The Collision-Broadened Line Shape of CO2via the Complex Robert-Bonamy Method:

The Complexity of Simplicity

Robert R. Gamache, Julien Lamouroux, Anne L. Laraia

Joint meeting of the 11th HITRAN Database Conference and 10th ASA conferenceJune 16-18, 2010

Why CO2?

Venus and MarsVenus Express with Venus IR ThermalImaging SpectrometerMars Reconnaissance Orbiter Missionwith the Mars Climate Sounder

Why CO2?

EarthA well mixed greenhouse gas with a very long lifetime.

MeasurementsAtmospheric Infrared Sounder (AIRS)Infrared Atmospheric Sounder Interferometer (IASI)Greenhouse Gas Observing Satellite (GOSAT)Orbiting Carbon Observatory (OCO-2) 1% precision

Previous calculations

Yamamoto et al. JQSRT 9, 371 (1969).Bouanich and Brodbeck, JQSRT 14, 141 (1974)Boulet et al. Can. J. Phys. 50, 2178 (1972).Arie et al., Appl Opt. 25, 2584 (1986).Rosenmann et al., J. Chem Phys 88, 2999 (1988);

JQSRT 40, 569 (1988), Appl. Opt 27, 3902 (1988).Gamache and Rosenmann, “N2-, O2-, air-, and self

broadening of CO2 transitions and temperaturedependence for HITRAN,” unpublished data, 1992.

Predoi-Cross et al., Can J. Phys. 87, 517 (2009);JQSRT 111, 1065 (2010).

Predoi-Cross , et al,.Can. J. Phys. 87, 517-535 (2009)

Semiclassical complex Robert-Bonamy theory

where n2 is the number density of perturbers and the average isover all trajectories given by impact parameter b and initialrelative velocity v and initial rotational state J2 of the collisionpartner.

The half-width and line shift are

! " i#( ) f$ i = n2 2 % c

J2 &2 J2 J2

' v f (v) dv 0

() 2%b 1" e" S1 + Im S2( ){ } e"Re S2( )[ ]0

(

) db

Radiator - Perturber trajectorydetermined via the potential

The isotropic component of the atom-atom potential is usedto define the trajectory of the collisions.

Trajectories: parabolic model or Hamilton’s Equations

b

v

V[R(t),Ψ(t)]

Potential Terms

V1,2elec = Vµ1µ2

+ Vµ1!2 + V

!1 µ2 + V

!1!2 + " " "

The electrostatic potential is given by an expansion ofthe charge distribution in terms of the electric momentsof the molecules

Atom-Atom Potential

Lennard-Jones (6-12) parametersCO2-N2

N-O N-Cσ/Å 2.55 2.77ε/ kB(K) 39.12 41.54

Vat!at = 4"ij # ij

rij

$

%&'

()

12

- # ij

rij

$

%&'

()

6*

+,,

-

.//ij

0

rij

Spherical Tensor Expansion of the Potential

where C(1 2 ; m1 m2 m) is a Clebsch-Gordan coefficient, Ω1=(α1, β1, γ1) and Ω 2=(α2, β2, γ2) are the Euler angles describing themolecular fixed axis relative to the space fixed axis. ω = (θ,φ)describes the relative orientation of the centers of mass.

Electrostatic interactions: q=1 and w=0

Atom-atom interactions: q=12 or 6 and w defined by the order ofthe expansion where Order=1+2+2w

V =U !1!2!,n1wq( )Rq+!1 +!2 +2w

w,q!

n1m1m2m

!!1!2!

!

"C !1!2!,m1m2m( ) Dm1n1

!1 #1( ) Dm2 0!2 #2( ) Y!m $( )

CO2

Wave functions - DJ00

Linear molecule - Quadrupole moment

D!2 !02 ,!!D0 !0

2 ,!D2 !02

Reduced matrix elements - CO2

! i ' !V !! i !d" !# ! D00Ji ' !!Dk0

l1' !D00Ji$ !d" !$

C(Ji’, l’, Ji; 0, k, 0)

Only potential term remaining is

D002

Trajectories

Parabolic Approximation

2!bdb"0

#

$ 2!rcrmin

#

$ drcvc'

v%&'

()*

2

For many choices of the Lennard-Jones parameterssolutions of the second order in time equations lead to imaginaryvc’. → integration of cross-section cannot be performed.

Very simple systemInitial results CO2-N2 at 296 K

Transition γ cm-1 atm-1

25 ← 24 0.0457 29 ← 28 0.0289 31 ← 30 0.0176 33 ← 32 0.0022 41 ← 40 -0.1540 45 ← 44 -0.3868 47 ← 46 -0.6069

CO2-N2 ν3 P20 transition, J2=0, parabolic trajectories, T=100 K

CO2-N2 ν3 P20 transition, J2=0, HE trajectories, T=100 K

Parabolic model trajectories

HE trajectories

Close collisions and thetrajectory model.

See talk V7 byJ. Lamouroux for details

Integrand

As Re{S2} gets large square bracket goes to one.

! f"i = n2 2 # c

J2 $2 J2 J2

% v f (v) dv 0

&

' 2#b 1( cos S1 + Im S2( )( ) e( Re S2( ))* +,0

&

' db

Test runs with electrostatic potential. We foundcases where the square bracket is larger than one- unphysical.

CO2-CO2 30012-00001 R50, J2=16, 296 K

CO2-CO2 30012-00001 R50, J2=16, 296 K

CO2-CO2 30012-00001 R50, J2=16, 296 K

J dependent - CO2-CO2 30012-00001 R12, J2=16, 296 K

CO2-CO2 30012-00001 R12, J2=16, 296 K

1=2, 2=2 resonance function at 296 K, b = 5.45 Å

Electrostatic potential calculations giving ε/kb=429.58 K and σ = 3.17 Å

1=2, 2=2 resonance function at 296 K, b = 5.45 Å

Atom-atom expansion → 8 2 2 giving ε/kb=158.06 K and σ = 4.22 Å

CO2-CO2 30012-00001 R50, J2=16, 296 K

Line Shift

J-M Hartmann, JQSRT 110 (2009) 2019–2026

! "1 !+!#"1,!"2 !+!#"2 ,!"3 !+!#"3,!J f !$ ! J( )%! "1,!"2 ,!"3,!Ji( )&' ()!=! Ji !$!J f( )!!Rot m( )!+! a1 !#"1 +!a2 !#"2 !+!a3 !#"3( )!!Vib m( )

Using Hartmann’s coefficients a1, a2, and a3 we form the ratios

r12 !=!a2a1,!!r13 !=!

a3a1,

We adjust a1 and set a2 = a1*r12 and a3 = a1*r13 by matching to the dataof Toth et al. and Devi et al. for the 30012-00001 band transitions.

Current valuesa1 = 0.1000, a2 = 0.0516, a3 = 0.2236

Atom-atom parameters

Future workPotential

powers, order, rank

→ Aim is that a single calculation will give thehalf-widths and their temperature dependence,the line shifts and their temperature dependence.

results are strongly dependent on the potential

Acknowledgement

The authors are pleased to acknowledge support ofthis research by the National Science Foundationthrough Grant No. ATM-0803135. Any opinions,findings, and conclusions or recommendationsexpressed in this material are those of the author(s)and do not necessarily reflect the views of theNational Science Foundation.