Causal Correlation Functions and Causal Correlation Functions and Fourier Transforms: Application in Fourier Transforms: Application in

Calculating Pressure Induced ShiftsCalculating Pressure Induced Shifts

Q. MaQ. MaNASA/Goddard Institute for Space Studies & NASA/Goddard Institute for Space Studies & Department of Applied Physics and Applied Department of Applied Physics and Applied

Mathematics, Columbia UniversityMathematics, Columbia University2880 Broadway, New York, NY 10025, USA2880 Broadway, New York, NY 10025, USA

R. H. TippingR. H. TippingDepartment of Physics and Astronomy, University Department of Physics and Astronomy, University

of Alabama, Tuscaloosa, AL 35487, USAof Alabama, Tuscaloosa, AL 35487, USA

N. N. LavrentievaN. N. LavrentievaV. E. Zuev Institute of Atmospheric Optics SB RAS, V. E. Zuev Institute of Atmospheric Optics SB RAS,

1, Akademician Zuev square, Tomsk 634021, 1, Akademician Zuev square, Tomsk 634021, RussiaRussia

I. General formalism in calculating the induced shiftI. General formalism in calculating the induced shift With the modified Robert-Bonamy (RB) formalism, the induced shift With the modified Robert-Bonamy (RB) formalism, the induced shift δδ is given by is given by

Usually, SUsually, S22 consists of three terms S consists of three terms S2,outer,I2,outer,I, S, S2,outer,f2,outer,f, and S, and S2,middle2,middle. For . For example, Sexample, S2,outer,I2,outer,I is given by is given by

One prefers to write the potential in terms of the spherical One prefers to write the potential in terms of the spherical expansions,expansions,

Thus, one needs to express the short range atom-atom component Thus, one needs to express the short range atom-atom component asas

As a result, to introduce cut-offs becomes necessary and it could As a result, to introduce cut-offs becomes necessary and it could cause convergence problems in practical calculations.cause convergence problems in practical calculations.

,min

1 2 1 2

0 0

2 2Re Re

( ) 2 sin( Im ) 2 ( ) sin( Im ) .2 2

c

b b

r

S Sn n dbvf v dv bdb S S e b dr S S e

c c dr

2 2

2

2 2

( )( )

2, , 2( )

2 2 2 2 2 2 2 2

1( )

(2 1)

| ( ( ))| | ( ( ')) | .

j j j ji i i i

i i

ti t t

outer i c jj j j mi

i i i i i i i i i i i i

S r dt dt ej

j m j m V R t j m j m j m j m V R t j m j m

1 2

1 1 2 { }

1 2

1 1 2

1 2

1 1 2 { }

2

* *1 2 1 2 0

( , , )( , , ( ))

( )

( , ) ( ) ( ) ( ( )).

ij

ijatom atom a b L L q w

L K L L n wq

L Lm K a m b Lm

m m m

U L K L L n wqV R t

R t

C L L L m m m D D Y t

1 2

1 1 2

1 1 2 1 2

* *1 2 1 1 2 1 2 0( ( )) ( ; ; ( )) ( , ) ( ) ( ) ( ( )).

L Lm K a m b Lm

L K L L m m m

V R t u L L L K R t C L L L m m m D D Y t

II. The Formalism in the Coordinate RepresentationII. The Formalism in the Coordinate RepresentationII-1. Introduction of the Coordinate RepresentationII-1. Introduction of the Coordinate Representation

In the coordinate representation, the basis set | In the coordinate representation, the basis set | αα > in Hilbert space are > in Hilbert space are

where Ωwhere Ωaaαα and Ω and Ωbbαα represent orientations of absorber molecule a and bath represent orientations of absorber molecule a and bath molecule b, respectively.molecule b, respectively.

The greatest advantage of the coordinate representation is the interaction The greatest advantage of the coordinate representation is the interaction potential V is diagonal and can be treated as an ordinary function. potential V is diagonal and can be treated as an ordinary function.

