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Argon-induced halfwidths and line shifts of water vapor transitions
Journal of Quantitative Spectroscopy & Radiative Transfer 64 (2000) 439}456
Argon-induced halfwidths and line shifts of water vapor transitions R.R. Gamache!,*, R. Lynch" !Department of Environmental, Earth, and Atmospheric Sciences, The University of Massachusetts Lowell, 1 University Avenue, Lowell, MA 01854, USA "Columbia University, Goddard Institute for Space Studies, 2880 Broadway, New York, NY 10025, USA
Abstract Pressure-broadened halfwidths and pressure-induced line shifts are determined for water vapor transitions with argon as the perturbing gas. Calculations based on the complex Robert}Bonamy (CRB) formalism are made for a number of vibrational bands, for which there are experimental data to compare with. The intermolecular potential is taken as a sum of Lennard}Jones (6}12) atom}atom, isotropic induction, and dispersion components. The dynamics of the collision process are correct to second order in time. The calculations demonstrate that the atom}atom potential for this system should be expanded to 12th order. A new feature in the CRB approach is that the real and imaginary components of the Liouville S matrix a!ect both the halfwidth and the line shift. It is shown here that the imaginary parts of the S matrix strongly a!ect the calculated H O}Ar halfwidths. The calculated values are compared with those obtained in a number of 2 experimental studies. In general, good agreement is observed between the CRB calculations and the measured values for both halfwidths and line shifts. It is also clear that some parameters describing the intermolecular potential need to be better determined. ( 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction Water is the principle absorber of longwave radiation in the terrestrial atmosphere, responsible for some 80% of `greenhousea warming of the Earth's surface [1]. The study and comprehension of the radiative-forcing e!ects and the interpretation of remote-sensing data of water vapor requires the use of atmospheric radiative-transfer models, which require high-precision parameters describing line positions, intensities, pressure-broadened halfwidths and line shifts. Although laboratory measurements can and do supply such parameters, it is di$cult to exhaustively cover the vast * Corresponding author. E-mail address: robert}[email protected] (R.R. Gamache) 0022-4073/99/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 9 9 ) 0 0 1 0 1 - 6
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spectral range and diversity of environmental conditions encountered in the actual atmosphere. Theoretical calculations are in principle an attractive alternative, depending on the accuracy requirements of the radiative transfer application one has in mind, and of course the credibility of the theory. Even when laboratory measurements are available, however, certain e!ects (such as line mixing [2]) may still require a sophisticated theoretical model in order to unravel observed spectra. Of the needed parameters the collision-broadened halfwidth is the least well-known parameter for atmospheric applications [3]. Our present focus is to evaluate the complex Robert}Bonamy (CRB) formalism [4] for the calculations of pressure broadening and pressure-induced line shifting parameters, c and d respectively, of water vapor transitions with argon as the bu!er gas. This is one of a series of papers considering the application of the CRB formalism to calculating c and d for water vapor transitions with a number of perturbing gases [5}11]. As described below and elsewhere [12}15], CRB theory uses a spherical tensor expansion in 1/R (R is the intermolecular separation) of a short-range, atom}atom potential. We have discussed previously [8,11] how this potential is combined with two others (electrostatic and induction}dispersion) in a natural and non-redundant fashion for the general XY }A system. Since the expansion in (1/R) must be 2 2 truncated, we also devote some discussion to questions of the convergence of calculated halfwidths and line shifts. A feature that is new in the CRB formalism is that the expressions for the halfwidth and line shift contain both real and imaginary components of the Liouville scattering matrix. This is an e!ect that is not present in Anderson}Tsao}Curnutte (ATC) theory [16}19] nor in previous applications of the theory of Robert and Bonamy to asymmetric rotor molecules [12}15,20}25], which used only the real components of the theory (labeled real Robert}Bonamy (RRB) in this work). We show below that for the H O}Ar system the e!ects of the imaginary terms on the calculated halfwidths 2 are large and should not be overlooked. To this purpose, we provide a comparison with halfwidths and line shifts for which recent, high-quality experimental data are available, namely those occurring in the 3l #l and 1 3 2l #2l #l bands reported by Grossmann and Browell [26], l data of Giesen et al. [27] as 1 2 3 2 well as several halfwidth measurements in the rotational band [21,23,28,29].
2. Complex Robert}Bonamy formalism 2.1. Introduction Within the complex Robert}Bonamy formalism, the line shift and halfwidth for the transition iQf are given by real and minus the imaginary parts of the diagonal elements of the relaxation matrix [4],
P
n l6 = " "!i 2 + SJ Do DJ T 2pb db[1!exp(!Mi(S #Im(S ))#Re(S )N)], (1) fi,fi 2 2 2 1 2 2 2pc 0 J2 where v6 is the mean relative thermal velocity, o and n are the density operator and number 2 2 density of perturbers and b is the impact parameter. S and S are the "rst- and second-order terms 1 2
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in the successive approximation expansion of the Liouville scattering matrix [30,31], described below. The complex Robert}Bonamy formalism was chosen for several reasons. (1) A cumulant expansion [32,33] incorporated in CRB formalism eliminates an awkward cuto! procedure [34] that characterized earlier theories. (2) The complex formalism yields halfwidths and line shifts from a single calculation. (3) A short-range atom}atom potential [12}15] (Lennard}Jones, 6}12 [35]) can be incorporated into the theory in a natural and non-redundant fashion. (4) The intermolecular dynamics are more realistic than in earlier theories, i.e. using curved rather than straight-line trajectories. In our implementation, these trajectories are determined by the isotropic part of the atom}atom potential. (5) Vibrational dependence of the isotropic part of the intermolecular potential can be included in a straightforward way. 2.2. Intermolecular potential For an atom interacting with a molecule there are a number of potentials that have been used [36]. In this work the potential is comprised of two types of terms. An atom}atom potential has been adopted, following the suggestion of Bauer et al. [21,23] and others [37], to model the anisotropic interactions and induction and dispersion potentials are used to model the isotropic interaction. The atom}atom potential is de"ned as the sum of pair-wise Lennard}Jones 6}12 interactions [35] between atoms of molecules 1 and the argon atom,
G
H
n p6 p12 i2 ! i2 . (u), (4) 12 1 2 m n 1 m 0 2 lm Rq`l1 `l2 `2w 1 n l1 l2 1 2 w,qq l mm m where C(l l l; m m m) is a Clebsch}Gordan coe$cient, ) "(a , b , c ) and ) "(a , b , c ) 12 1 2 1 1 1 1 2 2 2 2 are the Euler angles describing the molecular "xed axis relative to the space "xed axis. Subscripts 1
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and 2 refer to the radiating molecule (H O) and perturbing atom (Ar), respectively, and u"(h, /) 2 describes the relative orientation of the centers of mass. As described earlier, R is the center of mass separation. The powers w and q (integers) depend upon the interaction assumed, and the coe$cients ;(2) are given in Ref. [5] or Ref. [40]. Before taking up discussion of the other interaction (induction}dispersion), three further points should be emphasized with regard to the atom}atom potential. (1) When expressed in the form of Eq. (4), the atom}atom potential can be understood as two simultaneous expansions. One is de"ned by the tensorial ranks l and l , which determine the 1 2 symmetry of the interaction [40,41]. A second expansion is de"ned by the sum l #l #2w 1 2 [39,42], which we call the order of the expansion. It hardly needs to be emphasized that any calculation of line width or shift should be converged with respect to both order and rank. The issue of convergence is discussed below. (2) The atom}atom potential contains an isotropic part (l "l "0) which in general is 1 2 comprised of terms of the form 1/R6, 1/R8, 1/R10, etc. In our implementation, these are used to de"ne the intermolecular trajectory. For the purpose of comparison, however, it is convenient to "t this summation to an e!ective isotropic (or `hetero-moleculara) Lennard}Jones 6}12 potential,
CA B A B D
<*40"4e
p 12 p 6 . ! R R
(5)
The approximation represented by Eq. (5) had no signi"cant impact on our calculations other than to simplify the trajectory calculations somewhat. (3) The remaining parts of the atom}atom potential are anisotropic (nonzero l or l ). These 1 2 terms may be discussed with the same lexicon that one applies to electrostatic interactions. For example, when l "1 and l "2, we refer to a `dipole}quadrupolea interaction. It should be kept 1 2 in mind that these are symmetry appellations only, and do not refer to electrostatics [11]. Thus it happens that between H O and Ar, the lowest anisotropic atom}atom term is `dipole}monopolea, 2 i.e. with l "1 and l "0. 1 2 In another category of interactions relevant to intermolecular collisions are the isotropic induction and dispersion potentials, k2 a <*/$"! 1 2 , *40 R6 (6) a a 3 E E 1 2 1 2, V$*4"! *40 2 E #E R6 1 2 where k is the dipole moment of water vapor and a and E are the polarizability and ionization 1 potentials for water vapor and its perturber (argon), labeled 1 and 2, respectively. By design, these interactions occur with the same power of 1/R, and in some sense represent the same forces, as the attractive part of Eq. (5). Thus, they play no role in our trajectory calculations, although they do serve as a convenient check. We "nd that the magnitude of the combined induction}dispersion interactions is typically about half the value of the corresponding atom}atom term. This is consistent with observations of Gray and Gubbins [40], who point out that Eqs. (6) are themselves merely the "rst terms in an expansion, and by Lynch et al. [5]. Because the induction and
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dispersion terms in Eqs. (6) are isotropic, they only contribute through the imaginary Liouville scattering matrix, S . These terms provide a convenient, experimentally constrained method of 1 computing the diwerence in isotropic potential for di!erent vibrational states, via the vibrational dependence of k and a (this di!erence appears in the S term of CRB theory and is discussed in 1 1 1 Refs. [5,6,11]). 2.3. Details of CRBF theory The expressions for the S and S terms in the CRB formalism are described in detail in 1 2 Refs. [6,8,11]. Here only the salient features are summarized. The "rst-order (imaginary) term, S , 1 depends only on the di!erence in the isotropic part of the interaction potential between the initial and "nal vibrational states of the radiator and is accounted for by the vibrational dependence of the dipole moment and polarizability of H O. The second-order terms are comprised of two basic 2 parts; one describing the internal states of the radiating and perturbing molecules and another describing the interaction and dynamics of the collision. In order to calculate these terms, information is needed for a number of molecular constants describing the colliding pair. All molecular parameters used in this work are the best available values from the literature. No molecular constants are adjusted to give better agreement with experiment. The vibrational dependence of the dipole moment of water vapor was investigated by Shostak and Muenter [43] and is given in Debyes by k"1.855#0.0051(v #1)!0.0317(v #1)#0.0225(v #1), 1 2 2 2 3 2
(7)
where v is the number of quanta in the ith normal mode. The vibrational dependence of the i polarizability of water vapor was obtained by Luo et al. [44] and is, in atomic units,1 a"9.86#0.29(v #1)!0.03(v #1)#0.28(v #1). 1 2 2 2 3 2
(8)
The heteronuclear atom}atom parameters used in the present work, derived from homonuclear} atom}atom parameters obtained by Bouanich [45] using the combination rules [38], (see Eqs. (3)), are reported in Table 1. Reduced matrix elements for the internal states of the radiator and perturber must be calculated. For water vapor, these are evaluated using wavefunctions determined by diagonalizing the Watson Hamiltonian [46] in a symmetric top basis for the vibrational states involved in the transition. For the ground, l , l , and l states of H O, the Watson constants derived by Flaud and Camy}Peyret 1 2 3 2 [47] are used. For the (301) and (221) vibrational states of H O, the constants of Grossmann et al. 2 [26] are used. In Refs. [5,6], techniques were presented for determining the real and imaginary parts of the resonance functions for an arbitrary potential that is in the form of an expansion in the center of mass separation in the parabolic trajectory approximation.
