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