Precision considerations of classical and semiclassical methods used in collision line broadening calculations: Different linear molecules perturbed by argon

Journal of Quantitative Spectroscopy & Radiative Transfer 119 (2013) 84–94

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Precision considerations of classical and semiclassical methods used in collision line broadening calculations: Different linear molecules perturbed by argon Sergey V. Ivanov a,n, Oleg G. Buzykin b a b

Institute on Laser and Information Technologies, Russian Academy of Sciences, 2 Pionerskaya Str., 142190 Troitsk, Moscow Region, Russia Central Aerohydrodynamic Institute (TsAGI), Zhukovski, Moscow Region 140160, Russia

a r t i c l e in f o

abstract

Article history: Received 4 September 2012 Received in revised form 13 December 2012 Accepted 21 December 2012 Available online 3 January 2013

The accuracies of the classical and semiclassical methods currently used in collisional spectral line broadening calculations of molecules are compared. The primary goal is to elucidate the validity of each particular method applied to linear molecules that have different rotational constants but which are perturbed by the same atom. Vibration– rotational electric dipole absorption spectra of CO2, C2H2, CO, HCl and HF molecules perturbed by an Ar atom are examined at room temperature. The most accurate vibrationally independent potential energy surfaces (PESs) are applied to all molecular pairs. The following theoretical approaches are involved in cross-examination: two classical schemes (exact C3D and C3Diso) and four semiclassical formalisms (NG—of Neilsen and Gordon, PA—peaking approximation, SGC—of Smith, Giraud and Cooper, and RB—of Robert and Bonamy). Identical ‘‘exact’’ isotropic trajectories, driven by only the isotropic part of the PES, are used in C3Diso and in all semiclassical calculations. The comparison is made with experimental data as well as with benchmark quantum dynamical results obtained in the same conditions within close coupling and coupled states schemes. Quantum results reproduced the experimental data excellently in all cases considered, which is as it should be if the interaction PES is accurate. An exact classical C3D approach displays good results for all molecules. By contrast, NG, PA and SGC semiclassical formalisms seriously underestimate line broadening for molecules with small rotational constants (CO2, C2H2 and, to a lesser extent, CO), mainly due to the use of isotropic trajectories in these schemes. The RB method strongly overestimates line broadening for the majority of values of the rotational quantum number J for all molecules studied. The only exception is the case of HF molecules at high J values, where PA, SGC and RB semiclassical schemes provide good results, coinciding well with quantum CC calculations for J43. Mean overall magnitudes of relative errors, taking an average over all J values and all five molecules studied and of the methods considered are C3D—11%, C3Diso—14%, NG—14%, PA—12%, SGC—14% and RB—21%. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Collisional line broadening Half-widths Accuracy Linear molecules Trajectory calculations Classical and semiclassical methods

1. Introduction Accurate knowledge of molecular spectral line shapes in different environments is crucial for various up-to-date n Corresponding author. Tel.: þ 8 495 334 0992; fax: þ8 495 334 0201. E-mail address: [email protected] (S.V. Ivanov).

0022-4073/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jqsrt.2012.12.021

applications, such as combustion diagnostics and remote sensing of terrestrial and planetary atmospheres. The knowledge of the broadening and shifting coefficients of rovibrational lines is required in spectroscopic models targeted at the solution of the inverse spectroscopic problem, for example, the retrieval of gas temperature, pressure, concentrations and the like. This process is very sensitive to both inaccuracies in input data, caused by

S.V. Ivanov, O.G. Buzykin / Journal of Quantitative Spectroscopy & Radiative Transfer 119 (2013) 84–94

instrumental errors or noise, amongst other things, and/or to model defects. Thus, the requirements for underlying spectroscopic data are very high. For example, it was observed in Ref. [1] that, if the error in the underlying broadening coefficients exceeds 5%, then this becomes the main source of uncertainty in the total error of determination of the Earth’s atmospheric parameters. A typical accuracy of better than 1–2% is now required for adequate atmospheric retrievals of various species [2,3]. An estimation of retrieval errors for the proposed satellite mission ACCURATE revealed that inaccuracies of spectroscopic line parameters predominate over all other sources of errors [3]. The role played by collisional broadening theory in this respect is essential. The theoretical line shape models should now reproduce available laboratory measurements within 1–2% or even better for strong lines, in order to be reliably used for the estimation of line parameters not yet measured or not accessible [4,5]. Remember that although the capabilities of modern experimental instruments are strong, they are not infinite. There are many situations where measurements of broadening and shift coefficients are difficult to perform, or the interpretation of experimental results in a correct and accurate way is extremely challenging, as in the case of the dense spectra of polyatomic molecules, especially at high temperatures. Also, the experimental data coming from different laboratories may differ drastically, despite narrow error bars indicated. When faced with deadlocks such as these, a good physical theory may serve as a clue. The term ‘‘good’’ means that such a theory (1) is independent of particular experimental data, in that it does not include any adjustable parameters already fitted to describe experimental results, either in the theory itself or in terms of intermolecular potential and (2) has sufficiently well-evaluated accuracy for different molecular pairs at broad temperature and pressure intervals. If it does not meet these criteria, the theory should be categorized as misleading. Two sources of error are obviously present in broadening and shifting calculations: the first is related to approximations of the theory itself and the second arises from the inaccuracy of the intermolecular potential energy surface (PES). Currently, there is a strong need for the application of computational schemes based on the most accurate and physically substantiated selfconsistent theoretical approaches, using robust and realistic PESs. Therewith, the rapid progress of computer facilities over the last two decades made such PESs accessible for many simple molecular pairs. Let us now consider the assumptions employed in existing computational schemes of impact broadening and the shift of molecular vibration–rotational spectral lines. There are three principal kinds of theoretical methods used in calculations: quantal, semiclassical, and classical. The most rigorous and accurate are full quantum-mechanical calculations within the exact close coupling (CC) method [6] and within the coupled states (CS) approximation [7]. However, both schemes, particularly CC, become extremely time-consuming when many rotational states are populated. Therefore, practical applications of these methods are, as a rule, restricted to simple molecular systems at not too

