Analysis of the formation of CH+ in collision of C2+ ions with molecular hydrogen

Chemical Physics Letters 583 (2013) 23–27

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Analysis of the formation of CH+ in collision of C2+ ions with molecular hydrogen Marie-Christine Bacchus-Montabonel a,⇑, Laurent Wiesenfeld b a b

Institut Lumière Matière, UMR5306 Université Lyon 1-CNRS, Université de Lyon, 69622 Villeurbanne cedex, France Université Joseph Fourier-Grenoble 1/CNRS-INSU, Institut de Planétologie et d’Astrophysique de Grenoble (IPAG), UMR 5274, Grenoble F-38041, France

a r t i c l e

i n f o

Article history: Received 4 June 2013 In final form 1 August 2013 Available online 8 August 2013

a b s t r a c t A theoretical treatment of the different processes involved in the collision of C2+ ions with molecular hydrogen is developed with consideration of charge transfer and formation of the CH+ molecular ion. Calculations of the potential energy surfaces and couplings are performed by means of ab initio quantum chemistry methods. Analysis of the different routes is detailed. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction The methylidyne cation CH+ has been shown to be abundant in the interstellar medium. This is in fact the first ion detected in interstellar space [1,2] and such a species is a fundamental step in the formation of larger hydrocarbons in diffuse interstellar clouds. However, its presence raises also a puzzling problem as the CH+ abundances predicted by steady-state, UV dominated models are several orders of magnitude lower than observed ones [3]. A number of routes have been investigated, in particular numerous studies have been devoted to the formation of CH+ by radiative association [4,5]:




Cþ 2 P3=2;1=2 þ H 2 S1=2 ! CHþ 1 Rþ þ hm

An accurate description of such a process at very low temperature requires the consideration of spin-orbit and rotational couplings and involves the consideration of a great number of resonances. However the rate constants remain too low to account for the formation of CH+ in the interstellar medium [5]. The exo+ thermic reaction C þ Hþ 3 ? CH + H2 has also been widely investigated and is assumed to be fast at the low temperature of dense interstellar clouds [6]. However, taking account of the ground C(3P) state of atomic carbon, reactants would approach trough a triplet potential energy surface to form 3P CH+ and relaxation to 1 + R CH+ has to be considered [7–9]. Alternative routes, such as evaporation of gas mantle [10], reactions at the surface of grains [11], degradation of larger molecules in the gas phase, reaction of C+ with molecular hydrogen vibrationnally excited [12], radiative association of C+ and H2 as an initial step [13] have been proposed but the mechanism of formation of CH+ remains still in question [14].

⇑ Corresponding author. Fax: +33 472431507. E-mail address: [email protected] (M.-C. Bacchus-Montabonel). 0009-2614/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2013.08.003

Considering the fast destruction of CH+ by hydrogenation [3,15], the only pathway efficient enough to balance such process is considered at present to be the reaction:

Cþ þ H2 ! CHþ þ H However the present reaction is highly endothermic with a rate constant k = 1.0  1010 exp(4640/T) cm3 s1 [16]. A number of theories have been proposed in order to explain a possible increase of such rate constant, including neutral shocks [17,18], or lowvelocity magnetohydrodynamic (MHD) shocks [19,20], as well as Alfvén waves [15] and turbulent mixing [21,22] or turbulent dissipation [23–25]. Among these scenarios, two models appear to be almost likely mechanisms with regard to observations. The high CH+ abundances could be the signature of shock waves propagating through the interstellar medium with a formation of CH+ mainly through ion-neutral friction [19,20]. An alternative route could be the turbulent dissipation region (TDR) model, where energy is evacuated by ion-neutral friction and viscous dissipation at the edge of many small-scale magnetized vortices [3,25]. But an alternative chemical process could be suggested as the coexistence of doubly charged C2+ ions with H2 could be under consideration in strong X-ray radiation field in the Central Molecular Zone (CMZ) and CH+ could thus be possibly formed through the reaction:

C2þ þ H2 ! CHþ þ Hþ We have thus developed a complete theoretical treatment of the C2+ + H2 collision process. Such reactions between charged ions and diatomics have been extensively investigated in the eV–keV collision energy range [26–29]. In particular the C2+ + H2 collision has been studied in relation with experimental data [30] and vibrational analysis after single-electron capture [31–33], but only charge exchange processes have been considered. The aim now is to investigate all the possible different channels which may be involved in the C2+ + H2 collision at very low energy, including the

