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Pair Identity and Smooth Variation Rules Applicable for the Spectroscopic Parameters of H 2 O Transitions Involving High J States Q. Ma NASA/Goddard Institute for Space Studies & Department of Applied Physics and Applied Mathematics, Columbia University 2880 Broadway, New York, NY 10025, USA R. H. Tipping Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA N. N. Lavrentieva V. E. Zuev Institute of Atmospheric Optics SB RAS, 1, Akademician Zuev square, Tomsk 634021, Russia

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I. Basic Idea in Analyzing Spectroscopic Parameters of H 2 O Lines A whole system consists of one absorber H 2 O molecule, bath molecules, and electromagnetic fields. By considering this system as a black box, its inputs are the H 2 O lines of interest and its outputs are the spectroscopic parameters. Basic assumptions: (1) The outputs depend on the inputs. (2) Identical inputs should yield identical outputs. (3) Similar inputs should yield similar outputs. The inputs are the energy levels and wave functions associated with the initial and final states of the H 2 O lines. A Black Box inputs outputs

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II. Properties of the energy levels and wave functions of H 2 O (1) One categories the H 2 O states into different sets of paired states {J 0,J, J 1,J }, {J 1,J-1, J 2,J-1 }, {J 2,J-2, J 3,J-2 }, ···, {J J-2,3, J J-2,2 }, {J J-1,2,J J-1,1 }, {J J,1,J J,0 }. (2) Within each of the sets, the energy levels of two paired states with high J are almost identical. For different pairs, their energy levels vary smoothly as J varies and these variation patterns are well organized. (2) Within each of the sets, the energy levels of two paired states with high J are almost identical. For different pairs, their energy levels vary smoothly as J varies and these variation patterns are well organized. (3) With respect to the H 2 O wave functions, they are given in terms of expansion coefficients over the symmetric top wave functions |JKM>, (4) Within each of the sets, the coefficients of two paired states with high J have almost identical magnitudes. For different pairs, patterns of their coefficients are very similar and the placements of the patterns shift smoothly as J varies. (5) The above are useful features of the inputs. By exploiting them, one can access important conclusions on the outputs without requiring to know what really happens inside the box.

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II-1 Properties of the Energy Levels of H 2 O Fig. 1 A plot to show energy levels of H 2 O states with J = 11 thru 20 in the vibrational ground state. For states with J + K a – K c = even, their energy levels are plotted by × and their values of K a – K c are presented on the right side of the symbols. Meanwhile, for states with J + K a – K c = odd, their energy levels are plotted by ∆ and values of K a – K c are on the left side of the symbols.

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II-2 Properties of the H 2 O Wave Functions Fig. 2 A plot to show properties of H 2 O wave functions in the I R representation for three sets of pairs of states: {J J,0,J J,1 }, {J J-1,1,J J-1,2 }, and {J J-2,2,J J2,3 }.

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II-2 Properties of the H 2 O Wave Functions Fig. 3 A plot to show properties of H 2 O wave functions in the III R representation for three sets of pairs of states: {J 0,J,J 1,J }, {J 1,J-1,J 2,J-1 }, and {J 2,J-2,J 3,J-2 }. :.

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II-3 Boundaries for individual sets of paired states In each of the sets, the pair identity and the smooth variation break down for J is below certain boundaries. By introducing a numerical measure ε defined by where J, 1 and J, 2 are paired states and = K a – K c, one can calculate how ε varies with J. The higher the J, the smaller the ε. By choosing ε is about 1 %, one can determine a boundary J bd for each of the sets. The results is listed in Table. set J J,0 J J,1 J J-1,1 J J-1,2 J J-2,2 J J-2,3 J J-3,3 J J-3,4 J J-4,4 J J-4,5 J J-5,5 J J-5,6 J J-6,6 J J-6,7 ··· J 4,J-4 J 5,J-4 J 3,J-3 J 4,J-3 J 2,J-2 J 3,J-2 J 1,J-1 J 2,J-1 J 0,J J 1,J J bd 3579101214···191613107 ε (%) 0.630.660.490.321.080.600.32···1.280.551.131.000.60

