Article pubs.acs.org/JPCA
Adiabatic Potential Energy Surfaces for the Low-Energy Collisional Dynamics of C+(2P) Ions with H2 Molecules Matteo Bonfanti,† Gian Franco Tantardini,†,‡ and Rocco Martinazzo*,†,‡ †
Dipartimento di Chimica, Università degli Studi di Milano, v. Golgi 19, 20133 Milano, Italy Istituto di Scienze e Tecnologie Molecolari, Centro Nazionale Richerche, 20133 Milano, Italy
‡
ABSTRACT: The low-energy electronic states of the CH2+ molecular ion are investigated with multireference configuration interaction calculations based on complete active space self-consistent field reference wave functions using a large C(6s5p4d3f)/H(8s6p3d1f) basis set. The focus is on the three lowest-lying states describing formation and destruction of the astrophysically relevant methylidine cation CH+. Both processes are discussed in light of the topology of the relevant potential energy surfaces and their intersections.
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CH 2+ + γ ′ → CH+ + H
INTRODUCTION The methylidine cation CH+ was the first molecular ion detected in the interstellar medium,1 more than 70 years ago, and is believed to be crucial for the formation of large hydrocarbons, which is initiated by the reactions
was long considered to potentially solve the above issue, but recent measurements of the rate coefficient of reaction 2 seem to rule out this possibility.7 CH+ may also form through atomexchange reaction
CH+ + H 2 → CH 2+ + H +
+
CH 2 + H 2 → CH3 + H
C+ + H 2 → CH+ + H
C+ + H → CH+ + γ
is exceedingly slow, except maybe close to low-energy resonances,5 and association with H2
Special Issue: Franco Gianturco Festschrift Received: March 31, 2014 Revised: June 26, 2014 Published: June 27, 2014
(2)
followed by photodissociation © 2014 American Chemical Society
(4)
but this reaction is endoergic by about 0.4 eV and thus requires either high translational temperatures or excited H2 molecules. H2(ν > 0) significantly influences the formation of methylidine cation and other hydrocarbons in general,8 but observations rule out a strict correlation between CH+ and electronically pumped, vibrationally hot H2 (CH+ is also found abundantly in cold media), though evidence was found about the role of rotational excitation.9,10 At the same time, a clear anticorrelation with CN species (which forms only in dense, dark molecular clouds) shows that CH+ can form abundantly only in diffuse clouds.11 In general, for reaction 4 to work, nonthermal chemistry must be invoked, otherwise the necessary temperature would also affect the neutral chemistry, particularly OH production through the analogous (endoergic) reaction with oxygen atoms. Thus, alternative explanations based on the streaming of C+ ions in magneto-hydrodynamic shocks or the passage of Alfvén waves in an otherwise cold medium become plausible.12,13 Protonation of neutral carbon atoms is possible through the exoergic, proton-exchange reaction
(1)
and continued by radiative association, carbon insertion, and condensation processes.2 Since its unambiguous observation, its abundance has been puzzling theoreticians and experimentalists, to the point that the expression “CH+ mystery” was coined to indicate the failure of current steady-state, UVdominated evolutionary models in reproducing the observed CH+ column densities along different lines of sight.3 Current models often underestimate the observed CH+ abundance by several orders of magnitude while reproducing well much of the neutral chemistry, including formation of the parent neutral CH molecule.4 Thus, either formation pathways have underestimated rates or destruction is less efficient than currently believed. In the interstellar medium, CH+ is formed in reactions involving the most abundant hydrogen species. Direct radiative association
C+ + H 2 → CH 2+ + γ
(3) 6
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C + H3+ → CH+ + H 2
but is limited by the abundance of the H3+ molecular ion, while C2 + + H 2 → CH+ + H+
is possible only in the presence of a strong X-ray radiation field.14 On the other hand, the methyilidine cation is efficiently destroyed by common routes, i.e. dissociative recombination (attachment) CH+ + e− → C + H
and reactions with the abundant H and H2 species, namely the reverse of reaction 4 CH+ + H → C+ + H 2
(5)
Figure 1. Correlation diagram showing the connection between lowlying atomic and diatomic states and the electronic states of the triatom CH2+ at the most important configurations, as indicated by the schematics at the bottom (black circles for C and white circles for H atoms).