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M.-C. Bacchus-Montabonel, L. Wiesenfeld / Chemical Physics Letters 583 (2013) 23–27

charge transfer process C2+ + H2 ? C+ + Hþ 2 , with possible dissocia+ + + tion of Hþ 2 to H + H , and the reactive collision CH + H in order to study which species might be formed. The potential energy surfaces and couplings have been determined by means of ab initio quantum chemistry molecular calculations and a detailed analysis of the different routes has been performed. 2. Computational method The geometry of the C2+ + H2 collision system is described using internal Jacobi coordinates {R, q, h} with the origin at the centre of mass of the target molecule, in the middle of H2 as defined in Figure 1. The orientation of the projectile toward the molecular target may be studied for different values of the angle h, from linear (h = 0°) to perpendicular geometry (h = 90°) taking account of a great number of q and R distances. The molecular calculations were carried out by means of the MOLPRO suite of ab initio programs [34]. Tests have been performed and the correlation-consistent AV5Z basis set of Dunning has been chosen for all atoms [35]. The adiabatic potential energy surfaces (PES) were computed at the state-average CASSCF (Complete Active Space Self Consistent Field) level of theory followed by multi-reference configuration interaction (MRCI) calculations without consideration of the spin-orbit coupling. Six electrons in six orbitals are included in the active space involving the 1s orbitals of both hydrogen atoms and 2s, 2p orbitals of carbon. The 1s orbital on carbon is frozen in the calculation. The asymptotic energies of the different channels have been calculated at the same MRCI level of theory and compared to the experimental values of separated species [36,37] showing a good agreement (Table 1). Considering the 1R+ symmetry of the entry channel C2+ + H2, three 1R+ molecular states have to be considered in the molecular calculation: the entry channel itself, of course, and as far as the charge transfer process is concerned, the two exit + + channels dissociating respectively into C+ + Hþ 2 and C + H + H þ + + with dissociation of H2 . But the formation of CH + H has also to be considered through the (CH2)2+ intermediate and appears to be a stabilized route in the asymptotic region. For that reason, we have optimized the (CH2)2+ intermediate which is shown to be close to a linear geometry with coordinates {R = 3.616580 a.u., q = 2.080114 a.u., h = 0.01 239°}. The process is driven mainly by non-adiabatic interactions in the vicinity of avoided crossings [38], the radial coupling matrix elements between all pairs of states of the same symmetry have thus been calculated using the finite difference technique [39]:

1 g KL ðRÞ ¼ hwK j@=@RjwL i ¼ hwK ðRÞ lim wL ðR þ DÞ  wL ðRÞi D!0 D

Figure 1. Internal Jacobi coordinates for the C2+–H2 molecular system.

Table 1 Comparison of calculated asymptotic energies from separated species at equilibrium distance (in a.u.); position of the optimized geometry of (CH2)2+.

C2+(2s2)1S + H2 (1R+) C(2p2)3P + H+ + H+ (CH2)2+ opt C+(2s22p)2P + H + H+ C+(2s22p)2P + Hþ 2 CH+(1R+) + H+

Symmetry

MRCI calculation

Experiment [36,37]

1

0.0

0.0 0.1458511

R+ 3 P 1 + R 1,3 P, 1,3 P, 1 + R

1,3

R+ 1,3 + R

0.1661544 0.2248751 0.3221922 0.374653

0.2320217 0.3293388

Taking account of the orthogonality of the eigenfunctions, radial coupling between states jwK ðRÞi and jwL ðRÞi may be expressed by:

g KL ðRÞ ¼ hwK j@=@RjwL i ¼ lim

1

D!0 D

hwK ðRÞjwL ðR þ DÞi;

