1. THEORETICAL CALCULATIONS OF THE N2 BROADENED HALF-WIDTHS OF H2O
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 calculated from a formula given by
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. (2) to limit sets of irreducible tensor indices l1, k1, k’1, and l2.
When one calculates matrix elements of Vatom-atom(t), one needs to express the latter as a spherical harmonic expansion
There are too many summation terms in the products of the matrix elements of Vatom-atom(t) appearing in S2,outer,i. As a result, it is necessary to limit the number of these terms to be considered by introducing two cut-offs.
(1) 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.
(2) to limit sets of irreducible tensor indices l1, k1, k’1, and l2.
In terms of our notation, it means to limit the number of correlations in the calculations. By setting 2 as the upper limit for both l1 and l2, 20 correlations are considered. By increasing the upper limit for l1 from 2 to 3, 38 correlations are considered.
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.
Thus, there are two representations available. In the state representation the Hamiltonian is diagonal and in the coordinate representation the potential is diagonal. It is easy to transfer from one representation to another by using the inner products

5. III. To check convergences resulting from two cut-offs
By using our new formalism developed in 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 published calculated results are converged or not.
As an example, we consider 501 strong H2O lines in the pure rotational band with line intensities larger than × cm-1 /(molecule cm-2) and calculate their half-widths with different choices of these two cut-offs. In our calculations, we use the same potential model as used by Gamache et al.
With respect to the first cut-off, calculated  from the 8-th order have convergence problems; those from the 14-th order are well converged; those from the 20-th order are completely converged.
With respect to the second cut-off, calculated  from 20 correlations have convergence problems. One needs, at least, to include 38 correlations.
Detailed comparisons are given in the following.

6. III A. Convergence check resulting from the first cut-off: a comparison of half-widths from the 8-th, 14-th, and 20-th orders
Fig. 1 Calculated N2-broadened half-widths of the 501 strong lines in the H2O pure rotational band. These results are derived with three different orders of the first cut-off and with including 20 correlation functions. Values obtained from the 8-th, the 14-th, and the 20-th order cut-offs are plotted by symbols ∆, +, and ×, respectively.

7. III A. Convergence check resulting from the first cut-off: a comparison of half-widths from the 8-th, 14-th, and 20-th orders
Fig. 2 Calculated half-widths for the first 150 lines with small values in the 501 strong lines. They are derived from three different first cut-offs and including 20 correlations. Values obtained from the 8-th, the 14-th, and the 20-th order are represented by symbols ∆, +, and ×, respectively.

8. Some sample lines whose convergences are the worst
Intensity ω
(cm-1) 
(HITRAN)
Calculated 
8-th
14-th
20-th
99,0← 88,1
1.39E-20
436.4
0.0193
0.0197
0.0260
0.0264
130,13← 121,12
2.54E-20
248.8
0.0130
0.0240
0.0303
0.0307
88,1← 77,0
7.23E-20
394.2
0.0246
0.0259
0.0317
0.0320
121,12← 110,11
6.42E-20
230.7
0.0172
0.0294
0.0346
0.0349
122,11← 111,11
2.29E-20
249.9
0.0210
0.0358
0.0361
98,1← 87,2
2.93E-20
419.1
0.0308
0.0331
0.0375
0.0378
77,0← 66,1
2.89E-19
349.8
0.0319
0.0340
0.0387
0.0390
108,3 ← 97,2
1.06E-20
443.7
0.0330
0.0354
0.0396
0.0399
110,11←101,10
1.45E-19
212.6
0.0212
0.0365
0.0403
0.0406
111,10 ← 102,9
5.59E-20
231.2
0.0258
0.0385
0.0423
0.0425
112,10← 101,9
1.88E-20
232.1
0.0295
0.0428
0.0430
87,2← 76,1
1.34E-19
374.5
0.0383
0.0424
0.0455
0.0456
97,2← 86,3
5.48E-20
399.0
0.0448
0.0477
0.0479
97,3← 86,2
1.83E-20
398.9
0.0404
0.0449
0.0478
0.0480
66,1← 55,0
8.61E-19
303.0
0.0418
0.0447
0.0482

9. III A. Convergence check resulting from the first cut-off: a comparison of half-widths from the 8-th, 14-th, and 20-th orders
Fig. 3 Relative convergence errors of the calculated half-widths resulting from adopting the 8-th and the 14-th order cut-offs versus calculated half-width values from the 20-th order cut-off. The errors associated with the 8-th order cut-off are represented by symbols ∆ and that from the 14-th order are given by ×.

