Formation of the simplest stable negative molecular ion H 3. in interstellar medium

Formation of the simplest stable negative molecular ion H 3 in interstellar medium V. Kokoouline 1,2, M. Ayouz 1, R. Guerout 1, M. Raoult 1, J. Robert 1, and O. Dulieu 1 1 Laboratoire Aimé Cotton, CNRS, Bat 505, Campus d Orsay, 91405 Orsay Cedex, France 2 Department of Physics, University of Central Florida, Orlando, Florida 32816, USA We consider spectroscopy, bound states and long-lived predissociated resonances of the simplest stable negative polyatomic ion, H 3. We determine energies and rotational constants for the bound states and lifetimes of predissociated vibrational states (Feshbach resonances) of the ion. We discuss the possibility to observe the ion in interstellar space in an environment with a large fraction of ionized molecular hydrogen and consider possible ways to form the ion. An observation of H 3 can also serve as a probe to detect H in interstellar medium. PACS numbers: Chemical reactions in the interstellar medium (ISM) are powered by cosmic rays: Atoms and molecules (mainly molecular hydrogen) are ionized by the radiation that provides sufficient energy to initiate a chain of chemical reactions in interstellar clouds leading to the synthesis of polyatomic molecules. Many positive ions have been found and identified in ISM. In particular, the simplest positive ion, H + 3 plays an important role in chemistry and evolution of interstellar clouds [1, 2], mainly due to its stability and abundance of molecular hydrogen in ISM. On the other hand, only a few negative ions have been detected so far in ISM: C 3 N [3], C 4 H [4], C 6 H [5], and C 8 H [6].The simplest negative triatomic ion, H 3 is also stable although its binding energy is significantly lower than in H + 3 and it has not been detected so far in ISM. In this article we consider spectroscopy, formation and destruction of H 3 in ISM. We argue that the ion is indeed formed in cold ISM with a relatively high degree of ionization of molecular hydrogen and can be observed. Formation and destruction in ISM. The chemistry of interstellar clouds is initiated by ionization of molecular hydrogen by cosmic rays. The rate constant for such ionization is ζ 3 10 17 s 1 [2] in diffuse (densities about 10 2 cm 3 ) and dense (densities about 10 4 cm 3 ) interstellar clouds. The ionized molecular hydrogen H + 2 quickly forms H + 3 in collisions with H 2. The escaped electron has a large kinetic energy and, therefore, before its rethermalization should undergo many elastic collisions with environmental H 2. Possible inelastic e +H 2 collisions should lead to a vibrational excitation of H 2 and the dissociative attachment (e +H 2 H+H ) [7, 8]. The cross-section of the dissociative attachment at collision energies above 3.7 ev is of order of σ DA 10 21 cm 2 [7, 8]. Therefore, if we assume that electrons ejected from H 2 by a cosmic ray have large enough initial kinetic energy, they will form H with a high probability. Moreover, because the reaction rate (per one H 2 molecule) for dissociative attachment, which is of order of v e σ DA n(h 2 ), is larger than the rate of electron production ζ, the rate of production of H is determined by ζn(h 2 ) similarly to the production of H + 3. [2]. However, the observation of H is more difficult in ISM than the observation of H + 3, because there is only one stable bound state of H. Similarly to H + 3 (proton donor), the H ion is chemically active (electron donor) and should initiate its own chain of chemical reactions in space. First collisions that H would experience if formed are H +H 2. In this study we consider formation of the H 3 ion in such collisions. An observation of H 3 in ISM would indicate that H is indeed formed in interstellar clouds. The H 3 ion is well described as a van der Waals complex consisting of H 2 and H. The molecule has several relatively weekly bound rovibrational states and a number of predissociated resonances. The binding energy of the lowest rovibrational state is about 100 cm 1 (see Table I). The predissociated H 3 resonances can be considered as excited rovibrational states (j, v d ) of H 2 coupled to the vibrational continuum correlated to a H 2 (j, v d )+H dissociation limit with energy of the dimer state (j, v d ) lower than the (j, v d) state. Lowest bound states of H 3 are listed in Table I. The widths of the broadest resonances are in the range of 0.2 1.5 cm 1, which corresponds to lifetimes of 3.5 26 ps. In order to form a bound H 3 molecule in H 2+H collisions in ISM two mechanisms are possible: Three-body recombination (TBR) or radiative association (RA): H 2 + H + X H 3 + X : TBR, (1) H 2 + H H 3 + ω : RA. (2) The decay of H 3 in diffuse clouds is determined by collisions with H + 3, other positive ions, and by interstellar radiation. The three-body rate constant k 3b can roughly be estimated considering just geometrical (van der Waals) sizes of reactants and procedure outlined in Ref. [9]. We obtained k 3b or the order of 10 26 10 30 cm 5 /s. With typical number densities n = 10 2 10 4 cm 3 in ISM, the three-body recombination as a way to form H 3 must be much slower than possible two-body processes involving H. The RA cross-section can be estimated using theory developed below. Our estimation gives the RA rate coefficient k RA of the order of 3 10 21 cm 3 /s (see below). Although the rate seems to be quite small, it gives an H survival lifetime τ RA = 1/(k RA n) towards to RA comparable to its lifetime towards to other competing processes:

2 J, j, Ω, v t, v d, Γ Energies, cm 1 B Z, cm 1 B X, cm 1 0, 0, 0, 0, 0, A 1-105.01 211.2 3.23 0, 0, 0, 0, 0, E -104.8 211. 3. 0, 1, 0, 0, 0, E -44.0 345. 3.7 0, 1, 0, 0, 0, A 2-43.90 345.1 3.67 0, 0, 0, 1, 0, A 1-17.86 187.5 1.8 0, 0, 0, 1, 0, E -17.7 188. 2. 0, 1, 0, 1, 0, A 2 73.24 311.9 2.36 TABLE I: Position and rotational constants of bound rovibrational states of H 3. The energies are given with respect to the lowest dissociation limit H 2 (0, 0)+H. ionization of H by cosmic rays (τ ξ = 1/ξ) and recombination of H with positive molecular ions in ISM, in particular, with H + 3. Therefore, we predict that in cold diffuse interstellar clouds consisted of molecular hydrogen, the H 3 ion can be detected. We propose to search for H 3 in the absorption spectra in millimeter wavelength range in diffuse clouds with a relatively high fraction of ionized gas. One has to look into the H 3 absorption spectrum for transitions between the ground and excited rotational levels. The rotational constants of the lowest rovibrational levels are given in Table I. Spectroscopy. The H 3 molecule consists of atoms with three identical nuclei. Therefore, the total molecular symmetry group that includes the exchange of nuclei between the dimer and the third atom (CNPI group [10]) is D 3h. Although all bound and resonant states calculated here transform according to the D 3h group, due to its van der Waals nature, the lowest rovibrational states of H 3 can approximately be characterized using the C 2v subgroup (the MS group according to terminology of [10]) and the corresponding quantum numbers. Therefore, in order to characterize the lowest rovibrational states and identify allowed H 3 states for ortho- and para-configurations of the total nuclear spin I (I = 3/2 and 1/2, correspondingly) we consider approximate wave functions and quantum numbers of the lowest rovibrational states. The H 3 molecule has a definite total angular momentum J and its projection m on the space-fixed z-axis. The H 2 dimer in the molecule is characterized by the vibrational quantum number v d and its angular momentum j with projection Ω on axis Z of the molecular coordinate system (MCS). The MCS Z-axis connects the center of the dimer with the nuclei of H (nuclei 3). The X-axis of MCS is in the molecular plane. Orientation of MCS with respect to the space-fixed coordinate system (SCS) is given by the Euler angles α, β and γ. With these notations and assuming that the vibrational motion is separable from rotation, the wave function of the system is represented by the product [ D J mω (α, β, γ) ] P Ω j (cos θ) v d v t, (3) where Pj Ω (θ) (an associated Legendre polynomial) is the rotational state of the dimer, θ is the azimuthal angle describing orientation of the dimer in MCS. DmΩ J (α, β, γ) is the Wigner function corresponding to the total angular momentum J of the system: Projection of J on the MCS Z-axis is equal to Ω, therefore, the rotational state of the system is described only one Wigner function, but not by a