATC) method (2. 3) and the more accurate semiclassical model developed by Robert and Bonamy (RB) (4). It may be noted that the contributions of vibrational effects should be small ( 1) and have been disregarded. Moreover, CzHz in the II, state (corresponding to the final level of the perpendicular u5 band) is not, strictly speaking, a linear molecule but is considered as linear since the two light H nuclei should rotate very fast around and close to the C-C line. For the intermolecular potential, we consider in addition to the electrostatic contribution (quadrupole-quadrupole interactions) an anisotropic dispersion contribution in the ATC calculations and an atom-atom (called also site-site) potential in the RB calculations. As was previously shown for C02-X systems (5)) the limited radial expansion of the atom-atom potential given in Ref. (4) is quite inaccurate at short distances. We have calculated exactly the main coefficients I!I.J~,~,,( r) of this expansion for CZH2-N2 as well as for CZH2-02, and shown that the radial function U&r) which was always neglected in the calculations has almost the same importance as U&r) or ~&a(r). Therefore, we have derived explicit formulas for the UdoOcontribution to the differential cross section. In order to use these expressions as well as those obtained by Bonamy and Robert (6) and to take into account the exact radial dependence of the atom-atom potential, we have fitted the atom-atom parameters in a given intermolecular range, following the method described by Rosenmann et al. (5). Thus, we have applied the RB model by including coherently the truncated and exact expansion of the atom-atom interaction potential and we have compared the theoretical results with experimental data. Finally, from these calculations performed for C2H2-N2 and C2H2-O2 at 297 K (experimental temperature) and 200 K, we have evaluated the coefficient n that defines the variation with temperature of the broadening coefficients through a simple analytic law. 2. EXPERIMENTAL
RESULTS
FOR Np-BROADENING
COEFFICIENTS
OF CzHz
The spectra of the lines R(26) and R(29) to R(33) in the v5 band of C2H2 were recorded on a TDL spectrometer interfaced to a minicomputer. The gas mixture was contained in a White-type cell with an optical path length of about 4 m. Nitrogen was supplied by 1’Air Liquide with a purity of 99.9% and acetylene by Air Products with a stated purity of 99.6%. The partial pressure of C2H2 ranged from 10e2 to 10-l mbar. Four pressures of N2 equal to about 50,65, 80, and 95 mbar were used. These pressures were measured using an MKS Baratron gauge. All the measurements were made at a room temperature of about 297 K. The relative wavelength calibration of the spectra was performed from a confocal etalon with a fringe spacing of 0.00977 11 cm-‘. Spectral purity of the laser modes was checked by recording a saturated line of CZHI. The procedure used in data reduction is similar to that described previously (I). The pressure-broadening coefficients y. (in cm-’ atm-‘) of the lines studied in this work as well as in Ref. (I) are listed in Table I with a relative error estimated to be about 5%. Within this accuracy, the halfwidths are only Im I -dependent [m = -J for P(J) lines and m = J + 1 for R(J) lines] and are represented as a function of (m 1 in Fig. 1. By using a least-squares procedure, we have obtained smoothed out data
197
BROADENING COEFFICIENTS IN C2H2 TABLE I
N,-Broadening Coefficients y0 ( IO-’ cm-’ atm-‘) in the v5 Band of CzH2 at 297 K, (the wavenumbers (cm-‘) of the unperturbed lines were measured by Hietanen a a/. ( 16))
T-
Wavenumbers
R
: :
5 !: 8 1’0 1:. 13 14 15 16 17 18 19 20 21 :: 24 Ei 27 28 ii 31 32 33 34
731.5084 733.8598 736.2130 738.5636 740.9146 743.2639 745.6135 747.9640 750.3 106 752.6580 755.0062 757.3511 759.6959 762.0396 764.3827 766.7241 769.0654 771.4048 713.7444 716.0822 778.4183 780.7531 783.0868 785.4191 787.7506 790.0803 792.4083 794.7345 797.0597 799.3835 801.7049 804.0250 806.3437 808.6601
R
P
724.4482 722.0948 719.7389 717.3861 715.0305 712.6757 710.3202 707.9657 705.6104 703.2556 700.9005 698.5436 696.1895 693.8339 691.4800 689.1254 686.7701 684.4164 682.0631 679.7094 677.3566 675.0040 672.6519 670.3000 667.9489 665.5980 663.2419 660.8979 658.5488 656.2013 653.8539 651.5072 649.1619
?(a P
109.0 102.0
104.2 91.6 91.7
86.2 83.6
84.1
82.5 80.3 80.0
76.8 75.0 71.7
72.8
62.1
smoolbed 116.6 109.1 103.1 98.4 94.6 91.7 89.4 87.6 86.2 85.0 83.9 82.9 81.9 80.9 79.7 78.5 77.1 75.6 73.9 72.1 70.3 68.3 66.3 64.2 62.2 60.1 58 56.1 54.2 52.4 50.7 49 1 47.6 46.1
1
58.5 52.7 53. I 51.3 49.8 46.8 46.1
(the curve of Fig. 1) which are more accurate at large ( m 1 values than those given in Ref. ( 1) . It may be seen that our results are in satisfactory agreement with the results of Devi el al. ( 7) obtained in the ( v4 + v5) band.