1 The choice of units for Eqs. (7) and (8) is made to keep the expressions as concise as possible. For conversion to other units see for example Ref. [40, p. 575].
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R.R. Gamache, R. Lynch / Journal of Quantitative Spectroscopy & Radiative Transfer 64 (2000) 439}456 Table 1 Atom}Atom parameters for the H O}Ar 2 system
p (As ) e/k (K) B
Ar}H
Ar}O
3.1 37.3
3.2 78.7
In the parabolic approximation the isotropic part of the interaction potential is taken into account in determining the distance, e!ective velocity, and force at closest approach [4]. To simplify the trajectory calculations the isotropic part of the atom}atom expansion is "t to an isotropic Lennard}Jones 6}12 potential giving e/k "107.98 K and p"3.496 As for the 12th-order B expansion of the atom}atom potential for the H O}Ar collision pair. 2 Finally, the ionization potential of water was taken to be 12.61 eV [48] independent of vibrational state. The polarizability of argon is taken from Ref. [49] and is 16.41]10~25 cm3. All calculations are made at the HITRAN [50] reference temperature, 296 K. The CRB formalism yields the halfwidth and line shift from a single calculation and there are no adjustable parameters or cuto! procedure in the formalism.
3. Calculations Using the CRB method, we have calculated both halfwidths and line shifts for transitions in the pure rotation, l , l , l , 3l #l , and 2l #2l #l bands of water vapor perturbed by argon. 1 2 3 1 3 1 2 3 In the calculations presented here, we examine the e!ect of the order of expansion, the e!ect of the real and imaginary terms on the halfwidth and line shift, and compare with experiment when possible. It was demonstrated by Labani et al. [13}15] that for H O}A systems (A "N , O , etc.) the 2 2 2 2 2 atom}atom potential in Eq. (3) needs to be expanded to at least fourth order in the center of mass separation. (Note, Labani et al. used a di!erent form of the expansion of the atom}atom potential [38] for which the order is de"ned by l #l .) The development of Labani et al. was 1 2 limited to a fourth-order expansion. In this work, we use the analytic form of the two-center expansion given by Sack [39], to generate higher orders of the expansion. The only limitation of our method is that of computational resources, which currently is a 12th-order expansion for the H O}Ar system. 2 To study the e!ects of the order of the expansion of the atom}atom potential, we made calculations where the order of the expansion, l #l #2w, is varied from 2 to 12 and the results 1 2 compared. In Figs. 1 and 2 the results of the CRB calculation are plotted for the halfwidths and line shifts, respectively, versus the order of the atom}atom potential for four transitions in the 3l #l 1 3 band: 4 Q5 ; 3 Q4 ; 5 Q5 ; and 2 Q3 . Also plotted in the "gures are the experi22 23 30 31 05 24 12 13 mental results of Grossmann and Browell [26]. It is clear that the potential has converged by 12th-order. All calculations from this point on use the 12th-order atom}atom potential.
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Fig. 1. Halfwidth versus order of the atom}atom potential and experimental results of Grossmann and Browell [26] in the 3l #l band. 1 3
Fig. 2. Line shift versus order of the atom}atom potential and experimental results of Grossmann and Browell [26] in the 3l #l band. 1 3
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R.R. Gamache, R. Lynch / Journal of Quantitative Spectroscopy & Radiative Transfer 64 (2000) 439}456 Table 2 Average percent di!erence between CRB and RRB calculations for several vibrational bands Vibrational band
Number of transitions
Average percent di!erence
Rotation l 1 l 2 l 3 3l #l 1 3 2l #2l #l 1 2 3
921 353 349 335 82 36
!0.30 2.12 0.17 2.72 15.52 15.66
As stated above, a new feature of these calculations is the real and imaginary components of the theory contributing to both the halfwidth and line shift. To study the importance of the imaginary components to the halfwidth, CRB and RRB calculations were made for a number of vibrational bands of water vapor; 921 rotation, 353l , 349l , 335l , 82 3l #l , and 33 2l #l #l 2 1 3 1 3 1 2 3 transitions. Table 2 presents the average percent di!erence, 100(CRB-RRB)/CRB, for each band studied. For bands with zero (rotation band) or 1 quanta of energy (l , l , and l ) the contribution 1 2 3 of the S and Im(S ) terms can be comparable. This is especially true of the l band, for which the 1 2 2 coe$cient in Eq. (8) is small. As the number of quanta increases, the contribution of the S term is 1 greater than that of the imaginary part of the S term of the interaction [6] for this system. Note, 2 the results of Table 2 correlate with the coe$cients of Eq. (8) indicating that the vibrational dependence of the polarizability is the principal contributor to the S term. This fact agrees with 1 the studies of Lynch [6]. As the number of vibrational quanta increases, especially in the l and l levels, the e!ects of the 1 3 imaginary components become more important. Note that even with a single quanta of energy in either the l and l level, the di!erence between the CRB and RRB calculations is of the same 1 3 magnitude as the desired uncertainty of the halfwidth [51}53]. Of course, a more important question is, does the more complete theory give better agreement with experiment. For the 3l #l band transitions studied, the CRB calculations agree with the measurements of 1 3 Grossmann and Browell [26] better than the RRB calculations, !5.8 and 11.1 average percent di!erence, respectively. This fact is in agreement with other studies [7,11]. In Fig. 3 the percent di!erence between the CRB and RRB calculations is plotted versus an energy-ordered index, JA(JA#1)#KaA!KcA#1. The importance of properly including the imaginary terms is apparent from the "gure. The percent di!erences range from about 8 to 37%. Note as the number of vibrational quanta increases the contribution of the imaginary terms increases. Thus for transitions in the 3l #l band, it is expected that the di!erence between the two calculations will be large. In 1 3 all the following calculations the CRB formalism is employed. CRB calculations are made at 296 K for the transitions studied by Grossmann and Browell [26] for the 3l #l and 2l #2l #l bands. A comparison of the measured and calculated 1 3 1 2 3 halfwidths and line shifts are shown in Figs. 4 and 5 for the 3l #l band. Good agreement is 1 3 realized for most of the transitions. Note that the largest di!erences occur for a few transitions with the measured values being lower than the calculated ones. Grossmann and Browell commented that they had observed narrowing in their measurements and needed to employ the hard-collision
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Fig. 3. Percent di!erence between CRB and RRB calculations for transitions in the 3l #l band. 1 3
Fig. 4. Halfwidths for transitions in the 3l #l band versus energy-ordered index: CRB calculations and the 1 3 measurements (with error bars) of Grossmann and Browell [26].
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Fig. 5. Line shifts for transitions in the 3l #l band versus energy-ordered index: CRB calculations and the 1 3 measurements (with error bars) of Grossmann and Browell [26].
model of Rautian and Sobel'man [54] to model the line shape. The CRB calculations make no account for narrowing. The average percent di!erence (APD) between the measured and calculated halfwidth is !5.8 for the 80 lines compared. Fig. 5 shows the measured and calculated line shifts versus the energy ordered index for the transitions in the same band. Note that all of the observed line shifts are negative as expected for a band with a large number of vibrational quanta [11]. While the agreement is reasonable, 7.2 average percent di!erence based on 80 transitions compared, the calculated shifts appear to be systematically above the measured ones. This can be related to the coe$cients in Eqs. (7) and (8) (particularly those in Eq. (8)). The coe$cients in Eq. (8) used in this work are from an ab initio calculation and their uncertainty is not well understood. Scaling of these values was not considered a worthwhile exercise at this time. Figs. 6 and 7 are the corresponding halfwidth and line shift plots for the 2l #2l #l 1 2 3 band. Similar features are seen; the calculated halfwidths agree fairly well with measurement, APD"!10.8 for the 32 transitions compared, and there are still problems with some narrow lines. The calculated line shifts, with one exception, are all greater than the measured values and all values are negative. The APD for the line shifts is 20 for the 33 transitions compared. Table 3 provides the halfwidth and line shifts calculated via the CRB formalism with the atom}atom potential expanded to 12th-order for all the transitions considered in this work. Also present in the table are the corresponding experimental values when available. Next, comparisons are made with measurements of the halfwidth and line shift for transitions in the l band of H O. The measurements are those of Giesen et al. [27] for seven transitions 2 2 at 298 K. Table 4 gives the CRB formalism calculations at 296 K as well as the values from
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Fig. 6. Halfwidths for transitions in the 2l #2l #l band versus energy-ordered index: CRB calculations and the 1 2 3 measurements (with error bars) of Grossmann and Browell [26].
Fig. 7. Line shifts for transitions in the 2l #2l #l band versus energy-ordered index: CRB calculations and the 1 2 3 measurements (with error bars) of Grossmann and Browell [26].