and reaction 1 above. Reaction 5 has recently attracted considerable interest, both theoretically15−18 and experimentally,19,20 as is considered to be the main loss channel of CH+. The measured thermal rate constant showed a maximum around 60 K, which is close to the Langevin limit, and decreased with decreasing reagent rotational excitation, thereby suggesting that nonrotating CH+ is protected against attacks of H atoms.20 More generally, reactions 4 and 5 and the interaction potentials governing them have been the subject of much theoretical and experimental work until the 1980s,21−28 motivated by CH2+ being the simplest complex-forming triatomic system involving p orbitals, at a time though in which both experiments and theoretical results could not reach the accuracy levels accessible nowadays. Surprisingly, while the first theoretical analysis of the energy landscape did properly take into account all the electronic states accessible in the low-energy collisional dynamics,22,23 the most recent studies15,17 applied accurate ab initio methods in conjunction with large basis sets but focused on the ground electronic state only and neglected the important symmetryallowed conical intersection (CI) occurring for C2v geometries of the triatom. For this reason, and motivated by the importance that reactions 2, 3, 4, and 5 may have in explaining measured CH+ abundances in the interstellar medium, we reconsider in this paper the low-energy energetics of CH2+ by applying high-level ab initio methods in a balanced way to the three lowest-lying states. This is a first step toward a complete characterization and description of the energetics, which will be complemented by a multisurface analytical representation of the adiabatic potential energy surfaces (PESs) and of the corresponding nonadiabatic couplings in a forthcoming publication. This is a necessary prerequisite to tackling the complicated dynamical processes mentioned above, especially in light of the complex topology expected for the relevant PESs, as described in the next section.
In the atomic limit describing the entrance channel of C+ + H2 collisions, the C+(2s22p;2P) + H2(1∑+g ) state lies lowest in energy and is mostly unreactive. However, the more reactive charge-transfer state C(2s22p2;3P) + H2+(2∑+g ) lies low in energy (lower than states involving C+ excited states) and correlates diabatically with stable configurations of the threeatom system, thereby giving rise to adiabatic, though activated, routes to the “molecular” region. Interaction with a H 2 molecule in a symmetric C 2v configuration splits the three degenerate C+(2s22p;2P) states into A1, B1, and B2 symmetry species which transform into A′, A″, and A′ species for generic Cs approaches. The first two describe stable CH2+ configurations with carbon−hydrogen bonds built between C sp-hybrids and H s orbitals and with an unpaired electron in a carbon p orbital. They become degenerate in D∞h (i.e., symmetric HCH) configurations where they span a Πu state, but for slightly distorted configurations the state with the unpaired p electron in the molecular plane lies lowest in energy, and gives rise to a bent CH2+ structure and CH bonds with some sp2 character. On the other hand, when the singly occupied p orbital is perpendicular to the plane, the linear structure is the most stable; thus, a typeC Renner−Teller intersection appears. The remaining B2 state is only weakly binding because the unpaired electron is forced to occupy the p orbital parallel to the H2 axis but necessarily crosses the A1 state (which, as mentioned above, is repulsive at large C−H2 distances), thereby giving rise to a symmetryallowed conical intersection. Asymmetrically stretching the CH2+ molecule in the above Π state and breaking one CH bond results in H + CH+(3Π), where two unpaired electrons occupy sp (σ) and p (π) orbitals. However, this is not the lowest asymptote because CH+(3Σ) with two electrons in sp (σ) orbitals lies lower in energy; thus, a Σ/Π intersection occurs in this region of the configurational space. The same occurs for CH+ + H approach, with the Π state lying lower in energy in the interaction region and higher otherwise. This concludes our overview of the system which shows that for a proper description of the energetics and of the dynamics a consistent and simultaneous investigation of the three lowestlying electronic states with an accurate setup is necessary.