10. III A. Comparisons of calculated half-widths from the 20-th order and 20 correlations and those listed in updated HITRAN
Fig. 4 Comparisons between calculated half-widths from the 20-th order cut-offs and air-broadened half-widths listed in updated HITRAN. The calculated N2-broadened half-widths have been adjusted to air-broadened ones by multiplying by a factor of 1/1.09. The former are plotted by symbols × and the latter are given by symbols ∆, respectively.

11. III B. Convergence checks resulting from the second cut-off; a comparison of half-widths from including 20 and 38 correlations
Fig. 5 Relative convergence errors of calculated half-widths resulting from adopting the 8-th order and the14-th order cut-offs, respectively, and 20 correlations versus calculated half-widths from the 20-th order cut-off and 38 correlations. The errors associated with the 8-th order cut-off and 20 correlations are represented by symbols ∆ and those from the 14-th order and 20 correlations are given by ×.

12. Category 2 (73 lines). Calculated  have convergence problems.
III C. Convergence behaviors and values of the half-width for the 501 strong lines of the H2O pure rotational band
category 1 2 3
# of lines
300 73
128
relative error
< 4 %
4 – 10 %
> 10 %
half-width 
 > 0.075
0.075 >  > 0.06
 < 0.06
The 501 lines whose intensities are above × cm-1/(molecule cm-2) can be grouped into three categories.
Category 1 (300 lines). Calculated  from lower cut-offs are well converged.
Category 2 (73 lines). Calculated  have convergence problems.
Category 3 (128 lines). Calculated  are completely not converged at all.

13. III C. Comparisons of calculated half-widths from the 20-th order and 38 correlations and those listed in updated HITRAN
Fig. 6 Comparisons between calculated half-widths from the 14-th order cut-offs and 38 correlations and air-broadened half-widths listed in the updated HITRAN. The calculated N2-broadened half-widths have been adjusted to air-broadened ones by multiplying by a factor of 1/1.09. The former are plotted by × and the latter are given by ∆, respectively.

14. III D. Explanations of why calculated half-widths for some lines converge more quickly than others
By ignoring negligible contributions from ImS2, the expression for the half-widths is given by
where the subscript c refers to the distance of closest approach.
The half-width is determined by the integrand
By analyzing behaviors of the integrand, one is able to find out why calculated  for some lines converge more quickly than others.
As an example, we consider two typical lines:
The first line is the transition of 21,1 ← 20,2 (ω = cm-1 and  = cm-1/atm).
The second line is the transition of 88,0 ← 77,1(ω = cm-1 and  = cm-1/atm).

15. III D. Why calculated half-width for the line (21,1 ← 20,2) converges more quickly than that for (88,0 ← 77,1)
Fig. 7 ReS2, exp(-ReS2), and the integrand of γ derived from the 14-th order are represented by solid, dash-dotted, and dotted lines, respectively. Those associated with the (88,0 ← 77,1) line are plotted by red and those for (21,1 ← 20,2) are given by green.

16. III D. Why calculated half-width for the line (21,1 ← 20,2) converges more quickly than that for (88,0 ← 77,1)
Fig. 8 ReS2 derived from the 8-th, 14-th, and 20-th order are presented by solid, dashed, and dotted lines, respectively. Among them, those associated with (88,0 ← 77,1) are plotted by red and those for (21,1 ← 20,2) are plotted by green. As shown by the figure, for both these two lines results derived from 14-th and 20-th order cut-offs are almost identical.

17. III D. Why calculated half-width for the line (21,1 ← 20,2) converges more quickly than that for (88,0 ← 77,1)
Fig. 9 The integrand of γ derived from the 8-th, 14-th, and 20-th order are presented by solid, dashed, and dotted lines, respectively. Among them, those associated with (88,0 ← 77,1) are plotted by red and those for (21,1 ← 20,2) are plotted by green. As shown by the figure, for both these two lines results derived from 14-th and 20-th order cut-offs are almost identical.

18. III D. Explanations of why calculated half-width for the line (21,1 ← 20,2) converges more quickly than that for (88,0 ← 77,1)
As shown in Fig. 8, there are significant differences between distributions of ReS2 over rc for these two lines: 21,1 ← 20,2 and 88,0 ← 77,1.
As shown in Fig. 9, the integrand for 21,1 ← 20,2 differs completely from that of 88,0 ← 77,1. The latter is limited within a region with rc < 3.75 Å and the former distributes more widely. It becomes obvious that why the half-width value of 21,1 ← 20,2 is larger than that of 88,0 ← 77,1.
The short range interaction plays a crucial role in determining the half-width for 88,0 ← 77,1. Meanwhile, the long range interaction plays a major role for 21,1 ← 20,2. Thus, to describe the short range interaction correctly is more important for 88,0 ← 77,1 than for 21,1 ← 20,2. This is the reason why they have quite different convergence behaviors.
Thus, we have found answers two questions: why for some lines their convergence behaviors are good and others are poor? why some lines have large half-width values and others have small values? It turns out that these two questions are somehow related.

