1. I thus hope that you have a good memory
and remember the two Newton’s equations
of classical mechanics !!
How 1rst year of university knowledge can
become useful many years later
Some complex pressure effects on spectra from simple classical mechanics
Jean-Michel Hartmann
Laboratoire Interuniversitaire des Systèmes Atmosphériques
CNRS and Université Paris-Est Créteil, Créteil, France
Most or the problems treated here have no first principle quantum treatment for the molecular systems considered because it is to CPU costly or because there is no rigorous method available.

2. The basic system (free gas)
We choose
- the number of molecules NM (typically 104)
- the pressure P and temperature T
For «free» gas (unconfined), we put the molecules inside opened cubic box of size L=[NMkBT/P]1/3 L

3. The basic system (spatially confined gas)
Infinite // planes (distance D)
L=[NMkBT/(P/D)]1/2 L
L=[NMkBT/P]/(pR2)
Opened cylinder (radius R) L

4. The periodic boundary conditions (free gas)
26 boxes surounding central one
Important criterion
L>RPotential intermolecualr potential range
So that a molecule in central box does not « feel » its sister images in surrounding boxes

5. The periodic boundary conditions (confined gas)
Important criterion: L>RPotential intermolecular potential range L

6. Molecular coordinates Four for linear moleculesx y z

7. Initializations Random
- CoM positions (with minimum distance between molecules)
- CoM velocities orientations
- Molecules orientations
- Molecules angular momentum direction (but with )
Maxwell Boltzmann
- CoM speed
- Angular speed

8. Intermolecular interactions site-site pairwise
One has to sum over all site s’ of all molecule M’ within
« interaction » sphere of molecule, i.e: such that
Force on site s of molecule M due to site s’ of molecule M’

9. { Time evolution Next time step (dt small enough) Force on molecule M:
Torque on molecule M:
Acceleration of CoM position:
Acceleration of molecule rotation {
Next time step
(dt small enough)

10. Applications Data used
Collision Induced Absorption
Far wings of absorption band
Depolarized Rayleigh scattering band
Broadening and shape of isolated lines (free and spatially confined gas)
Line mixing effects
- Collisional dissipation of laser-induced alignement
Data used
All parameters from literature (not a single one adjusted)
- Molecules geometries
- Interaction induced dipole, polarizabilities, mulipoles
- Intermolecular potentials
- Molecules-surface interaction potentials
- Pore sizes (for confined gases)

11. Collision Induced Absorption (CIA)
We consider here « pure » CIA, ie situations in which there is no (permanent or vibrating) dipole tied to the molecule

12. The induced dipole z z y x x q2 j2 q
The presence of molecule M’
nearby molecule M induces in M a dipole hrough various mechanisims (eg the polarizability of M in the electric field of the quadrupole of M’ x y z q j2 R M M’ x
In this « molecular frame » frame, induced dipole is a function of R, q, q1, j2
Known from ab initio calculations or expressed (long range expansion) in terms of the molecules polarizabilities and electric multipoles
Easily transformed to coordinates of the CoMs and
axis orientations of molecules M and M’ in order to
compute dipole induced in M by M’ in Laboratory frame so that
is known

13. The CIA spectrum
Compute the induced dipole autocorrelation function (ACF)
Compute the Laplace-Fourier transform of the ACF
Compute the Absorption coefficient a(w)
Normalize by square of molecular density r(P,T) (for pure gas)
→ a(w)/r2 for comparisons with measurements

14. Pure CO2 CIA in the far IR
Hartmann et al. J. Chem. Phys. 134, (2011).
Measured values by Ho et al. J. Chem. Phys. 55, 1028 (1971).