with the parameter D = 0.0012 a.u. as previously tested [40] and using the three-point numerical differentiation method for reasons of numerical accuracy. The rotational coupling matrix elements hwK ðRÞjiLy jwL ðRÞi between states of angular moment D^ = ±1 are calculated directly from the quadrupole moment tensor from the @ @ expression iLy ¼ x @z  z @x with the centre of mass of the system chosen as origin of electronic coordinates. 3. Results and discussion The potential energy surfaces for the angle h = 45° are presented in Figure 2. A strong interaction, corresponding to a curve seam, is observable between the highest excited state, correlated asymptotically to the C2+ + H2 entry channel and the charge transfer state + + with dissociation of Hþ 2 reaching asymptotically C + H + H . A smoother one appears at lower R distances with the ground molecular state which presents a significant potential well assuming the possible formation of CH+ + H+. Such process may thus proceed through a two-step mechanism as clearly exhibited on the 2D cuts of the potential energy surfaces (PES) at h = 22.5° presented in Figure 3a,b and on the radial coupling matrix elements displayed in Figure 4. Effectively, a very strong avoided crossing may be pointed out between the entry channel and the dissociating charge transfer channel for the H2 distance q = 2.6 a.u., around R = 8 a.u., in correspondence to a peaked o/oR radial coupling. Such interaction corresponding to a quasi diabatic process smoothens when reaching larger q = H–H distances, beyond the seam, as for example for q = 2.8 a.u. shown in Figure 3b and Figure 4, with a wider avoided crossing and a smoother hw2 j@=@Rjw3 i radial coupling. However, the hw1 j@=@Rjw2 i radial coupling matrix elements driving the second step toward the ground molecular state remain rather smooth

Figure 2. Potential energy surfaces of the three 1R+ states of the C2+–H2 molecular system at h = 45°. For clarity, the lower potential energy surface has been lowered by 0.5 a.u.

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2 þ Figure 3. Cuts of the potential energy surfaces of the 1R+ states of the C2+–H2 molecular system at h = 22.5°. (a) q = 2.6 a.u.; (b) q = 2.8 a.u. (1) Red, C+(2s22p)2P + Hþ 2 ( Rg ); (2) 2+ 2 1 2 þ 1 þ green, C+(2s22p)2P + Hþ ( R ); (3) blue, C (2s ) S + H ( R ) entry channel. (For interpretation of the references to colour in this figure legend, the reader is referred to the web 2 2 u g version of this article.)

Figure 4. Cuts of the o/oR radial coupling matrix elements between the 1R+ states of the C2+–H2 molecular system at h = 22.5°. hw2|o/oR|w3i: black, q = 2.6 a.u.; blue, q = 2.8 a.u. hw1|o/oR|w2i: magenta, q = 2.4 a.u.; green, q = 2.6 a.u.; red, q = 2.8 a.u. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

for all geometries, with a maximum height about 0.3 a.u. in all cases. Furthermore, the analysis of the o/oq radial coupling matrix elements with regard to the q = H–H distance presented in Figure 5 provides almost the same feature for the interaction toward the ground 1R+ state, with the same order of magnitude for the couplings. A smooth peak is observed for the hw1 j@=@ qjw2 i coupling matrix element, reaching never more than 0.3 a.u. for any R distance. Such interactions, although quite significant, seem however unlikely to form CH+ with a great efficiency in the C2+ + H2 collision. Anyway, the clear potential well observed shows a stabilization of the CH+ + H+ species which could then evolve to the formation of CH+. A very similar analysis may be performed in the linear approach presented in Figure 6a with a significant well in the lowest potential. However, a significant evolution of the PES may be pointed out

Figure 5. Cuts of the o/oq radial coupling matrix elements between the 1R+ states of the C2+–H2 molecular system at h = 22.5°. hw1|o/oq|w2i: black, R = 2.2 a.u.; red, R = 2.4 a.u.; green, R = 3.0 a.u.; blue, R = 3.6 a.u.; yellow, R = 4.0 a.u.; magenta, R = 5.0 a.u. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

when going toward the perpendicular geometry. The well associated to the formation of CH+ appears less and less stable, and the main interaction moves toward the formation of CH2þ + H, characterized by a shoulder on the ground potential energy surface at shorter R internuclear distances. Such evolution is exhibited on the cuts of the PES for h = 67.5° and h = 90° presented in Figure 6b, c. Very clearly the energy barrier between the CH+ + H+ well toward the dissociative C+ + Hþ 2 asymptotic level around R = 4.0 a.u. is reduced at h = 67.5° to almost disappear in the perpendicular geometry. Simultaneously, a shoulder in the repulsive part of the potential of the ground state may be pointed out, which moves to a sharp interaction with the 21R+ state. The formation of CH+ + H+ seems thus more likely to occur close to the linear geometry, corroborated also by the optimized geometry of the (CH2)2+ intermediate. This point is also supported by looking at the