It is easy to make transformations between the state and the coordinate It is easy to make transformations between the state and the coordinate representations by using the inner productsrepresentations by using the inner products

where and are wave functions of the absorber where and are wave functions of the absorber and bath molecules at their orientations. and bath molecules at their orientations.

With the coordinate representation, one is able to overcome the With the coordinate representation, one is able to overcome the convergence challenge because one can select higher cut-offs to convergence challenge because one can select higher cut-offs to guarantee the complete convergence. guarantee the complete convergence.

| | ( ) | ( ) , a a b b

( , , ) | ( , , ) | .

a b a bV R V R

2 2 22| ( ) ( ),

i i i ji i i j j m a j m bj m j m

2 2( )

jj m b( ) i i ij m a

II-2. Irreducible Correlation Functions of the II-2. Irreducible Correlation Functions of the ŜŜ Matrix Matrix

With the coordinate representation, one introduces the irreducible With the coordinate representation, one introduces the irreducible correlation functions of the Ŝ matrix which contain all dynamical correlation functions of the Ŝ matrix which contain all dynamical information about the collisional processes and are defined byinformation about the collisional processes and are defined by

where is given bywhere is given by

One can introduce two functions which are independent of the potential One can introduce two functions which are independent of the potential and trajectory models and defined byand trajectory models and defined by

where where and are expansion coefficients of the Hand are expansion coefficients of the H22O wave functions.O wave functions.

Then, Then,

1 1 1 2 1 1 1 2( ) ( /2, /2),

L K K L L K K LF t dt G t t t t

1 1 1 2( , ) L K K LG t t

1 1 1 2

1 1 1 2

( )2 2 2

1 2

1 2 1 1 2 1 '

1( , ) ( 1) ( 1) (2 1)

4 (2 1) (2 1)

( ; ; ( )) ( ; ; ( )) (cos ).

K K L L L

L K K LL

L t t

G t t LL L

u L L L K R t u L L L K R t P

1 1 1 2 1 1 1 2

1 1 1 2

( ) ( )2, ,

0

( ) ( ) ( ) ( ).

a bouter i c L K K L L K K L

L K K L

S r dtW t W t F t

1 1 1

( )1 1 1 1( ) (2 1) ( ; ) ( ; ) ,

j ji i i i

i i

i taL K K i i i i i i i i i

j

W t j D j j L K D j j L K e

2 2

2 2

2 2

( ) 22 2 2 2 2( ) (2 1)(2 1) ( ,000) ,

j ji tbL j

j j

W t j j C j j L e

( ; ) ( 1) ( , ) k j j

k k Kk

D j j LK U U C j j L kK kK

jkU

II-3. Fourier Transforms of correlation functions and subsequent II-3. Fourier Transforms of correlation functions and subsequent Hilbert TransformsHilbert Transforms

There are two further steps required to calculate ReSThere are two further steps required to calculate ReS22 and ImS and ImS22. First of . First of all, by carrying out Fourier transforms of the correlation functionsall, by carrying out Fourier transforms of the correlation functions

One is able to obtain ReSOne is able to obtain ReS22 such as such as

The next step is to perform subsequent Hilbert transforms defined byThe next step is to perform subsequent Hilbert transforms defined by

where P means the principal value. where P means the principal value. Then, one can find ImSThen, one can find ImS22 such as such as

ReSReS22 = ReS = ReS2,outer,i 2,outer,i + ReS+ ReS2,outer,f 2,outer,f + S+ S2,middle 2,middle andand ImSImS22 = - ImS = - ImS2,outer,i 2,outer,i + ImS+ ImS2,outer,f2,outer,f..