superposition of several Wigner functions as it is the case for tightly-bound C 2v molecules. On the other hand, the fact that only one Wigner function determines the rotational state of the system makes the considered system to behave as the symmetric rotor, which is not surprising because two of the three moments of inertia are almost equal. The normalization factor is omitted in Eq. (3), because it is not important for the present discussion. For simplicity, we will use the notation J, j, Ω, v t, v d for this function. The quantum number M is omitted because it does not influence symmetry or energy of the state. The energy of the state J, j, Ω, v t, v d is E vd +E vt +B Z j(j +1)+ [ B X J(J + 1) + (B Z B X )Ω 2]. (4) where B X and B Z are the rotational constants (about the X- and Z-axes) of the trimer and dimer correspondingly. The first two terms give energies of the rotationless vibrational levels of the dimer and the trimer. The third term is the rotational energy of the dimer. The last term describes the rotational energy of the trimer. Deriving the third term we assumed that the trimer is approximately a symmetric top with two equal moment of inertia (about X and Y axes) [11]. The wave functions J, j, Ω, v t, v d transform in the following way under C 2v operators (the dimer is in the 1 Σ + g and the trimer is in the 1 A 1 electronic states correspondingly): (12) J, j, Ω, v t, v d = ( 1) j J, j, Ω, v t, v d E J, j, Ω, v t, v d = ( 1) J J, j, Ω, v t, v d. (5) The final step is the construction of approximate wave functions of H 3 is a symmetrization with respect to exchange of identical nuclei. It is made using operators ˆP Γ of projection on a particular irreducible representation Γ of the complete D 3h group of the system. The symmetrized states will be referred as J, j, Ω, v t, v d, Γ : J, j, Ω, v t, v d, Γ = ˆP Γ J, j, Ω, v t, v d. (6) For certain combination of quantum numbers J, j and Ω, some of the projections ˆP Γ are zero. It means that the corresponding irreducible representation Γ is not allowed for this set of quantum numbers. The general rules are derived from Eqs. (5): The both (even and odd) parities are allowed for non-zero Ω. If Ω = 0, the parity is given by ( 1) J. The E irreducible representation is allowed for any combination of J, j, Ω (assuming that the parity E or E is given by the above rule). The A 1 irreducible representation is allowed for even j, the A 2 irreducible representation is allowed for odd j (the parity is given by the above rule). Having constructed the wave functions transforming according to the irreducible representations of D 3h, we

3 can determine the allowed states for para- and ortho- H 3. Because the three nuclei are identical fermions, the total wave function (including the nuclear spin factor) of H 3 can only be of A 2 or A 2 irreducible representations. Since the para-h 3 nuclear spin part of the wave function transforms as E in D 3h, the allowed spatial irreducible representation could be E or E. For ortho-h 3 (A 1 irreducible representation of the nuclear spin), only A 2 and A 2 coordinate (rovibrational) wave functions are allowed. The nuclear spin multiplicities g I are 2I + 1: g I = 2 for para- and g I = 4 for ortho-h 3. The lowest rovibrational state J = 0, j = 0, Ω = 0, v t = 0, v d = 0, Γ = A 1 is not allowed for H 3 (but allowed for D 3 ). The lowest allowed state is J = 0, j = 0, Ω = 0, v t = 0, v d = 0, Γ = E is the para-h 3 state. The lowest A 2 rotational state (lowest ortho-h 3 ) is J = 0, j = 1, Ω = 0, v t = 0, v d = 0, Γ = A 2. We have performed the numerical calculation of bound states, their rotational constants, predissociated resonances, and cross-sections taking into account the the full D 3h CNPI symmetry group of the molecule. We used the H 3 potential from Ref. [12]. We solved numerically the three-dimensionall Schrödinger equation for the molecule in hyper-spherical coordinates, separating hyper-angles from hyper-radius and using the slow variable discretization. Widths of predissociated resonances are calculated by placing a complex absorbing potential at large values of hyper-radius. The widths of the broadest resonances in the low energy spectrum are 0.2 1.5 cm 1. The details of the numerical procedure are given in Refs. [13, 14]. For an