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high values of the rotational quantum number J, mainly at low and room temperatures. Semiclassical methods are less cumbersome since they treat translational motion classically, but the internal motions of vibration and rotation are modeled within quantum mechanics. The first defect of all these methods is the inability to account exactly for interactions of the translational and internal motions of colliding molecules. In all semiclassical methods, the influence of internal motion on the classical path and back-influence are simply ignored. At best, the trajectories are calculated numerically, using the isotropic part of the intermolecular potential - the so-called ‘‘exact’’ isotropic trajectories. Such trajectories are planar and symmetrical with respect to the point of closest approach. As a result, some essential features connected with rotation–translation coupling are completely omitted. For example, the formation of collision complexes is impossible (see Ref. [8] and the references therein). The second kind of semiclassical simplifications are related to the quantum mechanical characterization of rotation. In the most rigorous formalism of Neilsen and Gordon (NG) [9,10], the scattering matrix for rotation is calculated exactly using a ¨ numerical solution of the time-dependent Schrodinger equation. NG formalism corresponds to the reference semiclassical method. Subsequent simplifications generate the peaking approximation (PA) [11] and Smith– Giraud–Cooper (SGC) methods [11–16]. The differences between the NG, PA and SGC methods are explained in more detail in Refs. [17,18] (see also the book Ref. [19]). SGC formalism may be regarded, although not historically, as a starting point for many coarser but simpler semi-analytical schemes, such as those of Anderson– Tsao–Curnutte (ATC) [20,21] and Robert and Bonamy (RB) [22] (for details, see Ref. [23]). However, it is clear that these schemes are incapable of producing more accurate results than the SGC method, to say nothing of a comparison with PA and NG formalisms. Note that there are few results in the literature devoted to the application of NG, PA or SGC methods in line broadening and shift calculations. The vast majority of computations in the past were made using the ATC scheme and, at present and for the last three decades, the most widely used method is that of Robert and Bonamy - the traditional formalism with parabolic trajectories (RBP) [22] and its two modified versions, namely complex implementation (RBC) [24] and the recent variant with ‘‘exact’’ isotropic trajectories (RBE) [23,25–28]. The classical approach, originally proposed as early as 1966 by Roy Gordon [29,30], ensures the exact 3D selfconsistent characterization of rotational and translational molecular motions. This method is visual and rapid in computations. Recently regenerated, the classical approach, via its application to several atom–diatom and diatom– diatom molecular systems, has acquired a reputation as a very efficient and quite accurate instrument [31–37]. In our recent work [18], we have compared the accuracy of classical and basic semiclassical methods, namely, NG, PA and SGC, in the calculations of impact line widths of C2H2 molecules perturbed by argon and helium. The NG, PA and SGC methods have been proven

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not to be as accurate as was generally posited in the literature. The classical approach has displayed better, if somewhat overestimated, results than the semiclassical. Implementation of isotropic trajectories into classical broadening calculations produced similarly appreciably lower line widths, as is the case in semiclassical NG, PA and SGC methods. The final conclusion of the article [18] asserted the general non-applicability of simplified trajectories in broadening calculations using accurate PES. The use of even ‘‘exact’’ isotropic trajectories, such as those driven solely by the isotropic part of accurate PES, has been shown to be inadequate and risky. The subject of the present study is similar to [18], but deals with different linear molecules perturbed by the same noble atom. Our primary goal is to elucidate the validity of each particular theoretical method for molecules with different rotational constants and which have different rates of RT-exchange in collisions. Electric dipole absorption spectra of CO2, C2H2, CO, HCl, and HF molecules perturbed by an Ar atom are examined at room temperature. The most accurate up-to-date PESs are applied to all the molecular pairs. The following theoretical approaches are involved in cross-examination in identical conditions: two classical schemes, exact C3D and C3Diso—with artificially imposed ‘‘exact’’ isotropic trajectories—and four semiclassical methods, NG, PA, SGC, and RB. In this examination, we included the RB method (RBE), since this formalism, especially in its RBC and RBE versions, is used on a worldwide scale. However, one must note that the RB method is not clearly allied with the NG, PA, and SGC methods, which are related to one another, forming the explicit hierarchy [17–19]. The comparison is then made with benchmark quantum dynamical CC/CS calculations, as well as with experimental data. The paper is organized as follows. In Section 2, we briefly reiterate the details of the present classical and semiclassical computations, including some modifications to the classical Gordon method and the description of the interaction potentials used. Section 3 contains the results of all the molecular systems considered and discussions thereof, as well as quantitative cross-evaluation of the accuracy of classical and semiclassical methods. The final section, Section 4, summarizes the conclusions and forecasts the further development of classical and semiclassical methods. 2. Details of computations The computations in this work have been made according to two classical methods and four semiclassical methods, NG, PA, SGC, and RBE, for five molecular systems: CO2–Ar, C2H2–Ar, CO–Ar, HCl–Ar and HF–Ar. All molecules were treated as rigid rotors; therefore only vibrationally independent interaction potentials were used. For this reason, no attempts at shift calculations were done in this work. The identical ‘‘exact’’ isotropic trajectories, driven only by the isotropic part of the PES, were applied in all semiclassical calculations. The same isotropic trajectories were also inserted into an auxiliary classical C3Diso scheme with imposed isotropic