1 1 1 2 1 1 1 2

1( ) ( ) ,

2

L K K L L K K Li tH e F t dt

1 1 1 2

2 1 1 1 2 2 2

2 2

2, , 1 1 1 1

22 2 2 2 2

Re ( ) (2 1) ( ; ) ( ; )2

(2 1)(2 1) ( ,000) ( ).

i i

i i i i

outer i c i i i i i i i i iL K K L j

j L K K L j j j jj j

S r j D j j L K D j j L K

j j C j j L H

1 1 1 2 1 1 1 2

1 1( ) ( ),

L K K L L K K LI P d H

1 1 1 2

2 1 1 1 2 2 2

2 2

2, , 1 1 1 1

22 2 2 2 2

Im ( ) (2 1) ( ; ) ( ; )2

(2 1)(2 1) ( ,000) ( ).

i i

i i i i

outer i c i i i i i i i i iL K K L j

j L K K L j j j jj j

S r j D j j L K D j j L K

j j C j j L I

II-4. Challenge in Performing the Hilbert TransformsII-4. Challenge in Performing the Hilbert Transforms

Relationships among are:Relationships among are:

Starting from the correlations their Fourier transforms areStarting from the correlations their Fourier transforms are Then, the Hilbert transforms of areThen, the Hilbert transforms of are

In practical calculations, the continuous Fourier transforms are replaced In practical calculations, the continuous Fourier transforms are replaced by the discrete Fourier transforms with proper samplings. The latter can by the discrete Fourier transforms with proper samplings. The latter can be easily carried out with the fast Fourier transforms (FFT) algorithm. As be easily carried out with the fast Fourier transforms (FFT) algorithm. As a result, there is no obstacle to derive a result, there is no obstacle to derive

A big challenge arises as one tries to perform the subsequent Hilbert A big challenge arises as one tries to perform the subsequent Hilbert transforms by carrying out the Cauchy principal integrations. The latter’s transforms by carrying out the Cauchy principal integrations. The latter’s subroutines are available, but their performances are not always subroutines are available, but their performances are not always satisfactory. In fact, their unstable performances do happen occasionally satisfactory. In fact, their unstable performances do happen occasionally and that could cause lager errors.and that could cause lager errors.

Mainly due to lack of reliable ways to derive we have not Mainly due to lack of reliable ways to derive we have not reported any calculated results involving evaluations of reported any calculated results involving evaluations of

In summary, to find an alternative way to evaluate becomes In summary, to find an alternative way to evaluate becomes mandatory. We began to wonder whether taking the two steps is the only 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 way to find or can one derive these functions directly from the correlations?correlations?

1 1 1 2 1 1 1 2 1 1 1 2( ), ( ), ( ) L K K L L K K L L K K LF t H and I

1 1 1 2( ),L K K LF t

1 1 1 2( ).L K K LH

1 1 1 2( )L K K LH

1 1 1 2( ).L K K LI

1 1 1 2( ).L K K LH

1 1 1 2( ),L K K LI

1 1 1 2( ).L K K LI

1 1 1 2( )L K K LI

1 1 1 2( )L K K LI

III. Causal Function and Fourier transformIII. Causal Function and Fourier transform

The solution has been found in signal processing. The solution has been found in signal processing. (1) Instead of starting from the function (1) Instead of starting from the function F(t)F(t) itself, one define its causal function itself, one define its causal function FF(t) (t) defined by defined by FF(t) = (t) = F(t) F(t) ×× θθ(t(t) ) where where θθ(t(t) is the unit step function.) is the unit step function.

(2) The Fourier transform of (2) The Fourier transform of θθ(t)(t) denoted by denoted by ΘΘ((ωω)) is well known is well known

(3) The causal function (3) The causal function FF(t) is not an even function. Its (t) is not an even function. Its Fourier transform denoted by Fourier transform denoted by HH((ωω) becomes) becomes complex and can be expressed as complex and can be expressed as

(4) Thus, by taking only one step and without involving the Cauchy principal (4) Thus, by taking only one step and without involving the Cauchy principal integrations, one is able to derive both integrations, one is able to derive both H(H(ωω)) and and I(I(ωω)) from from FF(t) such that(t) such thatH(H(ωω)) = 2Re = 2ReHH((ωω) and ) and I(I(ωω)) = 2Im = 2ImHH((ωω). ).

F(t)The correlation

function

H(ω)The Fourier

transform of F(t)

I(ω)The Hilbert

transform of H(ω)

With Fourier Transform

With Hilbert Transform

?

