Transcript qma-Ohio-presentation.ppt

Refinement of the RB Formalism:
Taking into Account Effects from
the Line Coupling
Q. Ma
NASA/Goddard Institute for Space Studies & Department
of Applied Physics and Applied Mathematics, Columbia
University
2880 Broadway, New York, NY 10025, USA
C. Boulet
Institut des Sciences Moléculaires d’Orsay
CNRS (UMR8214) and Université Paris-Sud Bât 350
Campus dOrsay F-91405, FRANCE
R. H. Tipping
Department of Physics and Astronomy, University of
Alabama, Tuscaloosa, AL 35487, USA
I. The Robert-Bonamy Formalism
Advantages and Weaknesses
• The RB formalism has been widely used in calculating half-widths
and shifts for many years.
• In comparison with the Anderson-Tsao-Curnutte formalism, it is
characterized by two features:
(1) A non-perturbative treatment of the Ŝ matrix through use of the
Linked-Cluster Theorem (i.e., the Cumulant Expansion).
(2) A convenient description of classical trajectories.
• Since the formalism was developed in 1979, there have been some
improvements.
(1) The “exact” trajectory model has been proposed in 1992.
(2) A derivation error in applying the linked cluster theorem has
been corrected in 2007.
But, its core part remains the same until now.
• There are several approximations whose applicability has not
been thoroughly justified. One of them is the isolated line
approximation.
I. The Robert Bonamy Formalism
The Isolated Line Approximation
• In developing their formalism, Robert and Bonamy have relied on
the isolated line approximation twice.
• First, in calculation the spectral density F(ω), they have only
considerd the diagonal matrix elements of the relaxation operator W,
F ( ) 
1

Im  | X if |2 i
if
1
.
  if  nb  if | W | if 
Effects from the line mixing are ignored.
• Second, when they evaluated the diagonal matrix of W, they have
assumed that
 iS1  S 2
if | iS1  S 2 |if 
 if | e
| if   e
Effects from the line coupling are ignored.
.
I. The Robert-Bonamy Formalism
Validity Criteria for Ignoring the Line Mixing and the Line Coupling
• These two simplifications relied on the same approximation, their
validity criteria are completely different and the latter is more
stringent than the former.
• For ignoring the line mixing, the criterion is ω – La >> nbW.
Roughly speaking, as frequency gaps between lines are much
larger than their half-widths, this criterion is valid at least in cores of
lines.
1 cm-1 separation or so is enough to neglect the line mixing.
• For ignoring the line coupling, the criterion is
‹‹ if| – iS1 – S2 |if ›› >> ‹‹ i′f′| – iS1 – S2 |if ››.
The criterion depends on systems. As shown later, for the Raman Q
lines of N2 – N2, 140 cm-1 separation or so is the minimum
requirement.
II. Criterion for Ignoring the Line Coupling
Analyzing Formulas for the Raman Q Lines of N2 – N2
• For the Raman Q lines of N2 – N2, if the potential does not depend
on the vibration, S1 is zero. In addition, because imaginary parts of
S2,outer,i and S2,outer,f cancel out exactly, the S2 matrix (= S2,outer,i +
S2,outer,f +S2,middle) becomes real.
• Both S2,outer,i and S2,outer,f are diagonal. But, S2,middle is off-diagonal.
S2,middle is the only source responsible for the line coupling.
, i f
S 2i,fmiddle
(rc )  2 (1)
j f  j f
(2 ji  1)( 2 j f  1)( 2 ji  1)( 2 j f  1)  (1)1 J  L1 W ( ji j f ji j f ; JL1 )
L1 L2
 C ( ji jiL1 ,000)C ( j f j f L1 ,000) (2i2  1)( 2i2  1) i2 C 2 (i2i2 L2 ,000) H L1L2 (ii  i2 i2 , rc ).
i2i2
The selection rule is determined by the product of two ClebschGordan coefficients. Because L1 must be even, the line coupling
occurs only among even j lines or among odd j lines.
II. Criterion for Ignoring the Line Coupling
Numerical Estimation for the Raman Q Lines of N2 – N2
• Magnitudes of the off-diagonal elements of S2,middle are mainly
determined by the Fourier transforms of H22(ω,rc).
• The profile of H22(ω,rc) is presented in Fig.1 where one uses
dimensionless k (= ωrc/v) to represent ω.
Its magnitudes decrease as k increases, but remain not negligible
until k ≥ 14 or so. The later corresponds to ω ≥ 140 cm-1 in the small
rc region.
For the Raman Q lines of N2 – N2, separations of nearby coupled
lines are around 4(2j + 3) cm-1. Thus, one must consider the line
coupling because the criterion is not satisfied here.
In evaluating the cumulant expansion, to apply the
isolated line approximation is not justified and it
could cause large errors.
II. Criterion for Ignoring the Line Coupling
Profile of H22(k,rc)
Fig. 1 Profile of the Fourier transform of H22(k,rc) (in ps-2) at T = 298 K
for the N2 - N2 pair as a two dimensional function of k and rc (in Å).
III. How to Consider the Line Coupling
Two Different Definitions of the Cumulant Expansion
The key is to evaluate all the matrix elements of exp(– iS1 – S2).
There are two ways to introduce the cumulant expansion. The one
used by Robert and Bonamy contains a mistake. By correcting the
mistake, a new way has been developed by us.
Their essential difference results from two different ways to definite
the average. The difference yields
S1 or 2,mod RB   (2i2  1) i2 S1 or 2, RB   S1 or 2, RB ib .
i2
Consequently, there are different expressions for the half-width:
 RB