15. Pure N2 CIA in the far IR
Bussery-Honvault et al. J. Chem. Phys. 140, (2014)
149 K
296 K
Measured values by by Stone et al., Can. J. Phys. 62, 338 (1984)
and Dagg et al., Can. J. Phys. 63, 85 (1985)

16. Far wings of « allowed » bands
We consider here the (wing) regions of bands generated through a tensor (dipole, polarizability) tied to the molecule

17. The spectrum
There is now an induced dipole and a “permanent” dipole (not collision-induced but tied to the isolated molecule, e.g. due to the molecule vibration)
Induced dipole: as explained before
Intrinsic (permanent) dipole:
Compute the total dipole autocorrelation function (ACF)
Compute the Laplace-Fourier transform C(w) of the ACF [eventually including the term exp(-iwVibt) due to vibration]
Then, as before, compute absorption coefficient and normalize by square of molecular density for comparisons with measurements

18. Pure CO2 absorption around the n3 band (4.3 mm)
Hartmann et al. J. Chem. Phys. 134, (2011).
Measured value by Tran et al. J. Quant. Spectrosc. Radiat. Transfer 112, 985 (2011)
and Lamouroux et al. J. Quant. Spectrosc. Radiat. Transfer 111, (2010)

19. The various dipoles contributions
Hartmann et al. J. Chem. Phys. 134, (2011).

20. Pure CO2 depolarized Rayleigh scattering
Hartmann et al. J. Chem. Phys. 134, (2011).
Measured values by Teboul et al. J. Chem. Phys. 103, 1384 (1995)

21. Individual Line shapes
We consider isolated lines vs pressure - For free gas - for gases confined in porous materials

22. A problem of the purely classical approach
Purely classical → the spectrum obtained after Laplace Fourier transform does not show any line structure
The dipole ACF does not show the periodic revivals
due to the quantum nature of rotation

23. Introducing the Doppler effect
« Requantization »
Once in a while (periodically or, alternatively, when the current
molecule is not strongly interacting with any other) we change the
molecule angular momentum from
to where JM is an
integer such that is as close as possible to
NOTES: (1) This is for P lines, JM being replaced by JM +1 for R
lines. (2) JM must be even for CO2 (odd for O2)
Introducing the Doppler effect
For a radiation at w0 propagating along z (wave vector
Compute dipole ACF taking molecules displacements, through

24. Dipole autocorrelation function
CO2, 0.4 atm, 296 K
No Doppler (s0=20000 cm-1)
Doppler for s0=20000 cm-1

25. Resulting spectrum

26. Broadening and line shapes of individual lines of free gases
We simulate the spectrum, fit the lines with Voigt profiles, then look at the broadening coefficient and fit residuals (non-Voigt effects)

27. Pressure broadening coefficients: pure CO2, 296 K
Hartmann et al. Phys. Rev. A 87, (2013)
Measured values by Predoi-Cross et al. J. Mol. Spectrosc. 245, 34 (2007).

28. Non Voigt effects: pure CO2, P(14) line, 296 K
Hartmann et al. Phys. Rev. A 87, (2013)
Experimental results obatined from CRDS spectra recorded in S.-M. Hu’s group

29. Non Voigt effects: pure CO2, 4 lines, 296 K
Hartmann et al. Phys. Rev. A 87, (2013)
P(16)
P(14)
R(12)
R(14)
Experimental values obtained using spectra recorded by 4 different groups

30. Pressure broadening coefficients: O2 in air, 296 K
Lamouroux et al. Phys. Rev. A 89, (2014)
Measured values from many sources

31. Non Voigt effects: pure O2 in air, 5 lines, 296 K
Lamouroux et al. Phys. Rev. A 89, (2014)
Experimental results obatined from CRDS spectra recorded in J. Hodges’s group

32. Broadening of individual lines of gases confined in the pores of xerogels and ceramics
Xerogel sample inserted inside cell
Small gap between sample and cell windows
→ Both free and confined gas contributions

33. R(6) line of CO at low P in two xerogel samples
CO gas in pores
Free CO gas
CO gas in pores
Free CO gas
Spectra measured by J. Van der Auwera. J. VdA et al, Phys Rev A 88, (2013)

34. Line widths of R(J) lines pure CO free versus confined gas
Hartmann et al. J. Chem. Phys. 140, (2014)
Exp. values for confined CO from Van der Auwera et al. , Phys Rev A 88, (2013)
Experimental values for free CO from HITRAN data base

35. Line widths of 2 lines of confined CO effect of pressure
Hartmann et al. J. Chem. Phys. 140, (2014)
Gas-gas collisions
Gas-wall and
Gas-gas collisions
Gas-wall collisions

