The H H 2 reaction: a dinamically-biased study for the ortho/para conversion

The H + 3 + H 2 reaction: a dinamically-biased study for the ortho/para conversion Susana Gómez Carrasco Department of Physical Chemistry, University of Salamanca Salamanca, Spain 1 Motivations 2 Related work 3 Theoretical Framework and Results 4 Conclusions 5 Acknowledgements 2 nd National Conference on Laboratory and Molecular Astrophysics

The H + 3 + H 2 reaction: motivations H is the most abundant element in the universe ( 90 %). Deuterated species are also observed. H + 3 is the most abundant cation in the Universe. It can initiate a chain of deuteration processes: H + 3 + HD H 2D + + H 2 H 2 D + + HD D 2 H + + H 2 D 2 H + + X XD + + HD and so on.. Chemical processes in interstellar medium are governed by hydrogenic species. Spectroscopic observations of ortho/para (o/p) ratios of H 2 and H + are used to measure temperatures, dimensions and densities 3 of interstellar clouds. H H + 3

The H + 3 + H 2 reaction Reaction Mechanisms (M) and Nuclear Spin (I) Selection Rules This reaction can proceed in three ways: Identity (M = 1) H + 3 + H 2 H + 3 + H 2 weight = 1/10 H + hop (M = 2) H 3 + + H 2 k hop HH 2 + + H 2 weight = 3/10 H + exc. (M = 3) H 3 + + H 2 k exc. HH 2 + + HH weight = 6/10 1 M. Quack, Mol. Phys. (1977); T. Oka, J. Mol. Spectrosc. (2004); Park&Light, J.Chem.Phys. (2007)

The H + 3 + H 2 reaction Reaction Mechanisms (M) and Nuclear Spin (I) Selection Rules This reaction can proceed in three ways: Identity (M = 1) H + 3 + H 2 H + 3 + H 2 weight = 1/10 H + hop (M = 2) H 3 + + H 2 k hop HH 2 + + H 2 weight = 3/10 H + exc. (M = 3) H 3 + + H 2 k exc. HH 2 + + HH weight = 6/10 Each mechanism is subject to nuclear spin constraints 1 (I = constant = 5/2, 3/2 or 1/2) s (I 3, I 2 ) s (I 3, I 2 ) 1 M. Quack, Mol. Phys. (1977); T. Oka, J. Mol. Spectrosc. (2004); Park&Light, J.Chem.Phys. (2007)

The H + 3 + H 2 reaction Reaction Mechanisms (M) and Nuclear Spin (I) Selection Rules This reaction can proceed in three ways: Identity (M = 1) H + 3 + H 2 H + 3 + H 2 weight = 1/10 H + hop (M = 2) H 3 + + H 2 k hop HH 2 + + H 2 weight = 3/10 γ hop s I=3/2 s o,o o,p p,o p,p o,o 4/9 0 5/9 0 o,p 0 0 1 0 p,o 5/9 1 4/9 0 p,p 0 0 0 0 H + exc. (M = 3) H 3 + + H 2 k exc. HH 2 + + HH weight = 6/10 Each mechanism is subject to nuclear spin constraints 1 (I = constant = 5/2, 3/2 or 1/2) s (I 3, I 2 ) s (I 3, I 2 ) γ sis = branching ratio matrix = γ id sis + γ hop sis + γ exc sis for each value of I (5/2, 3/2, 1/2). 1 M. Quack, Mol. Phys. (1977); T. Oka, J. Mol. Spectrosc. (2004); Park&Light, J.Chem.Phys. (2007)

The H + 3 + H 2 reaction Reaction Mechanisms (M) and Nuclear Spin (I) Selection Rules This reaction can proceed in three ways: Identity (M = 1) H + 3 + H 2 H + 3 + H 2 weight = 1/10 H + hop (M = 2) H 3 + + H 2 k hop HH 2 + + H 2 weight = 3/10 γ hop s I=3/2 s o,o o,p p,o p,p o,o 4/9 0 5/9 0 o,p 0 0 1 0 p,o 5/9 1 4/9 0 p,p 0 0 0 0 H + exc. (M = 3) H 3 + + H 2 k exc. HH 2 + + HH weight = 6/10 Each mechanism is subject to nuclear spin constraints 1 (I = constant = 5/2, 3/2 or 1/2) s (I 3, I 2 ) s (I 3, I 2 ) α = k hop /k exc { α = 0.5 only statistical behaviour?? 1 M. Quack, Mol. Phys. (1977); T. Oka, J. Mol. Spectrosc. (2004); Park&Light, J.Chem.Phys. (2007)