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THE SYSTEM The intricate connection between the relevant atomic and molecular electronic states is described by the correlation diagram given in Figure 1, with energy values obtained in this work. This diagram was first suggested by Mahan and Sloane on the basis of molecular beam results of the title reaction21 and later confirmed by Liskow et al.,22 who first applied largescale configuration interaction methods to investigate the system energetics. 6596
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The next section gives details of the adopted computational approach, and the two subsequent sections describe the results and their relevance for the CH+ formation and destruction processes; finally, the last section summarizes and concludes.
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COMPUTATIONAL METHOD Ground- and excited-state potentials have been determined by means of multireference configuration interaction (MRCI) calculations performed with the help of GAMESS-US code29,30 and including singly and doubly excited configurations out of a complete active space generated by distributing the five valence electrons into seven active orbitals, one more than the valence set, to correctly include any potentially important higher-lying states. The orbitals were optimized for each point in configuration space by complete active space self-consistent (CASSCF) calculations upon uniformly averaging over the first three roots of the secular problem. The active space for the latter was chosen to be slightly larger than the one used in the subsequent MRCI calculations and included three additional orbitals in the active set, i.e., CASSCF(5,10). Excitations were allowed over the whole set of active orbitals, as determined by the adopted single-electron basis. For the latter, we made nonstandard choices of the atom-centered basis sets, as described below. For hydrogen, we adopted the original set of Siegbahn and Liu31 augmented with diffuse functions as described in ref 32; the resulting set comprises 13s6p3d1f Gaussian spherical orbitals contracted to 8s6p3d1f and proved to be slightly superior than the aug-cc-pV5Z set of Dunning et al.33,34 (at a comparable computational cost) in describing hydrogen atom properties (electron affinity and polarizability of the anion) and hydrogen molecule energetics; in particular, at the full configuration interaction (CI) level, it guarantees that the ground-state energy values of H2 lie within 100 cm−1 from the numerically exact results of Kolos and Wolniewicz35 for any internuclear distance larger than 1.0 a0 (see ref 32). For carbon, we used the aug-cc-pVQZ set of Dunning et al.33,34 and omitted g orbitals to alleviate the computational burden, thereby resulting in 22s7p4d3f orbitals contracted to 6s5p4d3f. We checked that this set gives, at the full CI level, a fair representation of the low-lying excited states of both the C atom and its positive singly charged ion, with root-mean-square deviations from the exact excitation energies below 50 meV over the set of 2Pu,4Pg,2Dg,2Sg, and 2Pg states for the ion and of 3 Pg,1Dg,1Sg, and 5Su states for the neutral (multiplet centers from ref 36). Figure 2 gives our “best estimate” of the asymptotic energetics, i.e., the above-mentioned numerically exact results for H2 and valence full CI results for CH+ obtained with the basis set above. These two-body potentials will be included in the final potentials in place of the MRCI asymptotes of our PESs, upon fitting the important three-body potential term to a convenient functional form; however, the MRCI asymptotes obtained with our setup, which were obtained in collinear calculations with one of the two internuclear distances set at a large value (50 a0 for CH and 500 a0 for H2), lie very close to the above curves, making us confident about the quality of our results (see Figure 3). Also shown in Figure 2 are the vibrational energy levels obtained by diagonalizing a Colbert−Miller discrete variable representation37 of the Hamiltonian of the diatomics, neglecting any coupling between orbital and rotational angular momenta. As is evident from this figure, and as mentioned in
Figure 2. Best-estimate asymptotic energetics, namely numerically exact results for H2 of Kolos and Wolniewicz (left) and valence full-CI results with the adopted basis set for CH+ (right), for the four states dissociating into C+(2P)+H(2S). Also shown are the first vibrational levels of H2(X1Σ+g ), CH+(X1Σ+) (solid horizontal lines) and CH+(a3Π) (dashed horizontal lines) and the corresponding reaction energetics.