Fig. 1 A diagram to show Fig. 1 A diagram to show the usual route to derive the usual route to derive the Fourier transform the Fourier transform H(ωH(ω) ) from the function from the function F(t)F(t) and a and a subsequent Hilbert subsequent Hilbert transform transform I(ω)I(ω) of of H(ω)H(ω). Is . Is there a way to establish a there a way to establish a direct link between direct link between F(t)F(t) and and I(ω)I(ω)??

1 1( ) ( ( ) ).

2iP

( )1 1 1( ) ( ) ( ) [ ( ) ].

2 2H

H d H i P d

H

III-1. Carrying out the Fourier Transform with the Sampling TheoryIII-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 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 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 handwidth Ωf and Δt is the Nyquist rate (= 1/(2Ωf), then F(t) can be recovered uniquely and exactly from the sequence {F(n)}, recovered uniquely and exactly from the sequence {F(n)},

where N is the number of sampling.where N is the number of sampling.

With FFT, one calculates the discrete Fourier transform of {F(n)} denoted With FFT, one calculates the discrete Fourier transform of {F(n)} denoted by by {F(n)}. Then, one can show that {H(m)}, the sampling sequence of {F(n)}. Then, one can show that {H(m)}, the sampling sequence of H(H(ωω) obtained with the Nyquist rate in frequency domain, can be ) obtained with the Nyquist rate in frequency domain, can be expressed asexpressed as

Thanks to the sampling theory again, {H(m)} can represent Thanks to the sampling theory again, {H(m)} can represent H(ω)H(ω) without without any distortions. Thus, the combination of the sampling theory and FFT any distortions. Thus, the combination of the sampling theory and FFT provides an effective way to derive the continuous Fourier transform. By provides an effective way to derive the continuous Fourier transform. By applying this method to the causal correlation functions, the challenge to applying this method to the causal correlation functions, the challenge to perform the Hilbert transform is completely overcome.perform the Hilbert transform is completely overcome.

1

0

( ) ( )sin (2 ( )).N

fn

F t F n c t n t

{ ( )} { ( )}.H m N t F n F

III-2. The accuracy Check of Calculated Hilbert TransformIII-2. The accuracy Check of Calculated Hilbert Transform

We consider a Gaussian function F(t) = exp(- tWe consider a Gaussian function F(t) = exp(- t22/2) whose Fourier transform is also a Gaussian /2) whose Fourier transform is also a Gaussian H(H(ωω) = exp(- ) = exp(- ωω22/2) and /2) and the subsequent Hilbert transform is the well known Dawson’s integral. the subsequent Hilbert transform is the well known Dawson’s integral.

By selecting different N and By selecting different N and ∆∆t, calculated values are listed below. As shown in the table, the 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 method works excellently. With the moderate choice of N = 131072, the errors at ωω = 10, 100, = 10, 100, and 1000 are 0.0002%, 0.024%, and 2.59%. and 1000 are 0.0002%, 0.024%, and 2.59%.

In general, the smaller In general, the smaller ωω is, the higher accuracy of I( is, the higher accuracy of I(ωω). In addition, the larger the N is, the ). In addition, the larger the N is, the higher the accuracy. higher the accuracy.

ωω Dawson’s Dawson’s IntegralIntegral

Calculated I(Calculated I(ωω))

N=2097152, N=2097152, ∆∆ t=0.1 t=0.1××1010-11-11 N=131072, N=131072, ∆∆ t=0.1 t=0.1××1010-7-7 N=65536, N=65536, ∆∆ t=0.1 t=0.1××1010-6-6