The H + 3 + H 2 reaction Reaction Mechanisms (M) and Nuclear Spin (I) Selection Rules This reaction can proceed in three ways: Identity (M = 1) H + 3 + H 2 H + 3 + H 2 weight = 1/10 H + hop (M = 2) H 3 + + H 2 k hop HH 2 + + H 2 weight = 3/10 γ hop s I=3/2 s o,o o,p p,o p,p o,o 4/9 0 5/9 0 o,p 0 0 1 0 p,o 5/9 1 4/9 0 p,p 0 0 0 0 H + exc. (M = 3) H 3 + + H 2 k exc. HH 2 + + HH weight = 6/10 Each mechanism is subject to nuclear spin constraints 1 (I = constant = 5/2, 3/2 or 1/2) s (I 3, I 2 ) s (I 3, I 2 ) α = k hop /k exc { α = 0.5 only statistical behaviour?? α = f(temperature) some dynamical effect 1 M. Quack, Mol. Phys. (1977); T. Oka, J. Mol. Spectrosc. (2004); Park&Light, J.Chem.Phys. (2007)

Related Work Experiments H + 3 + H 2 reaction: K. N. Crabtree et al. (2011): α = 0.5 ± 0.1 at T = 135 K α 1.6 at T = 350 K M. Cordonnier et al. (2000): α = 2.4 ± 0.6 at T = 400 K Isotopic variants of the H + 3 + H 2 reaction: D. Gerlich (1993): α 1.6 at T = 340 K (D + 3 + H 2) Smith et al. (1981, 1992): rate coefficients for T = 80-100 K D. Gerlich (2002, 2006): rate coefficients E. Hugo et al.(2009): state-to-state rate coefficients for T < 50K

Related Work Theory H + 5 PES Kraemer et al. (1994) Prosmiti et al. (2001) (+ 1 st perturbative DIM) Moyano and Collins (2003) (Interpolated pes) Xie, Bowman et al. (2005) (+ZPE) Barragan et al. (2010) (DFT-like pes) Aguado et al. (2010) (this work) Collisions H + 3 +H 2 reaction Park and Light (2007) Statistical Model isotopic variants of H + 3 +H 2 Moyano and Collins (2003) (QCT) Xie, Bowman et al. (2005) Hugo et al. (2009) Quantum calculations are still challenging

Theoretical Framework: the statistical model We follow the statistical treatment of Park and Light a, but introducing some introducing some dynamical effect in the so-called Scrambling Matrix, S M.. a K. Park and J. C. Light, JCP 126 (2007) 044305

Potential energy surface of H 5 + H + 3 + H 2 (H + 5 )* H 2 + H + 3 Full dimensional PES a CCSD(T) / cc-pvqz (MOLPRO) V H+ 5 = V TRIM + V 5-body charge-induced dipole and quadrupole interactions included We take into account the anisotropy of the interaction potential H H H H H H Figure 1: Energy (cm-1) diagram of H + using zero-point energies calculated using a 5 7D model H H H hop barrier < 100 cm 1 a Aguado et al., JCP 133 (2010) 024306 H H H scrambling barrier < 1500 cm 1

Theoretical Framework: the statistical model Some equations... State-to-state canonical reaction rate coefficient: ( ) h 2 1 3/2 K sr,m s r = [j d ] [j t ] [I 2 ] [I 3 ] 2πµk B T where M = (id, hop, exchange) [J] = 2J + 1 s (I 2, I 3 ) de e (E Esr)/K BT N sr,m s r (E) r (j d, j t, ω t ). State-to-state cumulative reaction probabilities (CRP): N sr,m s r (E) = [J] [I] Psr,M s r JI (E) JI

Theoretical Framework: the statistical model State-to-state reaction probabilities: g Is γ M W JΩ Psr,M s r JIΩ (E) = sis srνω t (E) W JΩ s r ν Ω (E) t γsis M W s r ν Ω JΩ t (E) M s r ν Ω t

Theoretical Framework: the statistical model State-to-state reaction probabilities: g Is γ M W JΩ Psr,M s r JIΩ (E) = sis srνω t (E) W JΩ s r ν Ω (E) t γsis M W s r ν Ω JΩ t (E) M s r ν Ω t 1. Capture probability of the H + 5 complex, W JΩ srνω t 2. Nuclear spin statistical weight matrix, g Is, s = (I 3, I 2 ) 3. Spin branching ratio, γ M sis γ M sis = S M Γ M sis /e se s a. Spin modification matrix, Γ M sis = I, I z, I 3, i 2, I 2 O M I, I z, I 3, i 2 3, I 2 i 2 i 2 b. Scrambling matrix, S M = {1/10, 3/10, 6/10} }{{}}{{}}{{} id hop exc.

