Refinement of the RB Formalism: Taking into Account Effects from the Line Coupling Q. Ma...

Refinement of the RB Formalism: Refinement of the RB Formalism: Taking into Account Effects from Taking into Account Effects from

the Line Couplingthe Line Coupling

Q. MaNASA/Goddard Institute for Space Studies & Department of Applied Physics and Applied Mathematics, Columbia

University2880 Broadway, New York, NY 10025, USA

C. BouletInstitut des Sciences Moléculaires d’Orsay

CNRS (UMR8214) and Université Paris-Sud Bât 350Campus dOrsay F-91405, FRANCE

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

Alabama, Tuscaloosa, AL 35487, USA

I. The Robert-Bonamy FormalismI. The Robert-Bonamy FormalismAdvantages and WeaknessesAdvantages 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 FormalismI. The Robert Bonamy FormalismThe Isolated Line ApproximationThe 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,

Effects from the line mixing are ignored.

• Second, when they evaluated the diagonal matrix of W, they have assumed that

Effects from the line coupling are ignored.

.||

1||Im

1)( 2

ifWifnXF

bifi

ifif

.|| || 2121 ifSiSifSiS eifeif

I. The Robert-Bonamy FormalismI. The Robert-Bonamy FormalismValidity Criteria for Ignoring the Line Mixing and the Line CouplingValidity 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 II. Criterion for Ignoring the Line Coupling Analyzing Formulas for the Raman Q Lines of 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.

The selection rule is determined by the product of two Clebsch-Gordan coefficients. Because L1 must be even, the line coupling occurs only among even j lines or among odd j lines.

).,()000,()12)(12()000,()000,(

);()1()12)(12)(12)(12()1(2)(

22212

22

21

1

2222

2211

11,

,2

ciiiiLLiii

ffii

fifiLL

LJfifi

jjc

fifimiddle

rHLiiCiiLjjCLjjC

JLjjjjWjjjjrS ff

II. Criterion for Ignoring the Line CouplingII. Criterion for Ignoring the Line CouplingNumerical Estimation for the Raman Q Lines of 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 CouplingII. Criterion for Ignoring the Line CouplingProfile of Profile of HH2222(k,r(k,rcc) )

Fig. 1 Profile of the Fourier transform of HH2222(k,r(k,rcc) (in ps) (in ps-2-2) at T = 298 K for ) at T = 298 K for the Nthe N22 - N - N22 pair as a two dimensional function of k and r pair as a two dimensional function of k and rcc (in Å). (in Å).

III. How to Consider the Line CouplingTwo 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

Consequently, there are different expressions for the half-width:

},))Im(cos(1{22

)Re(

21mod22

22

min,

i

c

S

iic

r c

bRB eSSdr

dr

dbb

c

n

,))Im(cos(122 2

2

min,

)Re(21 i

Sc

r c

bRB eSSdr

dr

dbb

c

n

c

.)12( ,21,212mod,21 2

2

biRBorRBorii

RBor SSiS

III. How to Consider the Line CouplingMore 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 LinesIV. Numerical Calculations for the Raman Q LinesFor 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 0.02

59 2.5924 4.8986 2.3701

0 0 0 0 0 0.0114 1.8560 3.6083

IV. Numerical Calculations for the Raman Q LinesIV. Numerical Calculations for the Raman Q Lines

A sample of <<i′f′| exp(– S2) |if>> with the isolated line approximation

A sample of <<i′f′| exp(– S2) |if>> without the 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

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


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.

Calculated half-widths will be significantly reduced because this factor is a major part of the integrand of the half-widths.

).||1(22

)(2

min,

ifeifdrdr

dbb

c

nc

c

rSc

r c

b


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.


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.


A 16 × 16 sub-matrix of the relaxation operator W with j = 0, 2, 4, ∙∙∙, 30

The diagonal matrix elements of W represent the calculated half-widths.

The Sum rule is satisfied, i.e.,

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 0.0

0 0.00 0.00 0.00 0.00 0.00 0.00 5.41 19.07

.0)12(

)12(

}{}{},{

fiiffi

i

i Wj

j

V. Discussions and ConclusionsV. 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 ConclusionsV. 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 RefinementVI. 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.

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