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Table 3 Halfwidth and line shift in units of cm~1/atm for transitions in the 3l #l and 2l #2l #l bands of H O. CRB 1 3 1 1 1 2 calculations compared with the data of Grossmann and Browell [26] Vibrational transition
J@ Ka{Kc{
JA A A Ka Kc
c (CRB)!
d (CRB)!
c [26]!
301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000
8 1 6 3 8 0 6 2 6 3 6 1 6 2 5 3 5 2 6 0 6 1 5 1 5 2 7 7 4 2 5 1 4 1 4 2 7 6 3 3 3 3 6 6 3 2 4 0 3 1 5 0 3 2 3 0 3 1 4 0 2 2 2 1 2 0 2 1 1 1 5 4 5 4 1 0 5 3 1 1 3 1 4 3 6 3 4 3
9 1 7 3 9 0 7 2 7 3 7 1 7 2 6 3 6 2 7 0 7 1 6 1 6 2 7 7 5 2 6 1 5 1 5 2 7 6 4 3 4 3 6 6 4 2 5 0 4 1 5 2 4 2 4 0 4 1 4 2 3 2 3 1 3 0 3 1 2 2 5 4 5 4 2 0 5 3 2 1 3 1 4 3 6 3 4 3
0.04376 0.05117 0.04302 0.05161 0.04793 0.04873 0.04736 0.04911 0.05199 0.04766 0.04758 0.05075 0.04919 0.03481 0.05205 0.04973 0.05215 0.05094 0.04022 0.05239 0.05217 0.03888 0.05311 0.05168 0.05278 0.05177 0.05297 0.05336 0.05337 0.05352 0.05462 0.05477 0.05518 0.05487 0.05581 0.04783 0.04783 0.05778 0.05031 0.05611 0.05509 0.05107 0.04990 0.05104
!0.03922 !0.03305 !0.04126 !0.03283 !0.03569 !0.03472 !0.03627 !0.03482 !0.03218 !0.03699 !0.03713 !0.03373 !0.03481 !0.04701 !0.03162 !0.03522 !0.03311 !0.03334 !0.04239 !0.03404 !0.03416 !0.04332 !0.03148 !0.03324 !0.03163 !0.03398 !0.03204 !0.03254 !0.03196 !0.03264 !0.03150 !0.02998 !0.03163 !0.03095 !0.02850 !0.03624 !0.03617 !0.03068 !0.03410 !0.02924 !0.03163 !0.03351 !0.03423 !0.03353
0.03540 0.04840 0.03120 0.05110 0.00000 0.04670 0.04760 0.04550 0.05150 0.04310 0.04280 0.05200 0.04970 0.02630 0.05150 0.04850 0.05390 0.05360 0.02520 0.04370 0.04480 0.02400 0.04980 0.05560 0.05410 0.04420 0.05290 0.05770 0.05740 0.00000 0.05090 0.05000 0.05870 0.06000 0.05840 0.03720 0.04130 0.06200 0.04460 0.06190 0.05740 0.04630 0.05170 0.04480
7 3 8 4 4 5 5 3 3 6 6 4 4 1 2 5 3 3 2 0 1 0 1 4 2 5 2 3 3 4 0 1 2 2 0 2 1 1 3 1 3 2 3 1
8 4 9 5 5 6 6 4 4 7 7 5 5 0 3 6 4 4 1 1 2 1 2 5 3 4 3 4 4 3 1 2 3 3 1 1 2 2 2 2 2 1 4 2
d [26]!
(1.4) (1.9) (1.2) (2.0) (0.0) (1.9) (1.9) (1.8) (2.1) (1.7) (1.7) (2.1) (2.0) (1.1) (2.1) (1.9) (2.2) (2.1) (1.0) (1.9) (1.8) (1.0) (2.0) (2.2) (2.2) (1.8) (2.1) (2.3) (2.3) (0.0) (2.0) (2.0) (2.3) (2.4) (2.3) (1.5) (1.7) (2.5) (1.8) (2.5) (2.3) (1.9) (2.1) (1.8)
!0.03980 !0.03390 !0.04210 !0.03360 !0.03640 !0.03600 0.00000 !0.03590 !0.03540 !0.04110 !0.04050 !0.03530 !0.03750 !0.03840 !0.03690 !0.04070 !0.03590 !0.03690 0.00000 !0.03260 !0.03290 !0.03840 !0.03730 !0.03850 !0.03820 !0.03650 !0.03520 !0.03660 !0.03490 !0.03320 !0.03530 !0.03850 !0.03570 !0.03340 !0.03490 !0.03720 !0.03760 !0.03230 !0.03610 !0.03420 !0.03050 !0.03590 !0.03570 !0.03630
R.R. Gamache, R. Lynch / Journal of Quantitative Spectroscopy & Radiative Transfer 64 (2000) 439}456 Table 3. (continued) Vibrational transition
J@ Ka{Kc{
JA A A Ka Kc
c (CRB)!
d (CRB)!
c [26]!
d [26]!
301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 221 000 221 000 221 000 221 000 221 000 221 000 221 000 221 000
3 3 3 3 3 2 0 0 2 1 2 2 2 2 3 2 1 1 4 2 1 1 2 1 6 2 1 0 4 1 2 1 2 0 2 1 3 1 3 0 3 2 6 1 4 3 4 1 4 2 4 0 5 1 5 3 4 2 5 0 6 1 6 0 6 3 6 2 7 1 8 1 9 2 8 2 3 1 3 2 2 1 3 3 2 0 1 1 3 2 2 2
3 3 3 3 3 2 1 0 2 1 2 2 2 2 3 2 1 1 4 2 1 1 2 1 6 2 0 0 4 1 1 1 1 0 1 1 2 1 2 0 2 2 6 1 3 3 3 1 3 2 3 0 4 1 4 3 3 2 4 0 5 1 5 0 5 3 5 2 6 1 7 1 8 2 7 2 4 1 4 2 3 1 4 3 3 0 2 1 3 2 2 2
0.05065 0.05066 0.05384 0.05748 0.05701 0.05421 0.05363 0.05284 0.05862 0.05151 0.05621 0.05469 0.04997 0.05737 0.05251 0.05539 0.05571 0.05703 0.05394 0.05365 0.05299 0.04916 0.05066 0.05280 0.05251 0.05234 0.05104 0.05031 0.05256 0.05125 0.04890 0.04919 0.04925 0.04909 0.04665 0.04676 0.04261 0.05070 0.05063 0.04965 0.05166 0.04880 0.05191 0.05395 0.05067 0.05125
!0.03364 !0.03373 !0.03187 !0.02880 !0.03147 !0.03122 !0.03030 !0.03027 !0.03065 !0.03142 !0.02578 !0.02742 !0.03312 !0.02841 !0.03142 !0.02886 !0.02751 !0.02865 !0.02974 !0.02954 !0.02963 !0.03347 !0.03204 !0.03113 !0.03164 !0.03088 !0.03273 !0.03375 !0.03113 !0.03256 !0.03444 !0.03423 !0.03508 !0.03492 !0.03637 !0.03705 !0.04010 !0.03378 !0.02634 !0.02635 !0.02443 !0.02847 !0.02585 !0.02434 !0.02624 !0.02565
0.04540 (1.8) 0.04490 (1.8) 0.05380 (2.2) 0.06260 (2.5) 0.06260 (2.5) 0.05420 (2.2) 0.05360 (2.1) 0.05310 (2.1) 0.06340 (2.5) 0.05330 (2.1) 0.05330 (2.1) 0.05730 (2.3) 0.05270(2.1) 0.05890(2.4) 0.05850(2.3) 0.05670(2.3) 0.05530(2.2) 0.05450(2.2) 0.05560(2.2) 0.05450(2.2) 0.05170(2.1) 0.05080(2.0) 0.04680(1.9) 0.05250(2.1) 0.05320(2.1) 0.05270(2.1) 0.04740(1.9) 0.04580(1.8) 0.05190(2.1) 0.04870(1.9) 0.04440(1.8) 0.04380(1.8) 0.04510(1.8) 0.04620(1.8) 0.03780(1.5) 0.04030(1.6) 0.03260(1.3) 0.04670(1.9) 0.05260(2.1) 0.04720(1.9) 0.04720(1.9) 0.04060(1.6) 0.05380(2.2) 0.05880(2.4) 0.04750(1.9) 0.04910(2.0)
!0.03450 !0.03490 !0.03670 !0.03110 !0.03120 !0.03340 !0.03330 !0.03730 !0.03430 !0.03620 !0.02970 !0.03240 !0.03400 !0.02800 !0.03100 !0.02930 !0.03000 !0.03420 !0.03240 !0.03300 !0.03800 !0.03550 !0.03800 !0.03460 !0.03730 !0.03530 !0.03660 !0.03810 !0.03870 !0.03660 !0.03690 !0.03790 !0.03700 !0.03490 !0.03930 !0.03750 !0.03860 !0.03450 !0.03490 !0.03310 !0.03600 !0.03100 !0.03340 !0.03280 !0.03280 !0.02870
1 0 2 0 2 1 0 1 1 2 0 1 4 1 3 2 2 1 3 3 1 5 1 4 3 4 5 3 2 5 6 6 4 5 7 7 8 6 3 2 1 1 2 1 2 1
0 1 1 1 1 0 1 2 0 3 1 2 5 0 4 1 1 0 2 2 0 6 0 3 2 3 4 2 1 4 5 5 3 4 6 6 7 5 4 3 2 2 3 2 1 0
451
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Table 3. (continued) Vibrational transition
J@ Ka{Kc{
JA A A Ka Kc
c (CRB)!
d (CRB)!
c [26]!
d [26]!