1.01.0 0.578290E+000.578290E+00 0.578290E+000.578290E+00 0.578290E+000.578290E+00 0.578290E+000.578290E+00

10.010.0 0.806116E-010.806116E-01 0.806116E-010.806116E-01 0.806118E-010.806118E-01 0.806124E-010.806124E-01

50.050.0 0.159641E-010.159641E-01 0.159640E-010.159640E-01 0.159651E-010.159651E-01 0.159681E-010.159681E-01

100.0100.0 0.797964E-020.797964E-02 0.797954E-020.797954E-02 0.798156E-020.798156E-02 0.798765E-020.798765E-02

200.0200.0 0.398952E-020.398952E-02 0.398940E-020.398940E-02 0.399344E-020.399344E-02 0.400566E-020.400566E-02

300.0300.0 0.265964E-020.265964E-02 0.265964E-020.265964E-02 0.266571E-020.266571E-02 0.268412E-020.268412E-02

400.0400.0 0.199472E-020.199472E-02 0.199475E-020.199475E-02 0.200285E-020.200285E-02 0.202755E-020.202755E-02

500.0500.0 0.159578E-020.159578E-02 0.159581E-020.159581E-02 0.160596E-020.160596E-02 0.163708E-020.163708E-02

600.0600.0 0.132981E-020.132981E-02 0.132985E-020.132985E-02 0.134206E-020.134206E-02 0.137978E-020.137978E-02

700.0700.0 0.113984E-020.113984E-02 0.113989E-020.113989E-02 0.115416E-020.115416E-02 0.119870E-020.119870E-02

800.0800.0 0.997357E-030.997357E-03 0.997419E-030.997419E-03 0.101378E-020.101378E-02 0.106537E-020.106537E-02

900.0900.0 0.886539E-030.886539E-03 0.886610E-030.886610E-03 0.905067E-030.905067E-03 0.964032E-030.964032E-03

1000.01000.0 0.797885E-030.797885E-03 0.797964E-030.797964E-03 0.818543E-030.818543E-03 0.885226E-030.885226E-03

III-3. The accuracy Check of Calculated Hilbert TransformIII-3. The accuracy Check of Calculated Hilbert Transform

For two linear molecules, the For two linear molecules, the resonance functions resonance functions associated with the Vassociated with the Vdddd, V, Vdqdq, , and Vand Vqq qq interactions are interactions are available in literary. We can available in literary. We can compare calculated results compare calculated results with them.with them.

Comments:Comments:(1) (1) H(k)H(k) are even and are even and I(k)I(k) are odd.are odd.(2) One has to evaluate (2) One has to evaluate I(k)I(k) in a lager range of k in a lager range of k because they decrease because they decrease more slowly than more slowly than H(kH(k). ). Therefore, the logarithmic Therefore, the logarithmic scale is used for them. scale is used for them. (3) Calculated results (3) Calculated results match the resonance match the resonance functions exactly.functions exactly.

Fig. 2 Calculated Fig. 2 Calculated HH1111(k), I(k), I1111(k), (k), HH1212(k), I(k), I1212(k), H(k), H2222(k), I(k), I2222(k)(k) from from the causal correlations the causal correlations FF1111(z), (z), FF1212(z), (z), FF2222(z). They are plotted (z). They are plotted in (a)-(f) by red dotted curves. in (a)-(f) by red dotted curves. The resonance functions are The resonance functions are given by black solid lines. given by black solid lines.

IV. Applications in Calculating NIV. Applications in Calculating N2 2 Induced ShiftsInduced Shifts

The main tasks to calculate NThe main tasks to calculate N22-broadened half-widths and induced shifts for -broadened half-widths and induced shifts for HH22O lines are evaluations of several dozens of the correlation functions O lines are evaluations of several dozens of the correlation functions labeled by one tensor rank L labeled by one tensor rank L1 1 with two subsidiary indices Kwith two subsidiary indices K11, K, K11΄ related ΄ related to Hto H22O and another tensor rank LO and another tensor rank L22 for N for N22. .

Because NBecause N22 is a diatomic molecule, L is a diatomic molecule, L22 must be even. If one chooses the II R must be even. If one chooses the II R representation to develop the Hrepresentation to develop the H22O wave functions where the two H atoms O wave functions where the two H atoms are symmetrically located in the molecular-fixed frame, Kare symmetrically located in the molecular-fixed frame, K1 1 and Kand K11΄ must also ΄ must also be even.be even.