Statistical Cumulative Reaction Probability (CRP), N sr s. Potential energy surface dependence of CRP full PES (Ours) anisotropic (P & L) isotropic (P & L) 6 4 j d =1, j t =1, ω t =0 (1,3/2) (0,1/2) j d =1, j t =1, ω t =0 (1,3/2) (1,1/2) j d =1, j t =1, ω t =0 (1,3/2) (1,3/2) j d =1, j t =1, ω t =0 (1,3/2) (0,3/2) N sr s / (2j d +1) x 1000 2 6 4 2 6 4 2 j d =1, j t =1, ω t =1 (1,1/2) (0,1/2) j d =0, j t =1, ω t =0 (0,3/2) (0,1/2) j d =1, j t =1, ω t =1 (1,1/2) (1,1/2) j d =0, j t =1, ω t =0 (0,3/2) (1,1/2) x 1/3 j d =1, j t =1, ω t =1 (1,1/2) (1,3/2) j d =0, j t =1, ω t =0 (0,3/2) (1,3/2) j d =1, j t =1, ω t =1 (1,1/2) (0,3/2) j d =0, j t =1, ω t =0 (0,3/2) (0,3/2) Anisotropy reduces CRP. ortho/para conversion data of H + 3 and H 2 are 6 4 j d =0, j t =1, ω t =1 (0,1/2) (0,1/2) j d =0, j t =1, ω t =1 (0,1/2) (1,1/2) j d =0, j t =1, ω t =1 (0,1/2) (1, 3/2) j d =0, j t =1, ω t =1 (0,1/2) (0,3/2) available. 2 0 200 400 200 400 200 400 Collision energy / cm 1 200 400

α = P hop / P exc. from QCT & dinamically-biased statistical method α= P hop /P exc 5 4 QCT 3 j d =0, j t =1, ω t =1 j d =1, j t =1, ω t =1 2 j d =0, j t =1, ω t =0 j d =1, j t =1, ω t =0 1 j d =0, j t =2, ω t =2 j d =1, j t =2, ω t =2 0 j d =0, j t =3, ω t =0 0 200 400 600 800 Energy / cm 1 Dynamically biased statistical j d =0, j t =1, ω t =1 j d =1, j t =1, ω t =1 j d =0, j t =1, ω t =0 j d =1, j t =1, ω t =0 j d =0, j t =2, ω t =2 j d =1, j t =2, ω t =2 0 200 400 600 800 Energy / cm 1 Experimental data: K. N. Crabtree et al. (2011): α = 0.5 ± 0.1 at T = 135 K α 1.6 at T = 350 K M. Cordonnier et al. (2000): α = 2.4 ± 0.6 at T = 400 K α (from QCT) is not statistical at low temperatures!! exchange mechanism is somehow underestimated.

Zero-point energy effect ZPE H + 3 + ZPE H2 = 6538 cm 1 ZPE H + 5 = 7167 cm 1

Zero-point energy effect ZPE H + 3 + ZPE H2 = 6538 cm 1 ZPE H + 5 = 7167 cm 1 Excess of energy unphysical fast disociation of the H + 5 complex. Exchange underestimated α too large.

Zero-point energy effect ZPE H + 3 + ZPE H2 = 6538 cm 1 ZPE H + 5 = 7167 cm 1 Method: ZPE reduction RRKM unimolecular rate constant: K (E) = N(E) hρ(e) N(E) = level number at the TST. ρ(e) = density of states of products. Excess of energy unphysical fast disociation of the H + 5 complex. Exchange underestimated α too large.

Zero-point energy effect ZPE H + 3 + ZPE H2 = 6538 cm 1 ZPE H + 5 = 7167 cm 1 Method: ZPE reduction RRKM unimolecular rate constant: K (E) = N(E) hρ(e) N(E) = level number at the TST. ρ(e) = density of states of products. E n N(E) = (classical) n! ω i N(E, γ) = N(E [1 γ]e ZPE ) Excess of energy unphysical fast disociation of the H + 5 complex. Exchange underestimated α too large.