n
db
 b  2b
drc  1  cos( S1  Im( S 2 ))e  Re( S 2 )  i2 ,
2c rc ,min
drc
 mod RB

n
db
 Re(  S 2  i2 )
 b  2b
drc {1  cos( S1  i2  Im(  S 2  i2 ))e
},
2c rc ,min
drc
III. How to Consider the Line Coupling
More Profound Consequence due to Two Different Choices
Within the RB formalism
1. The operator iS1 – S2 depends on states of the bath molecule.
2. The matrix size of iS1 – S2 is determined by (# of lines) × (# of the
bath states).
3. One needs to diagonalize a huge size matrix of iS1 – S2 for each
of collisional trajectories.
4. Computational burdens have forced people to give up attempts to
consider the line coupling, unless extra approximations are
introduced.
Within the modified RB formalism
1. The operator iS1 – S2 is independent of the bath states.
2. The matrix size of iS1 – S2 is only determined by # of lines.
3. One needs to diagonalize a smaller size matrix of iS1 – S2 for
each of collisional trajectories.
4. Computational burdens would be reduced by several dozen
thousand times and become very reasonable.
IV. Numerical Calculations for the Raman Q Lines
For the Raman Q lines of N2 – N2, accurate potential models are
available. The most accurate full quantum calculations match
measured half-width data well.
The RB formalism overestimates the half-widths by large amounts. This
implies the RB formalism is not able to yield reliable results.
By considering the line coupling, one is able to make an
improvement.
First, one calculates <<i′f′| –S2 |if>> for each of the trajectories.
A sample of <<i′f′| –S2 |if>> with j = 0, 2, ∙∙∙, 14 and at rc = 3.677 Å
14.1192
5.9477
0.3832
0
0
0
0
0
5.8249
10.3909
5.7690
0.2221
0
0
0
0
0.3648
5.5228
10.8162
5.5545
0.1302
0
0
0
0
0.1993
5.1939
9.9161
4.9012
0.0739
0
0
0
0
0.1064
4.4496
8.3002
4.0079
0.0396
0
0
0
0
0.0539
3.4982
6.5061
3.1225
0.0190
0
0
0
0
0.0259
2.5924
4.8986
2.3701
0
0
0
0
0
0.0114
1.8560
3.6083
IV. Numerical Calculations for the Raman Q Lines
A sample of <<i′f′| exp(– S2) |if>> with the isolated line approximation
0.00000
0
0
0
0
0
0
0
0
0.00003
0
0
0
0
0
0
0
0
0.00002
0
0
0
0
0
0
0
0
0.00005
0
0
0
0
0
0
0
0
0.00025
0
0
0
0
0
0
0
0
0.00149
0
0
0
0
0
0
0
0
0.00746
0
0
0
0
0
0
0
0
0.02710
A sample of <<i′f′| exp(– S2) |if>> without the approximation
0.02587 0.05518 0.06669 0.06758 0.05946 0.04442 0.02267 0.01151
0.05410 0.11592 0.14144 0.14546 0.13073 0.10056 0.06269 0.02821
0.06263 0.13547 0.16868 0.17899 0.16818 0.13738 0.09238 0.04504
0.05931 0.13020 0.16726 0.18612 0.18678 0.16632 0.12414 0.06728
0.04730 0.10606 0.14244 0.16929 0.18555 0.18464 0.15685 0.09644
0.03077 0.07104 0.10134 0.13131 0.16087 0.18326 0.18144 0.12916
0.01530 0.03670 0.05646 0.08124 0.11330 0.15046 0.17692 0.14879
0.00516 0.01291 0.02153 0.03444 0.05451 0.08383 0.11647 0.11680
IV. Numerical Calculations for the Raman Q Lines
Fig. 2 Two sets of the factor (1 - <<if| exp(– S2) |if>>) for the four Q lines with j =
0 (black), 4 (red), 12 (green), and 20 (blue). The sets derived with and without
the isolated line approx. are plotted by dotted-dash and solid lines, respectively.