221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221
5 3 2 2 2 1 6 3 4 3 3 3 5 4 5 4 2 0 3 1 4 0 3 2 3 3 5 0 6 1 6 0 4 1 7 0 4 2 4 3 6 1 7 2 8 1 6 3 6 4
5 3 2 2 2 1 6 3 4 3 3 3 5 4 5 4 1 0 2 1 3 0 2 2 4 1 4 0 5 1 5 0 3 1 6 0 3 2 3 3 5 1 6 2 7 1 5 3 5 4
0.04644 0.05092 0.05191 0.04560 0.04745 0.04722 0.04350 0.04352 0.05295 0.05155 0.04991 0.05066 0.05340 0.04854 0.04567 0.04611 0.05021 0.04336 0.05024 0.04741 0.04794 0.04329 0.04326 0.04712 0.04298
!0.02823 !0.02472 !0.02186 !0.02803 !0.02744 !0.02759 !0.03014 !0.03003 !0.02175 !0.02416 !0.02500 !0.02444 !0.02355 !0.02662 !0.02815 !0.02795 !0.02611 !0.02952 !0.02557 !0.02569 !0.02743 !0.03042 !0.03093 !0.02745 !0.02978
0.04220(1.7) 0.04890(2.0) 0.05500(2.2) 0.04310(1.7) 0.03830(1.5) 0.04350(1.7) 0.03130(1.3) 0.02950(1.2) 0.05140(2.1) 0.05220(2.1) 0.04860(1.9) 0.04800(1.9) 0.05250(2.1) 0.04550(1.8) 0.03810(1.5) 0.03850(1.5) 0.05050(2.0) 0.03440(1.4) 0.04750(1.9) 0.03990(1.6) 0.04210(1.7) 0.03390(1.4) 0.03670(1.5) 0.04090(1.6)
!0.03400 !0.03060 !0.02920 !0.03420 !0.03440 !0.03190 !0.03710 !0.03560 !0.02880 !0.03140 !0.03260 !0.03240 !0.02100 !0.03450 !0.03610 !0.03650 !0.03490 !0.03980 !0.03640 !0.03700 !0.03310 !0.03380 !0.03630 !0.03350 !0.04210
000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
3 0 1 3 1 1 2 1 2 3 4 2 1 5 6 6 3 7 2 1 5 6 7 3 3
2 1 2 4 2 0 1 2 1 2 3 1 4 4 5 5 2 6 1 0 4 5 6 2 2
!units of cm~1 atm.
Giesen et al. The agreement between the calculated and measured halfwidth is not good. Ref. [27] states that `the collisional narrowing e!ect has been observed clearly in the high-J transitions of H O seeded in Ara. The argon-broadening measurements of Giesen et al. range from JA"9 to 14. 2 It is clear from Table 4 that none of the transitions compared demonstrate good agreement between the calculations and the measurements of Giesen et al. However, as the rotational quantum number increases from JA"9 to 12, 14, and 15 the agreement becomes much worse indicating that a large part of the poor agreement may be due to collisional narrowing e!ects not accounted for in the current calculations. The comparison of the line shift do not show the same level of agreement as demonstrated for the higher vibrational bands recorded by Grossmann and Browell [26]. This is due, in part, to the comparison of numbers of smaller magnitude. For the rotational and l bands, there are a number of examples in the literature [55,56] where the 2 uncertainty of the line shift, for some transitions, is of the same order of magnitude as the line shift. It is also noted that in the paper of Giesen et al. [27] the line shift for air-broadening could not be determined by the weighted mean of the shifts measured for nitrogen and oxygen separately. In a following paper [57], the same group reports measurements using the same instrument with
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453
Table 4 Halfwidth and line shift in units of 10~3 cm~1 atm for transitions in the 3l #l and 1 3 2l #2l #l bands of H O. CRB calculations compared with the data of Giesen et al. [27] 1 1 1 2 Upper
Lower
c (CRB)!
d (CRB)!
c"
14 1 13 2 13 1 11 2 9 2 9 3 9 5
15 0 14 1 14 2 12 1 10 2 10 4 10 6
13.2 15.5 15.5 22.0 35.8 38.1 29.7
!5.48 !5.84 !5.95 !5.10 !4.80 !3.06 !4.08
3.45(5) 5.98(5) 5.77(5) 7.80(3) 18.74(9) 24.62(3) 14.59(7)
14 12 12 10 7 6 5
15 13 13 11 8 7 4
d" !3.56(10) !3.57(24) !3.87(35) !3.67(5) !5.10(58) !3.12(24) !3.23(19)
!CRB results at 296 K in units of 10~3 cm~1 atm. "Giesen et al. [27] at 298 K in units of 10~3 cm~1 atm.
Table 5 Halfwidth in units of 10~3 cm~1 atm for transitions in the rotational band of H O, CRB 2 calculations compared with measured values at 300 K Line
Upper
Lower
c!
Ref.
CRB"
22 GHz 183 GHz
5 23 3 13
6 16 2 20
4 14
3 21
[29] [23] [28] [21]
44.8 50.0
380 GHz
52.2 50.2(0.76) 49.9(1.52) 45.4(0.8); 46.4#
47.7
!Measured values in 10~3 cm~1 atm. "CRB calculations in 10~3 cm~1 atm at 296 K. #Calculated value by RRB formalism.
increased accuracy and sensitivity for which the line shift data do obey the linear relationship. Unfortunately, there were no results for broadening by argon reported in this work. Finally in Table 5, measured halfwidth for 3 rotational transitions of water vapor broadened by argon are presented with the corresponding CRB calculations. Note, one RRB calculation by Bauer et al. [21] is also presented in the table. The agreement ranges from 14 to !5% with an average of 2.3%. For the one transition for which two measurements are available, the agreement between measurements and calculation are less than a percent.
4. Conclusions These calculations demonstrate that for the H O}Ar system, an intermolecular potential 2 comprised of an anisotropic atom}atom part and an isotropic induction and London dispersion part yields good values for the halfwidth and line shift. With the exception of the study of Giesen et al., where collisional narrowing may be a factor, good agreement between theory and
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experiment is observed. The expansion of the atom}atom potential in terms of the center of mass intermolecular separation needed to be made to 12th order to insure convergence of the potential. The induction and London dispersion isotropic potentials yield line shifts that are good agreement with measurement. Considering Figs. 1 and 2, it is interesting to note that increasing the order of the atom}atom potential has a di!erent e!ect on each line considered. The "nal results agree well with experiment, however there appears to be a shift between calculations and the measured values. For example, considering the line shift, Fig. 2 shows that the relative ordering of the values for the 4 lines considered is correct but that better agreement would be observed if the calculations were decreased slightly in magnitude. A decrease in magnitude of the calculated halfwidths would also give better agreement with the experiment of Grossmann and Browell. The calculations presented here, as well as those of other studies [7,11], indicate that the imaginary components of the Liouville scattering matrix strongly a!ect the determination of the halfwidth. This e!ect can vary greatly from transition to transition. For the H O}Ar system 2 di!erences as large as 37% are observed. Clearly, this e!ect must be accounted for if the calculations are to be meaningful. The results from the experiments indicate that velocity changing collisions are important for the H O}Ar collision system. This e!ect is observed when the di!erence between the mass of 2 the radiating and perturbing molecules is large, as is the case here. It was also pointed out by Dicke [58] and by Rao and Oka [59] that collisional narrowing is signi"cant if the quantum state of the radiating molecule is not a!ected by collisions and the mean free path is smaller than the wavelength of the probing radiation. Certainly, the "rst of these conditions is met by high JA states of H O for which collisionally induced transitions to allowed states involves large energy changes. 2 In conclusion, it is noted that the CRB formalism gives good agreement with measured halfwidths and line shifts for many transitions. As the rotational quantum number of water vapor increases collisional narrowing is observed in the measurements. To account for this the semiclassical model of Robert and Bonamy should consider approximate corrections as indicated by Robert et al. [60] and by Billing [61]. It should be noted that evidence for collisional narrowing has also been noted for other perturbing species (N , O ) and large J++. Hence, it is hoped that such 2 2 a correction should improve the results for other perturbing molecules.
Acknowledgement The authors are pleased to acknowledge support of this research by the National Science Foundation through Grant No. ATM-9415454.
References [1] Hartmann DL. Global physical climatology. Boston: Academic Press, Inc., 1994. [2] Levy A, Lacome N, Chackerian Jr C. In: Rao KN, Weber A, editors. Spectroscopy of the earth's atmosphere and interstellar molecules. Boston: Academic Press, Inc., 1992. [3] Atmospheric Spectroscopy Applications Workshop, ASA REIMS 96, UniversiteH de Reims, Champagne Ardenne, September 4}6, 1996.
R.R. Gamache, R. Lynch / Journal of Quantitative Spectroscopy & Radiative Transfer 64 (2000) 439}456 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51]
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Robert D, Bonamy J. J Phys 1979;40:923. Lynch R, Gamache RR, Neshyba SP. J Chem Phys 1996;105:5711. Lynch R. PhD dissertation, Physics Department, University of Massachusetts Lowell, June 1995. Gamache RR, Lynch R, Plateaux JJ, Barbe A. JQSRT 1997;57:485. Lynch R, Gamache RR, Neshyba SP. JQSRT 1998;59:595. Lynch R, Gamache RR, Neshyba SP. JQSRT 1998;59:615}26. Gamache RR, Lynch R, Brown LR. JQSRT 1996;56:471. Gamache RR, Lynch R, Neshyba SP. JQSRT 1998;59:319. Neshyba SP, Gamache RR. JQSRT 1993;50:443. Labani B, Bonamy J, Robert D, Hartmann JM, Taine J. J Chem Phys 1986;84:4256. Hartmann JM, Taine J, Bonamy J, Labani B, Robert D. J Chem Phys 1987;86:144. Labani B, Bonamy J, Robert D, Hartmann JM. J Chem Phys 1987;87:2781. Anderson PW. PhD dissertation, Harvard University, 1949. Anderson PW. Phys Rev 1949;76:647. Anderson PW. Phys Rev 1950;80:511. Tsao CJ, Curnutte Jr B. JQSRT 1962;2:41. Gamache RR, Neshyba SP, Plateaux JJ, Barbe A, ReH galia L, Pollack JB. J Mol Spectrosc 1995;170:131. Bauer A, Gordon M, Kheddar M, Hartmann JM, Bonamy J, Robert D. JQSRT 1987;37:531. Hartmann JM, Camy-Peyret C, Flaud J-M, Bonamy J, Robert D. JQSRT 1988;40:489. Bauer A, Godon M, Kheddar M, Hartmann JM. JQSRT 1989;41:49. Bauer A, Godon M, Carlier J, Gamache RR. JQSRT 1998;59:273. Bauer A, Godon M, Carlier J, Gamache RR. J Mol Spectrosc 1996;176:45. Grossmann BE, Browell EV. J Mol Spectrosc 1989;138:562. Giesen T, Schieder R, Winnewisser G, Yamada KMT. J Mol Spectrosc 1992;153:406. Rusk JR. J Chem Phys 1965;42:493. Liebe HJ, Dillon TA. J Chem Phys 1969;50:727. Baranger M. Phys Rev 1958;112:855. Ben-Reuven A. Spectral line shapes in gases in the binary-collision approximation. In: Prigogine I, Rice SA, editors. Adv Chem Phys, vol. 20. New York: Academic Press, 1975. p. 235. Kubo R. J Phys Soc Jpn 1962;17:1100. Bloch C. Nucl Phys 1958;7:451. Tipping RH, Herman RM. JQSRT 1970;10:881,897. Lennard-Jones JE. Proc Roy Soc 1924;A106:463. Birnbaum G. In: Hirschfelder JO, editor. Intermolecular forces. New York: Interscience, 1967. p. 487. Bonamy J, Khayar A, Robert D. Chem Phys Lett 1981;83:539. Hirschfelder JO, Curtiss CF, Bird RB. Molecular theory of gases and liquids. New York: Wiley, 1964. Sack RA. J Math Phys 1964;5:260. Gray CG, Gubbins KE. Theory of molecular #uids. Oxford: Clarendon Press, 1984. Gray CG. Can J Phys 1968;46:135. Downs J, Gray CG, Gubbins KE, Murad S. Mol Phys 1979;37:129. Shostak SL, Muenter JS. J Chem Phys 1991;94:5883. Luo Y, Agren H, Vahtras O, Jorgensen P, Spirko V, Hettema H. J Chem Phys 1993;98:7159. Bouanich J-P. JQSRT 1992;47:243. Watson JKG. J Chem Phys 1967;46:1935. Flaud J-M, Camy-Peyret C. Private communication. University of Pierre and Marie Curie, Paris, France, 1994. Lias SG, Bartness JE, Liebman JF, Holmes JL, Levin RD, Mallard WG. J Phys Chem Ref Data 1988;17(S1):1. Orcutt RH, Cole RH. J Chem Phys 1967;46:697. Rothman LS, Gamache RR, Goldman A, Flaud J-M, Tipping RH, Rinsland CP, Smith MAH, Toth RA, Brown LR, Devi VM, Benner DC. JQSRT 1992;48:469. Proceedings Introduction and Overview, ASA Workshop, Moscow, 1990.
