Theoretical investigation of rotationally inelastic collisions of CH(⁠$X2Π$⁠) with molecular hydrogen Free

There has been significant interest in rotationally inelastic and reactive collisions of the methylidyne (CH) radical because of the role it plays in combustion¹ and its presence in interstellar environments.^2–5 This radical has been of interest in combustion chemistry since it is the simplest carbon-containing radical. It was one of the first molecules observed in the interstellar medium and is one of the most abundant diatomic species (except for H₂) in molecular clouds.⁶ Determination of CH abundance is of special interest since its concentration has been found to track the concentration of the most abundant astrophysical molecule, namely, H₂, which is difficult to detect directly.

Detection of interstellar molecules relies on the observation of radiative transitions (in absorption or emission) of the molecules of interest. The intensity of the transition is related to the column densities of the initial and final levels of the transition. Relating these to the total CH concentration is, in principle, straightforward if local thermodynamic equilibrium applies (provided the temperature of the cloud can be estimated). Otherwise, a collisional-radiative model must be applied to account for the collision-induced and radiative transitions.^7,8 The radiative transition rates for CH are well known. However, knowledge of rate constants for collisional (de-)excitation of CH by the most abundant species, specifically H, H₂, and electrons is required but, to our knowledge, these are not available.

As a surrogate for CH–H/H₂ energy transfer rates, several theoretical investigations of the CH–He system have been carried out. Alexander et al.⁹ carried out the first calculations of CH–He collisional rate constants, using a potential computed by Wagner et al.¹⁰ Recently, Marinakis et al.¹¹ performed a coupled cluster [RCCSD(T)] calculation of the potential and calculated cross sections and rate constants for the rotational/fine-structure excitation of CH in collisions with He. The only state-resolved experimental study of rotationally inelastic collisions of CH(⁠$X2Π$⁠) with He was carried out by Macdonald and Liu.¹² Here, a CH molecular beam, almost exclusively in its ground rotational level, with equal populations in the Λ-doublet levels, collided at variable intersection angles with a target He beam. Relative cross sections for excitation to specific final rotational levels as a function of collision energy were determined. These measured cross sections agreed well with the two sets of calculations, especially those computed in the more recent work.

We have recently determined quasi-diabatic potential energy surfaces (PES’s) describing the interaction of CH(⁠$X2Π$⁠) with H₂,¹³ for which the CH and H₂ bond lengths were kept fixed. We employed an electronic basis related to the isolated CH molecule, namely, in terms of the two components of the CH orbital angular momentum along the CH internuclear axis, with projections $Λ=±1$⁠. This quasi-diabatic basis can then be conveniently employed in scattering calculations. In this paper, we present cross sections and rate constants for the collisional (de-)excitation of CH in collisions with H₂, computed with this potential.

One complication in the calculation of the CH–H₂ potential is that these collision partners can react or combine to form the methyl radical

Indeed, the room-temperature low-pressure bimolecular [Eq. (1)] rate constant for the reaction of CH with H₂ has been measured to equal ca. $2×10−11$ cm⁻¹ molecule⁻¹ s⁻¹.¹⁴ The corresponding high-pressure rate constant, corresponding to recombination [Eq. (2)] equals $2.0×10−10$ cm⁻¹ molecule⁻¹ s⁻¹.¹⁵ The abstraction reaction [Eq. (1)] cannot occur in our computed potential since the diatomic bond lengths are fixed. However, we did find a deep well at small intermolecular separations R, corresponding to the recombination reaction [Eq. (2)], albeit with a smaller well depth than the global well depth [6200 and 37 000 cm⁻¹, respectively].