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III. Categorizations of H 2 O lines and discovery of two rules applicable for all spectroscopic parameters To categorize H 2 O lines such that within individual groups, the inputs of lines have similarities. Then, one expects their outputs have similarities too. The procedure is carried out by dividing all lines into the P, Q, and R branches first and then, by categorizing them into sets of paired lines. As example, a group consists of paired lines J′ 0,J’ ← J″ 1,J" and J′ 1,J′ ← J″ 0,J" and a group of J′ J′,0 ← J″ J″,1 and J′ J′,1 ← J″ J″,0. It turns out that the similarities of the outputs do exist. Thus, two rules can be established. These rules hold within certain accuracy tolerances. Corresponding to 1 % accuracy of the inputs, we choose 5 % as the accuracy tolerances. The pair identity rule: Two paired lines whose J values are above certain boundaries in the same groups have almost identical spectroscopic parameters. The smooth variation rule: For different pairs of states in the same groups, values of their spectroscopic parameters vary smoothly as their J values vary.

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IV. Demonstrations of the two rules by Measurements Fig. 4 Demonstrations of the pair identity and smooth variation rules for line strengths, air- and N 2 -broadened half-widths, and induced shifts in a group of paired lines J′ 3,J’-2 ← J″ 0,J" and J′ 2,J’-2 ← J″ 1,J" in the R branch. The measured values by Toth are plotted by × and ∆, respectively. The spin degeneracy factor is excluded for the strength. The boundary of this group is J bd = 13.

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IV. Demonstration of the two rules by Measurements Fig. 5 Demonstration of the pair identity and smooth variation rules in a group of paired lines of J' 0,J' ← J" 1,J"-1 and J' 1,J' ← J" 2,J"-1 in the Q branch of the ν 2 band. The air-broadened half-widths (in cm -1 /bar) are from measurements by Yamada. The boundary of this group is about J bd = 10.

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V. Comparison between HITRAN H 2 O 2006 and 2009 Fig. 6 Comparisons between air-broadened half-widths in HITRAN H 2 O 2006 and 2009. They are plotted by × and ∆, respectively. The 1639 lines of the H 2 O pure rotational band are arranged according to the ascending order of the half-width values of HITRAN H 2 O 2009.

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V. Comparison between HITRAN H 2 O 2006 and 2009 Fig. 7 The same as Fig. 6 except for values of the temperature exponent (T exponent) n. There are dramatically differences between these two versions of n. Among 1639 lines, there are 245 lines whose T exponent n become negative.

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V. Comparison between HITRAN H 2 O 2006 and 2009 Table 2. Relative differences of parameters in HITRAN H 2 O 2006 and 2009. Rough estimations of uncertainties associated with all the spectroscopic parameters. (1) For the line position and intensity, their uncertainties are less than the accuracy tolerance. (2) For the half-widths, shift, and T exponent, their uncertainties are larger than the accuracy tolerance. Conclusions: For positions and intensities of lines, two rules enable one to identify errors. For other parameters, they enable one to identify errors and to improve accuracies. Number of lines differenceIntensityAir-widthSelf-widthshiftT-expon. > 50 % 25143246572630 30 - 50 % 8165280121360 10 – 30 % 13369601420421 < 10 % 1593962512526228 unchanged158515010715820

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VI-1. Screening HITRAN H 2 O 2009 (Example 1 in the R branch) Fig. 8 Six spectroscopic parameters for a group of paired lines J′ J′,1 ← J″ J″,0 and J′ J′,0 ← J″ J″,1 in the R branch. Their values in HITRAN H 2 O 2009 are plotted by × and ∆, respectively. The boundary of this group is J bd = 3. Parameter Questionable Lines PositionNone IntensityNone Air- Half-width Half-width 9 9,0 ← 8 8,1, 9 9,1 ← 8 8,0, 10 10,0 ← 9 9,1, 10 10,1 ← 9 9,0, 11 11,0 ← 10 10,1, 11 11,1 ← 10 10,0 Self- Half-width Half-width 13 13,0 ← 12 12,1,13 13,1 ← 12 12,0 Induced shift 9 9,0 ← 8 8,1, 9 9,1 ← 8 8,0, 10 10,0 ← 9 9,1, 10 10,1 ← 9 9,0, 11 11,0 ← 10 10,1, 11 11,1 ← 10 10,0, 12 12,0 ← 11 11,1, 12 12,1 ← 11 11,0, 13 13,0 ← 12 12,1, 13 13,1 ← 12 12,0, 14 14,0 ← 13 13,1, 14 14,1 ← 13 13,0 T exponent 9 9,0 ← 8 8,1, 9 9,1 ← 8 8,0, 10 10,0 ← 9 9,1, 10 10,1 ← 9 9,0, 11 11,0 ← 10 10,1, 11 11,1 ← 10 10,0

