Alexandre Faure, Claire Rist, Yohann Scribano,

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Potential energy surfaces for inelastic collisions

• Alexandre Faure, Claire Rist, Yohann Scribano,
• Pierre Valiron, Laurent Wiesenfeld
• Laboratoire dAstrophysique de Grenoble
• Mathematical Methods for Ab Initio Quantum
  Chemistry, Nice, 14th november 2008

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Outline

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

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1. Molecules in space
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New windows on the  Molecular Universe 
Herschel (2009) 490?5000 GHz
ALMA (2010) 30?950 GHz
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RTN FP6  Molecular Universe  (2004-2008)
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Astrochemistry ?

• 1. 90 hydrogen
• 2. Low temperatures
• (T 10 1,000K)
• 3. Ultra-low densities
• (nH 103-1010 cm-3).

Astronomers periodic table, adapted from
Benjamin McCall
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A very rich chemistry !
Smith (2006)
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Molecules as probes of star formation
Lada et al. (2003)
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Challengemodelling non-LTE spectra

• Electric-dipolar transitions obey strict
  selection rules
• ?J ?1
• Collisional transitions obey  propensity 
  rules
• ?J ?1, ?2, etc.

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WantedCollisional 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.

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2. Computing PESs
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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

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Electronic structure calculations
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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 !

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State-of-the-art R12 theory
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CO-H2R12 versus basis set extrapolation
Wernli et al. (2006)
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H2O-H2Towards the basis set limit
Double ? quality
R12
Faure et al. (2005) Valiron et al. (2008)
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H2O-H2ab initio convergence
Ab initio minimum of the H2O-H2 PES as a function
of years
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Computational strategy

• where

Faure et al. (2005) Valiron et al. (2008)
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Expanding 5D PES
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• Scalar products
• Sampling  estimator  
• Mean error

In preparation
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Convergence of S-1 (48 basis functions)
Rist et al.,in preparation
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Convergence of ei(48 basis functions)
Rist et al.,in preparation
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Application to H2O-H2wavenumber accuracy !
Valiron et al. (2008)
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2D plots of H2O-H2 PES
Valiron et al. (2008)
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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)
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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)

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H2CO-H2
Troscompt et al. (2008)
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NH3-H2
Faure et al., in preparation
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SO2-H2
Feautrier et al. in preparation
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HC3N-H2

• 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)
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Isotopic effects HDO-H2
?
?21.109o
Scribano et al., in preparation
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Isotopic effects significant ?
Scribano et al., in preparation
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2. Scattering calculations
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Close-coupling approach
Schrödinger (time independent) equation
Born-Oppenheimer
PES
Total wavefunction
Cross section and S-matrix S2 transition
probability
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Classical approach
Hamiltons equations
Cross section and impact parameter
Statistical error
Rate coefficient (canonical Monte-Carlo)
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CO-H2 Impact of PES inaccuracies
Wernli et al. (2006)
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CO-H2 Impact of PES inaccuracies

• Inaccuracies of PES are NOT dramatically
  amplified
• Wavenumber accuracy sufficient for computing
  rates at Tgt1K
• Note the current CO-H2 PES provides
  subwavenumber accuracy on rovibrational spectrum
  ! (see Jankowski Szalewicz 2005)

Lapinov, private communicqtion, 2006
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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)
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H2O-H2 Ultra-cold collisions
Scribano et al., in preparation
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Isotopic effects
Yang Stancil (2008)
Scribano et al., in preparation
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HC3N-H2Classical mechanics as an alternative to
close-coupling method ?
T10K
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• 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

T10K
T100K
Wernli et al. (2007), Faure et al., in
preparation
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(No Transcript)
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Faure et al. (2006)
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Experimental tests

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

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CO as a benchmark
T294K
T294K
T15K
T15K
Carty et al. (2004)
Jankowski Szalewicz (2005)
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H2O-H2total cross sections
Cappelletti et al., in preparation
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H2O-H2differential cross sections
max
min
Ter Meulen et al., in preparation
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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