trajectories. Quantum-mechanical results needed for the comparison were taken from other articles. 2.1. Semiclassical methods The main formulas and algorithms of basic semiclassical formalisms, namely NG, PA, and SGC, have been presented previously and are not discussed here again. Note that all the methods used here are of the standard level as described in the literature. The computational details of the Neilson– Gordon method are specified in Refs. [9,10,17,18] and the peaking approximation in Refs. [17,18]. As for the SGC method, we do not use its initial variant [11], because it employs several unnecessary simplifications aiming to obtain analytical results. These were later relaxed by Hutson and Howard [15,16]. In fact, our variant of the SGC algorithm is that of Hutson and Howard [15]. We have also described it in detail in Ref. [18]. Since the molecules studied here have different rotational constants, we used different dimensions Jmax of scattering matrices in the present calculations. In the case of the CO2 molecule, we set Jmax ¼71 for the PA and SGC methods. For the NG method in the case of CO2, we were forced to restrict ourselves by setting Jmax ¼32 due to the extremely timeconsuming computation. In addition, Jmax ¼ 22 was used for the comparison with a greater number of trajectories, giving better Monte Carlo statistics. For the C2H2 molecules, we set Jmax ¼32 in the case of the PA and SGC methods and Jmax ¼15 and 9 in the case of NG. For other molecules, we set Jmax r21 (CO) and Jmax r9 (HCl, HF). As for the Robert–Bonamy semiclassical method, here we test its RBE version, with ‘‘exact isotropic trajectories’’ (for details, see articles [31–33] and the book [23]). We have developed an independent semiclassical RBE computer code, following the Eqs. (1), (3–5) of Ref. [32]. Test calculations showed a close fit with the results of Refs. [31–33]. However, in our present calculations, we did not use Eqs. (6) and (7) of Ref. [32], but prepared an isotropic classical path directly from the trajectory equations, as is the case for the NG, PA, and SGC methods. Another distinctive feature of our RBE calculations is the application of the Monte Carlo method for averaging over the impact parameter and relative velocity, instead of the regular method. In short, our RBE code is the result of simplification of the SGC code, retaining its structure, and therefore all of the semiclassical methods examined here have the same software architecture, the quality of which is extremely practical for the comparison. 2.2. Classical methods The equations of exact classical trajectory 3D dynamics (C3D) for the atom-rigid rotor system can be found in Ref. [38]. In this section, we reiterate only the classical expression for electric dipole absorption impact halfwidth [29,30] D  aE g ¼ nb vrel 1Pel cos Zcos2 : ð1Þ 2 b,vrel ,0 Here, nb is the number density of perturbing atoms; Pel is the probability of the collision to be elastic, calculated via

S.V. Ivanov, O.G. Buzykin / Journal of Quantitative Spectroscopy & Radiative Transfer 119 (2013) 84–94

the box-quantization procedure (see Eqs. (7,8) of Ref. [34]). In Eq. (1) vibrational dephasing is neglected, since molecules are treated as rigid. The average /yS is taken over the impact parameter b, relative velocity vrel and over the initial 3D orientations, denoted here by 0, of the vectors ! ! of the molecular axis r and its angular velocity o . The variable Z characterizes ‘‘rotational dephasing’’ and a is the angle between the initial and final orientations of ! vector o , or ‘‘rotational reorientation’’. The values of Z and a are extracted from the classical dynamics of each particular collision. In this article, we present a clear description of the algorithm for the rotational phase shift Z determination originally proposed by Gordon [29,30], with some essential modifications (see Appendix A). Test calculations show that the application of this corrected algorithm, as compared to our previous classical calculations, does not change the linewidths for molecules with small rotational constants like CO2 and C2H2, where inelastic collisions play a dominant role in line broadening, but greatly improves the results for molecules with large rotational constants, like HCl, and HF, making them noticeably closer to the measurement than before. One must note that this correction in the algorithm does not influence the spectra, which are completely independent of ‘‘rotational dephasing’’ Z, namely, the isotropic Raman scattering when neglecting centrifugal distortion.

2.3. Trajectory calculations Two variants of classical calculations of line broadening are made: (1) C3D—an exact self-consistent calculation which uses realistic trajectories obtained by employing full interaction PES and (2) C3Diso—classical broadening calculations allied with artificially implemented ‘‘isotropic trajectories’’ (ISO). ISO trajectories are obtained by using only the isotropic part Viso(R) of an interaction potential (see below). They are planar (i.e., 2D) and symmetrical with respect to the point of the closest approach of particles, since they neglect the back-influence of the RT-exchange on the classical path. The details of this idea, useful illustrations and some examples for different molecular pairs, are given in Refs. [32,33,35–37]. As for the semiclassical methods, it should be emphasized that they are all based on using ISO trajectories as their principal feature. Note that in the present semiclassical and classical ISO computations, identical trajectories were used. The initial intermolecular distance was set large enough ˚ to exclude starting interaction between mole(Rmax ¼15 A) cules in all of the cases considered. Monte Carlo sampling ˚ as was applied to the impact parameter b (brbmax ¼ 12 A), well as in classical calculations to the initial orientations of

87

r vectors in 3D space. The mean thermal velocity o and !

!