Experimental evidence for the role of a methyl complex in inelastic collisions of CH comes from the crossed beam experiments by Macdonald and Liu. They also investigated collisions of CH with D₂, along with their study of CH–He collisions. In addition to observing rotationally excited CH radicals,¹⁶ they also observed a small population of CD radicals, which came from the dissociation of the transiently formed CHD₂ complex.^17,18

In the usual approach to the calculation of cross sections for inelastic transitions,^19,20 the scattering wave function is expanded in a set of internal states of the system. The resulting radial differential equation is integrated out from the small-R classically forbidden region, with the application of appropriate boundary conditions. Because of the deep well on the lower CH(⁠$X2Π$⁠)–H₂ PES,¹³ the starting value of R needs to be very small (⁠$R≈1.5a0$⁠), and the radial step size must also be very small (⁠$≤0.01a0$⁠). An additional complication is the presence of an avoided crossing of the second CH(⁠$X2Π$⁠)–H₂ state with a higher state near $R≈3a0$⁠,¹³ so that the quasi-diabatization procedure becomes invalid at small R. We did carry out preliminary calculations of cross sections for rotationally inelastic transitions in CH–H₂ collisions, invoking the usual inelastic scattering boundary conditions.¹³ However, even with a large CH rotational basis and small radial step size, it was not possible to obtain converged cross sections without an excessive computational time.

In this work, we employ the quantum statistical method of Manolopoulos and co-workers^21–23 to compute CH–H₂ inelastic cross sections. This theory, which is based on earlier work by Pechukas and Light,²⁴ Miller,²⁵ and Clary and Henshaw,²⁶ was developed for quantum mechanical calculations for chemical reactions on PES’s that have deep wells. For such systems, a statistical approximation should be valid. Here, inelastic scattering within each reaction arrangement is treated rigorously, and radial equations are integrated from a capture radius, with appropriate boundary conditions, to determine the S matrix for each arrangement. The S matrix can then be used to compute the capture probability for each scattering channel, from which the reaction cross section can be determined.

The quantum statistical method has been employed to predict product fine-structure branching in the O(⁠$D1$⁠) + H₂ → OH(⁠$Π2$⁠) + H reaction, and to treat vibrational relaxation in OH–H collisions²⁷ and isotope exchange in OD–H collisions,²⁸ as well as to compute transport properties for the H–O₂ collision pair.²⁹ For the latter system, the $A″2$ state correlates with the deeply bound HO₂(⁠$X∼2A″$⁠) state. A similar treatment has been applied to the study of NH(⁠$X3Σ−$⁠)–NH(⁠$X3Σ−$⁠) collisions at low temperature.³⁰ This statistical treatment will allow us to include both direct and indirect scattering in CH + H₂ collisions. The latter describes the formation and decay of a transient complex and allows the estimation of isotope exchange in CH–D₂ collisions, as observed by Macdonald and Liu.^17,18

This paper is organized in the following way: We briefly review quantum statistical theory in Sec. II, with special application to $Π2$–$Σ+1$ molecule-molecule collisions. Sec. III presents the details about the scattering calculations, while Sec. IV presents the computed cross sections and rate constants for CH–H₂ collision-induced rotational transitions. The paper concludes with a discussion in Sec. V.

For reference, we briefly describe the level structure within the CH(⁠$X2Π$⁠) electronic state. This electronic state falls very close to the Hund’s case (b) limit. This is illustrated in Fig. 1, in which the energies of the lower rotational/fine-structure levels are plotted. The energies were computed from the spectroscopic constants $B= 14.192 33$ cm⁻¹, $A=28.146 80$ cm⁻¹, $p=0.0339$ cm⁻¹, and $q=0.038 66$ cm⁻¹.³¹ The angular momentum exclusive of the electron spin s is denoted as n, which can take on the magnitude 1, 2, etc. The CH total angular momentum j is the vector sum of n and s and whose magnitude can equal $n+12$ (⁠$F1$ fine-structure manifold) or $n−12$ (⁠$F2$⁠). Because of the orbital degeneracy of the Π state, each $Fi$ fine-structure level consists of two nearly degenerate levels called Λ-doublets, with a symmetry index ϵ equal to +1 and −1 for the e and f levels, respectively.^32,33 In our scattering calculations, we assumed that the spin-orbit interaction was equal to that of the isolated CH radical and independent of nuclear geometry.