Figure 3. Computed MRCI energy curves for linear H−H−C configurations with RCH = 50 a0 (left) and RHH = 500 a0 (right). Lower and upper panels on the right correspond to the 1Σ and 3Π states of CH+, respectively. Also shown for comparison are the “bestestimate” curves of Figure 2.
the introduction, the reaction C+(2P) + H2(1∑+g ; v = 0) → CH+(1∑+; v = 0) + H is endoergic by 426 meV but becomes slightly exoergic by 89 meV if H2 molecules are in the first vibrationally excited state. The computed value compares well with the experimental endoergicity of the reaction (0.3932 ± 0.0001 eV), which is based on the measured dissociation energies of CH+ and H2 (D0 = 4.0849 ± 0.0001 eV and D0 = 4.47807 ± 0.00001 eV, respectively38), the residual ∼29 meV discrepancy being mainly related to the size of the basis-set used in CH+ calculations (a similar, 23 meV large difference is found when comparing the computed De of the methyilidine cation (4.227 eV) with similar calculations5 but using a much larger aug-cc-pV6Z set, without including core-correlation and relativistic effects). Much higher vibrational excitation (v > 3) is needed to overcome the reaction endoergicity (1.603 eV) to form CH+(3Π), but it is not unreasonably large. Thus, both these processes are potentially relevant for forming CH+ molecules. The three lowest-lying PESs of CH2+ were computed for a reasonably large grid of configurations of the triatom with different values of the distance R between C and the midpoint of the H−H bond, the distance r between the two H atoms, and the angle θ between them. These include C2v symmetric configurations (θ = 90°) and a number of distorted Cs 6597
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Figure 4. Overview of the PESs of the title system in C2v configurations as functions of half the distance r between the two H atoms and the distance R between C and the midpoint of the HH bond (in a0). Panel (a): the A1/B2 states (left, in cyan and green, respectively) and A1/B1 states (right, in cyan and orange, respectively) with their intersections. Also shown are the corresponding maps, A1 in the bottom planes and B2 (left) and B1(right) in the upper planes. (b) From left to right, countour maps of the A1, B2, and B1 state. Black energy levels are 0.5 eV spaced in the interval [−10.0, 0.0] eV (referenced to the break-up channel C+ + H + H) ; blue lines are at the energy of the asymptotes C+ + H2 and CH+ + H.
long-range and displays a minimum which lies −1.172 eV below the entrance channel, at a relatively large value of the distance R between the carbon ion and the molecule (R ∼ 2.40 a0), with the H−H distance r very similar to the values it takes in the free molecule (r ∼ 1.75 a0). Similarly, 2B1, which displays a minimum at even larger distance R ∼ 2.88 a0 with a HH bond about 1.52 a0 long, is only ∼0.79 eV more stable than the reagents. On the other hand, 2A1 is strongly repulsive at long-range, until an avoided crossing with a higher-lying state occurs (R ∼ 2.87 a0), which produces a barrier ∼2.18 eV higher in energy than the reagents. For shorter distances, the adiabatic state becomes increasingly binding because the carbon atom has now hybrid sp character, until the region of complex CH 2+ formation is reached. Similarly, for the 2B1 state, the avoided crossing (barrier) occurs at a shorter distance (R ∼ 2.00 a0) and at a smaller energy, Eb ∼ 1.28 eV. Of particular interest is the crossing between 2A1 and 2B2 because it becomes avoided for the slightly distorted Cs configuration, where mixing of the in-plane p orbitals of the carbon atom occurs. This originates a well-known symmetryallowed conical intersection, whose seam is already evident in Figure 4a and will be made clearer below in conjunction with the analysis of the slightly distorted Cs geometries. The lowestenergy point on this curve lies ∼0.89 eV below C+ + H2 and
configurations (θ = 80°, 70°, 50° and 40°), apart from the collinear and quasi-collinear C−H−H and H−C−H configurations which were sampled for several values of the internuclear distances. For representative purposes only we performed a preliminary fit of the computed data to the modified Aguado−Paniagua ansatz used in ref 32, which includes the long-range contributions to the potential. The adiabatic surfaces were fitted separately for C2v and collinear geometries and globally for all the considered Cs geometries. Each resulting fit had a small standard deviation of less than 100 cm−1 and can thus be considered a faithful representation of the computed ab initio points.
