_{2}gas in the nonadiabatic regime

^{1,a)}and C. Boulet

^{2}

### Abstract

A quantum approach and classical molecular dynamics simulations (CMDS) are proposed for the modeling of rotational relaxation and of the nonadiabatic alignment of gaseous linear molecules by a nonresonant laser field under dissipative conditions. They are applied to pure CO_{2} and compared by looking at state-to-state collisional rates and at the value of ⟨cos^{2}[*θ* _{z}(t)]⟩ induced by a 100 fs laser pulse linearly polarized along . The main results are: (i) When properly requantized, the classical model leads to very satisfactory predictions of the permanent and transient alignments under non-dissipative conditions. (ii) The CMDS calculations of collisional-broadening coefficients and rotational state-to-state rates are in very good agreement with those of a quantum model based on the energy corrected sudden (ECS) approximation. (iii) Both approaches show a strong propensity of collisions, while they change the rotational energy (i.e., J), to conserve the angular momentum orientation (i.e., M/J). (iv) Under dissipative conditions, CMDS and quantum-ECS calculations lead to very consistent decays with time of the “permanent” and transient components of the laser-induced alignment. This result, expected from (i) and (ii), is obtained *only if* a properly J- and M-dependent ECS model is used. Indeed, rotational state-to-state rates and the decay of the “permanent” alignment demonstrate, for pure CO_{2}, the limits of a M-independent collisional model proposed previously. Furthermore, computations show that collisions induce a decay of the “permanent” alignment about twice slower than that of the transient revivals amplitudes, a direct consequence of (iii). (v) The analysis of the effects of reorienting and dephasingelastic collisions shows that the latter have a very small influence but that the former play a non-negligible role in the alignment dynamics. (vi) Rotation-translation collisionally induced transfers have also been studied, demonstrating that they only slightly change the alignment dissipation for the considered laser energy conditions.

J.M.H. thanks the *Institut du Développement et des Ressources en Informatique Scientifique* (IDRIS) for giving access to the IBM Blue Gene/P parallel computer. The authors thank Velisa Vesovic for providing data on rotation-translation relaxation and J.-M. Launay and F. Thibault for their efficient routines for 3J and 6J calculations. O. Atabek, F. Chaussard, O. Faucher, and B. Lavorel are acknowledged for useful discussions and their careful reading of this manuscript and the referees are thanked for their help in improving this paper.

I. INTRODUCTION

II. CLASSICAL MOLECULAR DYNAMICS CALCULATIONS

A. Model

B. Data used and computational procedure

C. Requantization

1. Energy

2. Raman frequencies

III. QUANTUM APPROACH

A. Alignment model

B. State-to-state collision-induced rotational transfers

C. Data used and computational procedure

IV. COMPARISON BETWEEN CLASSICAL AND QUANTUM RESULTS

A. Alignment under non-dissipative conditions (no collisions, low pressure)

B. Collisional relaxation

1. Broadening coefficients

2. Rates for J changing (inelastic) collisions

3. Rates for J and M changing (inelastic) collisions

4. Rates for J conserving and M changing (elastic-reorienting) collisions

C. Alignment under dissipative conditions

V. VIBRATION-ROTATION-TRANSLATION EXCHANGES

VI. CONCLUSION

### Key Topics

- Carbon dioxide
- 28.0
- Collision theories
- 21.0
- Elastic collisions
- 17.0
- Dephasing
- 8.0
- Elasticity
- 8.0

## Figures

Time dependence of CO_{2} alignments for a 100 fs pulse of 150 TW/cm^{2} predicted by the quantum model (red) and re-quantified classical calculation (black).

Time dependence of CO_{2} alignments for a 100 fs pulse of 150 TW/cm^{2} predicted by the quantum model (red) and re-quantified classical calculation (black).

Detailed views of the results of Fig. 1 with, from left to right, the alignment peak and the first four revivals.

Detailed views of the results of Fig. 1 with, from left to right, the alignment peak and the first four revivals.

Effect of the (100 fs) laser power on the amplitudes and shapes of the alignment pic (a) and (b), and first revival (a′) and (b′). The (requantized) classical (CMDS) and the quantum results are displayed on the left (a) and (a′) and right (b) and (b′) plots, respectively. The curves correspond to laser powers of 25 (green), 50 (blue), 100 (black), and 150 (red) TW/cm^{2}.