The number of correlations required in calculations is determined by the The number of correlations required in calculations is determined by the cut-offs for Lcut-offs for L1 1 and Land L22. Due to symmetries, some of the correlations are . Due to symmetries, some of the correlations are identicalidentical. . For different cut-offs, the numbers of correlations and For different cut-offs, the numbers of correlations and independent ones are listed below. In the present study, we have selected independent ones are listed below. In the present study, we have selected the highest cut-offs. the highest cut-offs.

With the new method, we have evaluate 39 independent correlations and With the new method, we have evaluate 39 independent correlations and converted them to their causal functions. Then, we have carried out the converted them to their causal functions. Then, we have carried out the Fourier transforms to derived all and Some samples are Fourier transforms to derived all and Some samples are presented here.presented here.

Cut-offsCut-offs LL1,max 1,max = L= L2,max 2,max =2=2 LL1,max 1,max =3, L=3, L2,max 2,max =2=2 LL1,max 1,max =4, L=4, L2,max 2,max =2=2 LL1,max 1,max = L= L2,max 2,max =4=4

# of Correlations# of Correlations 2020 3838 8888 132132

# of Independent # of Independent 88 1414 2626 3939

1 1 1 2( , )L K K L cF t r

1 1 1 2( , )L K K L cH r

1 1 1 2( , ).L K K L cI r

IV-1. Contributions from ImSIV-1. Contributions from ImS22(r(rcc) to calculated Half-Widths) to calculated Half-Widths

People have assumed that ImSPeople have assumed that ImS22(r(rcc) can be ignored in calculating the half-width ) can be ignored in calculating the half-width

such that the formula can be simplified assuch that the formula can be simplified as

We can show that this assumption is an acceptable and justified approximation. Of We can show that this assumption is an acceptable and justified approximation. Of course, if one knows how to accurately evaluate ImScourse, if one knows how to accurately evaluate ImS22(r(rcc) which are necessary for ) which are necessary for

calculations of the shifts, it is better to take into account of ImScalculations of the shifts, it is better to take into account of ImS22(r(rcc).).

Fig. 3 Comparisons between the calculated N2-broadened half-widths obtained from excluding and including contributions from ImS2(rc). 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 ImS2(rc).

,min ,min

22 2

( ) ( )] ]

Re Re2 ( )[1 cos(Im ( )) 2 ( )[1 .

2 2

c c

b bc c c

c cr r

r rc cS Sn ndb dbb S r e dr b e dr

c dr c dr

IV-2. Comparison between Shifts in HITRAN and our ResultsIV-2. Comparison between Shifts in HITRAN and our Results

(1) There are significant differences between ours and that in HITRAN 2008. (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 Among the 1639 lines, there are 649 lines with relative differences above 50 %, 746 lines within 10 – 50 %, and 244 lines less than 10 %. %, 746 lines within 10 – 50 %, and 244 lines less than 10 %.

(2) Most of the values in HITRAN 2008 come from theoretical calculations. (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 (3) Our values are obtained from the same potential model used in deriving HITRAN values. This implies these two theoretical calculations with the HITRAN values. This implies these two theoretical calculations with the same potential model differ markedly from each other.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 H2O pure rotational band are arranged according to the ascending order of the calculated shift values.

IV-3. Modification of the RB formalismIV-3. Modification of the RB formalism

In developing the RB formalism, there is a subtle derivation error In developing the RB formalism, there is a subtle derivation error in applying the Linked-Cluster Theorem.in applying the Linked-Cluster Theorem.After making the correction, the expressions for the half-width After making the correction, the expressions for the half-width and shift differ from the original ones. and shift differ from the original ones.

In the original RB formalismIn the original RB formalism

In the modified RB formalism In the modified RB formalism

where <A >where <A >j2 j2 is a notation foris a notation for

2

21 2

0 0

Re( )( ) 2 1 cos( Im( )) ,

2

b

RB j

n Sv f v dv bdb S S e

c

2

2 2

)2mod 1 2

0 0

Re(( ) 2 [1 cos ( Im( )) ].