(MTHV) approximation was applied in most of the calculations, whereby the initial relative speed offfi molecules in all pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi collisions was set equal to v ¼ 8 kB T=pm (where kB is the Boltzmann constant, m is the reduced mass of a colliding pair and T is an absolute temperature). The MTHV approximation is very efficient in reducing the time of computations, but needs verification when the results are compared with the measured data. It has been shown in many studies (first in Ref. [39], see also Refs. [33,35,36]), that the difference between the MTHV line-widths and the proper integral ones depends on the molecular system, interaction potential, varies with J and with temperature, having increased error as the temperature decreases. The lengths of the active molecules and their rotational constants were set equal to their values in a ground vibrational state (see Table 1). The initial rotation frequency of an active molecule in classical C3D and C3Diso methods was determined via its rotational quantum   number J by the rigid rotor formula o0 ¼ 2B=_ J þ 1=2 , often called a Langer prescription [40]. It is much closer to reality for J¼0, 1 p than the usual quantum-mechanical ffiffiffiffiffiffiffiffiffiffiffiffiffiffi formula o0 ¼ 2B=_ J ðJ þ1Þ gives; for other J values the difference between these two formulae is negligible. Also, in the present Monte Carlo calculations, we applied the very effective algorithm of b-sampling as described in the article of Chapman and Green [41], which speeds up our computations approximately twofold. Classical trajectory equations, both ISO and exact C3D, were integrated numerically by the implicit BDF Gear method [42]. For C3D, we have 11 first-order differential equations in body-fixed coordinates [38], and for C3Diso, 9 equations. In C3Diso, the molecular rotation is 3D, while the classical path is 2D – it is the so-called planar approximation of Pattengill [38]. All the calculations were conducted using double precision with a typical tolerance parameter of 10  9 and a variable step of integration within fixed time-grid intervals Dt ¼0.05  10  13 s. The statistical error of line width computations, an RMS error of Monte Carlo averaging, was, in most cases, kept at the level of  1%. 2.4. Potential energy surfaces The PES of the atom-rigid linear rotor interaction was expressed in terms of Legendre polynomials, Pl(cosy) by the expansion V ðR, yÞ ¼

lmax X

V l ðRÞP l ðcos yÞ ¼ V iso ðRÞ þ V aniso ðR, yÞ,

l¼0

V iso ðRÞ ¼ V 0 ðRÞ,

V aniso ðR, yÞ ¼

lX max

V l ðRÞP l ðcos yÞ:

ð2Þ

l40

Table 1 The parameters of the studied active molecules (length r and rotational constant B) in their ground vibrational states. Parameter

12 16

12

12 16

˚ r (A) B (cm  1)

2.329633

3.323286

1.130843

0.3902

1.176642

1.923

C O2

C2H2

C O

H35Cl 1.2908 10.439826

H19F 0.925574 20.559

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In the case of a symmetric molecule—in the present study, CO2 and C2H2—only even Legendre polynomials l ¼0, 2, 4, y appear in the series for the PES. Radial functions Vl(R) are extracted from the exact PES, as follows: Z 2l þ 1 p V l ðRÞ ¼ V ðR, yÞP l ðcosyÞsinydy ð3Þ 2 0 The PESs in this work and the limiting number lmax in their Legendre expansion were selected for the reasons of precision and adequate comparison with available quantum CC/CS results. Note that all the PESs we use here are of benchmark accuracy. The need-to-know information on these PESs is collated in Table 2.

Some words regarding the apparent oscillations of NG, PA and SGC broadening coefficients (see figures (a)) for the CO2 and C2H2 molecules should be mentioned. These oscillations do not have a physical nature, and originate from the inevitable constraint of the scattering matrix dimension Jmax. Note, however, that when calculating relative errors (figure (b)), we usually smoothed 5–10 last points where oscillations are apparent, in order to diminish maximum error and make the comparison with other methods more adequate and realistic. The examples of NG calculations made with two different Jmax values are presented in Fig. 1 (CO2) and Fig. 2 (C2H2). Remember that the time of calculations in the NG method is roughly proportionate to J 6max [19]. For example, in the case of CO2

3. Results and discussion

CO2-Ar T = 296 K

HWHM (cm-1 atm-1)

0.10

CC CS Expt

NG Jmax = 22 NG Jmax = 32 PA SGC RB C3D C3Diso

0.09 0.08 0.07 0.06 0.05 0.04 0.03 0

Relative error. %

In this section, we present the results of calculations starting from a CO2 molecule, which has the smallest rotational constant, and finish with the HF molecule, which has the largest rotational constant. All the results are shown in Figs. 1–5. Each figure consists of two parts: (a) the values of the broadening coefficient (HWHM half-widths at half-maximum) as a function of J value for different methods, and (b) the relative errors of each particular method versus J. Note, firstly, that the accuracy of a particular PES may be examined only by using a perfect method and, strictly speaking, only within a fully quantum CC scheme, which solely can be treated as irreproachable. Even quantum CS formalism is approximate -the defects of this scheme were thoroughly discussed in Refs. [50–52]. However, at the high energies of both translational and rotational molecules, the results of the CS method are really close to those of the exact CC method (see Refs. [35–37]). The experimental data projected onto the figures ascertain that the quantum CC/CS results coincide closely with measured values. This confirms that the PESs chosen for the molecular pairs considered are really good. Therefore, we decided to set the combined CC/CS results as benchmark data for the further intercomparison of other theoretical methods. It should be noted that CS values were used only if CC data were absent. The experimental data are put into figures only for the completeness of the picture, mainly to confirm CC/CS calculations and, in particular, to reveal the scatter of experimental data produced by different authors, as shown in the figures for the C2H2 and HCl molecules. The scatter of measured data is the main reason for us not using them as a benchmark. Our purpose here is not to fit measured data, but to compare quantitatively different methods in identical conditions using CC/CS benchmark results -note that most of them are obtained within MTHV approximation.