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RESULTS Symmetric C2v Configurations. We start by considering the symmetric C2v approach of the C+ ion to the H2 molecule. As mentioned above, in this case the three states which are relevant for the low-energy collision dynamics are the 2A1, 2B1, and 2B2 states. The corresponding potential energy surfaces are represented as three-dimensional graphs in Figure 4a and as contour maps in Figure 4b, along with the two contour levels (in blue) marking the energy of the entrance and the exit channel of reaction 4 computed without accounting for the zero-point energy. As anticipated, 2B2 is mildly attractive at 6598
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occurs at R = RMEX ∼ 1.99 a0 and r = rMEX ∼ 2.42 a0. This is the “low-energy passage” first found by Pearson and Roueff23 at exactly the same geometry (though at the slightly different energy of −0.65 eV because of the different computational setup adopted) which makes it possible for C+ species to form CH2+ at vanishing collision energy. The overall situation is depicted in Figure 5, where the energy curves for the 2A1, 2B1, and 2B2 are plotted along a
Figure 6. Close up of the A1 (left panel) and B1 (right panel) PESs in the Renner−Teller region as functions of half the distance r between the H atoms and the position Y of the C atom along the H−H normal. Spacing between the contour levels is 0.25 eV.
the radiative association reaction C+ + H2 → CH2+ + γ and the photodissociation of the CH2+ molecular ion, but they are reported here only briefly. Figure 5 suggests that the 2A1 minimum can be accessed radiatively from the 2B1 state only. The latter, however, is not directly accessible from the separated partners; rather, it requires that Renner−Teller mixing occurs soon after the nonadiabatic B2 → A1 transition takes place (see e.g. ref 39 and references therein for the theory and its application to the CH2+ system). Thus, radiative association actively involves all the lowest-lying energy states. Nonsymmetric Cs Configurations. Next we consider slight distortions of the C2v configurations where the C atom approaches the HH molecule at a quasi-normal angle. Figure 7 depicts the evolution of the conical intersection by showing the ground- and first excited-state potentials for symmetry species A′ (bottom and upper row, respectively) for the angles θ = 90, 80, 70, 60, 50°, as functions of R and r/2, similar to Figure 4. The case θ = 90° (C2v configurations) has been included to show the location of the seam of the conical intersection, which follows approximately the straight line defined by R/a0 = 1.104 + 0.356 r/a0, in good agreement with what was found by Pearson and Roueff23 (R/a0 = 1.094 + 0.417 r/a0). Clearly, for off-normal approaches the seam evolves in a barrier in the ground-state potential which separates the outer minimum (of B2 symmetry in C2v) from the deep well of the stable CH2+ molecular ion. The height of this barrier increases only slightly for decreasing values of θ, as can be guessed from Figure 8, in which we report the behavior of the energy as a function of θ for a given point in (R,r) space, which was chosen to be the minimum energy intersection point (i.e., R = RMEX and r = rMEX; see the white dot in the bottom left panel of Figure 7). The increase of the barrier height is moderate until the angle reaches a value where the projectile starts experiencing a repulsive interaction with the closest H. Ultimately, the barrier evolves into a repulsive wall at R = r/2 for quasi-collinear and collinear approaches, which prevents the system from reaching the region of HCH configurations (R < r/2). Also shown in Figure 7 are the asymptotic energy levels for reactions 4 and 5 (blue), which define the classically allowed regions for C+ + H2 (inner levels) and CH+ + H (outer levels) collisions at vanishing collision energies; notice that for cold C+ projectiles impinging on H2 molecules, complex formation is already closed at θ = 50°.