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[52] Smith MAH, editor. NASA Conference Publication 2396. NASA, Scienti"c and Technical Information Branch, 1985. [53] Reber CA. NASA Goddard Space Flight Center Publication 430-1003-001. 1985. [54] Rautian SG, Sobel'man II. Sov Phys Usp 1967;9:701. [55] Toth RA, Brown LR, Plymate C. JQSRT 1998;59:529. [56] Malathy Devi V, Benner DC, Smith MAH, Rinsland CP. J Mol Spectrosc 1997;182:221. [57] SchmuK cker N, Trojan Ch, Giesen T, Yamada KMT, Winnewisser G. J Mol Spectrosc 1997;184:250. [58] Dicke RH. Phys Rev 1953;89:472. [59] Rao DR, Oka T. J Mol Spectrosc 1987;122:16}27. [60] Robert D, Bonamy J, Marsault-Herail F, Levi G, Marsault J-P. Chem Phys Lett 1980;74:467}71. [61] Billing GD. PhD thesis, Copenhagen, 1978.
Argon-induced halfwidths and line shifts of water vapor transitions R.R. Gamache!,*, R. Lynch" !Department of Environmental, Earth, and Atmospheric Sciences, The University of Massachusetts Lowell, 1 University Avenue, Lowell, MA 01854, USA "Columbia University, Goddard Institute for Space Studies, 2880 Broadway, New York, NY 10025, USA
Abstract Pressure-broadened halfwidths and pressure-induced line shifts are determined for water vapor transitions with argon as the perturbing gas. Calculations based on the complex Robert}Bonamy (CRB) formalism are made for a number of vibrational bands, for which there are experimental data to compare with. The intermolecular potential is taken as a sum of Lennard}Jones (6}12) atom}atom, isotropic induction, and dispersion components. The dynamics of the collision process are correct to second order in time. The calculations demonstrate that the atom}atom potential for this system should be expanded to 12th order. A new feature in the CRB approach is that the real and imaginary components of the Liouville S matrix a!ect both the halfwidth and the line shift. It is shown here that the imaginary parts of the S matrix strongly a!ect the calculated H O}Ar halfwidths. The calculated values are compared with those obtained in a number of 2 experimental studies. In general, good agreement is observed between the CRB calculations and the measured values for both halfwidths and line shifts. It is also clear that some parameters describing the intermolecular potential need to be better determined. ( 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction Water is the principle absorber of longwave radiation in the terrestrial atmosphere, responsible for some 80% of `greenhousea warming of the Earth's surface [1]. The study and comprehension of the radiative-forcing e!ects and the interpretation of remote-sensing data of water vapor requires the use of atmospheric radiative-transfer models, which require high-precision parameters describing line positions, intensities, pressure-broadened halfwidths and line shifts. Although laboratory measurements can and do supply such parameters, it is di$cult to exhaustively cover the vast * Corresponding author. E-mail address: robert}[email protected] (R.R. Gamache) 0022-4073/99/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 9 9 ) 0 0 1 0 1 - 6
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spectral range and diversity of environmental conditions encountered in the actual atmosphere. Theoretical calculations are in principle an attractive alternative, depending on the accuracy requirements of the radiative transfer application one has in mind, and of course the credibility of the theory. Even when laboratory measurements are available, however, certain e!ects (such as line mixing [2]) may still require a sophisticated theoretical model in order to unravel observed spectra. Of the needed parameters the collision-broadened halfwidth is the least well-known parameter for atmospheric applications [3]. Our present focus is to evaluate the complex Robert}Bonamy (CRB) formalism [4] for the calculations of pressure broadening and pressure-induced line shifting parameters, c and d respectively, of water vapor transitions with argon as the bu!er gas. This is one of a series of papers considering the application of the CRB formalism to calculating c and d for water vapor transitions with a number of perturbing gases [5}11]. As described below and elsewhere [12}15], CRB theory uses a spherical tensor expansion in 1/R (R is the intermolecular separation) of a short-range, atom}atom potential. We have discussed previously [8,11] how this potential is combined with two others (electrostatic and induction}dispersion) in a natural and non-redundant fashion for the general XY }A system. Since the expansion in (1/R) must be 2 2 truncated, we also devote some discussion to questions of the convergence of calculated halfwidths and line shifts. A feature that is new in the CRB formalism is that the expressions for the halfwidth and line shift contain both real and imaginary components of the Liouville scattering matrix. This is an e!ect that is not present in Anderson}Tsao}Curnutte (ATC) theory [16}19] nor in previous applications of the theory of Robert and Bonamy to asymmetric rotor molecules [12}15,20}25], which used only the real components of the theory (labeled real Robert}Bonamy (RRB) in this work). We show below that for the H O}Ar system the e!ects of the imaginary terms on the calculated halfwidths 2 are large and should not be overlooked. To this purpose, we provide a comparison with halfwidths and line shifts for which recent, high-quality experimental data are available, namely those occurring in the 3l #l and 1 3 2l #2l #l bands reported by Grossmann and Browell [26], l data of Giesen et al. [27] as 1 2 3 2 well as several halfwidth measurements in the rotational band [21,23,28,29].
2. Complex Robert}Bonamy formalism 2.1. Introduction Within the complex Robert}Bonamy formalism, the line shift and halfwidth for the transition iQf are given by real and minus the imaginary parts of the diagonal elements of the relaxation matrix [4],
P
n l6 = " "!i 2 + SJ Do DJ T 2pb db[1!exp(!Mi(S #Im(S ))#Re(S )N)], (1) fi,fi 2 2 2 1 2 2 2pc 0 J2 where v6 is the mean relative thermal velocity, o and n are the density operator and number 2 2 density of perturbers and b is the impact parameter. S and S are the "rst- and second-order terms 1 2
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in the successive approximation expansion of the Liouville scattering matrix [30,31], described below. The complex Robert}Bonamy formalism was chosen for several reasons. (1) A cumulant expansion [32,33] incorporated in CRB formalism eliminates an awkward cuto! procedure [34] that characterized earlier theories. (2) The complex formalism yields halfwidths and line shifts from a single calculation. (3) A short-range atom}atom potential [12}15] (Lennard}Jones, 6}12 [35]) can be incorporated into the theory in a natural and non-redundant fashion. (4) The intermolecular dynamics are more realistic than in earlier theories, i.e. using curved rather than straight-line trajectories. In our implementation, these trajectories are determined by the isotropic part of the atom}atom potential. (5) Vibrational dependence of the isotropic part of the intermolecular potential can be included in a straightforward way. 2.2. Intermolecular potential For an atom interacting with a molecule there are a number of potentials that have been used [36]. In this work the potential is comprised of two types of terms. An atom}atom potential has been adopted, following the suggestion of Bauer et al. [21,23] and others [37], to model the anisotropic interactions and induction and dispersion potentials are used to model the isotropic interaction. The atom}atom potential is de"ned as the sum of pair-wise Lennard}Jones 6}12 interactions [35] between atoms of molecules 1 and the argon atom,
G
H
n p6 p12 i2 ! i2 . (u), (4) 12 1 2 m n 1 m 0 2 lm Rq`l1 `l2 `2w 1 n l1 l2 1 2 w,qq l mm m where C(l l l; m m m) is a Clebsch}Gordan coe$cient, ) "(a , b , c ) and ) "(a , b , c ) 12 1 2 1 1 1 1 2 2 2 2 are the Euler angles describing the molecular "xed axis relative to the space "xed axis. Subscripts 1
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and 2 refer to the radiating molecule (H O) and perturbing atom (Ar), respectively, and u"(h, /) 2 describes the relative orientation of the centers of mass. As described earlier, R is the center of mass separation. The powers w and q (integers) depend upon the interaction assumed, and the coe$cients ;(2) are given in Ref. [5] or Ref. [40]. Before taking up discussion of the other interaction (induction}dispersion), three further points should be emphasized with regard to the atom}atom potential. (1) When expressed in the form of Eq. (4), the atom}atom potential can be understood as two simultaneous expansions. One is de"ned by the tensorial ranks l and l , which determine the 1 2 symmetry of the interaction [40,41]. A second expansion is de"ned by the sum l #l #2w 1 2 [39,42], which we call the order of the expansion. It hardly needs to be emphasized that any calculation of line width or shift should be converged with respect to both order and rank. The issue of convergence is discussed below. (2) The atom}atom potential contains an isotropic part (l "l "0) which in general is 1 2 comprised of terms of the form 1/R6, 1/R8, 1/R10, etc. In our implementation, these are used to de"ne the intermolecular trajectory. For the purpose of comparison, however, it is convenient to "t this summation to an e!ective isotropic (or `hetero-moleculara) Lennard}Jones 6}12 potential,
CA B A B D
<*40"4e
p 12 p 6 . ! R R
(5)
The approximation represented by Eq. (5) had no signi"cant impact on our calculations other than to simplify the trajectory calculations somewhat. (3) The remaining parts of the atom}atom potential are anisotropic (nonzero l or l ). These 1 2 terms may be discussed with the same lexicon that one applies to electrostatic interactions. For example, when l "1 and l "2, we refer to a `dipole}quadrupolea interaction. It should be kept 1 2 in mind that these are symmetry appellations only, and do not refer to electrostatics [11]. Thus it happens that between H O and Ar, the lowest anisotropic atom}atom term is `dipole}monopolea, 2 i.e. with l "1 and l "0. 1 2 In another category of interactions relevant to intermolecular collisions are the isotropic induction and dispersion potentials, k2 a <*/$"! 1 2 , *40 R6 (6) a a 3 E E 1 2 1 2, V$*4"! *40 2 E #E R6 1 2 where k is the dipole moment of water vapor and a and E are the polarizability and ionization 1 potentials for water vapor and its perturber (argon), labeled 1 and 2, respectively. By design, these interactions occur with the same power of 1/R, and in some sense represent the same forces, as the attractive part of Eq. (5). Thus, they play no role in our trajectory calculations, although they do serve as a convenient check. We "nd that the magnitude of the combined induction}dispersion interactions is typically about half the value of the corresponding atom}atom term. This is consistent with observations of Gray and Gubbins [40], who point out that Eqs. (6) are themselves merely the "rst terms in an expansion, and by Lynch et al. [5]. Because the induction and