2

jbRB j j

Snvf v dv bdb S S e

c

2

2

22 2

( ) /(2 1) ( ) / . j bj

E j kTA j e A j Q

2

21 2

0 0

Re( )( ) 2 sin( Im( )) .

2

b

RB j

n Sv f v dv bdb S S e

c

2

2 2

)2mod 1 2

0 0

Re(( ) 2 sin ( Im( )) .

2

jbRB j j

Snvf v dv bdb S S e

c

IV-4. Effect on Shifts from the Modification of the RB FormalismIV-4. Effect on Shifts from the Modification of the RB Formalism

(1) The comparisons between calculated shifts of 1639 lines from the (1) The comparisons between calculated shifts of 1639 lines from the original and modified RB formalisms show there are 384 lines with errors 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 %.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 (2) One can conclude that effects on the calculated shifts from the modification of the RB formalism are important.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 H2O pure rotational band are arranged according to the ascending order of the calculated shift values with the modified RB formalism.

V. Applying the Two Rules to Calculated ShiftsV. Applying the Two Rules to Calculated ShiftsV-1. The Pair Identity and Smooth Variation RulesV-1. The Pair Identity and Smooth Variation Rules

One considers a whole system consisting of one absorber HOne considers a whole system consisting of one absorber H22O molecule, O molecule, bath molecules, and electromagnetic fields as a black box. Its outputs are bath molecules, and electromagnetic fields as a black box. Its outputs are the spectroscopic parameters and its inputs are the Hthe spectroscopic parameters and its inputs are the H22O lines of interest. O lines of interest. The latter is represented by the energy levels and the wave functions The latter is represented by the energy levels and the wave functions associated with their initial and final Hassociated with their initial and final H22O states.O states.

One can categorize One can categorize HH22OO lines into different groups such that for the lines lines into different groups such that for the lines of interest of interest within individually defined groups, their inputs have identity within individually defined groups, their inputs have identity and similarity properties. Then, one should expect their outputs to have and similarity properties. Then, one should expect their outputs to have similar properties too.similar properties too.

Two rules are established. Two rules are established. The pair identity rule: two paired lines whose j values are above The pair identity rule: two paired lines whose j values are above certain boundaries have almost identical spectroscopic certain boundaries have almost identical spectroscopic parameters. parameters. The smooth variation rule: for different pairs in the same groups, The smooth variation rule: for different pairs in the same groups, values of their spectroscopic parameters vary smoothly as their j values of their spectroscopic parameters vary smoothly as their j values vary.values vary.

By screening calculated shifts with these two rules, one can check By screening calculated shifts with these two rules, one can check whether the results contain mistakes.whether the results contain mistakes.

V-2. Screening induced shifts listed in HITRAN 2008V-2. Screening induced shifts listed in HITRAN 2008

Fig. 6 The shifts for three Fig. 6 The shifts for three groups {j′groups {j′0,j0,j'' ← j″ ← j″1,j1,j″ ″ , j′, j′1,j1,j'' ← j″ ← j″0,j0,j″ ″ }, }, {j′{j′jj',0',0 ← j″ ← j″jj″,1″,1, j′, j′jj',1',1 ← j″ ← j″jj″,0″,0}}, and {j, and {j′′3,j3,j‘-2‘-2 ← j″ ← j″0,j0,j″ ″ , j′, j′2,j2,j‘-2‘-2 ← j″ ← j″1,j1,j″″} in the } in the R branch, two groups {j′R branch, two groups {j′jj',0',0 ← j″ ← j″jj

″-1,1″-1,1, j′, j′jj',1',1 ← j″ ← j″jj″-1,2″-1,2}}, and {j′, and {j′2,j2,j‘-2‘-2 ← j ← j″″1,j1,j″-1 ″-1 , j′, j′3,j3,j‘-2‘-2 ← j″ ← j″2,j2,j″-1″-1} in the Q } in the Q branch, and one group {j′branch, and one group {j′2,j2,j‘-2‘-2 ← ← j″j″1,j1,j″ ″ , j′, j′3,j3,j‘-2‘-2 ← j″ ← j″0,j0,j″″} in the P } in the P branch. The values of these branch. The values of these groups in HITRAN 2008 are groups in HITRAN 2008 are plotted by plotted by ×× and and ∆∆ in Figs. (a) - in Figs. (a) - (f). Meanwhile, our calculated (f). Meanwhile, our calculated results are given by results are given by ++ and and □□ which are connected by two which are connected by two solid color lines. solid color lines.

