1. Causal Correlation Functions and Fourier Transforms: Application in Calculating Pressure Induced Shifts 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

2. I. General formalism in calculating the induced shift With the modified Robert-Bonamy (RB) formalism, the induced shift δ is given by Usually, S 2 consists of three terms S 2,outer,I, S 2,outer,f, and S 2,middle. For example, S 2,outer,I is given by One prefers to write the potential in terms of the spherical expansions, Thus, one needs to express the short range atom-atom component as As a result, to introduce cut-offs becomes necessary and it could cause convergence problems in practical calculations.

3. II. The Formalism in the Coordinate Representation II-1. Introduction of the Coordinate Representation In the coordinate representation, the basis set | α > in Hilbert space are where Ω aα and Ω bα 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 where and are wave functions of the absorber and bath molecules at their orientations. With the coordinate representation, one is able to overcome the convergence challenge because one can select higher cut-offs to guarantee the complete convergence.

4. II-2. Irreducible Correlation Functions of the Ŝ Matrix With the coordinate representation, one introduces the irreducible correlation functions of the Ŝ matrix which contain all dynamical information about the collisional processes and are defined by where is given by One can introduce two functions which are independent of the potential and trajectory models and defined by where and are expansion coefficients of the H 2 O wave functions. Then,

5. II-3. Fourier Transforms of correlation functions and subsequent Hilbert Transforms There are two further steps required to calculate ReS 2 and ImS 2. First of all, by carrying out Fourier transforms of the correlation functions One is able to obtain ReS 2 such as The next step is to perform subsequent Hilbert transforms defined by where P means the principal value. Then, one can find ImS 2 such as ReS 2 = ReS 2,outer,i + ReS 2,outer,f + S 2,middle and ImS 2 = - ImS 2,outer,i + ImS 2,outer,f.

6. II-4. Challenge in Performing the Hilbert Transforms Relationships among are: Starting from the correlations their Fourier transforms are Then, the Hilbert transforms of are Then, the Hilbert transforms of are In practical calculations, the continuous Fourier transforms are replaced by the discrete Fourier transforms with proper samplings. The latter can be easily carried out with the fast Fourier transforms (FFT) algorithm. As a result, there is no obstacle to derive A big challenge arises as one tries to perform the subsequent Hilbert transforms by carrying out the Cauchy principal integrations. The latter’s subroutines are available, but their performances are not always satisfactory. In fact, their unstable performances do happen occasionally and that could cause lager errors. Mainly due to lack of reliable ways to derive we have not reported any calculated results involving evaluations of In summary, to find an alternative way to evaluate becomes mandatory. We began to wonder whether taking the two steps is the only way to find or can one derive these functions directly from the correlations?

7. III. Causal Function and Fourier transform The solution has been found in signal processing. (1) Instead of starting from the function F(t) itself, one define its causal function F (t) defined by F (t) = F(t) × θ(t) where θ(t) is the unit step function. (2) The Fourier transform of θ(t) denoted by Θ(ω) is well known (3) The causal function F (t) is not an even function. Its Fourier transform denoted by H (ω) becomes complex and can be expressed as (4) Thus, by taking only one step and without involving the Cauchy principal integrations, one is able to derive both H(ω) and I(ω) from F (t) such that H(ω) = 2Re H (ω) and I(ω) = 2Im H (ω). With Fourier Transform With Hilbert Transform ? Fig. 1 A diagram to show the usual route to derive the Fourier transform H(ω) from the function F(t) and a subsequent Hilbert transform I(ω) of H(ω). Is there a way to establish a direct link between F(t) and I(ω)?

8. III-1. Carrying out the Fourier Transform with the Sampling Theory With a sampling rate Δt, one converts F(t) to a sequence {F(n)}. According to the Whittaker-Shannon sampling theorem, if F(t) is band limited with handwidth Ωf and Δt is the Nyquist rate (= 1/(2Ωf), then F(t) can be recovered uniquely and exactly from the sequence {F(n)}, where N is the number of sampling. With FFT, one calculates the discrete Fourier transform of {F(n)} denoted by  {F(n)}. Then, one can show that {H(m)}, the sampling sequence of H(ω) obtained with the Nyquist rate in frequency domain, can be expressed as Thanks to the sampling theory again, {H(m)} can represent H(ω) without any distortions. Thus, the combination of the sampling theory and FFT provides an effective way to derive the continuous Fourier transform. By applying this method to the causal correlation functions, the challenge to perform the Hilbert transform is completely overcome.