3. GENERAL FORMULATION
The collisional
halfwidth
OF
‘HE LINEWIDTH
of an isolated vibration-rotation
line f -
i may be written
as
where n2 is the number density of the colliding molecules; 03 is the partial optical cross section; pJ2 is the relative population of the perturber 1Jz, v2 = 0) state; F(v) is the distribution of the relative velocities v; S’i is the diffusion operator or the differential cross section representing the collisional efficiency; and b is the impact parameter. The integration over the velocity distribution is not generally performed and the calculations are made for 5, the mean collisional velocity. In the ATC theory, a straightpath trajectory described at constant velocity is assumed; S(b) is expanded through
198
BOUANICH ET AL.
0 01
”
“I’.
5 ‘.
10 ’
”
”
15
20
25
30
FIG. 1. Experimental N2-broadening coefficients y. for IR lines of CzHZ (for measurements in P(J) and R(J + 1) lines, only the average is represented): (0) our results (~5 band) (the smooth curve fits our experimental data); (0) results of Devi et Q/. ( 7) (~6 + u5 band); ( X ) results of Podolske et a/. (8) (“4 + vg band).
a second-order perturbation treatment which requires avoid divergence of S for small values of b) so that
a cutoff procedure
(in order to
i)ri a/i = 7rbi +
f b0
2irb&( b)db,
(2)
where b. is evaluated for each Jz from &( bo) = 1. If b. is smaller than the distance of closest approach rO, b. is replaced by r. in Eq. (2). In the RB model, the diffusion operator is expressed in a closed form through the linked cluster theorem, which allows a partial resummation of all V-orders, such as S(b)
= 1 - expl-(&
+ 5’2~2 +
&,&I,
(3)
where &,a, &,fz, and Sz,/ziz are respectively equal to Sq>ter, Sqy, and Syiddle, the second order terms of the perturbation treatment of S(b) in the ATC theory. The trajectory model includes the influence of the isotropic potential vi,, in energy conservation and of the isotropic force F, (F, = -grad I&) in the equation of motion around the distance of closest approach rc. The actual trajectory is replaced near Y, by a parabolic trajectory described with an apparent velocity vi given by
BROADENING
COEFFICIENTS
199
IN &HZ
where v),is the relative velocity at the closest approach and m is the reduced mass of the colliding partners. In our calculations, the equivalent straight-path trajectory (with velocity v’,) defined in Ref. ( 9) is used instead of the parabolic trajectory. This model greatly simplifies the expressions of the resonance functions given in Ref. (6) and should lead to nearly identical results of yo. 4. CALCULATED
RESULTS
FROM
A SIMPLE
INTERMOLECULAR
POTENTIAL
In the ATC approach, only electrostatic interactions are accounted for in the intermolecular potential, Here, we consider in addition an anisotropic dispersion contribution according to (10) f hPz(cos 0,),
V= Vo,gz - 4ty, 0
(5)
where Q1 and Q2 are the quadrupole moments of molecule I (absorber) and molecule 2 (perturber); yl denotes here the polarizability anisotropy of C2H2 defined by ( ali - aI)/( a// + 2aL); Pz is the second Legendre polynomial; and 8, is the angle between the axis of molecule 1 and the intermolecular axis. The Lennard-Jones (LJ) parameters t and c for the molecular pairs CZH*-N2 and CZHZ-O2 are determined from the usual combination rules, t = (E~c~)~” and u = ((T] + u2)/2. The quadrupole moment of C2H2 is evaluated from self-broadening linewidths measured in the 5~~ ( 11) and ( v, + q) ( 12) bands. Using the ATC and the RB formulations with I/defined by Eq. ( 5 ), we obtain approximately 4.3 and 3.5 D * A, respectively. We have adopted Q, = 4 D * A, a value notably smaller than the result ( 5.42 D - A) derived recently from collision-induced spectra of C2HZ-Ar ( 13). For the quadrupole moment of N2, we have considered the value - 1.4 D - A obtained experimentally ( 14), and we have used the O2 quadrupole moment recommended by Stogryn and Stogryn ( 15), Q2 = -0.39 D - A. The molecular parameters introduced in the computations are summarized in Table II. The coefficients y. at 297 K were computed by including the contributions from perturbing molecules having J? values up to 42 for N2 and 5 1 for 02, weighted by the TABLE 11 Molecular
Parameters
Used in the Calculations
wb M (g.mol-1)
26.04
B, (cm’)
1.1766406 a
B; @ml)
1.176432 1
N2
9
28.0134
31.9988
1.989622 b 5.763 x lo-6 b
D, (cm-‘)
1.6221 x 10-6 a
Q@,A)
4.0
-1.4d
199.2 f
95.1 f
E (K) 0 (A)
4.523 f
n
0.18Og
3.75 f
a Reference (16)
dRef-
b Reference (17)
e Reference (15)
c Reference (18)
fRefmnce(19)
(14)
g Calculated from polarizabilities values given in Ref. (20 p. 581).
