Transcript PP talk Ohhio-causal-function. final..ppt

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
I. General formalism in calculating the induced shift
With the modified Robert-Bonamy (RB) formalism, the induced
shift δ is given by

 Re S2
 Re S2
nb 
nb 
db

vf
(
v
)
dv
2

bdb
sin(
S

Im
S
)
e

2

b
(
)
dr
sin(
S

Im
S
)
e
.
1
2
1
2
0

2 c 0
2 c rc ,min
dr
Usually, S2 consists of three terms S2,outer,I, S2,outer,f, and S2,middle.
For example, S2,outer,I is given by

t
1
i ( ji i ji i  j2 j2 )( t t  )

S2,outer,i (rc )  2

dt
dt
e
 j   
(2 ji 1) j2 j2 2 ji i ( m ) 

 ji i mi j2 m2| V (R (t ))| ji imi j2 m2   ji imi j2 m2 | V ( R (t ')) | ji i mi j2 m2  .
One prefers to write the potential in terms of the spherical
expansions,
V (R(t )) 

u ( L1 L2 L; K1 ; R(t ))
L1K1L2 L
 C ( L L L, m m m ) D
1 2
1
2
L1 *
m1K1

(a ) DmL22 0*(b )YLm
( (t )).
m1m2 m
Thus, one needs to express the short range atom-atom
component as
U ( L1 K1 L2 L, n{ij} , wq)
Vatomatom (a , b , R(t )) 
 
L1K1L2 L n{ ij } wq

 C ( L L L, m m m ) D
1 2
1
2
L1 *
m1K1
R L1  L2 q 2 w (t )

(a ) DmL22 0*(b )YLm
( (t )).
m1m2 m
As a result, to introduce cut-offs becomes necessary and it could
cause convergence problems in practical calculations.
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
|    |  (a  a )   |  (b  b ) ,
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.
V ( R, a , b) |    V ( R, a , b ) |   .
It is easy to make transformations between the state and the coordinate
representations by using the inner products
  | ji i mi j2 m j2    ji  imi (a )   j2m j (b ),
2
where  ji  i mi (a ) and  j2m j (b ) are wave functions of the absorber
2
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.
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

FL1K1K1L2 (t ) 
 dt G
L1K1K1L2
(t   t/2, t   t/2),

where GL K K L (t , t ) is given by
1
GL1K1K1L2 (t , t ) 
(1) K1  K1  (1)( L1  L2  L ) (2 L  1)
2
2
2
4 (2 L1  1) (2 L2  1)
L
 u ( L1 L2 L; K1 ; R(t )) u ( L1 L2 L; K1; R(t )) PL (cos t t ' ).
1 1 1 2
One can introduce two functions which are independent of the potential
and trajectory models and defined by
WL(1aK)1K1 (t )   (2 ji  1) D( ji i ji i; L1 K1 ) D( ji i ji i; L1K1) e
i ji i ji i t
,
ji i
WL(2b) (t )   (2 j2  1)(2 j2  1)  j2 C 2 ( j2 j2 L2 ,000) e
i j2 j2 t
,
j2 j2
k
j
j



D
(
j

j

;
LK
)

(

1)
U
U
k
k k  K  C ( j j L, k K  k K )
where
and U kj are expansion coefficients of the H2O wave functions.
Then,
S2,outer,i (rc ) 


(a)
(b )
dtW
(
t
)
W
 L1K1K1 L2 (t ) FL1K1K1L2 (t ).
L1 K1K1L2 0
II-3. Fourier Transforms of correlation functions and subsequent
Hilbert Transforms
There are two further steps required to calculate ReS2 and ImS2. First of
all, by carrying out Fourier transforms of the correlation functions
1
H L1K1K1L2 ( ) 
2

e
it
FL1K1K1L2 (t ) dt ,

One is able to obtain ReS2 such as
Re S2,outer,i (rc ) 

  (2 j   1) D( j 
2 L1K1K1L2
i
ji i
j   ; L1 K1 ) D( ji i ji i ; L1 K1)
i i i i
  (2 j2  1)(2 j2  1)  j2 C 2 ( j2 j2 L2 ,000) H L1K1K1L2 ( ji i ji i   j2 j2 ).
j2 j2
The next step is to perform subsequent Hilbert transforms defined by
I L1K1K1L2 ( ) 
1


P  d 

1
H L1K1K1L2 ( ),
  
where P means the principal value.
Then, one can find ImS2 such as
Im S2,outer,i (rc ) 

  (2 j   1) D( j 
2 L1K1K1L2
ji i
i
j   ; L1 K1 ) D( ji i ji i ; L1 K1)
i i i i
  (2 j2  1)(2 j2  1)  j2 C 2 ( j2 j2 L2 ,000) I L1K1K1L2 ( ji i ji i   j2 j2 ).
j2 j2
ReS2 = ReS2,outer,i + ReS2,outer,f + S2,middle and ImS2 = - ImS2,outer,i + ImS2,outer,f.
II-4. Challenge in Performing the Hilbert Transforms
Relationships among FL1K1K1L2 (t ), H L1K1K1L2 ( ), and I L1K1K1L2 ( ) are:
Starting from the correlations FL1K1K1L2 (t ), their Fourier transforms are
H L1K1K1L2 ( ). Then, the Hilbert transforms of H L1K1K1L2 ( ) are I L1K1K1L2 ( ).
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 H L K K L ( ).
1 1 1 2
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 I L1K1K1L2 ( ), we have not
reported any calculated results involving evaluations of I L1K1K1L2 ( ).
In summary, to find an alternative way to evaluate I L1K1K1L2 ( ) becomes
mandatory. We began to wonder whether taking the two steps is the only
way to find I L K K L ( ) or can one derive these functions directly from the
correlations? 1 1 1 2
III. Causal Function and Fourier transform
F(t)
The correlation
function
With Fourier
Transform
?
I(ω)
H(ω)
The Fourier
transform of F(t)
With Hilbert
Transform
The Hilbert
transform of H(ω)
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)?
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
() 
1
1
( ()  iP ).
2