nb
db
 S 2 ( rc )

2

b
dr
(
1


if
|
e
| if ).
c

2c rc ,min
drc
Calculated half-widths will be significantly reduced because this factor is
a major part of the integrand of the half-widths.
IV. Numerical Calculations for the Raman Q Lines
Fig. 3 Two sets of profiles of b(db/drc)[1 – exp(– S2(rc))] associated with three Q
lines with j = 4 (black), 12 (red), and 20 (green). They are derived with
excluding and including the line coupling and plotted by dotted-dash and solid
lines, respectively.
Calculated half-widths will be reduced significantly because the
factor is the integrand of the half-widths.
IV. Numerical Calculations for the Raman Q Lines
Fig. 4 Comparison of calculated half-widths and measured data. Values
derived with excluding and including the line coupling are plotted by +
and ∆. Results obtained from the close coupling calculations are given
by ○. Two different measured values are plotted by □ and × ,
respectively.
IV. Numerical Calculations for the Raman Q Lines
A 16 × 16 sub-matrix of the relaxation operator W with j = 0, 2, 4, ∙∙∙, 30
75.44
10.22
4.59
3.34
2.72
2.18
1.69
1.23
0.83
 0.00 0.00
10.05
61.80
12.22
7.78
6.20
4.96
3.83
2.80
1.89
 0.00 0.00
4.27
11.63
56.75
12.16
8.80
6.90
5.31
3.88
2.64
 0.01 0.00
2.92
6.89
11.13
53.94
12.77
8.99
6.77
4.95
3.39
 0.01 0.00
2.19
5.07
7.42
11.25
52.78
12.92
8.70
6.24
4.29
 0.01 0.00
1.59
3.66
5.24
7.13
10.90
51.77
12.83
8.23
5.57
 0.02 0.00
1.09
2.49
3.55
4.72
6.41
10.32
50.50
12.71
7.74
 0.04 0.01
0.68
1.57
2.23
2.96
3.94
5.60
9.72
48.76
12.64  0.07 0.01
0.39

0.89

1.28

1.70

2.27

3.16

4.84

9.22

46.44

 0.14 0.02



0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.01
0.02
 22.60 9.46
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
 5.41 19.07
The diagonal matrix elements of W represent the calculated half-widths.
The Sum rule is satisfied, i.e.,

{if }
(2 ji  1)
W{if },{if }  0.
(2 ji  1)
V. Discussions and Conclusions
• Without justification, to apply the isolated line approximation in
evaluating exp(– iS1 – S2) could cause large errors because the
approximation is more likely not applicable.
• With the modified RB formalism, one is able to consider the line
coupling in practical calculations.
• For the Raman Q lines of N2 – N2, the RB formalism overestimates
the half-widths by a large amount.
• By including the line coupling, our new calculated half-widths are
significantly reduced and become closer to measurements.
• By overcoming one of the main weaknesses of the RB
formalism, our refinement effort goes in the right direction.
V. Discussions and Conclusions
• The new calculated half-widths still do not match measured data.
We don’t consider the differences as a bad sign of our refinement.
• There are other main weaknesses remaining in the RB formalism.
The gaps provide room for further refinements.
• The method can be applied for other molecular systems, such as
such as the N2 and CO mixtures, CO2 – Ar, C2H2 – Ar, CO- Ar, HCl –
Ar, HF – Ar. In all these systems, the RB formalism significantly
overestimates the half-widths.
• For H2O – N2, S2,middle is the only source responsible for off-diagonal
matrix elements of S2. But, the leading correlation functions with L1 =
1 makes major contributions. We expect that for strongly coupled
H2O lines, effects on calculated half-widths from the line
coupling could be significant.
VI. Remaining Challenges in the Refinement
(1) To give up the assumption that the translation and internal
motions are not connected and the trajectories are only
determined by isotropic potentials.
Benefit: Couplings between the translation and internal motions are
taken into account.
Challenge: As a semi-classical theory, the translational motion is
treated classically and the internal motion is treated quantum
mechanically. To consider them together is a very difficult job.
(2) To consider contributions from the third-order expansion of
the Ŝ matrix.
Benefit: One can get higher order contributions and make sure that
the results are converged.
Challenge: One has to include many more terms in calculations.
But, at least for two linear molecules, it is possible to solve this
problem.