Zero-point energy effect ZPE H + 3 + ZPE H2 = 6538 cm 1 ZPE H + 5 = 7167 cm 1 Method: ZPE reduction RRKM unimolecular rate constant: K (E) = N(E) hρ(e) N(E) = level number at the TST. ρ(e) = density of states of products. E n N(E) = (classical) n! ω i N(E, γ) = N(E [1 γ]e ZPE ) 100 level number, N(E) 50 H 5 + ZPE Clasical γ=0.6 γ=0.7 γ=0.8 Quantum 0 7000 7500 8000 8500 9000 Energy / wavenumbers

Zero-point energy effect ZPE H + 3 + ZPE H2 = 6538 cm 1 ZPE H + 5 = 7167 cm 1 Method: ZPE reduction RRKM unimolecular rate constant: K (E) = N(E) hρ(e) N(E) = level number at the TST. ρ(e) = density of states of products. E n N(E) = (classical) n! ω i N(E, γ) = N(E [1 γ]e ZPE ) 100 γ = 0.65 used ZPE: 0.25 ZPE reactants level number, N(E) 50 H 5 + ZPE Clasical γ=0.6 γ=0.7 γ=0.8 Quantum 0 7000 7500 8000 8500 9000 Energy / wavenumbers

Zero-point energy effect 5 QCT, 25% ZPE Statistical, 25% ZPE 4 α= P hop /P exc 3 2 1 0 5 4 QCT j d =0, j t =1, ω t =1 Statistical Exp. data: Crabtree et al. ( 11): α = 0.5 ± 0.1 (T = 135 K) 3 j d =0, j t =1, ω t =1 j d =1, j t =1, ω t =1 2 j d =0, j t =1, ω t =0 j d =1, j t =1, ω t =0 1 j d =0, j t =2, ω t =2 j d =1, j t =2, ω t =2 j 0 d =0, j t =3, ω t =0 0 200 400 600 800 Energy / cm 1 j d =0, j t =1, ω t =1 j d =1, j t =1, ω t =1 j d =0, j t =1, ω t =0 j d =1, j t =1, ω t =0 j d =0, j t =2, ω t =2 j d =1, j t =2, ω t =2 0 200 400 600 800 Energy / cm 1 α decreases when using corrected ZPE.

Rate constants state-to-state rate constants, k sr s statistical biased ZPE biased Park&Light 1 j d =1, j t =1, ω t =0 (1,3/2) (0,1/2) j d =1, j t =1, ω t =0 (1,3/2) (1,1/2) j d =1, j t =1, ω t =0 (1,3/2) (1,3/2) j d =1, j t =1, ω t =0 (1,3/2) (0,3/2) k sr s (T) x 10 9 (cm 3 /s) 1 1 j d =1, j t =1, ω t =1 (1,1/2) (0,1/2) j d =0, j t =1, ω t =0 (0,3/2) (0,1/2) j d =1, j t =1, ω t =1 (1,1/2) (1,1/2) j d =0, j t =1, ω t =0 (0,3/2) (1,1/2) j d =1, j t =1, ω t =1 (1,1/2) (1,3/2) j d =0, j t =1, ω t =0 (0,3/2) (1,3/2) j d =1, j t =1, ω t =1 (1,1/2) (0,3/2) j d =0, j t =1, ω t =0 (0,3/2) (0,3/2) 1 j d =0, j t =1, ω t =1 (0,1/2) (0,1/2) j d =0, j t =1, ω t =1 (0,1/2) (1,1/2) j d =0, j t =1, ω t =1 (0,1/2) (1, 3/2) j d =0, j t =1, ω t =1 (0,1/2) (0,3/2) 0 0 200 400 200 400 200 400 Temperature / Kelvin 200 400

Comparison with the experiment 4 statistical biased ZPE biased Park&Light Expt. 1 Expt. 2 α = k hop /k exc. Expt. 1: K. N. Crabtree et al. (2011) Expt. 2: M. Cordonnier et al. (2000) α(t) 2 1 ZPE plays an important role for T> 300 K, ZPE-biased works fine for T< 200 K, statistical behaviour Need of quantum treatments 0 0 200 400 Temperature / Kelvin This Work: J. Chem. Phys. 137 (2012) 094303

Conclusions We presented a dynamically-biased statistical/qct treatment for the H + 3 + H 2 reaction. We account for the anisotropy of the H + 3 reactant and the H+ 5 well depth. We obtain a more realistic, energy-dependent scrambling matrix by means of QCT calculations. Transition from a statistical-like behaviour at low temperatures to a direct mechanism at high termperatures. We get a reasonable description of α(t ) = k hop / k exchange We analyze the role of the ZPE on the QCT calculations. Quantum methods are needed. We plan to study the isotopic variants of this reaction.