45 40 35 30 25 20 15 10 5 0 -5 -10 -15 -20 -25 -30 -35 -40

10

20

30

40 J

CO2- Ar T = 296 K NG Jmax = 22 NG Jmax = 32 PA SGC RB

0

10

20

30 J

50

60

70

80

C3D C3Diso

40

50

60

Fig. 1. (a) Half-widths g of the n3 IR dipole absorption lines of CO2 molecule buffered by Ar at T¼ 296 K. CC/CS values and the experimental data taken from Ref. [51] (see also Ref. [53]); (b) Intercomparison of the relative errors e ¼(g  gCC/CS)/gCC/CS of different methods with respect to full quantum CC/CS values.

Table 2 The need-to-know information on the interaction potential energy surfaces used in the present calculations. CO2–Ar

C2H2–Ar

CO–Ar

HCl–Ar

HF–Ar

Hutson et al. [43], split repulsion, lmax ¼ 24

Yang et al. [44], lmax ¼8

Toczy"owski and Cybulski [45], see also [46,47], lmax ¼15

Hutson [48], H6(4,3,0), lmax ¼8

Hutson [49], H6(4,3,2), lmax ¼ 8

S.V. Ivanov, O.G. Buzykin / Journal of Quantitative Spectroscopy & Radiative Transfer 119 (2013) 84–94

0.11

0.09 0.08 0.07 0.06

C2H2 - Ar T = 297 K expt 1 (P, LS1) expt 1 (P, LS2) expt 1 (R, LS1) expt 1 (R, LS2) expt 2, P expt 2, R

CC NG Jmax = 9 NG Jmax = 15 PA SGC RB C3D C3Diso

0.05 0.04

CO - Ar T = 296 K

0.09

CC MTHV CC thermally averaged CS thermally averaged Expt of Luo et al. NG PA SGC RB C3D C3Diso

0.08 HWHM (cm-1atm-1)

HWHM (cm-1 atm-1)

0.10

89

0.07 0.06 0.05

0.03 0.04

35 C H - Ar T = 297 K 2 2 30 NG Jmax = 9 25 NG Jmax = 15 PA 20 SGC 15 RB C3D 10 C3Diso 5 0 -5 -10 -15 -20 -25 -30 0 2 4 6 8 10 12 14 16 18 20 22 J

Fig. 2. (a) Half-widths of the IR dipole absorption lines of C2H2 molecule buffered by Ar at T¼ 297 K. Experimental data expt1 correspond to n5 band [54] and expt2 to n1 þ n3 band [55]. Symbols P and R denote P- and R- branches. LS1 and LS2 mean different line shapes used in the experimental spectral fitting procedure (LS1 – Voigt line shape; LS2 – general Rautian line shape). CC values are taken from the Ref. [56]. (b) Intercomparison of the relative errors of different methods.

molecules, our NG calculations took 2 months for Jmax ¼22 at a Monte Carlo error of o2%, and the same time for Jmax ¼32 at a Monte Carlo error of  5%. No problem of this sort arises in the classical approach, to say nothing of the speed of calculations, which is faster by an order of magnitude. Let us now compare the results of different methods. Fig. 1 concerns the CO2 molecule. The results of quantum CC/CS calculations are from Ref. [51]. Experimental data correspond to its n3 absorption band [53]. From Fig. 1, it is clear that the exact classical C3D method demonstrates excellent accuracy, similar to CC. For the majority of J values, the error of C3D is no more than few percent. The average error is 5%, and the maximum error is  22% (therewith it is solely for J¼0). NG, PA, and SGC methods fail, since they make the half-width too low and PA and SGC give a wrong J- dependence. The worst are the PA and SGC schemes, where the error reaches 36%. As for the RB method, it works decently, but makes the broadening slightly too large, with an error of o16%, where the mean value is 11%, and produces the somewhat wrong J- dependence for J o20. Note that the application of the isotropic path to the classical method significantly

0

30

2

4

6

8

10 12 14 16 18 20 22 J

CO-Ar T = 296 K

25 20 Relative error, %

Relative error, %

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 J

NG PA SGC RB C3D C3Diso

15 10 5 0 -5 -10 -15 0

2

4

6

8

10 J

12

14

16

18

20

Fig. 3. (a) Half-widths of the IR dipole absorption R-lines of CO molecule buffered by Ar at T¼296 K. Full quantum CC and CS results are taken from Ref. [57]. The experimental data corresponding to CO 0–1R- lines are from Ref. [58]. (b) Intercomparison of the relative errors of different methods.

impairs the results, making them too low (see C3Diso curves). It is interesting that, except for small J, the results of C3Diso practically coincide with those of NG and PA. Also, one should note that the artificial application of isotropic trajectories leads at J 425 to results similar to the semiclassical ones in frames of PA and SGC formalisms and, possibly, NG, but not for the RB. Recall that in all semiclassical schemes, the same isotropic trajectories are used. Fig. 2 concerns the C2H2 molecule. Experimental data ‘‘expt1’’ put in this figure correspond to n5 absorption band [54] and ‘‘expt2’’ to n1 þ n3 band [55]. One should note that the experimental data presented in Fig. 2 differ significantly, despite small and often non-overlapping error bars. These differences may be caused by the effects of nonrigidity and vibrations (measured data are for different vibrational bands and rotational branches), the use of an inadequate spectral line shape in the fitting procedure, and/or other reasons. In any case, at the present state of line broadening theory, such diverse experimental data cannot serve as benchmark information for the comparison of theoretical methods. This once

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0.1

HCl-Ar T = 296 K

HWHM (cm-1 atm-1)

HWHM (cm-1 atm-1)