9. III-2. The accuracy Check of Calculated Hilbert Transform We consider a Gaussian function F(t) = exp(- t 2 /2) whose Fourier transform is also a Gaussian H(ω) = exp(- ω 2 /2) and the subsequent Hilbert transform is the well known Dawson’s integral. By selecting different N and ∆t, calculated values are listed below. As shown in the table, the method works excellently. With the moderate choice of N = 131072, the errors at ω = 10, 100, and 1000 are 0.0002%, 0.024%, and 2.59%. In general, the smaller ω is, the higher accuracy of I(ω). In addition, the larger the N is, the higher the accuracy. ω Dawson’s Integral Calculated I(ω) N=2097152, ∆ t=0.1×10 -11 N=131072, ∆ t=0.1×10 -7 N=65536, ∆ t=0.1×10 -6 1.00.578290E+000.578290E+000.578290E+000.578290E+00 10.00.806116E-010.806116E-010.806118E-010.806124E-01 50.00.159641E-010.159640E-010.159651E-010.159681E-01 100.00.797964E-020.797954E-020.798156E-020.798765E-02 200.00.398952E-020.398940E-020.399344E-020.400566E-02 300.00.265964E-020.265964E-020.266571E-020.268412E-02 400.00.199472E-020.199475E-020.200285E-020.202755E-02 500.00.159578E-020.159581E-020.160596E-020.163708E-02 600.00.132981E-020.132985E-020.134206E-020.137978E-02 700.00.113984E-020.113989E-020.115416E-020.119870E-02 800.00.997357E-030.997419E-030.101378E-020.106537E-02 900.00.886539E-030.886610E-030.905067E-030.964032E-03 1000.00.797885E-030.797964E-030.818543E-030.885226E-03

10. III-3. The accuracy Check of Calculated Hilbert Transform For two linear molecules, the resonance functions associated with the V dd, V dq, and V qq interactions are available in literary. We can compare calculated results with them. Comments: (1) H(k) are even and I(k) are odd. (2) One has to evaluate I(k) in a lager range of k because they decrease more slowly than H(k). Therefore, the logarithmic scale is used for them. (3) Calculated results match the resonance functions exactly. Fig. 2 Calculated H 11 (k), I 11 (k), H 12 (k), I 12 (k), H 22 (k), I 22 (k) from the causal correlations F 11 (z), F 12 (z), F 22 (z). They are plotted in (a)-(f) by red dotted curves. The resonance functions are given by black solid lines.

11. IV. Applications in Calculating N 2 Induced Shifts The main tasks to calculate N 2 -broadened half-widths and induced shifts for H 2 O lines are evaluations of several dozens of the correlation functions labeled by one tensor rank L 1 with two subsidiary indices K 1, K 1 ΄ related to H 2 O and another tensor rank L 2 for N 2. Because N 2 is a diatomic molecule, L 2 must be even. If one chooses the II R representation to develop the H 2 O wave functions where the two H atoms are symmetrically located in the molecular-fixed frame, K 1 and K 1 ΄ must also be even. The number of correlations required in calculations is determined by the cut-offs for L 1 and L 2. Due to symmetries, some of the correlations are identical. For different cut-offs, the numbers of correlations and independent ones are listed below. In the present study, we have selected the highest cut-offs. With the new method, we have evaluate 39 independent correlations and converted them to their causal functions. Then, we have carried out the Fourier transforms to derived all and Some samples are presented here. Cut-offs L 1,max = L 2,max =2 L 1,max =3, L 2,max =2 L 1,max =4, L 2,max =2 L 1,max = L 2,max =4 # of Correlations 203888132 # of Independent 8142639

12. IV-1. Contributions from ImS 2 (r c ) to calculated Half-Widths People have assumed that ImS 2 (r c ) can be ignored in calculating the half-width such that the formula can be simplified as We can show that this assumption is an acceptable and justified approximation. Of course, if one knows how to accurately evaluate ImS 2 (r c ) which are necessary for calculations of the shifts, it is better to take into account of ImS 2 (r c ). Fig. 3 Comparisons between the calculated N 2 -broadened half-widths obtained from excluding and including contributions from ImS 2 (r c ). They are plotted by ∆ and ×, respectively. The 1639 lines in the pure rotational band are arranged according to the ascending order of the calculated half-width values without ImS 2 (r c ).

13. IV-2. Comparison between Shifts in HITRAN and our Results (1) There are significant differences between ours and that in HITRAN 2008. Among the 1639 lines, there are 649 lines with relative differences above 50 %, 746 lines within 10 – 50 %, and 244 lines less than 10 %. (2) Most of the values in HITRAN 2008 come from theoretical calculations. (3) Our values are obtained from the same potential model used in deriving HITRAN values. This implies these two theoretical calculations with the same potential model differ markedly from each other. Fig. 4 A comparison between the shifts listed in HITRAN 2008 and our calculated values. They are plotted by ∆ and ×, respectively. The 1639 lines in the H 2 O pure rotational band are arranged according to the ascending order of the calculated shift values.

