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 Potential energy surfaces for inelastic collisions

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

1. Molecules in space

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

RTN FP6 « Molecular Universe » ( )

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

A very rich chemistry ! Smith (2006)

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

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

Wanted: Collisional rate coefficients M(j, v) + H 2 (j 2, v 2 )  M(j’, v’) + H 2 (j 2 ’, v 2 ’) Collision energies from ~ 1 to 1,000 cm -1, i.e. rotational excitation dominant As measurements are difficult, numerical models rely on theoretical calculations.

2. Computing PESs

Born-Oppenheimer approximation 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

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

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 !

State-of-the-art: R12 theory

CO-H 2 R12 versus basis set extrapolation Wernli et al. (2006)

H 2 O-H 2 Towards the basis set limit Double  quality R12 Faure et al. (2005); Valiron et al. (2008)

H 2 O-H 2 ab initio convergence Ab initio minimum of the H 2 O-H 2 PES as a function of years

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

Expanding 5D PES

Scalar products : Sampling « estimator »: Mean error: In preparation

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

Convergence of e i (48 basis functions) Rist et al.,in preparation

Application to H 2 O-H 2 wavenumber accuracy ! Valiron et al. (2008)

2D plots of H 2 O-H 2 PES Valiron et al. (2008)

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)

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 basis functions) Cubic spline radial extrapolation (for short and long-range)

H 2 CO-H 2 Troscompt et al. (2008)

NH 3 -H 2 Faure et al., in preparation

SO 2 -H 2 Feautrier et al. in preparation

HC 3 N-H 2 «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) Wernli et al. (2007)

Isotopic effects: HDO-H 2  = o Scribano et al., in preparation 

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

2. Scattering calculations

Close-coupling approach Schrödinger (time independent) equation + Born-Oppenheimer PES Total wavefunction Cross section and S-matrix S 2 = transition probability

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

CO-H 2 Impact of PES inaccuracies Wernli et al. (2006)

Inaccuracies of PES are NOT dramatically amplified Wavenumber accuracy sufficient for computing rates at T>1K Note: the current CO-H 2 PES provides subwavenumber accuracy on rovibrational spectrum ! (see Jankowski & Szalewicz 2005) Lapinov, private communicqtion, 2006 CO-H 2 Impact of PES inaccuracies

H 2 O-H 2 Impact 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)

H 2 O-H 2 Ultra-cold collisions Scribano et al., in preparation

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

HC 3 N-H 2 Classical mechanics as an alternative to close-coupling method ? T=10K

Wernli et al. (2007), Faure et al., in preparation T=10K T=100K o-H 2 /p-H 2 selectivity due to interferences Rotational motion of H 2 is negligible at the QCT level As a result, o-H 2 rates are very similar to QCT rates

Faure et al. (2006)

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

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

Cappelletti et al., in preparation H 2 O-H 2 total cross sections

para 0 00 → 1 11 H2OH2O H2H2 min max Ter Meulen et al., in preparation H 2 O-H 2 differential cross sections

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. CH 3 OCH 3 ) Vibrational excitation, in particular « floppy » modes