Acknowledgements Coworkers Octavio Roncero, CSIC Alfredo Aguado, Univ. Autónoma Madrid Cristina Sanz, CSIC Alexandre Zanchet, CSIC Lola González Sánchez, Univ. Salamanca Financial Support Ministerio de Ciencia y Tecnología (Spain) CSD2009-00038, "Molecular Astrophysics: The Herschel and Alma era" THANK YOU FOR YOUR ATTENTION

The H + 3 + H 2 reaction Reaction Mechanisms (M) and Nuclear Spin (I) Selection Rules This reaction can proceed in three ways: Identity (M = 1) H 3 + + H 2 H 3 + + H 2 weight = 1/10 H + hop (M = 2) H 3 + + H 2 k hop HH 2 + + H 2 weight = 3/10 Γ id ss = Γ hop ss = o,o o,p p,o p,p o,o 6/5 0 0 0 o,p 0 2/5 0 0 p,o 0 0 6/5 0 p,p 0 0 0 2/5 o,o o,p p,o p,p o,o 12/5 0 6/5 0 o,p 0 0 6/5 0 p,o 6/5 6/5 3/5 3/5 p,p 0 0 3/5 3/5 H + exc. (M = 3) H 3 + + H 2 k exc. HH 2 + + HH weight = 6/10 Γ exc ss = o,o o,p p,o p,p o,o 19/5 1 8/5 4/5 o,p 1 3/5 4/5 0 p,o 8/5 4/5 19/5 1 p,p 4/5 0 1 3/5 Each mechanism is subject to nuclear spin constraints 2 2 M. Quack, Mol. Phys. 34 (1977) 477; T. Oka, J. Mol. Spectrosc. 228 (2044) 635

Nuclear Spin Combinations of H 2 and H + 3 H i = +1/2 z I = 1/2 00 11 0000 1111 000 111 0000 1111 000 111 00 11 0000 1111 0 1 000 111 I 3= 1/2 (E ) + para H 3 H + 3 00 11 0000 1111 000 111 0000 1111 000 111 00 11 0000 1111 000 111 I = 3/2 3 (A ) 1 0000 1111 000 111 i = 1/2 z 00 11 0000 1111 000 111 00 11 0000 1111 0 1 000 111 H 2 00 11 0000 1111 000 111 00 11 0000 1111 000 111 I = 0 I = 1 2 2 (antisymmetric) (symmetric) + ortho H para H ortho H 3 2 2 j, w > J >

Rate constants rotationally thermalized rate constants, k s (T ) 1 (I 2,I 3 )= (0,1/2) (I 2,I 3 )= (1,1/2) k s (T) x 10 9 (cm 3 /s) 0 1 (I 2,I 3 )=(0,3/2) (I 2,I 3 )=(1, 3/2) statistical biased ZPE biased Park&Light 0 0 200 400 0 200 400 Temperature / Kelvin

From Park & Light to Oka branching ratio matrices Oka Γ M ss Park & Light Γ M sis Γ M ss = I g si Γ M sis g I

Capture probabilities 1. Capture probability of the H + 5 complex, W JΩ { srνω t WsrνΩ JΩ 0 E < V max t (E) = 1 E > V max 800 V (J,j,Ω,r) (R) = Y Jm J eff jωr V Y Jm J jωr + Ejd + E jt ω t + J(J + 1) + j d (j d + 1) + j t (j t + 1) 2Ω 2 + 2νΩ t 2 Energy / cm 1 0 j d =1,ν=0, Ω t =0, J=0 Effective Potential J, j, Ω, r V eff 800 3 9 15 R / a.u. j t =1,ω t =1 j t =1,ω t =0 j t =2,ω t =2 j t =2,ω t =1 j d =1,ν=0, Ω t =0, J=15 j t =3,ω t =3 j t =3,ω t =2 j t =3,ω t =1 j t =3,ω t =0 3 9 15 R / a.u.

Park and Light potential energy surface H = H rot + L2 2µr 2 + V int + H kin V int = V 0 + V 2 P 2 (cosθ) isotropic part anisotropic part V 2 = qq r 3 V 0 = q2 α 2r 4 q2 (α par α perp) 3r 4

QCT probability identity hop exchange 1 0.8 j d =0, j t =3, ω t =0 j d =1, j t =3, ω t =0 0.6 0.4 0.2 1 Probability 0.8 0.6 0.4 0.2 j d =0, j t =2, ω t =2 j d =1, j t =2, ω t =2 hop & exchange decreases. No dependence on the initial states. 1 0.8 0.6 0.4 0.2 0 j d =0, j t =1, ω t =1 j d =1, j t =1, ω t =1 0 200 400 600 800 0 200 400 600 800 Energy / cm 1