1.437654 C 4.837 x 10-6 C -0.39 = 116f 3.57 f
200
BOUANICH
ET AL.
Boltzmann factor and the nuclear spin factor (( - 1)J2 + 3)/2 for N2 and ( 1 - ( -l)J2)/2 for Oz. As shown in Fig. 2, the results for C2H2-N2 derived from the ATC model are in reasonable agreement, especially at large [ml values, with the experimental data. The agreement is notably poorer for C2H2-02 in the range 7 < )m 1 < 30 (Fig. 3 ) . It may be shown that at low 1m 1, the predominant contribution arises from the dispersion terms of the potential and, at large 1m 1, the calculated yo- values derive mainly from the cutoff procedure S, ( b < ro) = 1. For C2HZ-02, this procedure leads to broadening coefficients nearly constant and independent of the lines with 1m ( > 14. It may be also noted that the consideration of a repulsive contribution in addition to the dispersion term [see Eq. (6) of Ref. (21)] or the inclusion of higher order electrostatic interactions involving the hexadecapole moments of C2Hz and the penurber would not notably change the results of yo. By using the RB model, we have computed the N2 - and 02-broadening coefficients from exactly the same interaction potential, and the molecular parameters given in Table II. As may be seen in Fig. 2, the results obtained for CZH2-N2 are close to the experimental data up to 1m I = 10 but decrease notably faster at higher I m (. The discrepancy is much more pronounced for CzH2-O2 and the theoretical results decrease dramatically with increasing (m ( (Fig. 3). In the latter case, the electrostatic contribution is weak owing to the smallness of the quadrupole moment of 02 and the interaction potential [ Eq. (5 )] appears to be quite insufficient: the short-range forces which are dominant in the broadening of the lines at large 1m) are greatly underes-
120
-0
5
10
20
15
25
30
35
Iml FIG. 2. N2 -broadening coefficients y. for IR lines of C2 H, . Experimental values ( l, 0, X ) . Results from ATC formalism applied with V defined by Eq. (5): curve (a); results from the RB model applied with the same potential: curve (b) .
BROADENING
o’.-..a....s 0
5
10
COEFFKIENTS
“‘*.““““”
15
20
201
IN C2Hz
25
“I..”
30
35
RG. 3. 02-broadening coeficients y0 for IR lines of C2H2. Experimental values given in Ref. (I): ( l ). Results from ATC formalism applied with I/ defined by Eq. (5): curve (a); results from the RB model applied with the same potential: curve (b).
ATC theory leads artificially to improved results at high Im I values only through the cutoff procedure used for b0 < ro. timated.
The
5. ATOM-ATOM
INTERACTION
MODEL
The interaction potential used in the RB model consists in adding to the electrostatic interactions V. the atom-atom interactions V,. V, is the sum of elementary atomatom contributions described by Lennard-Jones expressions:
Here, the indices 1i and 2 j refer to the ith atom of molecule I and the jth atom of molecule 2; rli,zj is the distance between these atoms and dij and eU are the atomic pair parameters. To our knowledge these parameters are not available for acetylene. For C-C interactions, we have considered the parameters derived from COz , and we have adopted the rather low values of e and d given by Murthy et al. (22). The H-H parameters are evaluated from eji = 4ta6 and dj, = 4~‘~ with t = 8.6 K and u = 2.8 13 A, LJ parameters derived from CH4 (Ref. (20, p. 112)). The parameters used for NN and O-O are the mean values determined by Oobatake and Ooi (23) from virial coefficients of NZ and 02. For the interaction between different atoms i and j, eii and dl, are calculated from the combination rules eiI = (eiiej,) “2, dg = ei,cT$,and alj = (oil
202
BOUANICH ET AL.