The quantum mechanical expression for the state-to-state integral cross section from a CH + H₂ initial level pair i = (⁠$j1$ $Fiϵ$⁠, $j2$⁠) to final level pair $f$ =(⁠$j1′$ $Fi′ϵ′$⁠, $j2$⁠) for total energy E can be written as

where $Ei$ is the internal energy of the initial level pair, μ is the collision reduced mass, J is the total angular momentum, and L and $L′$ are the initial and final orbital angular momenta, respectively. The angular momenta $𝒋12$ and $𝒋12′$ are vector sums of $𝒋1+𝒋2$ and $𝒋1′+𝒋2′$⁠, respectively. In the exact expression for the cross section, the probability $PfL′,iLJ(E)$ in Eq. (3) is the square modulus of the T matrix element between the initial and final scattering channels,

where the T matrix element is related to the corresponding S matrix element as follows:

In the quantum statistical theory,^21,22 the expression employed to compute the probability in Eq. (4) is replaced by

where

In Eq. (6), $pij12LJ$ is the capture probability, i.e., the probability of forming the collision complex from the initial level pair i with intermediate angular momentum $j12$ and orbital angular momentum L; $pfj12′L′J(E)$ and $pfj12″L″J(E)$ in Eq. (7) are similarly defined. The term $ffj12′L′J(E)$ in Eq. (7) is the fraction of collision complexes with angular momentum J and energy E that dissociate into the final level pair f with intermediate angular momentum $j12$ and orbital angular momentum $L′$⁠.

Because of the possibility of forming a complex, the S matrix is no longer unitary. The capture probability from channel $iL$ and total angular momentum J can thus be computed through

The probability $Pfj12′L′,ij12LJ(E)$ in Eq. (6) should, in principle, also include in the denominator capture probabilities $pf′i12″L″J(E)$ for the formation of the complex from CH₂ + H since the CH + H₂ → CH₂ + H reaction is only slightly endoergic [$ΔH0°=+14.2$ kJ mol⁻¹].

The CH₂ + H channel was not included in our calculations. Neglect of this channel will, in principle, lead to an overestimate of the contribution to the cross section from rotational transitions involving the formation and decay of a collision complex at total energies above the energetic threshold (1190 cm⁻¹) for the formation of CH₂ + H. Inclusion of this channel was beyond the scope of the present study.

The capture probability is computed by integrating the close-coupling scattering equations out from the so-called capture radius $Rc$⁠, at which the collision complex exists. As in treating inelastic and reactive collisions generally, it is convenient to use the log-derivative method.³⁴ We implemented Airy boundary conditions, corresponding to a linear reference potential at the capture radius.³⁵ It should be noted that the log-derivative matrix is complex, and not real as in the usual inelastic scattering equations, because of the boundary conditions. The radial differential equation for the log-derivative matrix was integrated using a linear reference potential in each sector. Unlike reactive scattering, a collision-induced inelastic transition can occur either through the formation and decay of the collision complex or through a direct collision. These two contributions to the inelastic cross section were computed using Eqs. (6) and (4), respectively.

The scattering equations were integrated in a body-fixed (BF) scattering basis since the matrix elements of the potential are expressed more simply than in a space-frame basis. The BF scattering basis functions are defined by the quantum numbers defining the levels [$j1$ $Fiϵ$ for CH and the rotational angular momentum $j2$ for H₂], the vector sum $𝒋12=𝒋1+𝒋2$⁠, and the projection K of the total angular momentum along the Jacobi vector R. To determine the S matrix, the BF log derivative vector was transformed to the space frame, and the scattering boundary conditions were applied.

Groenenboom et al.³⁶ have given the expression for the matrix element of the potential for a $Π2$–$Σ1$ molecule-molecule interaction in the primitive diabatic body-frame basis. We have checked this formula. This result was used to derive the matrix element involving symmetrized, intermediate-case wave functions, which describe the rotational levels of CH.