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VI-1. S uggested values of the air-broadened half-width for lines in the e xample 1 of the R branch Fig. 9 Based on the two rules, suggested air-broadened half-widths for the group of paired lines J′ J′,1 ← J″ J″,0 and J′ J′,0 ← J″ J″,1 in the R branch are plotted by +. Meanwhile, the original values in HITRAN H 2 O 2009 are given by × and ∆, respectively. Lines HITRAN 2009 Suggest- ed values 8 9,0 ← 7 8,1 8 9,1 ← 7 8,0 0.02720.02720.0264 9 9,0 ← 8 8,1 9 9,1 ← 8 8,0 0.01340.01340.0195 10 9,0 ← 9 8,1 10 9,1 ← 9 8,0 0.01240.01240.0161 11 9,0 ← 10 8,1 11 9,1 ← 10 8,0 0.02030.02030.0156

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VI-2. Screening HITRAN H 2 O 2009 (Example 2 in the R branch) Fig. 10 The same as Fig. 8 except for a group of paired lines J′ 3,J′-2 ← J″ 0,J″ and J′ 2,J′-2 ← J″ 1,J″ in the R branch. The boundary of this group is J bd = 13. Parameter Questionable lines PositionsNone Intensity 21 3,19 ← 20 0,20, 21 2,19 ← 20 1,20 Air- Half-width Half-width 17 3,15 ← 16 0,16, 17 2,15 ← 16 1,16, 18 3,16 ← 17 0,17, 18 2,16 ← 17 1,17, 19 3,17 ← 18 0,18, 19 2,17 ← 18 1,18 Self- Half-width Half-width 18 3,16 ← 17 0,17, 18 2,16 ← 17 1,17, 19 3,17 ← 18 0,18, 19 2,17 ← 18 1,18, 20 3,18 ← 19 0,19, 20 2,18 ← 19 1,19 Induced shift 14 3,12 ← 13 0,13, 14 2,12 ← 13 1,13, 15 3,13 ← 14 0,14, 15 2,13 ← 14 1,14, 16 3,14 ← 15 0,15, 16 2,14 ← 15 1,15, 18 3,16 ← 17 0,17, 18 2,16 ← 17 1,17, 19 3,17 ← 18 0,18, 19 2,17 ← 18 1,18 T exponent 15 3,13 ← 14 0,14, 15 2,13 ← 14 1,14, 16 3,14 ← 15 0,15, 16 2,14 ← 15 1,15, 21 3,19 ← 20 0,20, 21 2,19 ← 20 1,20

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VI-2. Suggested values of the induced shift for lines in the example 2 of the R branch Fig. 11 Based on the two rules, suggested induced shifts (in units of 10 -3 × cm -1 atm -1 ) for the group of paired lines J′ 3,J′-2 ← J″ 0,J″ and J′ 2,J′-2 ← J″ 1,J″ in the R branch are plotted by +. Meanwhile, the original values in HITRAN H 2 O 2009 are given by × and ∆, respectively. Lines HITRAN 2009 Suggest- ed values 14 3,12 ← 13 0,13 14 2,12 ← 13 1,13 3.001.142.11 15 3,13 ← 14 0,14 15 2,13 ← 14 1,14 3.121.742.34 16 3,14 ← 15 0,15 16 2,14 ← 15 1,15 2.911.612.46 17 3,15 ← 16 0,16 17 2,15 ← 16 1,16 2.662.642.50 18 3,16 ← 17 0,17 18 2,16 ← 17 1,17 3.356.982.03 19 3,17 ← 18 0,18 19 2,17 ← 18 1,18 0.480.471.22 20 3,18 ← 19 0,13 20 2,18 ← 19 1,19 0.860.861.02 21 3,19 ← 20 0,20 21 2,19 ← 20 1,20 1.551.551.41