+ Uij)/2. The intramolecular distances Tii for CzH2 are taken from Ref. (24). The parameters used for the atom-atom potential are listed in Table III. In Ref. (4), rii,zj is expanded in power series of the intermolecular distance r to include the fourth-order contribution in r-‘o-r-‘6 (for even I, and Z2), and V, is expressed in terms of the intramolecular distances rli, rzj and the spherical harmonics Y, tied to each molecule, as
where 8,) 4, and &, 42 are the polar angles for the two linear colliding molecules (Fig. 4). Here, the total order of the spherical harmonics 1, + Z2is limited to 4 in accordance with the electrostatic potential limited to quadrupolar interactions. Rosenmann et al. (5) have shown that the expressions given in Ref. (4) for the coefficients U,,& r) are generally not valid for small values of r. We have calculated the exact radial functions UI,,,,( r) for C2H2-N2 and C2H2-O2 in two different ways. The first is based on the properties of spherical harmonics (see Ref. (20, p, 450)). If w and w’ denote the polar angles 04, 19’4’of the two linear molecules, these radial functions are given by
Ul,i,dr) =
V(r, w, 0’)
J-s
4?r
Y,:,(w)Y;-,(w')dwdw'
with
U!,I~, may be computed as a function of r by expressing rii,zj in terms of r, rli, rzj, 0,) 19,,c#P,,ti2, and perfornring a triple integral over ~8,) d&, and d( 4, - $2). The second
method is based on the expressions obtained by Downs et al. (25) for the harmonic expansion of the site-site potential. In the coordinate axes of Fig. 4 where the two 2 axes coincide, the spherical harmonic expansion of V, may be expressed as V, = 2
21+ 1 “2
2 ~~(1~1~1,W(lJ2l;
m
-
m
O)Y/,,YI,-,
l,l~hn ij
( i --yg-
,
TABLE III Atom-Atom
Parameters, Intramolecular Distances, and LJ (6-n) Parameters Fitting U,(r) for CZHz-N2 and C, Hz-02 Cij (m-l’+A~) C.&N= 0.154 C2Hz
Nz
E(K)
o(A)
n
1, (A)
61.44
3.886
16
3.687
3.685
16.5
3.471
dc_R = 0.133
CH_N= 0.0862 dw, = 0.0764 ec_o = 0.122
CzH2 - 02
rli,r2i
d(w7c;Aq
r,.H = 1.6614
d%xJ= 0.0682
.ZHXJ = 0.0682 dw, = 0.0394
75.63
(A)
r,_c = 0.604 n-2-N = 0.550 r*xJ = 0.605
(8)
BROADENING
FIG. 4. Orientational
COEFFICIENTS
and radial coordinates
203
IN C2H2
for two interacting
linear molecules.
where C( lIIIl; mlm2m) is a Clebsch-Gordan coefficient and 1 varies from 11, - lz 1 to 1, + 12.(Only terms with even { 1,121} are nonzero for the symmetric linear molecules considered here.) By comparing Eqs. ( 7) and ( 8), we obtain Ul,i,&r) = (4~)~~~~ C C uij(l1121,r)C(I,lJ; I
m - m O))tGi
ij
with
where the coefficientsS,( l,lJ, Y) have been given in Eq. (4) of Ref. (25). This determination of the Ul,,,,,,(r) functions is more accurate and needs much less computing time than the first method. As shown in Figs. 5 and 6, the exact and truncated radial functions U,,,,,(Y), calculated from the atom-atom parameters of C2HZ-N2 and C2H2-02 given in Table III, are very different for r < 4.5 A. It appears also that the Uz2, and U2z2contributions may be neglected in the linewidth calculations but the exact Udoocontribution has almost the same importance as the I&O and Uz20contributions. This result is probably caused by the relatively long tetraatomic molecule C2H2. Therefore, starting from the truncated expression of U400( r) ,
204
BOUANICH ET AL.
1400
c
tf
-u, 1200 1000
-._
u tr
-_-__.
”
--
u,tr
tr
600
Go -_- Go ______ UZ
23 c@O 3 400
-_
200
.
“u,
0 -200 3.00
4.50
5.00
5.50
r CA> FIG.5. Exact and truncated radial functions U,,& r) of the spherical harmonic expansion of the atomatom potential for C2H2-N2.