CC S.Green CC F.Thibault NG PA SGC RB C3D C3Diso Expt 1 Expt 2 Expt 3

CC MTHV CC Maxwell average Expt of Grigoriev et al. NG PA SGC RB C3D C3Diso

HF-Ar T = 296 K

0.1

0.01

0.01 1E-3

0

1

2

3

4

5

6

7

8

0

9

1

2

3

HCl-Ar T = 296 K

Relative error, %

Relative error, %

35 30 25 20 15 10 5 0 -5 -10 -15 -20 -25 -30 -35 -40

NG PA SGC RB C3D C3Diso 0

1

2

3 J

4

5

6

Fig. 4. (a) Half-widths of the pure rotational absorption lines of HCl molecule buffered by Ar at T¼ 296 K. All data presented are velocity averaged. Full quantum CC values are taken from Fig. 4 of Ref. [59] and from Ref. [61]. Expt1, Expt2 and Expt3 correspond to experimental data of Refs. [62, 63] and [64–66], respectively. (b) Intercomparison of relative errors of different methods.

again demonstrates the necessity of rigorous and accurate quantum-mechanical CC calculations, upon which we may rely. We imported the results of quantum CC calculations from Ref. [56]. The accuracy of the NG, PA, and SGC formalisms in the C2H2 case is appreciably better than for CO2. However, the RB results here are worse – the largest error reaches 34% and the mean error is 25%. The RB method also makes broadening noticeably too large and, as before, gives the wrong J- dependence. NG and PA are good at low J ( o3), but fail at large J values. On the other hand, the SGC method is especially poor at J o5, but gets much better at large J. All semiclassical schemes, except RB, seriously underestimate broadening. As for the C3D method, its error is generally o10%. However, its maximum error reaches 24%, but solely at J¼0. The results for the CO molecule are collected in Fig. 3. Full quantum CC and CS results are taken from Ref. [57]. Note that velocity averaged CC results presented in Fig. 3 nearly coincide with the experimental data of Luo et al. [58], corresponding to R- lines of the fundamental

4

5

6

7

J

J 50 40 30 20 10 0 -10 -20 -30 -40 -50 -60 -70 -80 -90

HF-Ar T = 296 K

NG PA SGC RB C3D C3Diso

0

1

2

3

4

5

6

7

J Fig. 5. (a) Half-widths of the pure rotational absorption lines of HF molecule buffered by Ar at T¼ 296 K. Full quantum CC results are taken from the Ref. [67] (both velocity average and MTHV (unaverage)). The experimental data of Grigoriev et al. are taken from Ref. [68]. (b) Intercomparison of the relative errors of different methods (against CC MTHV).

absorption band. However, CS results are not accurate for Jo10. Note also that the RB method is the worst among the others, with an average error of 21%, since for J44 it strongly overestimates broadening and produces the wrong J-dependence. Note also that the classical method also gives the somewhat incorrect J- dependence of broadening at intermediate and high J values. C3D results are too large for J o12, with an error of 5–22%, but become excellent for J 412. The NG, PA and SGC methods noticeably underestimate broadening for J42 with a mean error of 9–10%, but they all, except PA at low J values, provide true J- dependence. Fig. 4 is devoted to HCl molecule. Here we use a logarithmic scale for better presentation of broadening coefficients, since they fall dramatically as J grows. We used the results of the quantum CC calculations of Green [59] which, however, employ the older H6(3) version of HCl–Ar PES of Hutson [60] in a Legendre expansion up to lmax ¼9 as a benchmark. This interaction potential is slightly different from that used in our calculations, namely H6(4,3,0) of Hutson [48], with a Legendre

S.V. Ivanov, O.G. Buzykin / Journal of Quantitative Spectroscopy & Radiative Transfer 119 (2013) 84–94

91

J-dependence of broadening, especially prominent in the case of C3Diso. Since C3D and C3Diso results are very similar at Jo4 - the relative difference is 0.6–4% - the conclusion could be drawn that the majority of HF–Ar trajectories for these low J values are well isotropic at the temperature considered. To be more precise, the small differences between real and isotropic trajectories do not influence line broadening at small J values. This leads to the conclusion that, in this case, NG results should coincide with quantum CC results, and this is indeed the case (see Fig. 5). However, starting from J¼4, despite the fact that real trajectories turn out to be more and more isotropic due to effective angular averaging of the PES through the rapid rotation, the influence of the aforementioned small difference begins to affect the line width appreciably, which gets very small at these high J values. This fact makes C3Diso values lower than exact C3D ones and NG values are up to 50% lower than quantum CC. The explanation of this interesting phenomenon deserves special consideration. However, it is clear that, since the trajectories in the case of HCl and HF molecules are closer to isotropic than for the other molecules studied here, this proves that the approximation of the isotropic path is good enough for HCl–Ar and HF–Ar, at least for low J values. Once again, this confirms that the use of the isotropic classical path is not the only, nor the principal defect of the RB model. It seems useful to collate the principal results of the relative precision examination of all the considered theoretical methods in one table (see Table 3). The exact classical C3D method provides the best results. Its mean overall magnitude of relative error (11%) is less than that of the semiclassical methods (12–21%). The maximum overall error of the C3D method (25%) has approximately the same magnitude as the other methods (22–38%). The C3Diso method has very similar accuracy to the semiclassical schemes, since it employs isotropic trajectories.

