1. Refinement of the RB Formalism: Taking into Account Effects from the Line Coupling Q. Ma NASA/Goddard Institute for Space Studies & Department of Applied Physics and Applied Mathematics, Columbia University 2880 Broadway, New York, NY 10025, USA C. Boulet Institut des Sciences Moléculaires d’Orsay CNRS (UMR8214) and Université Paris-Sud Bât 350 Campus dOrsay F-91405, FRANCE R. H. Tipping Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA

2. I. The Robert-Bonamy Formalism Advantages and Weaknesses The RB formalism has been widely used in calculating half-widths and shifts for many years. In comparison with the Anderson-Tsao-Curnutte formalism, it is characterized by two features: (1) A non-perturbative treatment of the Ŝ matrix through use of the Linked-Cluster Theorem (i.e., the Cumulant Expansion). (2) A convenient description of classical trajectories. Since the formalism was developed in 1979, there have been some improvements. (1) The “exact” trajectory model has been proposed in 1992. (2) A derivation error in applying the linked cluster theorem has been corrected in 2007. But, its core part remains the same until now. There are several approximations whose applicability has not been thoroughly justified. One of them is the isolated line approximation.

3. I. The Robert Bonamy Formalism The Isolated Line Approximation In developing their formalism, Robert and Bonamy have relied on the isolated line approximation twice. First, in calculation the spectral density F(ω), they have only considerd the diagonal matrix elements of the relaxation operator W, Effects from the line mixing are ignored. Second, when they evaluated the diagonal matrix of W, they have assumed that Effects from the line coupling are ignored.

4. I. The Robert-Bonamy Formalism Validity Criteria for Ignoring the Line Mixing and the Line Coupling These two simplifications relied on the same approximation, their validity criteria are completely different and the latter is more stringent than the former. For ignoring the line mixing, the criterion is ω – L a >> n b W. Roughly speaking, as frequency gaps between lines are much larger than their half-widths, this criterion is valid at least in cores of lines. 1 cm -1 separation or so is enough to neglect the line mixing. For ignoring the line coupling, the criterion is ‹‹ if| – iS 1 – S 2 |if ›› >> ‹‹ i′f′| – iS 1 – S 2 |if ››. The criterion depends on systems. As shown later, for the Raman Q lines of N 2 – N 2, 140 cm -1 separation or so is the minimum requirement.

5. II. Criterion for Ignoring the Line Coupling Analyzing Formulas for the Raman Q Lines of II. Criterion for Ignoring the Line Coupling Analyzing Formulas for the Raman Q Lines of N 2 – N 2 For the Raman Q lines of N 2 – N 2, if the potential does not depend on the vibration, S 1 is zero. In addition, because imaginary parts of S 2,outer,i and S 2,outer,f cancel out exactly, the S 2 matrix (= S 2,outer,i + S 2,outer,f +S 2,middle ) becomes real. Both S 2,outer,i and S 2,outer,f are diagonal. But, S 2,middle is off-diagonal. S 2,middle is the only source responsible for the line coupling. The selection rule is determined by the product of two Clebsch- Gordan coefficients. Because L 1 must be even, the line coupling occurs only among even j lines or among odd j lines.

6. II. Criterion for Ignoring the Line Coupling Numerical Estimation for the Raman Q Lines of II. Criterion for Ignoring the Line Coupling Numerical Estimation for the Raman Q Lines of N 2 – N 2 Magnitudes of the off-diagonal elements of S 2,middle are mainly determined by the Fourier transforms of H 22 (ω,r c ). The profile of H 22 (ω,r c ) is presented in Fig.1 where one uses dimensionless k (= ωr c /v) to represent ω. Its magnitudes decrease as k increases, but remain not negligible until k ≥ 14 or so. The later corresponds to ω ≥ 140 cm -1 in the small r c region. For the Raman Q lines of N 2 – N 2, separations of nearby coupled lines are around 4(2j + 3) cm -1. Thus, one must consider the line coupling because the criterion is not satisfied here. In evaluating the cumulant expansion, to apply the isolated line approximation is not justified and it could cause large errors.

7. II. Criterion for Ignoring the Line Coupling Profile of H 22 (k,r c ) H 22 (k,r c ) (in ps -2 ) at T = 298 K for the N 2 - N 2 pair as a two dimensional function of k and r c (in Å). Fig. 1 Profile of the Fourier transform of H 22 (k,r c ) (in ps -2 ) at T = 298 K for the N 2 - N 2 pair as a two dimensional function of k and r c (in Å).

