alexandre faure, claire rist, yohann scribano, pierre valiron, laurent wiesenfeld laboratoire...
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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)4905000 GHz
ALMA (2010)30950 GHz
RTN FP6 « Molecular Universe » (2004-2008)
Astrochemistry ?
1. 90% hydrogen
2. Low temperatures
(T = 10 – 1,000K)
3. Ultra-low densities
(nH ~ 103-1010 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.
Rota
tion
al en
erg
y
0
6B
12B
2B
J=0
J=2
J=1
J=3
J(J+1)B
radiative collisional
Aij ~ Cij
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.
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
Infinitebasis
Improving electronic correlation
Imp
rovi
ng
the
bas
is s
et
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-H2
R12 versus basis set extrapolation
Wernli et al. (2006)
H2O-H2
Towards the basis set limit
Double quality
R12
Faure et al. (2005); Valiron et al. (2008)
H2O-H2
ab initio convergence
Ab initio minimum of the H2O-H2 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 ei(48 basis functions)
Rist et al.,in preparation
Application to H2O-H2
wavenumber accuracy !
Valiron et al. (2008)
2D plots of H2O-H2 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 100-200 basis functions)
Cubic spline radial extrapolation (for short and long-range)
H2CO-H2
Troscompt et al. (2008)
NH3-H2
Faure et al., in preparation
SO2-H2
Feautrier et al. in preparation
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)
Isotopic effects: HDO-H2
=21.109o
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-matrixS2 = transition probability
Classical approach
Hamilton’s equations
Cross section andimpact parameter
Statistical error
Rate coefficient (canonical Monte-Carlo)
CO-H2 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-H2 PES provides subwavenumber accuracy on rovibrational spectrum ! (see Jankowski & Szalewicz 2005)
Lapinov, private communicqtion, 2006
CO-H2 Impact of PES inaccuracies
H2O-H2
Impact of PES inaccuracies
Phillips et al.equilibrium geometries
CCSD(T) atequilibrium geometries
CCSD(T)-R12 at equilibrium geometries
CCSD(T)-R12at averaged geometries
Dubernet et al. (2006)
H2O-H2 Ultra-cold collisions
Scribano et al., in preparation
Isotopic effects
Scribano et al., in preparation
Yang & Stancil (2008)
HC3N-H2
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-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
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
H2O-H2
total cross sections
para 000→ 111
H2O
H2
min
max
Ter Meulen et al., in preparation
H2O-H2
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. CH3OCH3) Vibrational excitation, in particular « floppy » modes