Close-coupling calculations, as described in Sec. II B, were carried out, using the diabatic CH(⁠$X2Π$⁠)–H₂ PES’s presented in Ref. 13. A total of 46 and 34 terms were included in the angular expansion of the diagonal and off-diagonal matrix elements of the potential, respectively. A sector width of $0.05a0$ was employed in the calculations. A capture radius $Rc=4.0a0$ was employed. We checked the effect of varying $Rc$ on the computed cross sections for direct scattering at a total energy $E=250$ cm⁻¹. For cross sections with magnitude greater than 1 Ų, cross sections computed with $Rc=3.8a0$ and $4.2a0$ differed by 3% and 0.8%, respectively, from those computed with $Rc=4.0a0$⁠. The scattering basis included CH rotational levels $j≤9.5$⁠. The H₂ rotational basis included levels $j=0$ and 2 for $para$-H₂ and $j=1$ for $ortho$-H₂. Inclusion of higher H₂ rotational levels was not practical because of the very long computational time required. One test calculation for CH–$ortho$-H₂ with the inclusion of $j=3$ in the basis was carried out for total energy $E=250$ cm⁻¹. Considering only direct cross sections with magnitude greater than 1 Ų, these cross sections differed by an average of 8% from cross sections obtained in a calculation with only $j=1$ in the H₂ rotational basis.

Cross sections were computed at 46 values of the total energy with $E≤1200$ cm⁻¹ for both nuclear spin modifications of H₂. The energy spacing was not uniform and was 1 cm⁻¹ around rotational thresholds. The cross sections were checked for convergence with respect to the sector width, the size of the CH rotational basis, and the range of total angular momentum J included in the calculations. At the highest energy (1200 cm⁻¹), total angular momenta $J≤173$ were included. The cross sections are estimated to be converged to within several percent. We did not carry out calculations at higher total energies because the computational time would be excessive, particularly for collisions of $para$-H₂.

Our quantum statistical scattering program for $Π2$–$Σ+1$ molecule-molecule interactions was tested by the calculation of cross sections for OH(⁠$X2Π$⁠)–H₂ cross sections and comparison of these computed cross sections with values reported in our previous investigation of collision-induced rotational transitions in this system.³⁷ This calculation is entirely equivalent to a conventional inelastic scattering calculation since the proper boundary conditions in the repulsive small-R region were applied. Our scattering program is not efficient for computing OH(⁠$X2Π$⁠)–H₂ cross sections, since there is no formation and decay of a collision complex for this system. Hence, the log-derivative matrix is real, and not complex, in this case.

It is interesting to see what fraction of the collision-induced rotational transitions occur directly, or indirectly, by the formation and decay of a collision complex. Figure 2 presents the contributions to the integral cross sections for two transitions out of the ground initial $n=1$ $F2e$ level in collisions with H₂$(j=1)$⁠. We see that just above the energy threshold for the formation of the final level, indirect scattering, involving the formation and decay of a complex, makes a substantial contribution to the total integral cross section. Indeed, for the $n=1$ $F2e→n′=2$ $F1e$ transition, indirect scattering contributes more than direct inelastic scattering at low energies. The contribution to the integral cross section from indirect scattering drops off rapidly as the energy is increased. At the highest energy plotted in Fig. 2, indirect scattering makes a very small contribution to the integral cross section. The contribution from direct scattering is seen either to decrease slowly with increasing energy, or even to increase slightly over the plotted energy range. The contributions from direct and indirect scattering to other transitions show a dependence upon energy similar to those plotted in Fig. 2.

We can analyze the collisions of an initial CH–H₂ level pair that yield a particular final level pair through direct and indirect collisions by investigating the partial cross sections, i.e., the contribution to the integral cross section from individual total angular momenta J. Figure 3 displays partial cross sections for direct and indirect scattering for one of the transitions plotted in Fig. 2 at total energies of 160 and 1000 cm⁻¹. At 160 cm⁻¹, the cross section reaches a maximum value as a function of energy (see Fig. 2). For this energy, we see that the partial cross sections for direct and indirect scatterings span a similar range of total angular momenta, and hence orbital angular momenta, and the partial cross section for indirect scattering is larger than that for direct scattering over a range of total angular momenta.