19. III E. Convergence check for the temperature exponent n resulting from using different cut-offs
Fig. 10 Calculated temperature exponent n derived from adopting different combinations of the cut-offs versus calculated half-width values from the 20-th order cut-off and 38 correlations. Those derived from the 8-th order and 20 correlations, from the 14-th order and 20 correlations, and from the 20-th order and 38 correlations are represented by symbols ∆, +, and ×, respectively.

20. III E. Convergence check for the temperature exponent n resulting from using different cut-offs
Fig. 11 Relative percentage errors of the calculated temperature exponent n resulting from adopting different combinations of the cut-offs versus calculated half-widths from the 20-th order cut-off and 38 correlations. The convergence errors associated with the 8-th order cut-off and 20 correlations, with the 20-th order and 20 correlations, and with the 14-th order and 38 correlations are represented by symbols ∆, +, and ×, respectively.

21. III E. Comparisons of calculated temperature exponents n from the 20-th order and 38 correlations and those in updated HITRAN
Fig. 12 Comparisons between calculated temperature exponent n derived from the 20-th order cut-off and 38 correlations and those listed in HITRAN. The former are plotted by × and the latter are given by ∆, respectively. As shown in the figure, large differences occur not only for lines with small half-widths, but also for those with large half-widths.

22. IV. MODIFICATION OF THE CRB FORMALISM
The RB formalism is a theory. Every step in the derivation must be sound.
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 simple notation for

23. IV. EFFECTS FROM MODIFICATION OF THE CRB FORMALISM
Fig. 13 Relative errors of calculated half-widths resulting from the subtle derivation error in developing the CRB formalism. The percentage errors are measured by (modifiedRB - RB) / modifiedRB % where the subscript modifiedRB and RB of  indicate these half-widths are derived from Eq. (1) and Eq. (11), respectively. In calculating , the 20-th order and 38 correlations are used. In the plot, the 501 strong lines are arranged in the order according to their half-width values from the smallest to the largest ones.

24. IV. EFFECTS FROM MODIFICATION OF THE CRB FORMALISM
Conclusions:
(1)The modified formulas are not only physically sound, they requires less numerical calculations.
(2) With respect to high accuracy required by atmospheric applications, the effect resulting from the modification could be important. Among the 501 strong lines of the H2O pure rotational band, there are 85 lines with relative errors above 5 %.
(3) It is worthwhile to use the modified formulas in the theoretical calculations.

25. V. HOW TO ESTIMATE HALF-WIDTHS WITHOUT COMPLICATED CALCULAIONS
(1) A simplified expression for the half-width is given by
(2) The Expression for Re(S2,outer,i) is given by
where HL1K1K1’L2(ω) are Fourier transforms of the correlations and D(…) is a simple notation for
(3) For lines of interest their values of  are mainly determined by the arguments ωii’’ appearing in HL1K1K1’L2 . Thus, one can use averaged ωii’’ to estimate .

26. V. HOW TO ESTIMATE HALF-WIDTHS WITHOUT COMPLICATED CALCULAIONS
(4) We introduce averaged energy differences of coupled states, defined by
were f(ω) is a weighting function to model profiles of HL1K1K1’L2(ω) and W is a normalization constant defined by
(5) We consider two major correlations with L1 = 1 (i.e., 1000 and 1001). One can derive values of for thousands lines with a few seconds of CPU. Then, one can estimate corresponding values of  with a simple fitting formula.
(6) By comparing two figures (a plot of  vs and a plot of  vs. (ji + jf)/2), one can see how the new parameter works.

27. V A. The pure rotational band of H2O
Fig. 14 Air-broadened half-widths of the 1639 lines of the pure rotational band in HITRAN versus (Ji + Jf)/2.

28. V A. The pure rotational band of H2O
Fig. 15 Air-broadened half-widths of the 1639 lines of the pure rotational band in HITRAN versus the averaged energy difference.

29. V B. The (010) ← (000) band of H2O
Fig. 16 Air-broadened half-widths of the 1903 lines of the (010) ← (000) band in HITRAN versus (Ji + Jf)/2.

30. V B. The (010) ← (000) band of H2O
Fig. 17 Air-broadened half-widths of the 1903 lines of the (010) ← (000) band in HITRAN versus the averaged energy difference.

31. V C. The (100) ← (000) band of H2O
Fig. 18 Air-broadened half-widths of the 1326 lines of the (100) ← (000) band in HITRAN versus (Ji + Jf)/2.

32. V C. The (100) ← (000) band of H2O
Fig. 19 Air-broadened half-widths of the 1326 lines of the (100) ← (000) band in HITRAN versus the averaged energy difference.