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Uncertainties Associated with Theoretically Calculated N 2 Broadened Half-widths of H 2 O Lines

Uncertainties Associated with Theoretically Calculated N 2 Broadened Half-widths of H 2 O Lines. Q. Ma NASA/Goddard Institute for Space Studies & Department of Applied Physics and Applied Mathematics, Columbia University 2880 Broadway, New York, NY 10025 R. H. Tipping

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Uncertainties Associated with Theoretically Calculated N 2 Broadened Half-widths of H 2 O Lines

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  1. Uncertainties Associated with Theoretically Calculated N2 Broadened Half-widths of H2O Lines Q. Ma NASA/Goddard Institute for Space Studies & Department of Applied Physics and Applied Mathematics, Columbia University 2880 Broadway, New York, NY 10025 R. H. Tipping Department of Physics and Astronomy, University of Alabama Tuscaloosa, AL 35487 R. R. Gamache Department of Environmental, Earth and Atmospheric Science, University of Lowell Lowell, MA 01854

  2. I. Calculations of the half-widths • With the modified Robert-Bonamy (RB) formalism, the half-width  is given by The integrand of  is • Usually, S2 consists of three terms S2,outer,I, S2,outer,f, and S2,middle. For example, S2,outer,I is given by • In order to consider more realistic potentials, it is necessary to include a short range atom-atom model Vatom-atom(t)

  3. When one calculates matrix elements of Vatom-atom(t), one needs to express the latter as a spherical harmonic expansion • It is necessary to limit the number of terms to be considered by introducing two cut-offs. (1) to limit sets of irreducible tensor indices l1 and l2. For l1 = 1,2 and l2 = 0,2 # of correlations = 20. (only 8 required to be derived) For l1 = 1,2,3 and l2 = 0,2 # of correlations = 38. (14 required …) For l1 = 1,2,3,4 and l2 = 0,2 # of correlations = 88. (26 required …) For l1 = 1,2,3,4 and l2 = 0,2,4 # of correlations = 132. (39 required …) (2) to set an upper limit of w appearing as R-2w. If one chooses 8 as the maximum of 2w, it is called as the 8-th order cut-off. In updating HITRAN 2006, theoretical calculations are derived with the 8-th order cut-off and 20 correlations. • It is the introduction of these two cut-offs that opens the possibility that the results derived are not converged.

  4. II. The Coordinate Representation In the coordinate representation, the basis set | α > in Hilbert space are where Ωaα and Ωbα represent orientations of absorber molecule a and bath molecule b, respectively. The greatest advantage of the coordinate representation is the interaction potential V is diagonal and can be treated as an ordinary function. It is easy to make transformations between the state and the coordinate representations by using the inner products With this new representation, one is able not only to derive converged results, but also to analyze uncertainties associated with theoretically calculated half-widths.

  5. III.Three Categories of the H2O Lines • With the coordinate representation, one can choose any cut-offs one wants. As a result, one can use this method as a powerful diagnostic tool to check whether values in the literature are converged or not. • For convenience, one can divide all H2O lines into three categories according to magnitudes of their half-widths. One can use this division to judge the convergence behavior and to estimate the uncertainty associated with them. For example, there are 268, 661, and 710 lines in categories 1, 2 and 3 (corresponding to  > 0.075, 0.075<  < 0.045, and  < 0.045 in HITRAN 2006). • Three typical lines in these categories and their convergence checks. In the table, c stands for correlations, p and e stand for parabolic and “exact” trajectories. Calculated N2 broadened half-widths are given in the units of cm-1 atm-1 .

  6. III-A Lines in Category 1 Fig. 1 A plot to show why calculated half-widths for 31,3 ← 22,0 converge very quickly. Re(S2(rc)) derived from 20, 38, and 88 correlations and using the 20-th cut-off are plotted by black solid, dashed, and dash-dotted lines. Exp(-Re(S2(rc))) and the integrand of  are given by blue and red curves. In order to show exp(-Re(S2(rc))) more clearly, their values are multiplied by 1000. There are significant differences among Re(S2(rc)) derived with different numbers of the correlations, the integrands are almost identical. This implies calculated half-widths with 20 correlations are converged.

  7. III-A Lines in Category 1 Fig. 2 The same as Fig. 1 except that results are derived from the 8-th, 14-th, and 20-th order cut-offs. The calculations are based on including 20 correlations and the “parabolic” trajectory model. The plot demonstrates that with respect to the second kind of cut-off, the calculated half-width of this line converges very quickly.