36. Line widths of confined O2 at 1 atm effect of confinement dimension
Hartmann et al. Phys. Rev. A 87, (2013)
Values measured in several porous materials in several papers T. Svensson (Lund, Sweden)

37. Line mixing
We consider conditions where due to their broadening, the lines strongly overlap
The spectrum is then sensitive to collision-induced transfers of populations between rotationl levels (ie transfers of intensity between the lines)

38. CO2 - CO2, 00031 00001 IR band, T=294K Experimental
Lamouroux et al. . J Chem Phys 138, (2013)
22.7Am
35.5Am
51.3Am
Experimental
rCMDS
Tran et al., JQSRT 112 (2011), 925
No LM (Voigt)
CMDS for
ECS calculations for the other bands

39. Pure CO2, 22 isotropic Raman Q branch, T=295K
Lamouroux et al. . J Chem Phys 138, (2013)
0.5Am
Experimental
rCMDS
Lavorel et al.,
J. Chem. Phys. 93 (1990), 2176
No LM (Voigt)
Striking agreement by the
Strong overestimation of the widths by the Voigt profile
2.0Am
10.0Am

40. Laser-kicked molecules
We consider the nonadiabatic (alignment) of molecule by a non-resonant very short and intense linearly polarized laser pulse And look at the collisional dissipation in the « laser-aligned » molecular gas system
Extensions have been made to molecules « prepared » by an « optical centrifuge » (rotating extremelly fast, many in the same direction and plane)

41. Introducing the laser pulse
Hartmann et al. J. Chem. Phys. 136, (2012).
Laser pulse creates dipole in molecule m through the laser electric field and molecule polarizability tensor
, known in molecular frame, transformed to laboratory fixed frame using rotation matrix (obtained from molecule orientation)
Then dipole in field creates torque that tends to align the molecule axis onto the filed polarization direction.
Requantize in order to match the |DJ|=2 Raman transitions involved
In considered experiments, alignment measured through the average A(t)=<cos2[q(t)]-1/3> where q(t) is angle between molecule axis and laser (linear) polarization [A(t)=0 when no alignment (isotropy)]

42. rCMDS vs measurements. Decay of transient alignment revivals
Hartmann et al, J. Chem. Phys. 139, (2013).
296 K, 2 bar of CO2,
laser peak intensity of 45 TW/cm2
Experiments by Dijon group. Vieillard et al. Phys. Rev. A 87, (2013)

43. rCMDS vs measurements. Decay of permanent alignment
Hartmann et al, J. Chem. Phys. 139, (2013).
Experiments by Dijon group. Vieillard et al. Phys. Rev. A 87, (2013)

44. Conclusion The good points of rCMDS - Easy to implement
- Enables to treat (no adjusted parameters, treats both R-R and R-T
transfers) problems untractable with first principle quantum mechanics
- Extendable to symmetric tops (to be done)
The limitations of rCMDS
- Not applicable (if requantized) to molecules other than
linear and symmetric tops
- Not applicable to molecules with large rotational constants
(eg H2) or very low T (when DERot not << kT)
- Line shifts not easy to take into account
Future studies
Dissipation (collision) effects on spectra and laser(s)-kicked
- Symmetric-top molecules
- Liquids

45. Thanks to AVERBUKH I. (Israel) BOUAZAOUI M. (France)
JACQUEMART D. (France)
LANDSHEERE X. (France)
LAMOUROUX J. (France)
LAVOREL B. (France)
LEPERE M. (Belgium)
NGO N.H.. (Vietnam)
SIRONNEAU V. (USA)
PANGUI E. (France)
SNELS M. (Italy)
SVENSSON T. (Sweden)
TEBOUL V. (France)
TRAN H. (France)
VAN DER AUWERA (Belgium)
XU C. (Sweden)
AVERBUKH I. (Israel)
BOUAZAOUI M. (France)
BOULET C. (France)
BUSSERY-HONVAULT B. (France)
CAPOEN B. (France)
CHAUSSARD F. (France)
EL HAMZAOUI H. (France
FAUCHER O. (France)
GIANFRANI L. (Italy)
HERTZ E. (France)
HODGES J. (USA)
HU S. (China)