Figure 5. Left panel: energy of the A1 (white symbols), B1 (red), and B2 (green) along an approximate minimum energy path on the ground-state (B2/A1) potential as a function of the distance of the carbon atom from the midpoint of the HH bond (see text for details). Also shown as dashed lines (black, red, and green, respectively) are the energies of the same states along similar paths, but defined on each PES. Right panel: the moduli of the transition moments A1/B1 (red) and A1/B2 (green) along the ground-state path.
common approaching path, here defined, for each R, by the minimum along r of the lowest-energy PES. Also shown in that figure as dashed lines is the energy of the above three states along paths similarly defined on each potential energy surface. The figure clearly shows the above-mentioned barriers in the A1 and B1 states, as well as the outer minima in B1 and B2 and the global minimum of the CH2+ molecular ion. The latter occurs for r/2 = 1.948 a0 and R = 0.731 a0, corresponding to a CH bond 2.081 a0 long and a HCH angle α of 138.9°, which are close to the experimental values17 R(CH) = 2.088 a0 and α = 139.8°. The molecule is 4.807 eV more stable than CH+ + H (4.811 eV if based on the ab initio energy at the above geometry), which compares well with recent theoretical results15,17 and has a barrier 0.1572 eV high for converting into the symmetrically equivalent configuration. Notice that the minimum arises because of a type-C Renner−Teller interaction, which splits the Π state at linear D∞h geometry into an A1/B1 pair, where the B1 state, with the singly occupied p orbital perpendicular to the molecular plane, experiences a repulsive interaction upon bending. A close-up of this Renner−Teller region of the configuration space is provided in Figure 6 in which the potential energy surfaces of the A1 and B1 states are plotted as functions of r/2 and Y (the position of the C atom along the normal of the H−H bond) for symmetric C2v displacements and display a doublewell structure for the ground state and a simple minimum for the excited one. Also shown in Figure 5 (right panel) are the transition dipole moments for the transitions B1 ← A1 and B2 ← A1 along the same path used for the left panel (the third transition, B2 ← B1, is symmetry-forbidden). Such moments were computed in the whole configuration space because they are of interest for both 6599
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Figure 7. Ground-state (bottom row) and excited-state (top row) potentials for A′ symmetry species, for angles θ = 90, 80, 70, 60, 50° (from left to right). Energy levels as in Figure 4, with the additional contour (red lines) at the lowest energy on the conical intersection seam, marked with a white dot in the bottom left panel.
Conversely, the excited-state potential shows a minimum in the funnel region, potentially accessible through nonadiabatic transitions, which is placed at progressively higher energies as the angle θ decreases, and the above classically allowed region shrinks and disappears at about the same angle of 50°. Collinear C∞v Configurations. Finally, we consider the collinear HHC approach which is the limiting case θ = 0°. In this case, formation of the stable complex CH2+ is prevented by the repulsion between C and a H atom, and only a direct route to the products H+CH+ is left, provided the projectile has enough energy to overcome the reaction endoergicity. An overview of the relevant interaction potentials is shown in Figure 9a where the PESs for the 2Σ+ and 2Π states are plotted as functions of the HH distance r and the distance R′ between C and the closest H. As is evident from that figure, and as was anticipated above, the two PESs cross because, different from Σ, the Π state is attractive but correlates with excited-state products H + CH+(a3Π). The crossing is actually a three-state crossing, whereby Σ and one of the components of Π evolve
Figure 8. Energy of the three lowest-lying states as a function of the angle θ for fixed values of R and r, corresponding to the minimum energy intersection point in C2v (white dot in the bottom left panel of Figure 7). Symbols are ab initio data, and curves are spline interpolations to guide the eyes. Energies are referenced to the C+ + H2 asymptote.
Figure 9. (a) Three-dimensional graphs of the Σ (red) and Π (cyan) PESs for collinear geometries C−H−H as functions of the distance r between the H atoms and the distance R′ between C and the closest H. (b) From left to right, contour maps of the resulting ground-state potential and of the constituent Σ and Π ones. Contour levels as in in Figure 4. 6600
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into A′ species in Cs geometries, whereas the remaining Π component spans A″; the intersection remains one-dimensional and is of conical type (upon bending, the Π/Σ degeneracy is lifted at first order, whereas the Π pair separates at second order only). Figure 9b shows the details of the Σ and Π states (middle and right panels) along with the resulting ground-state potential, which evolves into the lowest-energy A′ PES for quasi-collinear approaches (left). The seam of the CI follows approximately the quadratic curve R′/a0 = −4.133 + 2.943 r/a0 − 0.227 (r/a0)2, and the minimim energy CI point occurs for R′ very close to the CH+ bond length and lies 74 meV above CH+ + H. This kink in the ground-state PES turns into a barrier in the lowest A′ state which is of decreasing height for increasing deviations from linearity, up to the point where it disappears.