Effect of the (100 fs) laser power on the amplitudes and shapes of the alignment pic (a) and (b), and first revival (a′) and (b′). The (requantized) classical (CMDS) and the quantum results are displayed on the left (a) and (a′) and right (b) and (b′) plots, respectively. The curves correspond to laser powers of 25 (green), 50 (blue), 100 (black), and 150 (red) TW/cm^{2}.

Effect of the (100 fs) laser power on the amplitudes of alignment. In blue (right scale) are values of the “permanent” component ⟨cos[*θ*(t)]^{2}⟩_{P}. The maximum and the minimum of ⟨cos[*θ*(t)]^{2}⟩_{T} in the alignment peak and first revival are given (left scale) by the upper and lower results in red, respectively. The symbols are quantum predictions while the lines have been obtained using the (requantized) classical (CMDS) approach.

Effect of the (100 fs) laser power on the amplitudes of alignment. In blue (right scale) are values of the “permanent” component ⟨cos[*θ*(t)]^{2}⟩_{P}. The maximum and the minimum of ⟨cos[*θ*(t)]^{2}⟩_{T} in the alignment peak and first revival are given (left scale) by the upper and lower results in red, respectively. The symbols are quantum predictions while the lines have been obtained using the (requantized) classical (CMDS) approach.

Relative populations of the rotational levels J after the extinction of the (100 fs) laser pulse (at t^{+} = 1 ps) for various laser powers. The symbols and lines are results obtained with the quantum and requantized classical (CMDS) approaches, respectively.

Relative populations of the rotational levels J after the extinction of the (100 fs) laser pulse (at t^{+} = 1 ps) for various laser powers. The symbols and lines are results obtained with the quantum and requantized classical (CMDS) approaches, respectively.

Average rotational energies just after the (100 fs) laser pulse. The symbols and lines are results obtained with the quantum and requantized classical (CMDS) approaches, respectively.

Average rotational energies just after the (100 fs) laser pulse. The symbols and lines are results obtained with the quantum and requantized classical (CMDS) approaches, respectively.

Collisional broadening coefficients of pure CO_{2} isotropic Raman Q(J) lines at 296 K. The ECS results are in red, the values from CMDS in blue, and the experimental values (see text) in black. The insert provides a comparison between ECS predictions for the Q(JM) (●) and Q(J) (o).

Collisional broadening coefficients of pure CO_{2} isotropic Raman Q(J) lines at 296 K. The ECS results are in red, the values from CMDS in blue, and the experimental values (see text) in black. The insert provides a comparison between ECS predictions for the Q(JM) (●) and Q(J) (o).

State-to-state rates *K* _{ J → J′ ≠ J } for J changes in pure CO_{2} at 296 K. Classical (CMDS) and quantum (ECS) results are in blue and red, respectively.

State-to-state rates *K* _{ J → J′ ≠ J } for J changes in pure CO_{2} at 296 K. Classical (CMDS) and quantum (ECS) results are in blue and red, respectively.

ECS calculated state-to-state rates *K* _{ J, M → J′, M′} for J = 20 (top) and J = 40 (bottom) with J′ = J−2 (left) and J′ = J−10 (right) plotted vs M/J and M′/J′.

ECS calculated state-to-state rates *K* _{ J, M → J′, M′} for J = 20 (top) and J = 40 (bottom) with J′ = J−2 (left) and J′ = J−10 (right) plotted vs M/J and M′/J′.

State-to-state rates for inelastic J and M changes (see text). Classical (CMDS) and quantum (ECS) results are in blue and red, respectively.

State-to-state rates for inelastic J and M changes (see text). Classical (CMDS) and quantum (ECS) results are in blue and red, respectively.

Same as Fig. 10 but for elastic reorienting (J conserving, M changing) collisions (see text).

Same as Fig. 10 but for elastic reorienting (J conserving, M changing) collisions (see text).

Time dependence of the alignment induced by a (100 ps) 50 TW/cm^{2} laser pulse in pure CO_{2} at 1 atm and 296 K. The classical (CMDS) results are in blue. Those of the quantum (ECS) approach with the M-dependent and M-independent models are in red and black, respectively.