8. III. How to Consider the Line Coupling Two Different Definitions of the Cumulant Expansion The key is to evaluate all the matrix elements of exp(– iS 1 – S 2 ). There are two ways to introduce the cumulant expansion. The one used by Robert and Bonamy contains a mistake. By correcting the mistake, a new way has been developed by us. Their essential difference results from two different ways to definite the average. The difference yields Consequently, there are different expressions for the half-width:

9. III. How to Consider the Line Coupling More Profound Consequence due to Two Different Choices Within the RB formalism 1.The operator iS 1 – S 2 depends on states of the bath molecule. 2.The matrix size of iS 1 – S 2 is determined by (# of lines) × (# of the bath states). 3.One needs to diagonalize a huge size matrix of iS 1 – S 2 for each of collisional trajectories. 4.Computational burdens have forced people to give up attempts to consider the line coupling, unless extra approximations are introduced. Within the modified RB formalism 1.The operator iS 1 – S 2 is independent of the bath states. 2.The matrix size of iS 1 – S 2 is only determined by # of lines. 3.One needs to diagonalize a smaller size matrix of iS 1 – S 2 for each of collisional trajectories. 4.Computational burdens would be reduced by several dozen thousand times and become very reasonable.

10. IV. Numerical Calculations for the Raman Q Lines For the Raman Q lines of N 2 – N 2, accurate potential models are available. The most accurate full quantum calculations match measured half-width data well. The RB formalism overestimates the half-widths by large amounts. This implies the RB formalism is not able to yield reliable results. By considering the line coupling, one is able to make an improvement. First, one calculates > for each of the trajectories. A sample of > with j = 0, 2, ∙∙∙, 14 and at r c = 3.677 Å

11. IV. Numerical Calculations for the Raman Q Lines A sample of > with the isolated line approximation A sample of > without the approximation

12. IV. Numerical Calculations for the Raman Q Lines Fig. 2 Two sets of the factor (1 - >) for the four Q lines with j = 0 (black), 4 (red), 12 (green), and 20 (blue). The sets derived with and without the isolated line approx. are plotted by dotted-dash and solid lines, respectively. Calculated half-widths will be significantly reduced because this factor is a major part of the integrand of the half-widths.

13. IV. Numerical Calculations for the Raman Q Lines Fig. 3 Two sets of profiles of b(db/dr c )[1 – exp(– S 2 (r c ))] associated with three Q lines with j = 4 (black), 12 (red), and 20 (green). They are derived with excluding and including the line coupling and plotted by dotted-dash and solid lines, respectively. Calculated half-widths will be reduced significantly because the factor is the integrand of the half-widths.

14. IV. Numerical Calculations for the Raman Q Lines Fig. 4 Comparison of calculated half-widths and measured data. Values derived with excluding and including the line coupling are plotted by + and ∆. Results obtained from the close coupling calculations are given by ○. Two different measured values are plotted by □ and ×, respectively.

15. IV. Numerical Calculations for the Raman Q Lines A 16 × 16 sub-matrix of the relaxation operator W with j = 0, 2, 4, ∙∙∙, 30 The diagonal matrix elements of W represent the calculated half-widths. The Sum rule is satisfied, i.e.,

16. V. Discussions and Conclusions Without justification, to apply the isolated line approximation in evaluating exp(– iS 1 – S 2 ) could cause large errors because the approximation is more likely not applicable. With the modified RB formalism, one is able to consider the line coupling in practical calculations. For the Raman Q lines of N 2 – N 2, the RB formalism overestimates the half-widths by a large amount. By including the line coupling, our new calculated half-widths are significantly reduced and become closer to measurements. By overcoming one of the main weaknesses of the RB formalism, our refinement effort goes in the right direction.

17. V. Discussions and Conclusions The new calculated half-widths still do not match measured data. We don’t consider the differences as a bad sign of our refinement. There are other main weaknesses remaining in the RB formalism. The gaps provide room for further refinements. The method can be applied for other molecular systems, such as such as the N 2 and CO mixtures, CO 2 – Ar, C 2 H 2 – Ar, CO- Ar, HCl – Ar, HF – Ar. In all these systems, the RB formalism significantly overestimates the half-widths. For H 2 O – N 2, S 2,middle is the only source responsible for off-diagonal matrix elements of S 2. But, the leading correlation functions with L 1 = 1 makes major contributions. We expect that for strongly coupled H 2 O lines, effects on calculated half-widths from the line coupling could be significant.

18. VI. Remaining Challenges in the Refinement (1) To give up the assumption that the translation and internal motions are not connected and the trajectories are only determined by isotropic potentials. Benefit: Couplings between the translation and internal motions are taken into account. Challenge: As a semi-classical theory, the translational motion is treated classically and the internal motion is treated quantum mechanically. To consider them together is a very difficult job. (2) To consider contributions from the third-order expansion of the Ŝ matrix. Benefit: One can get higher order contributions and make sure that the results are converged. Challenge: One has to include many more terms in calculations. But, at least for two linear molecules, it is possible to solve this problem.