(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
H () 
1
2


H ( )
1
1



H
(

)

(



)
d


[
H
(

)

i
P

     d  ].
2
 
(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) = 2ReH(ω) and I(ω) = 2ImH(ω).
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)},
F (t ) 
N 1
 F (n ) sinc (2  (t  n t )).
n 0
f
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
{H (m )}  N t  F {F (n )}.
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.
III-2. The accuracy Check of Calculated Hilbert Transform
We consider a Gaussian function F(t) = exp(- t2/2) whose Fourier transform is also a Gaussian
H(ω) = exp(- ω2/2) 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
N=2097152, ∆ t=0.1×10-11
N=131072, ∆ t=0.1×10-7
N=65536, ∆ t=0.1×10-6
1.0
0.578290E+00
0.578290E+00
0.578290E+00
0.578290E+00
10.0
0.806116E-01
0.806116E-01
0.806118E-01
0.806124E-01
50.0
0.159641E-01
0.159640E-01
0.159651E-01
0.159681E-01
100.0
0.797964E-02
0.797954E-02
0.798156E-02
0.798765E-02
200.0
0.398952E-02
0.398940E-02
0.399344E-02
0.400566E-02
300.0
0.265964E-02
0.265964E-02
0.266571E-02
0.268412E-02
400.0
0.199472E-02
0.199475E-02
0.200285E-02
0.202755E-02
500.0
0.159578E-02
0.159581E-02
0.160596E-02
0.163708E-02
600.0
0.132981E-02
0.132985E-02
0.134206E-02
0.137978E-02
700.0
0.113984E-02
0.113989E-02
0.115416E-02
0.119870E-02
800.0
0.997357E-03
0.997419E-03
0.101378E-02
0.106537E-02
900.0
0.886539E-03
0.886610E-03
0.905067E-03
0.964032E-03
1000.0
0.797885E-03
0.797964E-03
0.818543E-03
0.885226E-03
Calculated I(ω)
III-3. The accuracy Check of Calculated Hilbert Transform
For two linear molecules, the
resonance functions
associated with the Vdd, Vdq,
and Vqq 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 H11(k), I11(k),
H12(k), I12(k), H22(k), I22(k) from
the causal correlations F11(z),
F12(z), F22(z). They are plotted
in (a)-(f) by red dotted curves.
The resonance functions are
given by black solid lines.
IV. Applications in Calculating N2 Induced Shifts
The main tasks to calculate N2-broadened half-widths and induced shifts
for H2O lines are evaluations of several dozens of the correlation
functions FL K K L (t , rc ) labeled by one tensor rank L1 with two subsidiary
indices K1, K1΄ related to H2O and another tensor rank L2 for N2.
1 1 1 2
Because N2 is a diatomic molecule, L2 must be even. If one chooses the II
R representation to develop the H2O wave functions where the two H
atoms are symmetrically located in the molecular-fixed frame, K1 and K1΄
must also be even.
The number of correlations required in calculations is determined by the
cut-offs for L1 and L2. 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 H L K K L (, rc ) and I L K K L (, rc ). Some samples
are presented here.
1 1 1 2
1 1 1 2
Cut-offs
L1,max = L2,max =2
L1,max =3, L2,max =2
L1,max =4, L2,max =2
L1,max = L2,max =4
# of Correlations
20
38
88
132
# of Independent
8
14
26
39
IV-1. Contributions from ImS2(rc) to calculated Half-Widths
People have assumed that ImS2(rc) can be ignored in calculating the half-width
such that the formula can be simplified as
 Re S2 ( rc )
 Re S2 ( rc )
nb 
nb 
db
db
 
2

b
(
)[1

cos(Im
S
(
r
))
e
]
dr

2

b
(
)[1

e
] drc .
2 c
c


2 c rc ,min
drc
2 c rc ,min
drc
We can show that this assumption is an acceptable and justified approximation. Of
course, if one knows how to accurately evaluate ImS2(rc) which are necessary for
calculations of the shifts, it is better to take into account of ImS2(rc).
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).
IV-2. Comparison between Shifts in HITRAN and our Results
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.
(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.
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
 RB

nb 
 Re ( S2 )

v
f
(
v
)
dv
2

bdb

1

cos(
S

Im(
S
))
e
 j2 ,
1
2
0
2 c 0
 RB

nb 
 Re ( S2 )

v
f
(
v
)
dv
2

bdb

sin(
S

Im(
S
))
e
 j2 .
1
2


2 c 0
0
In the modified RB formalism
 mod RB

 Re ( S2  j )
nb 
2

vf
(
v
)
dv
2

bdb
[1

cos
(

S


Im(

S

))
e
].
1 j2
2 j2


2 c 0
0
 mod RB

 Re ( S2  j )
nb 
2

vf
(
v
)
dv
2

bdb
sin
(

S


Im(

S

))
e
.
1 j2
2 j2


2 c 0
0
where <A >j2 is a notation for
 A  j2   (2 j2  1) e E ( j2 ) / kT A( j2 ) / Qb .
j2
IV-4. Effect on Shifts from the Modification of the RB Formalism
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.
(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.
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 H2O molecule,
bath molecules, and electromagnetic fields as a black box. Its outputs are
the spectroscopic parameters and its inputs are the H2O lines of interest.
The latter is represented by the energy levels and the wave functions
associated with their initial and final H2O states.
One can categorize H2O 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.
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.
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 H2O 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.
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 N2 induced shifts of H2O 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 halfwidths 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.