14. IV-3. Modification of the RB formalism In developing 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. In the original RB formalism In the modified RB formalism where j2 is a notation for

15. IV-4. Effect on Shifts from the Modification of the RB Formalism (1) The comparisons between calculated shifts of 1639 lines from the original and modified RB formalisms show there are 384 lines with errors above 30 %, 767 lines within 5 – 30 %, and 488 lines with less than 5 %. (2) One can conclude that effects on the calculated shifts from the modification of the RB formalism are important. Fig. 5 Comparisons between the calculated shifts from the original RB formalism and from the modified version. They are plotted by ∆ and ×, respectively. The 1639 lines in the H 2 O pure rotational band are arranged according to the ascending order of the calculated shift values with the modified RB formalism.

16. V. Applying the Two Rules to Calculated Shifts V-1. The Pair Identity and Smooth Variation Rules One considers a whole system consisting of one absorber H 2 O molecule, bath molecules, and electromagnetic fields as a black box. Its outputs are the spectroscopic parameters and its inputs are the H 2 O lines of interest. The latter is represented by the energy levels and the wave functions associated with their initial and final H 2 O states. One can categorize H 2 O lines into different groups such that for the lines of interest within individually defined groups, their inputs have identity and similarity properties. Then, one should expect their outputs to have similar properties too. Two rules are established. The pair identity rule: two paired lines whose j values are above certain boundaries have almost identical spectroscopic parameters. The smooth variation rule: for different pairs in the same groups, values of their spectroscopic parameters vary smoothly as their j values vary. By screening calculated shifts with these two rules, one can check whether the results contain mistakes.

17. V-2. Screening induced shifts listed in HITRAN 2008 Fig. 6 The shifts for three groups {j′ 0,j' ← j″ 1,j″, j′ 1,j' ← j″ 0,j″ }, {j′ j',0 ← j″ j″,1, j′ j',1 ← j″ j″,0 }, and {j′ 3,j‘-2 ← j″ 0,j″, j′ 2,j‘-2 ← j″ 1,j″ } in the R branch, two groups {j′ j',0 ← j″ j″-1,1, j′ j',1 ← j″ j″-1,2 }, and {j′ 2,j‘- 2 ← j″ 1,j″-1, j′ 3,j‘-2 ← j″ 2,j″-1 } in the Q branch, and one group {j′ 2,j‘-2 ← j″ 1,j″, j′ 3,j‘-2 ← j″ 0,j″ } in the P branch. The values of these groups in HITRAN 2008 are plotted by × and ∆ in Figs. (a) - (f). Meanwhile, our calculated results are given by + and □ which are connected by two solid color lines. The majority of the shift data in HITRAN 2008 follow the pair identity rule, but there are severe violations of the smooth variation rule. In contrast, our calculated shifts follow the pair identity and the smooth variation rules consistently and accurately.

18. V-3. Comments on the shift data in HITRAN 2008 (1) The two rules are derived and established from the properties of the energy levels and wave functions of H 2 O states. Thus, all the spectroscopic parameters involving high j states must follow the rules whether they are measured data, or represent theoretically calculated values. (2) Unless one has made mistakes in deriving energy levels and wave functions or made inconsistent errors somewhere else, calculated results from any self-consistent theories should automatically follow these rules. (3) Most of shift values in HITRAN 2008 are theoretically calculated results. The severe violations of the rules definitely mean that the calculations contain large mistakes. Poorly evaluated resonance functions may play a role. (4) To support the above claim, we have presented our calculated shift values based on the same potential model. Our results follow the pair identity and smooth variation rules well.

19. VI. Conclusions The concept of causal function from signal processing and the sampling theory enabled us to discover a powerful and useful tool in evaluating the Hilbert transforms without performing the Cauchy principal integrations. With this new method, we are able to effectively and accurately calculate converged values of the N 2 induced shifts of H 2 O lines. Thus, the challenge to calculate converged line shifts with the formalism developed using the coordinate representation has finally been overcome. Thus, one is able to calculate both pressure broadened half- widths and pressure induced shifts to the accuracy of the approximations in the interaction-potential and trajectory models without containing convergence errors within the current framework of the modified RB formalism. By comparing our results with those listed in HITRAN 2008, most of which are theoretically calculated values using the same potential model, we have shown how large their differences are. Furthermore, by screening both calculated results with the two rules, we can conclude that shift data in HITRAN 2008 contain large errors and they should be updated.