Potential energy surfaces for inelastic collisions

1. Potential energy surfaces for inelastic collisions Alexandre Faure, Claire Rist, Yohann Scribano, Pierre Valiron, Laurent Wiesenfeld Laboratoire d’Astrophysique de Grenoble Mathematical Methods for Ab Initio Quantum Chemistry, Nice, 14th november 2008

2. Outline 1. Astrophysical context 2. Determining, monitoring and fitting multi-dimensional PESs 3. Computing scattering cross sections 4. Conclusions

3. 1. Molecules in space

4. New windows on the « Molecular Universe » Herschel (2009) 4905000 GHz ALMA (2010) 30950 GHz

5. RTN FP6 « Molecular Universe » (2004-2008)

6. 1. 90% hydrogen 2. Low temperatures (T = 10 – 1,000K) 3. Ultra-low densities (nH~ 103-1010 cm-3). Astrochemistry ? Astronomer’s periodic table, adapted from Benjamin McCall

7. A very rich chemistry ! Smith (2006)

8. Molecules as probes of star formation Lada et al. (2003)

9. Electric-dipolar transitions obey strict selection rules: J = 1 Collisional transitions obey « propensity » rules: J = 1, 2, etc. Aij ~ Cij J(J+1)B J=3 12B radiative collisional J=2 6B Rotational energy J=1 2B 0 J=0 Challenge:modelling non-LTE spectra

10. Wanted:Collisional rate coefficients • M(j, v) + H2(j2, v2)  M(j’, v’) + H2(j2’, v2’) • Collision energies from ~ 1 to 1,000 cm-1, i.e. rotational excitation dominant • As measurements are difficult, numerical models rely on theoretical calculations.

11. 2. Computing PESs

12. Electronic problem Orbital approximation Hartree-Fock (variational principle) Electronic correlation (configuration interaction) Nuclear problem « Electronic » PES Quantum dynamics: close-coupling, wavepackets Semi or quasi-classical dynamics: trajectories Born-Oppenheimer approximation

13. Hartree- Fock Improving electronic correlation Full CI Improving the basis set Infinite basis Hartree- Fock limit « Exact » solution Electronic structure calculations

14. van der Waals interactions • The interaction energy is a negligible fraction of molecular energies: E(A-B) = E(AB) – E(A) –E(B) • For van der Waals complexes, the bonding energy is ~ 100 cm-1 • Wavenumber accuracy (~ 1 cm-1) required !

15. State-of-the-art: R12 theory

16. CO-H2R12 versus basis set extrapolation Wernli et al. (2006)

17. H2O-H2Towards the basis set limit Double  quality R12 Faure et al. (2005); Valiron et al. (2008)

18. H2O-H2ab initio convergence Ab initio minimum of the H2O-H2 PES as a function of years

19. where Computational strategy Faure et al. (2005); Valiron et al. (2008)

20. Expanding 5D PES

21. Scalar products : • Sampling « estimator  »: • Mean error: In preparation

22. Convergence of ||S-1|| (48 basis functions) Rist et al.,in preparation

23. Convergence of ei(48 basis functions) Rist et al.,in preparation

24. Application to H2O-H2wavenumber accuracy ! Valiron et al. (2008)

25. 2D plots of H2O-H2 PES Valiron et al. (2008)

26. Equilibrium vs. averaged geometries The rigid-body PES at vibrationally averaged geometries is an excellent approximation of the vibrationally averaged (full dimensional) PES Faure et al. (2005); Valiron et al. (2008)

27. Current strategy • Monomer geometries: ground-state averaged • Reference surface at the CCSD(T)/aug-cc-pVDZ (typically 50,000 points) • Complete basis set extrapolation (CBS) based on CCSD(T)/aug-cc-pVTZ (typically 5,000 points) • Monte-Carlo sampling, « monitored » angular fitting (typically 100-200 basis functions) • Cubic spline radial extrapolation (for short and long-range)

28. H2CO-H2 Troscompt et al. (2008)

29. NH3-H2 Faure et al., in preparation

30. SO2-H2 Feautrier et al. in preparation

31. «Because of the large anisotropy of this system, it was not possible to expand the potential in a Legendre polynomial series or to perform quantum scattering calculations. »  (S. Green, JCP 1978) HC3N-H2 Wernli et al. (2007)

32. Isotopic effects: HDO-H2  =21.109o Scribano et al., in preparation

33. Isotopic effects: significant ? Scribano et al., in preparation

34. 2. Scattering calculations

35. Close-coupling approach Schrödinger (time independent) equation + Born-Oppenheimer PES Total wavefunction Cross section and S-matrix S2 = transition probability

36. Classical approach Hamilton’s equations Cross section and impact parameter Statistical error Rate coefficient (canonical Monte-Carlo)

37. CO-H2Impact of PES inaccuracies Wernli et al. (2006)

38. Inaccuracies of PES are NOT dramatically amplified Wavenumber accuracy sufficient for computing rates at T>1K Note: the current CO-H2 PES provides subwavenumber accuracy on rovibrational spectrum ! (see Jankowski & Szalewicz 2005) CO-H2Impact of PES inaccuracies Lapinov, private communicqtion, 2006

39. H2O-H2Impact of PES inaccuracies Phillips et al. equilibrium geometries CCSD(T) at equilibrium geometries CCSD(T)-R12 at equilibrium geometries CCSD(T)-R12 at averaged geometries Dubernet et al. (2006)

40. H2O-H2Ultra-cold collisions Scribano et al., in preparation

41. Isotopic effects Yang & Stancil (2008) Scribano et al., in preparation

42. HC3N-H2Classical mechanics as an alternative to close-coupling method ? T=10K

43. o-H2/p-H2 selectivity due to interferences Rotational motion of H2 is negligible at the QCT level As a result, o-H2 rates are very similar to QCT rates T=10K T=100K Wernli et al. (2007), Faure et al., in preparation

45. Faure et al. (2006)

46. Experimental tests • Total (elastic + inelastic) cross sections • Differential cross sections • Pressure broadening cross sections • Second virial coefficients • Rovibrational spectrum of vdW complexes

47. CO as a benchmark T=294K T=294K T=15K T=15K Carty et al. (2004) Jankowski & Szalewicz (2005)

48. H2O-H2total cross sections Cappelletti et al., in preparation

49. para 000→ 111 H2 H2O H2O-H2differential cross sections max min Ter Meulen et al., in preparation

50. Conclusions • Recent advances on inelastic collisions • PES • Ab initio: CCSD(T) + CBS/R12 • Fitting: Monte-Carlo estimator • Cross section and rates • Wavenumber accuracy of PES is required but sufficient • Success and limits of classical approximation • Future directions • « Large » polyatomic species (e.g. CH3OCH3) • Vibrational excitation, in particular « floppy » modes