expansion up to lmax ¼8. Since the results of calculations presented in Ref. [59] are thermally averaged, we were also led to make thermally averaged calculations for all the methods, in order to provide an adequate comparison. For the sake of the detection of differences between broadening coefficients obtained with old H6(3) and new H6(4,3,0) HCl–Ar PESs, we added the recent data of Thibault [61],which employs the new H6(4,3,0) PES, to Fig. 4. It can be observed that the mentioned differences are, in fact, small. Experimental data for HCl–Ar were taken graphically from Ref. [59]. They correspond to the measurements of Refs. [62–66]. Fig. 4 shows that NG, PA and SGC results nearly coincide with each other and are quite accurate, with the exception of SGC at J ¼0, so that their mean error is 7–9%. The RB method is significantly worse, displaying the mean error  25%. The classical C3D approach behaves decently, displaying the mean error  12%. The largest error is  23%, in the cases of the lowest and highest J values. This mean error, however, is somewhat larger than the NG, PA and SGC errors. The last figure, Fig. 5, deals with the HF molecule that has the largest rotational constant (B ¼20.559 cm  1) among the considered molecules and, hence, has the largest probability of elastic collisions and, consequently, the smallest probability of RT- exchange. In this case, we also used a logarithmic scale to better discern small differences. Full quantum CC results and the experimental data are taken from Ref. [67] and Ref. [68], respectively. In order to coordinate our semi-classical calculations with the best ones available in the literature, we first had to compare our thermally averaged NG results with those of Hartmann and Boulet [17], obtained with exactly the same PES, and observed an excellent agreement (not shown in Fig. 5). Note also that for J 43, the MTHV approximation gives noticeably lower results. Fig. 5 shows that the results of the NG and PA methods correlate well at low J values. RB and SGC results are nearly the same, giving a large error due to the wrong J-dependence at low J values (Jo4). Note that the PA, SGC and RB semiclassical schemes provide very good results for J4 4, coinciding well with quantum CC calculations. In comparison, the NG method is excellent for Jo4, but fails at higher J. With regards to the classical C3D method, it provides a relatively stable and accurate result, with an average error of 22%, the same as for semiclassical methods. However, here one may notice a slightly wrong

4. Conclusions In this study, we have thoroughly examined the validity of the classical and semiclassical methods that are currently used in the impact broadening calculations of the vibration– rotational spectral lines of molecules. The results of fully quantum CC/CS calculations were used as a benchmark. The prime study of such kind was launched in Ref. [18] for C2H2–Ar, He systems. Now, for the intercomparison,

Table 3 Cross-comparison of relative accuracy of the semiclassical (NG, PA, SGC, RB) and the classical (C3D, C3Diso) methods used in calculations for different molecules perturbed by the same argon atom. Indicated: the mean and maximum (in brackets) values of the magnitude (absolute value) of the relative error (in %) of a particular computational method for a given molecular pair. More detailed information is given in Figs. 1–5. Molecular pair

Theoretical method NG

CO2–Ar C2H2–Ar CO–Ar HCl–Ar HF–Ar Mean overall

18 13 9 9 22 14

PA (25) (16) (11) (13) (51) (23)

19 17 10 7 9 12

(34) (28) (15) (12) (19) (22)

SGC

RB

19 14 9 9 20 14

11 25 21 25 23 21

(36) (28) (13) (35) (66) (36)

(16) (34) (26) (29) (83) (38)

C3D

C3Diso

5 6 8 12 22 11

14 10 9 11 28 14

(22) (24) (22) (23) (32) (25)

(23) (23) (21) (25) (69) (32)

92

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we have chosen several linear molecules that have different rotational constants (CO2, C2H2, CO, HCl, HF), but which are perturbed by the same Ar atom. Only electric dipole absorption spectra are examined at the same room temperature. The most accurate up-to-date potential energy surfaces are applied for all the molecular pairs. The following theoretical approaches are involved in the examination: two 3D classical schemes, exact C3D and C3Diso with imposed ’’exact’’ isotropic trajectories, and four semiclassical methods (NG—of Neilsen and Gordon, PA—peaking approximation, SGC—of Smith, Giraud and Cooper, RB—of Robert and Bonamy, in its RBE version). The identical ‘‘exact’’ isotropic trajectories, those driven by only the isotropic part of the PES, exactly as in C3Diso, are used in all semiclassical calculations. First of all, it should be emphasized that the results of fully quantum CC/CS calculations reproduce experimental data excellently in all the cases considered. This confirms the high quality of the intermolecular potential energy surfaces that we selected for our comparative study. The exact classical C3D approach displays fair accuracy for all the molecules studied, except for HF. Note that the C3D accuracy gets better for the molecules with small rotational constants B. By contrast, the NG, PA, and SGC semiclassical methods seriously underestimate broadening for the molecules with small rotational constants (CO2, C2H2 and, to a lesser extent, CO). The results of additional classical isotropic calculations (C3Diso) reveal that this fact originates from the use of isotropic trajectories in these schemes. The RB method strongly overestimates line broadening for all the molecules considered. The only exception is the case of HF molecule at high J values, where the PA, SGC, and RB schemes provide good results, coinciding well for J43 with quantum CC calculations. Table 3 shows that the Neilsen– Gordon method, the most rigorous semiclassical method, sometimes displays less accuracy than the other semiclassical ones. This is clearly due to the partial cancellation of inaccuracy caused by the use of isotropic trajectories by the additional errors of the PA, SGC and RB schemes, which produce an illusion of better quality. For example, in PA, when calculating the evolution operator, the intermolecular ! axis R ðtÞ is assumed not to be rotating during the collision. In terms of NG formalism, this means dropping all Wigner D- matrices in Eq. (11) of Ref. [17] (see also Eq. (14) of Ref. [18]). The peaking approximation is justified if the kinetic energy of the collision is high and the impact parameter is low [17]. In proper conditions, such as for some values of J and relative speed, the use of PA may appear to create an ‘‘improvement’’. In the SGC method, in addition to PA, the time ordering in the evolution operator (see Eq. (11) in Ref. [17] or Eq. (14) in Ref. [18]) is neglected, so that the scattering S- matrix is simply approximated as the first term in the Magnus expansion of time-ordered exponentials. This spoils basic physics, but leads to the expression of the S- matrix in a closed form. The C3Diso method—a classical method with artificially imposed isotropic trajectories—is worse, sharing similar accuracy with the semiclassical schemes. As well as Ref. [18], our present study reveals that the use of simplified trajectories in broadening calculations, even if they are ‘‘exact’’ isotropic, seems to be inadequate. Realistic