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dispersion terms in Eqs. (6) are isotropic, they only contribute through the imaginary Liouville scattering matrix, S . These terms provide a convenient, experimentally constrained method of 1 computing the diwerence in isotropic potential for di!erent vibrational states, via the vibrational dependence of k and a (this di!erence appears in the S term of CRB theory and is discussed in 1 1 1 Refs. [5,6,11]). 2.3. Details of CRBF theory The expressions for the S and S terms in the CRB formalism are described in detail in 1 2 Refs. [6,8,11]. Here only the salient features are summarized. The "rst-order (imaginary) term, S , 1 depends only on the di!erence in the isotropic part of the interaction potential between the initial and "nal vibrational states of the radiator and is accounted for by the vibrational dependence of the dipole moment and polarizability of H O. The second-order terms are comprised of two basic 2 parts; one describing the internal states of the radiating and perturbing molecules and another describing the interaction and dynamics of the collision. In order to calculate these terms, information is needed for a number of molecular constants describing the colliding pair. All molecular parameters used in this work are the best available values from the literature. No molecular constants are adjusted to give better agreement with experiment. The vibrational dependence of the dipole moment of water vapor was investigated by Shostak and Muenter [43] and is given in Debyes by k"1.855#0.0051(v #1)!0.0317(v #1)#0.0225(v #1), 1 2 2 2 3 2
(7)
where v is the number of quanta in the ith normal mode. The vibrational dependence of the i polarizability of water vapor was obtained by Luo et al. [44] and is, in atomic units,1 a"9.86#0.29(v #1)!0.03(v #1)#0.28(v #1). 1 2 2 2 3 2
(8)
The heteronuclear atom}atom parameters used in the present work, derived from homonuclear} atom}atom parameters obtained by Bouanich [45] using the combination rules [38], (see Eqs. (3)), are reported in Table 1. Reduced matrix elements for the internal states of the radiator and perturber must be calculated. For water vapor, these are evaluated using wavefunctions determined by diagonalizing the Watson Hamiltonian [46] in a symmetric top basis for the vibrational states involved in the transition. For the ground, l , l , and l states of H O, the Watson constants derived by Flaud and Camy}Peyret 1 2 3 2 [47] are used. For the (301) and (221) vibrational states of H O, the constants of Grossmann et al. 2 [26] are used. In Refs. [5,6], techniques were presented for determining the real and imaginary parts of the resonance functions for an arbitrary potential that is in the form of an expansion in the center of mass separation in the parabolic trajectory approximation.
1 The choice of units for Eqs. (7) and (8) is made to keep the expressions as concise as possible. For conversion to other units see for example Ref. [40, p. 575].
444
R.R. Gamache, R. Lynch / Journal of Quantitative Spectroscopy & Radiative Transfer 64 (2000) 439}456 Table 1 Atom}Atom parameters for the H O}Ar 2 system
p (As ) e/k (K) B
Ar}H
Ar}O
3.1 37.3
3.2 78.7
In the parabolic approximation the isotropic part of the interaction potential is taken into account in determining the distance, e!ective velocity, and force at closest approach [4]. To simplify the trajectory calculations the isotropic part of the atom}atom expansion is "t to an isotropic Lennard}Jones 6}12 potential giving e/k "107.98 K and p"3.496 As for the 12th-order B expansion of the atom}atom potential for the H O}Ar collision pair. 2 Finally, the ionization potential of water was taken to be 12.61 eV [48] independent of vibrational state. The polarizability of argon is taken from Ref. [49] and is 16.41]10~25 cm3. All calculations are made at the HITRAN [50] reference temperature, 296 K. The CRB formalism yields the halfwidth and line shift from a single calculation and there are no adjustable parameters or cuto! procedure in the formalism.
3. Calculations Using the CRB method, we have calculated both halfwidths and line shifts for transitions in the pure rotation, l , l , l , 3l #l , and 2l #2l #l bands of water vapor perturbed by argon. 1 2 3 1 3 1 2 3 In the calculations presented here, we examine the e!ect of the order of expansion, the e!ect of the real and imaginary terms on the halfwidth and line shift, and compare with experiment when possible. It was demonstrated by Labani et al. [13}15] that for H O}A systems (A "N , O , etc.) the 2 2 2 2 2 atom}atom potential in Eq. (3) needs to be expanded to at least fourth order in the center of mass separation. (Note, Labani et al. used a di!erent form of the expansion of the atom}atom potential [38] for which the order is de"ned by l #l .) The development of Labani et al. was 1 2 limited to a fourth-order expansion. In this work, we use the analytic form of the two-center expansion given by Sack [39], to generate higher orders of the expansion. The only limitation of our method is that of computational resources, which currently is a 12th-order expansion for the H O}Ar system. 2 To study the e!ects of the order of the expansion of the atom}atom potential, we made calculations where the order of the expansion, l #l #2w, is varied from 2 to 12 and the results 1 2 compared. In Figs. 1 and 2 the results of the CRB calculation are plotted for the halfwidths and line shifts, respectively, versus the order of the atom}atom potential for four transitions in the 3l #l 1 3 band: 4 Q5 ; 3 Q4 ; 5 Q5 ; and 2 Q3 . Also plotted in the "gures are the experi22 23 30 31 05 24 12 13 mental results of Grossmann and Browell [26]. It is clear that the potential has converged by 12th-order. All calculations from this point on use the 12th-order atom}atom potential.
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Fig. 1. Halfwidth versus order of the atom}atom potential and experimental results of Grossmann and Browell [26] in the 3l #l band. 1 3
Fig. 2. Line shift versus order of the atom}atom potential and experimental results of Grossmann and Browell [26] in the 3l #l band. 1 3
446
R.R. Gamache, R. Lynch / Journal of Quantitative Spectroscopy & Radiative Transfer 64 (2000) 439}456 Table 2 Average percent di!erence between CRB and RRB calculations for several vibrational bands Vibrational band
Number of transitions
Average percent di!erence
Rotation l 1 l 2 l 3 3l #l 1 3 2l #2l #l 1 2 3
921 353 349 335 82 36
!0.30 2.12 0.17 2.72 15.52 15.66
As stated above, a new feature of these calculations is the real and imaginary components of the theory contributing to both the halfwidth and line shift. To study the importance of the imaginary components to the halfwidth, CRB and RRB calculations were made for a number of vibrational bands of water vapor; 921 rotation, 353l , 349l , 335l , 82 3l #l , and 33 2l #l #l 2 1 3 1 3 1 2 3 transitions. Table 2 presents the average percent di!erence, 100(CRB-RRB)/CRB, for each band studied. For bands with zero (rotation band) or 1 quanta of energy (l , l , and l ) the contribution 1 2 3 of the S and Im(S ) terms can be comparable. This is especially true of the l band, for which the 1 2 2 coe$cient in Eq. (8) is small. As the number of quanta increases, the contribution of the S term is 1 greater than that of the imaginary part of the S term of the interaction [6] for this system. Note, 2 the results of Table 2 correlate with the coe$cients of Eq. (8) indicating that the vibrational dependence of the polarizability is the principal contributor to the S term. This fact agrees with 1 the studies of Lynch [6]. As the number of vibrational quanta increases, especially in the l and l levels, the e!ects of the 1 3 imaginary components become more important. Note that even with a single quanta of energy in either the l and l level, the di!erence between the CRB and RRB calculations is of the same 1 3 magnitude as the desired uncertainty of the halfwidth [51}53]. Of course, a more important question is, does the more complete theory give better agreement with experiment. For the 3l #l band transitions studied, the CRB calculations agree with the measurements of 1 3 Grossmann and Browell [26] better than the RRB calculations, !5.8 and 11.1 average percent di!erence, respectively. This fact is in agreement with other studies [7,11]. In Fig. 3 the percent di!erence between the CRB and RRB calculations is plotted versus an energy-ordered index, JA(JA#1)#KaA!KcA#1. The importance of properly including the imaginary terms is apparent from the "gure. The percent di!erences range from about 8 to 37%. Note as the number of vibrational quanta increases the contribution of the imaginary terms increases. Thus for transitions in the 3l #l band, it is expected that the di!erence between the two calculations will be large. In 1 3 all the following calculations the CRB formalism is employed. CRB calculations are made at 296 K for the transitions studied by Grossmann and Browell [26] for the 3l #l and 2l #2l #l bands. A comparison of the measured and calculated 1 3 1 2 3 halfwidths and line shifts are shown in Figs. 4 and 5 for the 3l #l band. Good agreement is 1 3 realized for most of the transitions. Note that the largest di!erences occur for a few transitions with the measured values being lower than the calculated ones. Grossmann and Browell commented that they had observed narrowing in their measurements and needed to employ the hard-collision
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447
Fig. 3. Percent di!erence between CRB and RRB calculations for transitions in the 3l #l band. 1 3
Fig. 4. Halfwidths for transitions in the 3l #l band versus energy-ordered index: CRB calculations and the 1 3 measurements (with error bars) of Grossmann and Browell [26].