up to v = 4−5 because recombination of hydrogen atoms on the surface of interstellar dust grains by means of Eley−Rideal reactions is known to produce vibrationally hot H 2 molecules.41−45 Recent ab initio molecular studies of the above surface process46 including energy dissipation to the surface and all competitive channels have shown that the large exoergicity (∼3.8 eV) of the reaction H(grain) + H(gas) → H2(gas) is equally shared between internal excitation of the nascent H2 molecules and surface heating, thereby suggesting the vibrationally excited v > 1 molecules are present in regions of intense hydrogen formation and possibly opening the CH+(3Π) + H channel. Interestingly, vibrational excitation has been shown to have such a weaker effect on the neutral chemistry8 that the observed overabundances of CH+ might be related to vibrationally hot H2 molecules (not necessarily arising from electronic pumping). Recently, reaction (i) has been investigated with exact, initial-state-selected quantum mechanical calculations for H2 in some rovibrational states (ν = 0,1 and j = 0,1) using the single adiabatic surface developed by Stoecklin and Halvick15 and has been found to be not completely statistical.47 Computed cross sections for H2(ν = 0) agree well with the experimental results at low energies19 but are significantly lower at higher energies. Similarly, the thermal rate constants for H2 in its first vibrationally excited state are found to be about two times smaller than the experimental value estimated by Hiertl et al.,40 possibly because of contributions from ν > 1 states in the experiments and/or the use of a single surface in the calculations. Despite this, inclusion of detailed state-to-state information for reaction (i) has been shown to considerably improve modeling of photondominated regions, thereby highlighting the role of chemical pumping in the observed CH+ excitation.47 The reverse, destruction process (iii) has recently seen a renewed interest both experimentally and theoretically. Recent measurements20 used cold ion traps in conjunction with effusive beams and obtained not only thermal rate coefficients but also state-specific rates for j = 0−2 in the astrophisically relevant regime T = 7−100 K. The measured rates showed a maximum at T = 60 K (k = 1.2 ± 0.5 × 10−9 cm3 s−1, to be compared with the Langevin result for this system kL = 2.0 × 10−9 cm3 s−1), beyond which the rate decays in agreement with the prediction of phase-space theory.15,16 However, at lower temperatures, the rate takes exceedingly small values (k = 5 ± 4 × 10−11 cm3 s−1 at the lower end of the T interval) because of the small reactivity of CH+(j = 0) molecules. Quasi-classical calculations showed the opposite, namely that rotational excitation decreases the low-temperature rate;16 in general, results from both quantum and quasi-classical calculations based on a single adiabatic surface15−17 agree only roughly with the above measured rate and fail to reproduce the pronounced decrease of the rate at the lowest temperature. Analysis of the adiabatic rotation states, in the presence of the long-range potential only, suggests that reagents in low-lying rotational states tend to align in the collinear C−H−H configuration where a barrier is present, and ad hoc corrected phase-space calculations that exclude contributions from such rotational states agree better with the experiment,18 at odds with exact (single-surface) quantum mechanical calculations.15,17 Hence, proper consideration of all quantum effects, including, e.g., nonadiabatic ones, would be highly desirable, even if their influence on the dynamics can only be indirect and reduces to a mere potential renormalization.