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VI-3. Screening HITRAN H 2 O 2009 (Example 3 in the R branch) Fig. 12 The same as Fig. 8 except for a group of paired lines J′ 3,J′-3 ← J″ 2J″-1 and J′ 4,J′-3 ← J″ 1,J″-1 in the R branch. The boundary of this group is J bd = 16. Parameter Questionable lines PositionsNone Intensity 19 3,16 ← 18 2,17, 19 4,16 ← 18 1,17, 20 3,17 ← 19 2,18, 20 4,17 ← 19 1,18 Air- Half-width Half-width 18 3,15 ← 17 2,16, 18 4,15 ← 17 1,16, 19 3,16 ← 18 2,17, 19 4,16 ← 18 1,17, 20 3,17 ← 19 2,18, 20 4,17 ← 19 1,18 Self- Half-width Half-width 18 3,15 ← 17 2,16, 18 4,15 ← 17 1,16 Induced shift 17 3,14 ← 16 2,15, 17 4,14 ← 16 1,15, 18 3,15 ← 17 2,16, 18 4,15 ← 17 1,16, 19 3,16 ← 18 2,17, 19 4,16 ← 18 1,17, 20 3,17 ← 19 2,18, 20 4,17 ← 19 1,18 T exponent 19 3,16 ← 18 2,17, 19 4,16 ← 18 1,17

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VI-4. Screening HITRAN H 2 O 2009 (Example 1 in the Q branch) Fig. 13 The same as Fig. 8 except for a group of paired lines J′ J′,0 ← J″ J″-1,1 and J′ J′,1 ← J″ J″-1,2 in the Q branch. The boundary of this group is J bd = 5.

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VI-5. Screening HITRAN H 2 O 2009 (Example 2 in the Q branch) Fig. 14 The same as Fig. 8 except for a group of paired lines J′ 2,J′-2 ← J″ 1,J″-1 and J′ 3J′-2 ← J″ 2,J″-1 in the Q branch. The boundary of this group is J bd = 13.

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VI-6. Screening HITRAN H 2 O 2009 (Example 1 in the P branch) Fig. 15 The same as Fig. 8 except for a group of paired lines J′ 2,J′-2 ← J″ 1,J″ and J′ 3,J′- 2 ← J″ 0,J″ in the P branch. The boundary of this group is J bd = 13.

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VI-7. Screening HITRAN 2009 (Example 1 in the Q branch of the v 2 band) Fig. 16 The same as Fig. 8 except for a group of paired lines J′ 0,J′ ← J″ 1,J″-1 and J′ 1,J′ ← J″ 2,J″ -1 in the Q branch of the v 2 band. The boundary J bd ≈ 10. Parameter Questionable lines PositionsNone IntensityNone Air- Half-width Half-width 15 0,15 ← 15 1,14, 15 1,15 ← 15 2,14, 16 0,16 ← 16 1,15, 16 1,16 ← 16 2,15, 17 0,17 ← 17 1,16, 17 1,17 ← 17 2,16, 18 0,18 ← 18 1,17, 18 1,18 ← 18 2,17, 19 0,19 ← 19 1,18, 19 1,19 ← 19 2,18 Self- Half-width Half-width 14 0,14 ← 14 1,13, 14 1,14 ← 14 2,13, 15 0,15 ← 15 1,14, 15 1,15 ← 15 2,14, 16 0,16 ← 16 1,15, 16 1,16 ← 16 2,15, 19 0,19 ← 19 1,18, 19 1,19 ← 19 2,18 Induced shift 15 0,15 ← 15 1,14, 15 1,15 ← 15 2,14, 16 0,16 ← 16 1,15, 16 1,16 ← 16 2,15, 17 0,17 ← 17 1,16, 17 1,17 ← 17 2,16, 18 0,18 ← 18 1,17, 18 1,18 ← 18 2,17 T exponent 19 0,19 ← 19 1,18, 19 1,19 ← 19 2,18

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VI-7. Suggested values of the air-broadened half-width for lines in the example 1 of the Q branch of the ν 2 band Fig. 17 Based on the two rules, suggested air-broadened half-widths (in cm -1 atm -1 ) for the group of paired lines J′ 0,J′ ← J″ 1,J″-1 and J′ 1,J′ ← J″ 2,J″ -1 in the Q branch of the v 2 band are plotted by +. Meanwhile, the original values in HITRAN H 2 O 2009 are given by × and ∆, respectively. Lines HITRAN 2009 Suggest- ed values 15 0,15 ← 15 1,14 15 1,15 ← 15 2,14 0.01400.01700.0150 16 0,16 ← 16 1,15 16 1,16 ← 16 215 0.01440.00900.0130 17 0,17 ← 17 1,16 17 1,17 ← 17 2,16 0.00920.01250.0115 18 0,18 ← 18 1,17 18 1,18 ← 18 2,17 0.01080.01080.0099 19 0,19 ← 19 1,18 19 1,19 ← 19 2,18 0.00640.00640.0073