  8. III-A. Lines in Category 1 • Magnitudes of Re(S2(rc)) for rc < σ are very large. Thus, its effects on the integrands of  are almost diminished for rc < σ. • The convergence of Re(S2(rc))at rc < σ is poor. But, the convergence of the integrands of  is very good. • Calculated half-widths for 31,3 ← 22,0 with low cut-offs are well converged. • Large Re(S2(rc))are associated with large . This implies that lines in category 1 would not suffer from the convergence difficulty. • Because calculated  are not sensitive to variations of the short range interaction, one should not choose lines in category 1 to optimize the site-site potential model. In contrast, they are good candidates to optimize the long range potential model. • Within this category, uncertainties associated with theoretically calculated  are the smallest.

  9. III-B Lines in Category 3 Fig. 3 The same as Fig. 1 except for the line of 172,15 ← 161,16 and multiplying exp(-Re(S2(rc)))by 10. Results are obtained with including 20, 38, and 88 correlations.

  10. III-B Lines in Category 3 Fig. 4 The same as Fig. 2 except for the line of 172,15 ← 161,16 and multiplying exp(-Re(S2(rc)))by 10. Results are obtained with adopting the 8-th, 14-th, and 20-th order cut-offs.

  11. III-B. Lines in Category 3 • In comparison with 31,3 ← 22,0 in category 1, magnitudes of Re(S2(rc))for 172,15 ← 161,16 areone order smaller. They decrease more quickly as rc increases, and are close to 0 after rc > σ.The convergence of Re(S2(rc))is poor. • Because the magnitudes of Re(S2(rc))are small, their effects on the integrand of  are well preserved. Thus, the latter’s convergence is poor. In addition, the latter’s magnitudes decrease very quickly and are close to 0 for rc > σ. This implies that  would be small. • The convergences of  are very poor such that with usual methods, one is not able to obtain converged results. • Dominant contributions to  come from nearly head-on collisions. This implies that both changes of the short range interaction and different depictions of the trajectories would cause significant changes of calculated . • One can choose lines in this category to optimize the site-site potential model and to test the trajectory model. • Uncertainties associated with theoretically calculated  are the largest.

  12. III-C Lines in Category 2 Fig. 5 The same as Fig. 1 except for the line of 99,0 ← 86,3and multiplying exp(-Re(S2(rc))) by 10. Results are obtained with including 20, 38, and 88 correlations.

  13. III-C Lines in Category 2 Fig. 6 The same as Fig. 2 except for the line of 99,0 ← 86,3and multiplying exp(-Re(S2(rc))) by 10. Results are obtained with adopting the 8-th, 14-th, and 20-th order cut-offs.

  14. III-C. Lines in Category 2 • In comparison with 31,3 ← 22,0 and 172,15 ← 161,16,magnitudes of Re(S2(rc))for 99,0 ← 8 6,3 are intermediate. • The convergence of Re(S2(rc) is poor. • Because magnitudes of Re(S2(rc))are intermediate, its effects on the integrand of  are partially preserved and partially suppressed for rc > σ. • The convergence of calculated  for 99,0 ← 8 6,3 is worse than 31,3 ← 22,0, but is better 172,15 ← 161,16. • Calculated  for this line is smaller than 31,3 ← 22,0, but larger than 172,15 ← 161,16. • Uncertainties associated with theoretically calculated  are the intermediate.

  15. III-D Comparisons among lines in three categories Fig. 7 A plot to show comparisons among calculated integrands of  for the three lines of interest. For each of them, we plots two results derived from the lowest choice of the cut-offs (i.e., the 8-th and 20 correlations) and from a higher one (i.e., the 20-th and 88 correlations) by a dot-dashed and a solid curves, respectively. Meanwhile, the curves for different lines are distinguished by three colors.

  16. III-D Comparisons among three lines in different categories Fig. 8 A plot to show percentage contributions to  at 296 K from collisions whose closet distances ranging from rc,min to rc. Results derived for 31,3 ← 22,0, 99,0 ← 86,3, and 172,15 ← 161,16 are represented by a black, red, and green curves, respectively. The calculations are carried out with the 20-th order and including 88 correlations.

  17. IV Convergence check Fig. 9 Calculated N2-broadened half-widths of the 1639 lines of the H2O pure rotational band. These results are derived by including 20, 38, 88, and 132 correlations and by adopting the 20-th order cut-off. They are plotted by symbols□, +, ×, and ∆ respectively.