(9) we have obtained (see the Appendix) the 4305’zcontribution to the differential cross section. The isotropic potential U,,(r) used in the trajectory model cannot be well represented by a 6- 12 LJ potential. We have fitted the truncated and exact U,( r) functions by a 6-n LJ potential, where n > 12, such as
By considering
the conservation
of angular
momentum
and energy, we obtain
from
Eq. (4)
The parameters E, 6, n for the 6-n potential, as well as ro( V), the minimum distance of approach corresponding to b = 0 at 297 K, are given in Tables III, IV, and V. In order to take into account the exact U2oo, &o, and U4m contributions within the RB formalism, we have used the approach described in Ref. (5 ): New parameters denoted by dg@‘, ego’, ds2’, eF”, d$w, ezm are obtained in four intermolecular ranges
205
BROADENING COEFFICIENTS IN CzH2
.
1400
tr
-U
1200
000
--I
”
tr
..11.1
”
tr
1000
tr
--u, 800 3 -600 s 3
-
u&
--_
u
20 _-----u zl -_ U.&
400
#
200 0 -200 3.50
3.00
4.00
FIG. 6. Exact and truncated radial functions Z/,,& atom potential for C2HZ-Oz.
4.50 r (A) r)
5.50
5.00
in the spherical harmonic expansion of the atom-
so that the truncated radial functions are fitted through a least-squares procedure to the exact radial functions such as U?W(&~, $0, r) r, Ugo(dij, eij, r), U&(dyO, ey, Y) % Uzo(dij, eij. r) and U&(d~oo, e$w, r) x U&(dij, eij, r). In this fit, we have limited U&(r) to the contributions in Y-’ and r-14. For C2Hz-XZ (X = N or TABLE IV Atom-Atom Parameters ei’2m(IO-” erg ti”) and dzhrn (IO-’ erg A’*), and LJ (6-n) Parameters for CZH2-N2 Deduced from the Fit to the Exact Radial Functions V,,(r), UzzO(r), U,& r), and C&(r)
rg
1.053
2.587
1.881
4.500
rg 5 r s 4.35
3.319
3.259
5.927
5.667
4.35 < I 5 4.85
0.324
1.161
0.578
2.019
4.3s < r 5 4.85
0.847
1SO4
1.513
2.615
4.85 < r S 6.55
0.151
0.509
0.269
0.884
4.85 < r < 6.35
0.198
0.606
0.353
1.054
r > 6.55
0.0862
0.0764
0.154
0.133
I > 6.35
0.0862
0.0764
0.154
0.133
~ntennolecular range
rgsrI;4.35
(A) 1.842
2.146
3.290
4.35 < I 5 4.85
0.549
1.078
0.980
1.876
4.85 < r z 5.75
0.174
0.469
0.310
0.816
r > 5.75
0.0862
0.0764
0.154
0.133
3.732
E(K)
och
30.92
4.370
n
ro (A)
21.5
4.071
206
BOLJANICH ET AL. TABLE V
Atom-Atom Parameters eihrn (lo-” erg A6) and df:“” ( IO-’ erg A I*), and LJ (6-n) Parameters for Cz Hz-02 Deduced from the Fit to the Exact Radial Functions l&(r) , U,,,(r) , U,,(r) , and u,(r) lntemolsularrange (.k)
e$j
dg
e
dzw co
lnterm~lecularran~c (A)
e$$'
d220 Ho
220
%a
d220 co 3.856
r#J5 I 5 4.35
0.799
1.777
1.430
3.074
q 5 r s 4.35
2.533
2.229
4.532
4.35 < I5 4.91
0.224
0.677
0.401
1.171
4.35 < r i 4.92
0.515
0.836
0.922
1.446
4.91
0.104
0.198
0.187
0.342
4.92 < r i 6.55
0.138
0.272
0.247
0.471
0.0682
0.0394
0.122
0.0682
I > 6.55
0.0682
0.0394
0.122
0.0682
7.05
r > 7.05
29.9 1.705
1.692
3.050
4.35 < r 5 4.92
rgsrs4.35
0.391
0.676
0.699
1.169
4.92 < r 5 6.95
0.120
0.217
0.214
0.376
r > 6.95
0.0682
0.0394
0.122
0.0682
4.256
23
3.981
2.928
0), the four energy parameters eu-x, dH-x, Q_~, de-x, are considered as adjustable variables. Since the truncated expressions of the u’s are given by the difference between two monomials in do and e,, it may be shown that only two variables are independent. Therefore the fit to the exact U(Y) functions is rather poor even if four regions have been considered. It should be noted that this approach involves another approximation because the integrations over z = vkt/rc (leading to the resonance functions in the RB formalism) imply that the atom-atom parameters are not r-dependent. To obtain the atom-atom parameters given in Tables IV and V, we have arbitrarily set
At long distance, these fitted parameters of V, is then convergent.