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VI-8. Screening HITRAN 2009 (Example 1 in the P branch of the v 2 band) Fig. 18 The same as Fig. 8 except for a group of paired lines J′ 1,J′ ← J″ 0,J″ and J′ 0,J′ ← J″ 1,J″ in the P branch of the v 2 band. The boundary J bd ≈ 7. Parameter Questionable lines PositionsNone IntensityNone Air- Half-width Half-width 12 1,12 ← 13 0,13, 12 0,12 ← 13 1,13, 13 1,13 ← 14 0,14, 13 0,13 ← 14 1,14, 14 1,14 ← 15 0,15, 14 0,14 ← 15 1,15, 16 1,16 ← 17 0,17, 16 0,16 ← 17 1,17, 17 1,17 ← 18 0,18, 17 0,17 ← 18 1,18, 19 1,19 ← 20 0,20, 19 0,19 ← 20 1,20 Self- Half-width Half-width 7 1,7 ← 8 0,8, 7 0,7 ← 8 1,8, 8 1,8 ← 9 0,9, 8 0,8 ← 9 1,9, 9 1,9 ← 10 0,10, 9 0,9 ← 10 1,10, 13 1,13 ← 14 0,14, 13 0,13 ← 14 1,14, 16 1,16 ← 17 0,17, 16 0,16 ← 17 1,17, 18 1,18 ← 19 0,19, 18 0,18 ← 19 1,19, Induced shift 13 1,13 ← 14 0,14, 13 0,13 ← 14 1,14, 14 1,14 ← 15 0,15, 14 0,14 ← 15 1,15, 18 1,18 ← 19 0,19, 18 0,18 ← 19 1,19, 19 1,19 ← 20 0,20, 19 0,19 ← 20 1,20 T exponent 18 1,18 ← 19 0,19, 18 0,18 ← 19 1,19, 19 1,19 ← 20 0,20, 19 0,19 ← 20 1,20

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VI-9. Screening HITRAN 2009 (Example 1 in the R branch of the v 2 band) Fig. 19 The same as Fig. 8 except for a group of paired lines J′ 1,J′ ← J″ 0,J″ and J′ 0,J′ ← J″ 1,J″ in the R branch of the v 2 band. The boundary J bd ≈ 7.

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VI-10. Screening HITRAN 2009 (Example 2 in the R branch of the v 2 band) Fig. 20 The same as Fig. 8 except for a group of paired lines J′ J′,0 ← J″ J″,1 and J′ J′,1 ← J″ J″,0 in the R branch of the v 2 band. The boundary J bd ≈ 3.

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VII. Predicting spectroscopic parameters for HITEMP Fig. 21 A plot to show N 2 -broadened half-widths for two groups of paired lines with J˝ = 26 ··· 50 predicted from an extrapolation method. One is a group of J′ 0,J’ ← J″ 1,J" and J′ 1J’ ← J″ 0J“ in the R branch and another is a group of J′ 2,J’-2 ← J″ 1,J"-1 and J′ 3J’-2 ← J″ 2J"-2 in the Q branch. For the former, theoretically calculated values with J″ = 6, ∙∙∙, 25 and those predicted ones are given by × and ∆. For the latter, calculated values with J″ = 12, ∙∙∙, 25 and predicted ones are given by + and □.

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VIII. Conclusions Two basic rules (i.e., the pair identity and the smooth variation rules) are applicable for all the spectroscopic parameters of H 2 O lines whose J values are above certain boundaries in the same groups. Different groups of lines have different boundaries. The latter can be calculated from the identity requirements for the energy levels and wave functions of paired H 2 O states. The rule hold within certain accuracy tolerance. The latter could vary as the parameter of interest varies. In general, the 5 % accuracy tolerance is suggested for the half-widths, shifts, and T exponents. The 5 % accuracy tolerance is poorer than the accuracies associated with line positions and intensities, but is better than those with other parameters. In addition, the current theoretical calculations and measurements for high J lines cannot reach such high accuracies. The present work can be extended for lines in vibrationaly excited states. Of course, one needs to check the properties of the energy levels and wave functions of vibrationaly excited states first and recalculate corresponding boundaries. The idea of the present work is simple and general. One can applied it for other molecules whose energy levels and wave functions have similarities.

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