insight into the H 2 bound and continuum states, Fig. 1 shows the hyperspherical adiabatic curves of H 3 calculated for J = 0. Each adiabatic curve at large values of hyper-radius correlated with a H 2 (v d, j)+h dissociation limit. The lowest bound states and resonances can be characterized by the approximate quantum numbers j, Ω, v t, v d. Theory of radiative association in dimer-atom collisions. In order to estimate the cross-section and the rate coefficient for RA, we develop a theoretical framework to treat the radiative association of a dimer and an atom. Our approach is based on theory developed by Herzberg [15], and later used by several authors [16 18] for diatomic molecules. In order to adapt the theory to triatomic systems, similarly to Ref. [18], we start with the Einstein coefficient A q,v t ;q,e for the photon emission from a continuum rovibrational state specified by quantum numbers q = {J, j, Ω, v d, Γ} and the energy E of collision between H 2 and H. After a photon of energy ω is emitted, the triatomic H 3 ion will be in a state specified by the quantum numbers q = {J, j, Ω, v d, Γ } and v t. We have singled out the quantum number v t in order to stress that it corresponds to the same degree of freedom as the kinetic energy E of the initial state. i.e. to the separation R between H 2 and H. The Einstein coefficient A q,v t ;q,e is given (in atomic units) A q,v t ;q,e = 4ω3 3c 3 r q,v t ;q,e 2, (7) where r q,v t ;q,e is the dipole moment matrix element (with three components r σ, σ = 1, 0, +1). The value r q,v t ;q,e 2 can be evaluated using a technique similar presented in Ref. [10]: We used Eq. (14-33) of Ref. [10] for the line strength an divide it with the total number of the initial states (in order to average over the initial states). Because, each rotational state of H 3 are characterized (in the approximation we use) by a single symmetric top rotational function, the two first sums in Eq. (14-33) of Ref. [10] are dropped out. Therefore, we obtain the following formula for the Einstein coefficient A q,v t ;q,e = 4ω3 3c 3 (2J + 1) (8) ( ) j, Ω, v t, v d, Γ µ σ J 1 J 2 j, Ω, E, v d, Γ Ω σ Ω, σ where µ σ is the σ component of the dipole moment calculated in MCS. The initial vibrational state in the above expression is energy normalized, the final state is the unity normalized. The probability P q,v t ;q,e of an RA event is given by the Einstein coefficient divided with the current in the flux of incident particles, which is equal to 1/(2π) for the energy normalized wave function. Finally, the RA cross-section is given by πp q,v t ;q,e /k 2 : σ q,v t ;q,e = 8π2 ω 3 3k 2 c 3 (2J + 1) (9) ( ) j, Ω, v t, v d, Γ µ σ J 1 J 2 j, Ω, E, v d, Γ Ω σ Ω. σ To obtain the cross section σ q (E) for the formation of any H 3 bound state {q, v t}, we have to sum over q and v t. Since the nuclear spin is conserved during the RA process, there is no need to include the nuclear spin degeneracy factor: After averaging over the initial state and summing up over final states, the factor will be one. The final step in calculation is to obtain the rate constant. It is made by a standard integration over a Boltzmann- Maxwellian distribution. In the integration, the nuclear spin degeneracy factors (2I + 1) as well as the rovibrational energy of the initial state of H 2 should be taken into account (see, for example, Ref. [19] for details of the averaging procedure). Having developed theory of RA in dimer-atom collisions, the rate constant can be calculated with a good precision if the dipole moment functions are known. Here we provide only a rough estimation of the rate constant to verify if the RA process is competitive with other processes in ISM that lead to the removal of H (if it is present) from the interstellar gas. The estimation is made in the following way. For given Ω and Ω, only one term in the sum of Eq. (9) is not zero. The 3j symbol together