6. CALCULATED
are taken as do and e,, since the radial expansion
RESULTS FROM THE RB MODEL
The RB formalism has been applied by considering, in addition to the electrostatic (quadrupole-quadrupole ) contributions, the atom-atom contributions derived from either truncated or exact radial functions UOoo(r), lJ2~ ( r), and U2zo( r). All the terms ‘1bS2[r,(b)] with even I, and I2 of the atom-atom and cross contributions given in Ref. (6) have been introduced in the computation. When truncated radial functions are considered, we have also taken into account all the terms arising from the r-I0 and r-l6 contributions in the potential according to Ref. (26). In both cases, we will show the influence of the U,, contribution on the broadening coefficients. Calculations were generally made for the mean relative velocity 3 = ( 8kBT/mn) ‘I*. However, for a few lines of CZ H2 perturbed by NZ , we have performed the average over the MaxwellBoltzmann distribution of velocities [see Eq. ( 1)] by using the method of integration
BROADENING
COEFFIClENTS
IN C1H2
207
described previously (IO), combined with that developed by Looney and Herman (27). Slightly larger results of y. ( 1 to 4%) were obtained. So the very costly consideration of the velocity distribution does not affect the calculated linewidths appreciably and was not generalized. 6.1. C2H2-N2 The collisional linewidths y. at 297 K have been computed with the spectroscopic constants given in Table II, the atom-atom parameters given in Tables III and IV. QGH~= 4 D-A, and QN, = -1.4 D *A. The broadening of the lines resulting only from quadrupole-quadrupole interactions depends strongly on the isotropic potential used in the trajectory model (Fig. 7 ): the exact VoW(r) function described by a 62 1.5 potential leads to a rather small electrostatic contribution. Using the truncated expansion of the atom-atom potential, we obtain calculated linewidths that are smaller than the experimental results at low ) m 1 values and significantly larger at great )m 1 values. For ) m 1 > 22, the difference is outside the range of the estimated experimental uncertainty. It appears also (Fig. 7) that the contribution arising from 4,0S2, for which analytic expressions are presented in the Appendix, is rather weak for all the lines. The consideration of the exact radial functions through the fitted atom-atom parameters yields somewhat larger results at intermediate or high ) m I values than the first
IL”
lml FIG. 7. N2-broadening coefficients yo for IR lines of C2Hz. Experimental values ( l, 0, X). results from the RB model: V = I/ala2 with U’&(r): curve (a); V = V,,, with Ue&,( r): V = Vo,a + V, with U&(r). U&&r). U&(r): curve (b); same interactions plus t&,(r): V= C’Q,Q~+ r\ with (i&(r). U%(r). and U&(r): curve(c); same interactions plus C&(r):
Calculated curve (a’); curve (b’): curve (c’).
208
BOUANICH ET AL.
0
5
10
15
20
25
30
35
IN
FIG. 8. 02-broadening coefficients y. for IR lines of C2H2. Experimental values (1): (0). Calculated results from the RB model: V = Vplo2 with U&(r): curve (a); V = Vala2 with U&(r): curve (a’); V = VQ,~ + V. with U&(r), C.&,(r), ugo(r): curve (b); same interactions plus U&,(r): curve (b’); V = ValQ + V, with Q&(r), UT&.,(r)and @&(r): curve (c); same interactions plus U&(r): curve (c’).
calculations. The theoretical results remain satisfactory for the lines in the range ( m 1 = 5 to 19 but the agreement with our experimental data is poorer at large (m ( values.
The linewidths of &Hz broadened by O2 are a good test for theoretical models because they are very sensitive to short-range interactions owing to the very small electrostatic contribution (Fig. 8). The calculation performed with the atom-atom parameters given in Table III involving the truncated UO,( r), Uzoo(r), U,,,(r), and U,,( r) functions provides satisfactory O2 -broadening coefficients. The consideration of the exact U(r) functions through the fitted parameters e$12” and df,l”“’ given in Table V leads to significantly larger results for 1m 1 > 10. We can explain this result as follows: the contributions to the differential cross section derived from the truncated radial functions Uzoo(r), U2,,( r), and Um( Y) are notably smaller for close collisions (dominating the widths at large (m I values) than those derived from these exact functions. The difference between these contributions is not entirely compensated by the respective influences, through the trajectory model, of the truncated and exact isotropic potentials Uoo~(r). From Fig. 8, it appears also that the 4,0S~contribution is important, especially for ) m I < 15, and cannot be neglected for CzH2-02.