Time dependence of the alignment induced by a (100 ps) 50 TW/cm^{2} laser pulse in pure CO_{2} at 1 atm and 296 K. The classical (CMDS) results are in blue. Those of the quantum (ECS) approach with the M-dependent and M-independent models are in red and black, respectively.

Detailed views of the results in Fig. 12 for successive revivals of the first and second transients.

Detailed views of the results in Fig. 12 for successive revivals of the first and second transients.

CMDS predicted average rotational (dashed lines) and translational (full lines) energies vs time for pure CO_{2} at 1 atm and initially at 296 K for 50 (red), 100 (blue), and 150 (black) TW/cm^{2}. The horizontal green lines indicate the asymptotic values calculated from Eq. (17).

CMDS predicted average rotational (dashed lines) and translational (full lines) energies vs time for pure CO_{2} at 1 atm and initially at 296 K for 50 (red), 100 (blue), and 150 (black) TW/cm^{2}. The horizontal green lines indicate the asymptotic values calculated from Eq. (17).

Rotational populations (top) and center of mass velocity distributions (bottom) 3 ps (black), 150 ps (red), and 300 ps (blue) after a (100 ps) 150 TW/cm^{2} laser pulse in pure CO_{2} at 1 atm and initially at 296 K. The symbols are CMDS results and the lines correspond to Boltzmann distributions calculated with the rotational and translational temperatures of Fig. 14.

Rotational populations (top) and center of mass velocity distributions (bottom) 3 ps (black), 150 ps (red), and 300 ps (blue) after a (100 ps) 150 TW/cm^{2} laser pulse in pure CO_{2} at 1 atm and initially at 296 K. The symbols are CMDS results and the lines correspond to Boltzmann distributions calculated with the rotational and translational temperatures of Fig. 14.

## Tables

Time constants (in ps at 1 atm and 296 K) of the exponential decays of the transient amplitudes (*τ* _{T}) and “permanent” (*τ* _{P}) components of ⟨cos^{2}[*θ*(t)]⟩ obtained from the requantized classical approach (CMDS) and the quantum (ECS) model using M-dependent and M-independent state-to-state rates. For comparison (see Appendix D), values of the averaged (using the populations after the pulse) Q(J)and Q(JM) line broadening coefficients are given in the last two columns.

Time constants (in ps at 1 atm and 296 K) of the exponential decays of the transient amplitudes (*τ* _{T}) and “permanent” (*τ* _{P}) components of ⟨cos^{2}[*θ*(t)]⟩ obtained from the requantized classical approach (CMDS) and the quantum (ECS) model using M-dependent and M-independent state-to-state rates. For comparison (see Appendix D), values of the averaged (using the populations after the pulse) Q(J)and Q(JM) line broadening coefficients are given in the last two columns.

Time constants (in ps at 1 atm) of the exponential decays of the transient amplitudes (*τ* _{T}) and “permanent” (*τ* _{P}) components of ⟨cos^{2}[*θ*(t)]⟩ obtained with the quantum (ECS) model by including or neglecting the effects of reorienting and/or dephasing elastic collisions. Results obtained for CO_{2} at 1 atm, 296 K, and a (100 fs) pulse of 50 TW/cm^{2}.

Time constants (in ps at 1 atm) of the exponential decays of the transient amplitudes (*τ* _{T}) and “permanent” (*τ* _{P}) components of ⟨cos^{2}[*θ*(t)]⟩ obtained with the quantum (ECS) model by including or neglecting the effects of reorienting and/or dephasing elastic collisions. Results obtained for CO_{2} at 1 atm, 296 K, and a (100 fs) pulse of 50 TW/cm^{2}.

Time constants (ps atm) of the exponential decays of the components of the alignment within the M-conserving and M-randomizing models; values obtained from a thermal average (at 296 K) of the broadening coefficient in the insert in Fig. 7.

Time constants (ps atm) of the exponential decays of the components of the alignment within the M-conserving and M-randomizing models; values obtained from a thermal average (at 296 K) of the broadening coefficient in the insert in Fig. 7.

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