By contrast, the partial cross section at 1000 cm⁻¹ for direct scattering spans a considerably larger range of total angular momenta than the partial cross section for indirect scattering at this energy. In fact, the partial cross sections for indirect scattering span a similar range of total angular momenta at the two energies. The difference is that the magnitudes of the partial cross sections at 1000 cm⁻¹ are much smaller than those for 160 cm⁻¹. As we show in detail below, the reason for this difference is that there are many more possible decay channels at the higher energy, so that the fraction $ffj12′L′J(E)$ [Eq. (7)] of collision complexes decaying to a particular final level pair is much smaller.

Of course, the contribution of indirect scattering to the integral cross section depends upon the magnitude of the capture probabilities [see Eqs. (6) and (7)]. Figure 4 presents the capture probabilities for the CH $n=2$ rotational/fine-structure levels in collisions with H₂$(j=1)$ at total energies of 160 and 1000 cm⁻¹. These capture probabilities have been averaged over the values for the scattering channels, defined by L and $j12$⁠, associated with each level. We see for 160 cm⁻¹ energy that the capture probabilities increase modestly as a function of J and reach a maximum near $J=12$ and drop to near zero by $J=17$⁠, in agreement with the value of J for which the partial cross sections drop to zero (see Fig. 3). The maximum capture probability is $≈0.3$ and 0.15 for the levels of $A′$ and $A″$ reflection symmetry,³⁸ respectively. Similar differences between levels of $A′$ and $A″$ symmetry are seen in the capture probabilities for CH levels.

The capture probabilities for 1000 cm⁻¹ energy are somewhat smaller than for 160 cm⁻¹ [compare Figs. 4(a) and 4(b)]. The capture probabilities for 1000 cm⁻¹ are approximately constant until $J=20$ and then drop quickly to zero. From Eqs. (6) and (7), the differences in the magnitudes of the cross sections for indirect scattering depends upon the capture probability $pij12LJ(E)$ for forming the complex in a given final level pair and the fraction $fij12LJ(E)$ of collision complexes decaying to the given level. The differences in the magnitudes of the capture probabilities for 160 and 1000 cm⁻¹ are much smaller (approximately a factor of 2) than the differences in the size of the partial cross sections for indirect scattering for these two energies [compare Figs. 3 and 4]. Hence, it is mainly the energy dependence of the decay fraction to the final level that causes the cross section for indirect scattering to drop substantially as a function of energy.

Neglect of the CH₂ + H channel in the decay of the CH₃ collision complex should not lead to a significant overestimate of the indirect contribution to the cross sections. This channel is open only at total energies greater than 1190 cm⁻¹. By this energy, the indirect contribution to the cross sections is a very small fraction of the total cross section (see Fig. 2).

Integral cross sections, including the sum of direct and indirect scattering, are displayed in Fig. 5 for transitions out of

the $n=1$ $F2e/f$ levels to the $n′=2$ $F1e/f$ and $F2e/f$ levels for collisions with H₂$(j=0,1)$⁠. We see that the cross sections rise rapidly above the energetic threshold and decrease slowly in magnitude as the collision energy is increased further. Unlike the case of OH–H₂,³⁹ where cross sections for collisions of H₂$(j=1)$ are generally significantly larger than those for H₂$(j=0)$⁠, the CH–H₂ cross sections for H₂$(j=1)$ and H₂$(j=0)$ are of comparable magnitude. This suggests that the long-range tail of the potential does not play a significant role in the inelastic scattering dynamics, unlike the case of OH–H₂.³⁹ We also observe in Fig. 5 that the cross sections for fine-structure conserving changing transitions (⁠$F2→F2$⁠) are somewhat larger than those for fine-structure changing transitions (⁠$F2→F1$⁠).

The cross sections displayed in Fig. 5 show some dependence of their magnitude upon the initial and final Λ-doublet levels. For the initial f Λ-doublet level, the cross sections are somewhat larger for the $f→e$ than for the $f→f$ transition in both fine-structure conserving and changing transitions. By contrast, the cross sections from the initial e level to the final e and f levels are almost identical. These relatively weak Λ-doublet preferences contrast sharply with the sharp Λ-doublet propensities observed in both CH–He and OH–H₂ collisions^11,39 and reflect the presence of the deep well and the considerable anisotropy of the potential.