448
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Fig. 5. Line shifts for transitions in the 3l #l band versus energy-ordered index: CRB calculations and the 1 3 measurements (with error bars) of Grossmann and Browell [26].
model of Rautian and Sobel'man [54] to model the line shape. The CRB calculations make no account for narrowing. The average percent di!erence (APD) between the measured and calculated halfwidth is !5.8 for the 80 lines compared. Fig. 5 shows the measured and calculated line shifts versus the energy ordered index for the transitions in the same band. Note that all of the observed line shifts are negative as expected for a band with a large number of vibrational quanta [11]. While the agreement is reasonable, 7.2 average percent di!erence based on 80 transitions compared, the calculated shifts appear to be systematically above the measured ones. This can be related to the coe$cients in Eqs. (7) and (8) (particularly those in Eq. (8)). The coe$cients in Eq. (8) used in this work are from an ab initio calculation and their uncertainty is not well understood. Scaling of these values was not considered a worthwhile exercise at this time. Figs. 6 and 7 are the corresponding halfwidth and line shift plots for the 2l #2l #l 1 2 3 band. Similar features are seen; the calculated halfwidths agree fairly well with measurement, APD"!10.8 for the 32 transitions compared, and there are still problems with some narrow lines. The calculated line shifts, with one exception, are all greater than the measured values and all values are negative. The APD for the line shifts is 20 for the 33 transitions compared. Table 3 provides the halfwidth and line shifts calculated via the CRB formalism with the atom}atom potential expanded to 12th-order for all the transitions considered in this work. Also present in the table are the corresponding experimental values when available. Next, comparisons are made with measurements of the halfwidth and line shift for transitions in the l band of H O. The measurements are those of Giesen et al. [27] for seven transitions 2 2 at 298 K. Table 4 gives the CRB formalism calculations at 296 K as well as the values from
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449
Fig. 6. Halfwidths for transitions in the 2l #2l #l band versus energy-ordered index: CRB calculations and the 1 2 3 measurements (with error bars) of Grossmann and Browell [26].
Fig. 7. Line shifts for transitions in the 2l #2l #l band versus energy-ordered index: CRB calculations and the 1 2 3 measurements (with error bars) of Grossmann and Browell [26].
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Table 3 Halfwidth and line shift in units of cm~1/atm for transitions in the 3l #l and 2l #2l #l bands of H O. CRB 1 3 1 1 1 2 calculations compared with the data of Grossmann and Browell [26] Vibrational transition
J@ Ka{Kc{
JA A A Ka Kc
c (CRB)!
d (CRB)!
c [26]!
301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000
8 1 6 3 8 0 6 2 6 3 6 1 6 2 5 3 5 2 6 0 6 1 5 1 5 2 7 7 4 2 5 1 4 1 4 2 7 6 3 3 3 3 6 6 3 2 4 0 3 1 5 0 3 2 3 0 3 1 4 0 2 2 2 1 2 0 2 1 1 1 5 4 5 4 1 0 5 3 1 1 3 1 4 3 6 3 4 3
9 1 7 3 9 0 7 2 7 3 7 1 7 2 6 3 6 2 7 0 7 1 6 1 6 2 7 7 5 2 6 1 5 1 5 2 7 6 4 3 4 3 6 6 4 2 5 0 4 1 5 2 4 2 4 0 4 1 4 2 3 2 3 1 3 0 3 1 2 2 5 4 5 4 2 0 5 3 2 1 3 1 4 3 6 3 4 3
0.04376 0.05117 0.04302 0.05161 0.04793 0.04873 0.04736 0.04911 0.05199 0.04766 0.04758 0.05075 0.04919 0.03481 0.05205 0.04973 0.05215 0.05094 0.04022 0.05239 0.05217 0.03888 0.05311 0.05168 0.05278 0.05177 0.05297 0.05336 0.05337 0.05352 0.05462 0.05477 0.05518 0.05487 0.05581 0.04783 0.04783 0.05778 0.05031 0.05611 0.05509 0.05107 0.04990 0.05104
!0.03922 !0.03305 !0.04126 !0.03283 !0.03569 !0.03472 !0.03627 !0.03482 !0.03218 !0.03699 !0.03713 !0.03373 !0.03481 !0.04701 !0.03162 !0.03522 !0.03311 !0.03334 !0.04239 !0.03404 !0.03416 !0.04332 !0.03148 !0.03324 !0.03163 !0.03398 !0.03204 !0.03254 !0.03196 !0.03264 !0.03150 !0.02998 !0.03163 !0.03095 !0.02850 !0.03624 !0.03617 !0.03068 !0.03410 !0.02924 !0.03163 !0.03351 !0.03423 !0.03353
0.03540 0.04840 0.03120 0.05110 0.00000 0.04670 0.04760 0.04550 0.05150 0.04310 0.04280 0.05200 0.04970 0.02630 0.05150 0.04850 0.05390 0.05360 0.02520 0.04370 0.04480 0.02400 0.04980 0.05560 0.05410 0.04420 0.05290 0.05770 0.05740 0.00000 0.05090 0.05000 0.05870 0.06000 0.05840 0.03720 0.04130 0.06200 0.04460 0.06190 0.05740 0.04630 0.05170 0.04480
7 3 8 4 4 5 5 3 3 6 6 4 4 1 2 5 3 3 2 0 1 0 1 4 2 5 2 3 3 4 0 1 2 2 0 2 1 1 3 1 3 2 3 1
8 4 9 5 5 6 6 4 4 7 7 5 5 0 3 6 4 4 1 1 2 1 2 5 3 4 3 4 4 3 1 2 3 3 1 1 2 2 2 2 2 1 4 2
d [26]!
(1.4) (1.9) (1.2) (2.0) (0.0) (1.9) (1.9) (1.8) (2.1) (1.7) (1.7) (2.1) (2.0) (1.1) (2.1) (1.9) (2.2) (2.1) (1.0) (1.9) (1.8) (1.0) (2.0) (2.2) (2.2) (1.8) (2.1) (2.3) (2.3) (0.0) (2.0) (2.0) (2.3) (2.4) (2.3) (1.5) (1.7) (2.5) (1.8) (2.5) (2.3) (1.9) (2.1) (1.8)
!0.03980 !0.03390 !0.04210 !0.03360 !0.03640 !0.03600 0.00000 !0.03590 !0.03540 !0.04110 !0.04050 !0.03530 !0.03750 !0.03840 !0.03690 !0.04070 !0.03590 !0.03690 0.00000 !0.03260 !0.03290 !0.03840 !0.03730 !0.03850 !0.03820 !0.03650 !0.03520 !0.03660 !0.03490 !0.03320 !0.03530 !0.03850 !0.03570 !0.03340 !0.03490 !0.03720 !0.03760 !0.03230 !0.03610 !0.03420 !0.03050 !0.03590 !0.03570 !0.03630
R.R. Gamache, R. Lynch / Journal of Quantitative Spectroscopy & Radiative Transfer 64 (2000) 439}456 Table 3. (continued) Vibrational transition
J@ Ka{Kc{
JA A A Ka Kc
c (CRB)!
d (CRB)!
c [26]!
d [26]!
301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 301 000 221 000 221 000 221 000 221 000 221 000 221 000 221 000 221 000
3 3 3 3 3 2 0 0 2 1 2 2 2 2 3 2 1 1 4 2 1 1 2 1 6 2 1 0 4 1 2 1 2 0 2 1 3 1 3 0 3 2 6 1 4 3 4 1 4 2 4 0 5 1 5 3 4 2 5 0 6 1 6 0 6 3 6 2 7 1 8 1 9 2 8 2 3 1 3 2 2 1 3 3 2 0 1 1 3 2 2 2
3 3 3 3 3 2 1 0 2 1 2 2 2 2 3 2 1 1 4 2 1 1 2 1 6 2 0 0 4 1 1 1 1 0 1 1 2 1 2 0 2 2 6 1 3 3 3 1 3 2 3 0 4 1 4 3 3 2 4 0 5 1 5 0 5 3 5 2 6 1 7 1 8 2 7 2 4 1 4 2 3 1 4 3 3 0 2 1 3 2 2 2
0.05065 0.05066 0.05384 0.05748 0.05701 0.05421 0.05363 0.05284 0.05862 0.05151 0.05621 0.05469 0.04997 0.05737 0.05251 0.05539 0.05571 0.05703 0.05394 0.05365 0.05299 0.04916 0.05066 0.05280 0.05251 0.05234 0.05104 0.05031 0.05256 0.05125 0.04890 0.04919 0.04925 0.04909 0.04665 0.04676 0.04261 0.05070 0.05063 0.04965 0.05166 0.04880 0.05191 0.05395 0.05067 0.05125
!0.03364 !0.03373 !0.03187 !0.02880 !0.03147 !0.03122 !0.03030 !0.03027 !0.03065 !0.03142 !0.02578 !0.02742 !0.03312 !0.02841 !0.03142 !0.02886 !0.02751 !0.02865 !0.02974 !0.02954 !0.02963 !0.03347 !0.03204 !0.03113 !0.03164 !0.03088 !0.03273 !0.03375 !0.03113 !0.03256 !0.03444 !0.03423 !0.03508 !0.03492 !0.03637 !0.03705 !0.04010 !0.03378 !0.02634 !0.02635 !0.02443 !0.02847 !0.02585 !0.02434 !0.02624 !0.02565
0.04540 (1.8) 0.04490 (1.8) 0.05380 (2.2) 0.06260 (2.5) 0.06260 (2.5) 0.05420 (2.2) 0.05360 (2.1) 0.05310 (2.1) 0.06340 (2.5) 0.05330 (2.1) 0.05330 (2.1) 0.05730 (2.3) 0.05270(2.1) 0.05890(2.4) 0.05850(2.3) 0.05670(2.3) 0.05530(2.2) 0.05450(2.2) 0.05560(2.2) 0.05450(2.2) 0.05170(2.1) 0.05080(2.0) 0.04680(1.9) 0.05250(2.1) 0.05320(2.1) 0.05270(2.1) 0.04740(1.9) 0.04580(1.8) 0.05190(2.1) 0.04870(1.9) 0.04440(1.8) 0.04380(1.8) 0.04510(1.8) 0.04620(1.8) 0.03780(1.5) 0.04030(1.6) 0.03260(1.3) 0.04670(1.9) 0.05260(2.1) 0.04720(1.9) 0.04720(1.9) 0.04060(1.6) 0.05380(2.2) 0.05880(2.4) 0.04750(1.9) 0.04910(2.0)
!0.03450 !0.03490 !0.03670 !0.03110 !0.03120 !0.03340 !0.03330 !0.03730 !0.03430 !0.03620 !0.02970 !0.03240 !0.03400 !0.02800 !0.03100 !0.02930 !0.03000 !0.03420 !0.03240 !0.03300 !0.03800 !0.03550 !0.03800 !0.03460 !0.03730 !0.03530 !0.03660 !0.03810 !0.03870 !0.03660 !0.03690 !0.03790 !0.03700 !0.03490 !0.03930 !0.03750 !0.03860 !0.03450 !0.03490 !0.03310 !0.03600 !0.03100 !0.03340 !0.03280 !0.03280 !0.02870
1 0 2 0 2 1 0 1 1 2 0 1 4 1 3 2 2 1 3 3 1 5 1 4 3 4 5 3 2 5 6 6 4 5 7 7 8 6 3 2 1 1 2 1 2 1
0 1 1 1 1 0 1 2 0 3 1 2 5 0 4 1 1 0 2 2 0 6 0 3 2 3 4 2 1 4 5 5 3 4 6 6 7 5 4 3 2 2 3 2 1 0
451
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Table 3. (continued) Vibrational transition
J@ Ka{Kc{
JA A A Ka Kc
c (CRB)!