BROADENING
6.3.
COEFFICIENTS
209
IN Cz Hz
Temperature Dependence of Linewidths
To our knowledge, no work has been yet devoted to the determination of the temperature dependence of y. for C2H2-Nz or CZH2-Oz. This temperature dependence can be accurately represented by using the simple relation TO(T) = YO(To)( Tl To)-“, where TO = 297 K (our reference temperature) and T has been arbitrarily taken as 200 K. By comparing for CZH2-NZ the calculated values of y. at 297 K and 200 K arising from the RB model including either the truncated or the exact radial expansion of the atom-atom potential (with or without the U4oocontribution ), we have derived (Figs. 9 and 10) for the n exponent values which are strongly 1m )-dependent. The decrease of n as ( m ( increases has been obtained theoretically and experimentally for other radiative molecules such as CO1 on collision with N2 or O2 (28). It may be seen (Figs. 9 and 10) that the influence of 4*oS2is rather small, but the exact atom-atom model provides values of n which are significantly smaller, for I m ( > 8, than those derived from the truncated model. Only accurate experimental values of broadening coefficients at low temperature would allow us to reach a conclusion about these theoretical results.
03L.““““’ 0
“’ 5
10
“““‘c”.‘n’.” 20
15
25
30
5
14 FIG. 9. variationof the temperatureexponentn with )m 1 for C2H2-N2. Calculated results from the RB model: V = VQla + V, with U&,(r), U&&r), U&,(r): curve (a): some interactions plus U&,(r):CUIYS (a’): I,’ = I’Q,~, + V, with U&(r), Uym( r), and @&(r): curve(b); same interactions plus U&(r): curve (b’).
210
BOUANICH ET AL. 0.9,
lml FIG. 10. Variation of the temperature exponent n with I m / for C,H,-0, Calculated results from the RB model: V = Vola2 + V, with U&+(r), uF&r), V&,(r): curve (a); same interactions plus u&,(r): curve (a’); I’= Vala2 + V. with U&(r), @&(r) and u;“,,(r): curve (b); same interactions plus Cl&(r): curve (b’).
6.4. Concluding Remurks By using the RB model, we have performed different calculations for the Nz- and 02-broadening coefficients of C2Hz. Contrary to our expectations, the truncated radial expansion of the atom-atom potential provides generally more satisfactory results than the exact expansion. We do not believe that this result is due to the way we have taken into account the exact radial spherical harmonics development of I’, by still using the RB formalism, although this approach may be criticized. We estimate that the discrepancy with experimental data may be caused by the atom-atom potential itself, which accounts approximately for short-range interactions but has some deficiencies related to the assumed pair-wise additivity (see Ref. (20, p. 34)). It should be noted that no parameters of the interaction potential (except the quadrupole moment of CZH2) were fitted to obtain the theoretical results; in particular the parameters dv and eii used for C2H2 arise from CO* (C) and CH4 (H). As for Qc2uZ, the value adopted, 4 D +A, is much lower than the other recent results (see Ref. (20, p. 581)) but represents a high limit for the RB calculations of self-broadening coefficients. Another explanation of this discrepancy may originate in the semiclassical approximation where the relative translational motion is assumed to be independent of the rational motion, which should limit the accuracy of the results, especially for the high J states. Moreover, as shown by Margottin-Maclou et al. (29)) a small modification in the trajectory model (which is derived from UO,( r) only) may induce large changes in the calculated values of yo. These authors have obtained a much better agreement
BROADENING
COEFFICIENTS
211
IN C>Hz
between experimental and calculated broadening coefficients of CO*-Nz and C02-O2 by slightly modifying the apparent velocity V: (F, in Eq. (4) was arbitrarily lowered by 15%). Finally, from our results, we estimate that the 4, 0, 0 contribution of the atomatom potential may be important for perturbers having a small quadrupole moment such as Oz. APPENDIX: THE 4, 0, 0 ATOM-ATOM CONTRIBUTION TO THE DIFFERENTIAL CROSS-SECTION
In the coordinate axes represented in Fig. Al of Ref. (27) (see also Fig, 2 of Ref. (3)) for the collision pair molecule 1 (radiator)-molecule 2 (perturber). the II = 4, fz = 0 truncated contribution to the atom-atom potential may be expressed as (25) 4’oVa = 4lr C 2 rt ij
m
~-~)YZ(n-il,1)~4m(I)l;o(2). (
with m varying from -4 to 4. Using the notations of Bonamy and Robert (6), we obtain the following expressions for the 4.0Sz[v,( b)] function:
+
c
cy “,“f$ k)
+
04)
-
z
(C
ii
di
elj)(
C
hi
x (C c)(4)“.“f$k Ji
) +
c
c:;“’
4.0f$k)
Gidij>
+
i,j
+
04)
+
2;(y;;;;3
Ji
with
“,““f ;:( k) = 13 ;9ik825 ( 14k12 + 252k” + 2466k” + 16 980k9 + 90
405k8+ 388 980k7+ 1 380 690k6 + 4 062 420k5 + 9 810 450k4 + 18918900k3+27584550k’+27
187650k+
13593825)
e-2k
4,0f;;(k)
=
47 760 763 050 + 1 156 437k”
( 14k15 + 504k14 + 9432k13 + 120 234k” + 8 826 894k”
+ 54 977 031k9 + 283 885 560k8
+1225445760k7+4430855520k6+13349311920k5 +33007768920k4+64972076400k3+96167055600k2 +95521526lOOk+47760763050)
BOUANICH
212
ET AL.