  18. IV Convergence check Fig. 10 Calculated N2-broadened half-widths of the 1639 lines of the H2O pure rotational band. These results are derived by adopting the 8-th, 14-th, and 20-th order cut-offs and by including 20 correlation functions. They plotted by symbols +, ×, and ∆, respectively.

  19. V. MODIFICATION OF THE RB FORMALISM • There is a subtle derivation error in applying the Linked-Cluster Theorem. • After making the correction, the expressions for the half-width and shift differ from the original ones. For example, in the RB formalism and in the modified RB formalism where <A >j2 is a notation for

  20. V. MODIFICATION OF THE RB FORMALISM Fig. 11 Relative errors of calculated  resulting from the subtle error in developing the RB formalism. The percentage errors are measured by (modifiedRB - RB) / modifiedRB %. The calculations are carried out with the 20-th order, 88 correlations, and the “parabolic” trajectory. In the plot, the 1639 lines are arranged in a ascending order of their half-width values.

  21. VI Comparison between calculated half-widths and HITRAN 2006 Fig. 12 Comparisons between calculated  from the 8-th order, 20 correlations, and the “parabolic” trajectory and air-broadened half-widths in HITRAN. The former are plotted by symbols × and the latter are given by symbols ∆, respectively. The calculated N2-broadened half-widths have been adjusted to air-broadened ones by multiplying by 1/1.09.

  22. VI Comparison between calculated half-widths and HITRAN 2006 Fig. 13 The same as Fig. 12 except that calculated half-widths are derived by using the 20-th order cut-off, including 88 correlations, and adopting the “exact” trajectory model.

  23. VII Comparison between HITRAN 2006 and 2009 Fig. 14 Comparisons between air-broadened half-widths in HITRAN 2006 and 2009. They are plotted by × and ∆, respectively. The H2O lines of the pure rotational band are arranged according to the ascending order of their air-broadened half-widths in HITRAN 2009. Among 1639 lines, there are 1489 lines whose half-widths have been modified. There are 677 lines with relative differences are above 10%, 313 lines within 4 – 10%, and 649 lines with less than 4 % including those without changes.

  24. VII Comparison between HITRAN 2006 and 2009 Fig. 15 A comparison of the temperature exponents n listed in the HITRAN 2006 and 2009. There are dramatically differences between these two versions of n. Among 1639 lines, there are 245 lines whose temperature exponent n become negative. It is worth mentioning that the lines with negative n values are those with high Ji and Jf and with small half-widths.

  25. An example to show the convergence problem associated with n Fig. 16 The T dependence of the half-width for the line 1614,2 ← 1513,3 in the range from 220 K – 340 K derived from adopting different cut-offs and including different numbers of the correlation functions. In the plot, a negative slope means a negative n and a positive slope means a positive n. In HITRAN 2006, n = 0.38 and in HITRAN 2009, n = - 0.79.

  26. VII Comments on HITRAN 2009 • There are significant differences between air-broadened half-widths listed in HITRAN 2006 and 2009. Among 1639 lines, there are 1489 lines whose half-widths have been modified. There are 677 lines with relative differences above 10 %, 313 lines within 4 – 10 %, and 649 lines with less than 4 % including those without changes. • The modification of HITRAN 2006 is necessary. But, in the 2009 version, many theoretically calculated results are not converged. In addition, the potential model used in these theoretical calculations is poor. • With respect to the temperature exponents n, there are dramatically differences between these two versions. Among 1639 lines, there are 245 lines whose temperature exponent n become negative. • In general, lines with negative n values are those with large Ji and Jf and with small half-widths. Thus, there are serious convergence problems associated with those calculated n. We are open to accept negative n values. But, we believe some of these negative values of n in HITRAN 2009 are not correct.

  27. IX. Conclusions • With the current RB formalism, calculated half-widths for lines in category 1 have small uncertainties. Thus, one can achieve the accuracy requirement set by HITRAN. • For lines in category 3, calculated half-widths contain larger uncertainties. It is impossible to meet this goal. • For lines in category 2, the situations are in between these two extremes. • When theorists optimize the potential models, they need to select lines properly. • When experimentalists determine the measurement priority, they can add another consideration that their measured data for lines in category 3 have an extra reliability advantage. • When people update HITRAN for lines in category 3, it would be helpful to know there are large uncertainties associated with theoretically calculated values.

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