d (CRB)!
c [26]!
d [26]!
221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221 221
5 3 2 2 2 1 6 3 4 3 3 3 5 4 5 4 2 0 3 1 4 0 3 2 3 3 5 0 6 1 6 0 4 1 7 0 4 2 4 3 6 1 7 2 8 1 6 3 6 4
5 3 2 2 2 1 6 3 4 3 3 3 5 4 5 4 1 0 2 1 3 0 2 2 4 1 4 0 5 1 5 0 3 1 6 0 3 2 3 3 5 1 6 2 7 1 5 3 5 4
0.04644 0.05092 0.05191 0.04560 0.04745 0.04722 0.04350 0.04352 0.05295 0.05155 0.04991 0.05066 0.05340 0.04854 0.04567 0.04611 0.05021 0.04336 0.05024 0.04741 0.04794 0.04329 0.04326 0.04712 0.04298
!0.02823 !0.02472 !0.02186 !0.02803 !0.02744 !0.02759 !0.03014 !0.03003 !0.02175 !0.02416 !0.02500 !0.02444 !0.02355 !0.02662 !0.02815 !0.02795 !0.02611 !0.02952 !0.02557 !0.02569 !0.02743 !0.03042 !0.03093 !0.02745 !0.02978
0.04220(1.7) 0.04890(2.0) 0.05500(2.2) 0.04310(1.7) 0.03830(1.5) 0.04350(1.7) 0.03130(1.3) 0.02950(1.2) 0.05140(2.1) 0.05220(2.1) 0.04860(1.9) 0.04800(1.9) 0.05250(2.1) 0.04550(1.8) 0.03810(1.5) 0.03850(1.5) 0.05050(2.0) 0.03440(1.4) 0.04750(1.9) 0.03990(1.6) 0.04210(1.7) 0.03390(1.4) 0.03670(1.5) 0.04090(1.6)
!0.03400 !0.03060 !0.02920 !0.03420 !0.03440 !0.03190 !0.03710 !0.03560 !0.02880 !0.03140 !0.03260 !0.03240 !0.02100 !0.03450 !0.03610 !0.03650 !0.03490 !0.03980 !0.03640 !0.03700 !0.03310 !0.03380 !0.03630 !0.03350 !0.04210
000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
3 0 1 3 1 1 2 1 2 3 4 2 1 5 6 6 3 7 2 1 5 6 7 3 3
2 1 2 4 2 0 1 2 1 2 3 1 4 4 5 5 2 6 1 0 4 5 6 2 2
!units of cm~1 atm.
Giesen et al. The agreement between the calculated and measured halfwidth is not good. Ref. [27] states that `the collisional narrowing e!ect has been observed clearly in the high-J transitions of H O seeded in Ara. The argon-broadening measurements of Giesen et al. range from JA"9 to 14. 2 It is clear from Table 4 that none of the transitions compared demonstrate good agreement between the calculations and the measurements of Giesen et al. However, as the rotational quantum number increases from JA"9 to 12, 14, and 15 the agreement becomes much worse indicating that a large part of the poor agreement may be due to collisional narrowing e!ects not accounted for in the current calculations. The comparison of the line shift do not show the same level of agreement as demonstrated for the higher vibrational bands recorded by Grossmann and Browell [26]. This is due, in part, to the comparison of numbers of smaller magnitude. For the rotational and l bands, there are a number of examples in the literature [55,56] where the 2 uncertainty of the line shift, for some transitions, is of the same order of magnitude as the line shift. It is also noted that in the paper of Giesen et al. [27] the line shift for air-broadening could not be determined by the weighted mean of the shifts measured for nitrogen and oxygen separately. In a following paper [57], the same group reports measurements using the same instrument with
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Table 4 Halfwidth and line shift in units of 10~3 cm~1 atm for transitions in the 3l #l and 1 3 2l #2l #l bands of H O. CRB calculations compared with the data of Giesen et al. [27] 1 1 1 2 Upper
Lower
c (CRB)!
d (CRB)!
c"
14 1 13 2 13 1 11 2 9 2 9 3 9 5
15 0 14 1 14 2 12 1 10 2 10 4 10 6
13.2 15.5 15.5 22.0 35.8 38.1 29.7
!5.48 !5.84 !5.95 !5.10 !4.80 !3.06 !4.08
3.45(5) 5.98(5) 5.77(5) 7.80(3) 18.74(9) 24.62(3) 14.59(7)
14 12 12 10 7 6 5
15 13 13 11 8 7 4
d" !3.56(10) !3.57(24) !3.87(35) !3.67(5) !5.10(58) !3.12(24) !3.23(19)
!CRB results at 296 K in units of 10~3 cm~1 atm. "Giesen et al. [27] at 298 K in units of 10~3 cm~1 atm.
Table 5 Halfwidth in units of 10~3 cm~1 atm for transitions in the rotational band of H O, CRB 2 calculations compared with measured values at 300 K Line
Upper
Lower
c!
Ref.
CRB"
22 GHz 183 GHz
5 23 3 13
6 16 2 20
4 14
3 21
[29] [23] [28] [21]
44.8 50.0
380 GHz
52.2 50.2(0.76) 49.9(1.52) 45.4(0.8); 46.4#
47.7
!Measured values in 10~3 cm~1 atm. "CRB calculations in 10~3 cm~1 atm at 296 K. #Calculated value by RRB formalism.
increased accuracy and sensitivity for which the line shift data do obey the linear relationship. Unfortunately, there were no results for broadening by argon reported in this work. Finally in Table 5, measured halfwidth for 3 rotational transitions of water vapor broadened by argon are presented with the corresponding CRB calculations. Note, one RRB calculation by Bauer et al. [21] is also presented in the table. The agreement ranges from 14 to !5% with an average of 2.3%. For the one transition for which two measurements are available, the agreement between measurements and calculation are less than a percent.
4. Conclusions These calculations demonstrate that for the H O}Ar system, an intermolecular potential 2 comprised of an anisotropic atom}atom part and an isotropic induction and London dispersion part yields good values for the halfwidth and line shift. With the exception of the study of Giesen et al., where collisional narrowing may be a factor, good agreement between theory and
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experiment is observed. The expansion of the atom}atom potential in terms of the center of mass intermolecular separation needed to be made to 12th order to insure convergence of the potential. The induction and London dispersion isotropic potentials yield line shifts that are good agreement with measurement. Considering Figs. 1 and 2, it is interesting to note that increasing the order of the atom}atom potential has a di!erent e!ect on each line considered. The "nal results agree well with experiment, however there appears to be a shift between calculations and the measured values. For example, considering the line shift, Fig. 2 shows that the relative ordering of the values for the 4 lines considered is correct but that better agreement would be observed if the calculations were decreased slightly in magnitude. A decrease in magnitude of the calculated halfwidths would also give better agreement with the experiment of Grossmann and Browell. The calculations presented here, as well as those of other studies [7,11], indicate that the imaginary components of the Liouville scattering matrix strongly a!ect the determination of the halfwidth. This e!ect can vary greatly from transition to transition. For the H O}Ar system 2 di!erences as large as 37% are observed. Clearly, this e!ect must be accounted for if the calculations are to be meaningful. The results from the experiments indicate that velocity changing collisions are important for the H O}Ar collision system. This e!ect is observed when the di!erence between the mass of 2 the radiating and perturbing molecules is large, as is the case here. It was also pointed out by Dicke [58] and by Rao and Oka [59] that collisional narrowing is signi"cant if the quantum state of the radiating molecule is not a!ected by collisions and the mean free path is smaller than the wavelength of the probing radiation. Certainly, the "rst of these conditions is met by high JA states of H O for which collisionally induced transitions to allowed states involves large energy changes. 2 In conclusion, it is noted that the CRB formalism gives good agreement with measured halfwidths and line shifts for many transitions. As the rotational quantum number of water vapor increases collisional narrowing is observed in the measurements. To account for this the semiclassical model of Robert and Bonamy should consider approximate corrections as indicated by Robert et al. [60] and by Billing [61]. It should be noted that evidence for collisional narrowing has also been noted for other perturbing species (N , O ) and large J++. Hence, it is hoped that such 2 2 a correction should improve the results for other perturbing molecules.
Acknowledgement The authors are pleased to acknowledge support of this research by the National Science Foundation through Grant No. ATM-9415454.
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