trajectories differ from the isotropic ones. They are 3D, not symmetric when R-T exchange takes place and are sometimes quite fanciful, such as when a collision complex is formed [8] (see illustrative examples in Ref. [32]). Naturally, all these have a bias on line widths and, possibly, on line shifts and spectral shapes. Consequently, the methods based upon the use of isotropic trajectories are not adequate, since they violate basic physics. Summing up, we note that all semiclassical methods in their present variations should be treated as only qualitative, because their poor accuracy does not satisfy modern requirements. The classical method is overall  1.5 times more accurate than the NG and PA schemes, not to mention the fact that it is much faster in computations and  2 times more accurate than the RB formalism. However, in some cases, its errors are intolerably high, especially for Jr1 for most of the molecules considered here, and this, consequently, strongly spoils the statistics presented in Table 3. Luckily, the perspectives of accuracy are increasing in classical and semiclassical approaches. In the classical approach, the ‘‘weak point’’ is the procedure of Pel determination. In the present variant, it is very roughly based on ‘‘box-quantization’’, but it is possible to apply some more sophisticated approach based on perturbation theory that will be possibly more effective for molecules with large rotational constants. The invention of combined procedures looks very attractive as well. Another disappointing thing in the present variant of the classical method is the too high results at J¼0 for electric dipole absorption lines of nearly all the molecules studied. This is another matter that can be improved in future. As for semiclassical methods, it would be interesting to embed in them the exact 3D classical trajectories, instead of isotropic ones. Meanwhile, one should recall that this will strongly increase CPU time, since trajectories will become J- and m- dependent. As a consequence, rigorous NG and PA methods have no prospects, at least for molecules with small B, due to terrible CPU time consumption at JZ20. It would seem that the mentioned hybrid approach is feasible only within the SGC or RB schemes, as it retains all their other intrinsic defects.

Acknowledgments The authors are grateful to Jeremy M. Hutson for providing them HCl, HF–Ar potentials in the form of FORTRAN code and to Franck Thibault for his many useful remarks, as well as for his very recent HCl–Ar close coupling data. Andrei Vigasin is acknowledged for his constant interest and Alexey Okhapkin for his careful reading of the manuscript. This work was partly supported by the RFBR grant 12-05-00802. Appendix A. Improved algorithm for the calculation of rotational phase shift g The general idea in calculating rotational phase shift is to somehow approximate the three-dimensional kinematics of an actual collision, specified by exact classical calculations, by means of the ‘‘instantaneous’’ one, as impact

S.V. Ivanov, O.G. Buzykin / Journal of Quantitative Spectroscopy & Radiative Transfer 119 (2013) 84–94

→ Ji → Jf

α

→ μ (tf)

→ μref (tf)

η

Θref Θ

→ i

Fig. 6. Geometry describing the reorientation of molecular angular ! momentum J (angle a) and formation of rotational phase shift Z as the result of a collision.

theory demands. The initial algorithm was described far back by Roy Gorgon in his two original articles [29,30] using ‘‘reference dipole’’, but it was somewhat vague. The main question is how to define the moment of time when the rotation plane of the reference dipole should instantly turn, if the molecule rotates many times before and after this turn? The algorithm presented below takes into account the multiple periods of molecular rotation before (m) and after (n) the moment of the instantaneous turn of the rotational plane and then minimizes the absolute value of rotational phase shift over (m, n) numbers. The rotational phase shift, Z, is determined by compar! ing the direction of the dipole m ðt f Þ at some time tf after the collision, with the reference direction which would have been reached at that time, had the dipole rotated freely with an initial angular frequency oi in the initial ! plane up to the line of intersection i in Fig. 6, and freely for the rest of the time up to tf in the final plane with final frequency of (see also figures in Refs. [29,30]). The collision starts and finishes at R¼ Rmax. Let the reference dipole make m and n complete revolutions before and ! after the line i respectively. The direction of the unit ! vector i of the intersection of the initial-final rotation planes is explicitly defined by the means of analytical ! ! geometry using the vectors of initial J i and final J f angular momenta. The angles y and Y (see Fig. 6) are obviously defined within a 2p interval, via means of ! ! analytical geometry using the vectors m ð0Þ, i and ! m ðtf Þ. The time elapsed from the start of the collision t ¼0 to the moment of the instantaneous turn of the rotational plane is t0 ¼((y þ 2pm)/oi). Since of(tf  t0 )¼ Yref þ2pn, the rotational phase shift now depends on two integer numbers m and n 



Zðm,nÞ ¼ YYref ¼ Y þ 2pnof tf t0 ¼ Yof tf þy





of of þ 2p n þm , oi oi

m, n ¼ 0,1,2, mmax , nmax ; mmax



oi t f of tf ¼ Int , nmax ¼ Int : 2p 2p

The key point of our idea is as follows: the integer numbers, mn and nn, should be determined for which the absolute value 9Z(m,n)9 takes its minimum. The final rotational phase shift then takes the value Z ¼ Z(mn,nn) (the sign included). The reason for such a step is simple: if a collision is exactly instantaneous, then Z must be zero. The values of m and n numbers are constrained by the natural inequality (see Fig. 6) mUT i þ

θ

→ μ (0)

93

y

oi

þ nUT f o t f ,

where Ti ¼ 2p/oi and Tf ¼2p/of are periods of rotation in initial and final planes. Thus, we arrive at the inequality that restricts the range of (m, n) definition   m n þ o t f y=oi =2p:

oi

of

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