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Figure 6. Cuts of the potential energy surfaces of the 1R+ states of the C2+–H2 molecular system at q = 2.6 a.u.. (a) h = 0°; (b) h = 67.5°; (c) h = 90°. (1) Red, þ 2 þ + 2 2 2+ 2 1 2 þ 1 þ C+(2s22p)2P + Hþ 2 ( Rg ); (2) green, C (2s 2p) P + H 2 ( Ru ); (3) blue, C (2s ) S + H2( Rg ) entry channel. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

minimum energy path along the dissociation of (CH2)2+ to CH+ + H+ presented on Figure 7 which is clearly favored in the linear geometry with a very low energy barrier of 3.55  103 a.u. (779 cm1) which could be passed over taking account of the zero point energy for (CH2)2+ [41]. If the coupling interactions would be efficient enough to stabilize the (CH2)2+ intermediate, the methylidyne cation CH+ could thus be formed through such mechanism. However the two-step mechanism proposed for this process goes through relatively smooth couplings and a complete dynamical treatment has to be performed in order to determine its rate constant and assess on its efficiency.

4. Concluding remarks

Figure 7. Minimum energy path along the dissociation of CH+ + H+. Blue, h = 0°; red, h = 22.5°; green, h = 45°, magenta, h = 67.5°. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

This letter provides accurate potential energy surfaces and couplings for the C2+ + H2 collision determined by means of ab initio quantum chemistry methods. A detailed analysis has been performed in order to investigate the possible formation of CH+ by such a process. A two-step mechanism is proposed, through a first diabatic charge transfer interaction, followed by the formation of the (CH2)2+ intermediate which could dissociate to form CH+ and H+. The process appears markedly favored in the linear approach, taking account of the stabilization of the (CH2)2+ intermediate, and of its dissociation into CH+ and H+. However, the couplings

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appear to be rather smooth to lead to a very efficient reaction. Further dynamic calculations are in progress. Acknowledgments We thank G. Pineau des Forêts (IAS Université Paris Sud, France) for analysis of possible mechanisms and C. Loison (ILM Université de Lyon, France) for fruitful discussions. This letter was granted access to the HPC resources of [CCRT/CINES/IDRIS under the allocation 2013-[i2010081655] made by GENCI [Grand Equipement National de Calcul Intensif]. The support of the COST action CM0805 ‘Chemical Cosmos’ is greatly acknowledged. References [1] T. Dunham Jr., PASP 49 (1937) 26. [2] A.E. Douglas, G. Herzberg, Astrophys. J. 94 (1941) 381. [3] B. Godard, E. Falgarone, M. Guerin, D.C. Lis, M. De Luca, J.H. Black, J.R. Goicoechea, J. Cernicharo, D.A. Neufeld, K.M. Menten, M. Emprechtinger, Astron. Astrophys. 540 (2012) A87. [4] M.M. Graff, J.T. Moseley, E. Roueff, Astrophys. J. 269 (1983) 796. [5] G. Barinovs, M.C. van Hemert, Astrophys. J. 636 (2006) 923. [6] E. Herbst, Annu. Rev. Phys. Chem. 46 (1995) 27. [7] D. Talbi, D.J. DeFrees, Chem. Phys. Lett. 179 (1991) 165. [8] D. Talbi, D.J. DeFrees, D.A. Egolf, E. Herbst, Astrophys. J. 374 (1991) 390. [9] R.P. Bettens, M.A. Collins, J. Chem. Phys. 108 (1998) 2424. [10] D.R. Bates, L. Spitzer Jr., Astrophys. J. 113 (1951) 441. [11] D. McNally, MNRAS 124 (1962) 155. [12] T.P. Stecher, D.A. Williams, MNRAS 168 (1974) 51P. [13] J.H. Black, A. Dalgarno, M. Oppenheimer, Astrophys. J. 199 (1975) 633. [14] E.F. van Dishoeck, in: T.W. Hartquist, D.A. Williams (Eds.), The Molecular Astrophysics of Stars and Galaxies, New York, Clarendon, 1998, p 53. [15] N. Indriolo, T. Oka, T.R. Geballe, B.J. McCall, Astrophys. J. 711 (2010) 1338.

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