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DISCUSSION The above description makes clear that interaction between a C+ ion and H2 molecules, and the reverse process involving the methylidine cation CH+ and hydrogen atoms, is complicated by possible nonadiabatic effects; only roughly may it be said to involve one-third of the available electronic states. Such interactions are relevant for both formation and destruction of CH+ ions and have been considered in discussing at least three different processes: (i) direct formation of the methylidine cation through the endothermic reaction 4, (ii) radiative association of C+ and H2 to form CH2+ (eq 2), and (iii) destruction of CH+ in collisions with H atoms through the exothermic reaction 5. In most cases, this kind of ion−molecule process can be treated to a good approximation by applying the Langevin capture model to estimate the rate of complex formation and statistical phase-space theory to investigate its breakup. The underlying assumption is that a complex forms through a simple barrierless reaction and lives long enough to undergo complete energy redistribution (randomization). In the cases mentioned above it is not obvious that these conditions hold and, importantly, it is not clear how many electronic states take part in the process. Experimental results for processes (i)−(iii) are still not conclusive in these respects, and a detailed quantum mechanical investigation including possibly nonadiabatic effects is highly desirable. Direct methylidine cation formation (i) has long been studied with thermal reagents, but a few initial state selected rate constants are available for H2 rotationally excited (j = 0− 7)19 or vibrationally excited (v = 1).40 Rotational excitation was investigated in the temperature range T = 200−1000 K, and it was found that it is used to diminish the reaction endothermicity; the rate constants indeed fit well to an Arrhenius-like expression in which rotational energy decreases the activation energy. For j = 7, reaction 4 is no longer activated and the rate attains the value k = 1.6 × 10−10 cm3 s−1. On the other hand, vibrational excitation is much more efficient in enhancing the rate, which is found to be constant in the temperature range T = 800−1300 K and 1 order of magnitude larger than the above value, k ∼ 1.6 × 10−9 cm3 s−1. This is close to the Langevin capture rate kL = 2π(α0/μ)1/2 (where α0 is the isotropic polarizability of the neutral and μ the reduced mass of the collision pair), which for C+ + H2 amounts to kL = 1.6 × 10−9 cm3 s−1, if the 3-fold degenerate electronic states of C+(2P) are given the same weight, in contrast with the topology of the PESs discussed above. Here, a detailed investigation of the role that vibrational excitation has in the above reaction would be desirable, including possibly higher vibrational states 6601
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Finally, radiative association (ii) is likely the most challenging process, both theoretically and experimentally. In the laboratory, ubiquitous three-body association reactions typically mask the slower radiative association48 unless buffer densities are exceedingly small. The rate of reaction 2 was measured to be very small at T = 10 K, but with a significant influence of the rotational energy,7 as signaled by the fact that the rate measured with normal-H2 (k = 6.8 ± 0.9 × 10−16 cm3 s−1) increases to k = 1.7 ± 0.2 × 10−15 cm3 s−1 when using para-H2. Because this effect is unlikely due to the radiative transition itself, either dynamics is more direct than currently believed or rotational motion of H2 plays some role in complex formation. This may occur even at a classical level, and this may also be of interest for process (i) above, because capture actually takes place on a more complicated (anisotropic) long-range potential than usually assumed, i.e., one including charge−quadrupole interactions that do not necessarily average out. Furthermore, and quite generally, even when capture occurs by means of the long-range part of the potential only, it can happen at a rate lower than that predicted by the classical model because threshold dynamics in the centrifugal barriers appearing for large values of the angular momentum is inherently quantum and likely shows overbarrier reflection (the opposite, potentially contributing effect, namely, tunneling for energies below the barrier, is likely to be of minor importance for such wide barriers).
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SUMMARY AND CONCLUSIONS We investigated the three lowest-lying states of the CH2+ molecular ion which are relevant for the formation and the destruction of the methylidine CH+ cation in interstellar space. We applied multireference configuration methods based on complete active space self-consistent optimizations of the reference functions, using a large basis-set, in a balanced way to describe the lowest-lying states and their interactions, as they are deemed to be relevant for the low-energy collisional dynamics involving either C+ ions and H2 molecules or CH+ and H species. This is a first, important step toward obtaining a multisheet analytical description of the adiabatic potentials (along with their nonadiabatic couplings) which can then be employed in full quantum dynamical studies of the above processes and possibly help solve some unsettled issues in the carbon chemistry of the interstellar medium.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
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