The majority of the shift The majority of the shift data in HITRAN 2008 data in HITRAN 2008 follow the pair identity follow the pair identity rule, but there are rule, but there are severe violations of the severe violations of the smooth variation rule. smooth variation rule. In contrast, our In contrast, our calculated shifts follow calculated shifts follow the pair identity and the the pair identity and the smooth variation rules smooth variation rules consistently and consistently and accurately.accurately.

V-3. Comments on the shift data in HITRAN 2008V-3. Comments on the shift data in HITRAN 2008

(1) The two rules are derived and established from the properties (1) The two rules are derived and established from the properties of the energy levels and wave functions of Hof the energy levels and wave functions of H22O states. Thus, all O states. Thus, all the spectroscopic parameters involving high j states must the spectroscopic parameters involving high j states must follow the rules whether they are measured data, or represent follow the rules whether they are measured data, or represent theoretically calculated values. theoretically calculated values.

(2) Unless one has made mistakes in deriving energy levels and (2) Unless one has made mistakes in deriving energy levels and wave functions or made inconsistent errors somewhere else, wave functions or made inconsistent errors somewhere else, calculated results from any self-consistent theories should calculated results from any self-consistent theories should automatically follow these rules. automatically follow these rules.

(3) Most of shift values in HITRAN 2008 are theoretically (3) Most of shift values in HITRAN 2008 are theoretically calculated results. The severe violations of the rules definitely calculated results. The severe violations of the rules definitely mean that the calculations contain large mistakes. Poorly mean that the calculations contain large mistakes. Poorly evaluated resonance functions may play a role. evaluated resonance functions may play a role.

(4) To support the above claim, we have presented our calculated (4) To support the above claim, we have presented our calculated shift values based on the same potential model. Our results shift values based on the same potential model. Our results follow the pair identity and smooth variation rules well. follow the pair identity and smooth variation rules well.

VI. ConclusionsVI. Conclusions

The concept of causal function from signal processing and the The concept of causal function from signal processing and the sampling theory enabled us to discover a powerful and useful tool sampling theory enabled us to discover a powerful and useful tool in evaluating the Hilbert transforms without performing the in evaluating the Hilbert transforms without performing the Cauchy principal integrations. Cauchy principal integrations.

With this new method, we are able to effectively and accurately With this new method, we are able to effectively and accurately calculate converged values of the Ncalculate converged values of the N22 induced shifts of H induced shifts of H22O lines. O lines. Thus, the challenge to calculate converged line shifts with the Thus, the challenge to calculate converged line shifts with the formalism developed using the coordinate representation has formalism developed using the coordinate representation has finally been overcome. finally been overcome.

Thus, one is able to calculate both pressure broadened half-Thus, one is able to calculate both pressure broadened half-widths and pressure induced shifts to the accuracy of the widths and pressure induced shifts to the accuracy of the approximations in the interaction-potential and trajectory models approximations in the interaction-potential and trajectory models without containing convergence errors within the current without containing convergence errors within the current framework of the modified RB formalism. framework of the modified RB formalism.

By comparing our results with those listed in HITRAN 2008, most By comparing our results with those listed in HITRAN 2008, most of which are theoretically calculated values using the same of which are theoretically calculated values using the same potential model, we have shown how large their differences are. potential model, we have shown how large their differences are. Furthermore, by screening both calculated results with the two Furthermore, by screening both calculated results with the two rules, we can conclude that shift data in HITRAN 2008 contain rules, we can conclude that shift data in HITRAN 2008 contain large errors and they should be updated. large errors and they should be updated.