4 Adiabatic potentials (cm -1 ) 6000 5000 4000 3000 2000 1000 0 A 1 A 2 4 8 16 32 hyper-radius (a.u.) (v d, j (1,2) (1,1) (1,0) (0,8) (0,7) (0,6) (0,5) (0,4) (0,3) (0,2) (0,1) (0,0) FIG. 1: (Color online) The figure shows the H 3 hyperspherical adiabatic curves calculated for J = 0. Only A 1 and A 2 vibrational symmetries are shown. The curves of the E symmetry are very similar to the A 1 and A 2 symmetries for energies below 4000 cm 1 (barrier for the proton exchange) and would be indistinguishable from A 1 and A 2 curves shown here. Horizontal lines indicate positions of several bound states and resonances. with symmetry Γ µ = A 2 E of the vector of dipole moment (µ 1, µ 0, µ +1 ) in the symmetry D 3h group determine selection rules: J J = J ±1; Ω Ω = Ω, Ω±1. In addition, parities of the initial and final states should be opposite. The largest vibrational dipole moment matrix element in Eq. (9) is expected when Ω = Ω (because µ 0 is much larger than µ ±1 ), j = j, and v d = v d. When all internal vibrational quantum numbers (v d, Ω, j) are the same except v t, the vibrational dipole moment matrix element can be estimated as µ 0 calculated at the equilibrium 1 A geometry times the vibrational Franck-Condon overlap, which can very roughly be approximated using the density of discrete states 1/ E calculated at the energy of the final bound state v t. The value of 3j can be taken to be 1 for the allowed transitions (J J ±1; Ω Ω = Ω) in the rough estimation. The equilibrium value of µ 0 is 4 a.u. [20], E 30 60 cm 1 or 1.5 3 10 4 a.u.; ω 100 cm 1 or 5 10 4 a.u.; k 2 /(2m) 1.3 10 4 a.u (it corresponds to 40K, a reasonable temperature for cold diffuse clouds), m 1200 a.u. is the reduced mass of the H 2 +H system. We took J = 0 and, correspondingly, J = 1. Plugging these values into Eq. (9), we obtain the value of 10 9 a.u. for the estimated crosssection. The rate coefficient can roughly be estimated as σ k/m 5 10 12 a.u. or 3 10 21 cm 3 /s. In the estimation, we have neglected all the Feshbach resonances present in the collisional spectrum of H and H 2. Such resonances should increase the total cross-setion and will be accounted in an accurate calculation in a separate publication. Summary and discussion. We have considered the formation of the H 3 ion in cold diffuse interstellar clouds. According to our results the ion can be detected in the absorption spectrum in the millimeter wavelength range. To estimate the rate of H 3 formation in H 2+H collisions, we have developed a theoretical approach to calculate the cross-section for the radiative association and determined exact and approximate quantum numbers that can be used to characterize the bound and resonant states of H 3. The developed theory is quite general and can be used to study similar dimer+atom van der Waals molecules. We have calculated energies of bound states, their rotational constants, position and lifetimes of Feshbach resonances. We have also calculated cross-sections of different inelastic processes in H 2 +H collisions. The results and more details about the bound and resonant states will be presented in a separate longer publication. Acknowledgments. We would like to thank Roland Wester for motivating us to study H 3. This work has been supported by the RTRA network Triangle de la Physique (France) and the National Science Foundation (USA) under grant No. PHY-0855622. [1] T. Geballe and T. Oka, Nature 384, 334 (1996). [2] T. Oka, Phil. Trans. R. Soc. Lond. A 364, 2847 (2006). [3] P. Thaddeus, C. A. Gottlieb, H. Gupta, S. Brunken, M. C. McCarthy, M. Agundez, M. Guelin, and J. Cernicharo, Astrophys. J. 677, 1132 (2008). [4] J. Cernicharo, M. Guelin, M. Agundez, K. Kawaguchi, and P. Thaddeus, A&A 61, L37 (2007). [5] M. C. McCarthy, C. A. Gottlieb, H. Gupta, and P. Thaddeus, Astrophys. J. Lett. 652, L141 (2006). [6] S. Brunken, H. Gupta, C. A. Gottlieb, M. C. McCarthy, and P. Thaddeus, Astrophys. J. Lett. 664, L43 (2007). [7] G. J. Schulz and R. K. Asundi, Phys. Rev. Lett. 15, 946 (1965). [8] J. Horáček, M. Čížek, K. Houfek, P. Kolorenč, and W. Domcke, Phys. Rev. A 70, 052712 (2004). [9] J. Glosík, R. Plašil, I. Korolov, T. Kotrík, O. Novotný, P. Hlavenka, P. Dohnal, J. Varju, V. Kokoouline, and C. H. Greene, Phys. Rev. A 79, 052707 (2009). [10] P. R. Bunker and P. Jensen, Molecular Symmetry and Spectroscopy (NRC Research Press, 1998). [11] L. Landau and E. Lifshitz, Quantum Mechanics: Nonrelativistic Theory (Burlington MA: Butterworth Heinemann, 2003). [12] A. N. Panda and N. Sathyamurthy, J. Chem. Phys. 121, 9343 (2004). [13] V. Kokoouline and F. Masnou-Seeuws, Phys. Rev. A 73,

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