e-2k
4q-;&k) =
(70k’* + 3780k17 + 104 760k16 + 1968
857266364354400
120k15
+ 27 953 424k14 + 317 538 144k13 -t 2 981 059 200k12 + 23597619
192k”+
159422475996k”+925
186646160k9
+4621044715560k8+19817845462800k7 +72462716144280k6+222971619356640k5 + 564969
197816
100k4+
1
137433809862350k3
+1711738724070375k2+1714532728708800k + 857 266 364 354 400). In these expressions,
C,!?’ are squared
Clebsch-Gordan
coefficients
according
to
c!4’ I’ = [ C(j4j’. >000) (2 and 04 =
(-1)““/2[(2j,
where Wis a Racah coefficient. ofj and m in Ref. (30).
+ 1)(2jf+
1)c:;4’c::4)]“*u/(jii/jri~;
The coefficients
14),
Ci?’ and D4 have been given in terms
ACKNOWLEDGMENTS
Support by the “Accords Culturels” between France and Belgium is gratefully acknowledged. We also thank Professor C. Boulet for many helpful discussions. REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9. IO. II. 12. 13. 14. 15. 16.
D. LAMBOT, G. BLANQUET, AND J. P. BOUANICH, J. Mol. Spectrosc. 136, 86-92 ( 1989). P. W. ANDERSON, Phys. Rev. 76,647-661 ( 1949). C. J. TSAO AND B. CURNUTTE, J. Quant. Spectrosc. Radiat. Transfer 2, 41-91 ( 1962). D. ROBERT AND J. BONAMY, J. Phys. 40,923-943 ( 1979). L. ROSENMANN, J. M. HARTMANN, M. Y. PERRIN, AND J. TAINE, J. Chem. Phys. 88, 2999-3006 (1988). J. B~NAMY AND D. ROBERT, unpublished. V. MALATHY DEVI, D. C. BENNER,C. P. RINSLAND,M. A. SMITH,AND B. D. SIDNEY,J. Mol. Spectrosc. 114,49-53 (1985). J. R. PODOLSKE,M. LOEWENSTEIN,AND P. VARANASI, J. Mol. Spectrosc. 107,241-249 ( 1984). J. BONAMY, L. BONAMY, AND D. ROBERT, J. Chem. Phys. 67,4441-4453 ( 1977). I. P. BOUANICH,C. CAMPERS,G. BLANQUET,AND J. WALRAND, J. Quant. Spectrosc. Radiat. Transfer 39,353-365 (1988). J. S. WONG, J. Mol. Spectrosc. 82,449-45 1 (1980). P. VARANASI AND B. R. P. BANGARU, J. Quant. Spectrosc. Radiat. Transfer. 15, 267-273 ( 1975). I. R. DAGG, A. ANDERSON,W. SMITH, M. MISSIO,C. G. JOSLIN,AND L. A. A. READ, Canad. J. Phys. 66,453-459 (1988). W. H. FLYGARE AND R. C. BENSON,Mol. Phys. 20,225-250 ( 1971). D. E. ST~CRYN ANDA. P. STOGRYN, Mol. Phys. 11, 371-393 (1966). J. HIETANEN AND J. KAUPPINEN, Mol. Phys. 42,41 I-423 ( 1981).
BROADENING
COEFFICIENTS
IN C2H2
213
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