We have also carried out scattering calculations for collisions of CH with D₂, in order to compare with the relative cross sections determined as a function of collision energy by Macdonald and Liu.^16,18 Since these cross sections were measured in a crossed beam experiment, absolute magnitudes could not be determined. They measured excitation functions, i.e., relative cross sections as a function of the collision energy. Figure 6 displays the experimentally measured excitation functions for the formation of the $F2e$ final levels for the various final rotational angular momenta n. The cross sections are seen to rise from their energetic threshold and reach a plateau at higher collision energies.

Also plotted in Fig. 6 are relative cross sections computed in this study for the production of the CH $n′=3–7$ $F2e$ final levels from CH–D₂ collisions as a function of collision energy. Here, our computed CH–H₂ potential can be used without modification since the center of mass of the hydrogen molecule does not shift with this isotopic substitution. The computed relative cross sections shown in Fig. 6 were obtained using Eq. (4) for the direct transition probability. We have neglected the indirect contribution to the cross sections. At the collision energies of the experiments of Macdonald and Liu, this contribution is relatively small compared to the direct contribution. Moreover, the probability that the CHD₂ complex decay to CH products will be approximately one third of the total probability, ignoring isotope effects leading to preferential production of the HD co-fragment.

Macdonald and Liu measured the initial level distribution in their incident CH beam.¹² The incident population distribution had equal population of the Λ-doublets, with 77% and 16% of the population in the $n=1$ $F2e/f$ and $F1e/f$ levels, respectively, and much smaller population in higher rotational levels. The computed cross sections in Fig. 6 are weighted averages of the theoretical cross sections out of the four $n=1$ initial levels. (Test calculations indicated that inclusion of higher levels did not materially affect the magnitudes of the cross sections displayed in Fig. 6.)

From measurements of the D₂ beam velocity, Macdonald and Liu estimated the incident D₂ level distribution to be 55%, 33%, and 12% in the $j=0$⁠, 1, and 2 rotational levels. We carried out separate calculations of cross sections for collisions of CH with D₂ $j=0$ and 1. The cross sections shown in Fig. 6 include averaging over the D₂ incident level distribution. We assumed that the cross sections for the D₂ $j=2$ initial level were the same as for $D2$ $j=1$⁠. For OH–H₂ collisions, cross sections for collisions of H₂ $j=1$ and $j=2$ were found to be approximately equal.³⁹ 

We see in Fig. 6 that the calculated relative cross sections provide a generally reasonable representation of the relative cross sections measured by Macdonald and Liu. The major discrepancy is that the cross section for the formation of the $n=4$ $F2e$ final level relative to the cross sections for the formation of the other levels is significantly underestimated. We discuss in Sec. V possible reasons for the disagreement between experiment and theory. Macdonald and Liu also measured the relative cross section into the $n′=8$ $F2e$ level. We did not include this level in Fig. 6 since our calculations for this final level were not converged.

Macdonald and Liu also investigated fine-structure propensities in the CH products from the inelastic scattering of CH with D₂. Here, we look at one such propensity. Figure 7 presents the fraction of the population in the four Λ-doublet/fine-structure levels as a function of n at a collision energy of 900 cm⁻¹, where the cross sections have reached a plateau. We see that there is a preferential formation of the $F2e$ and $F1f$ levels, as compared to the formation of the $F2f$ and $F1e$ levels. Macdonald and Liu found the same propensity in their experiments. As Macdonald and Liu noted, in the high-j limit, the $F2e$ and $F1f$ levels have $A″$ symmetry with reflection of the electronic wave function through the plane of rotation, while the $F2f$ and $F1e$ levels have $A′$ symmetry. There was no evidence in the computed fine-structure probabilities for the oscillation as a function n noted by Macdonald and Liu in their experimental values.

Macdonald and Liu also measured the rotational/fine-structure distribution of CD product from isotope exchange collisions of CH with D₂.¹⁸ We did not compute the state distribution for this channel. The CH–H₂ potential would have to be transformed in order to describe the CD–HD potential because of the shift in the centers of masses of both collision partners. For low-$n′$ final levels, they found that the excitation function showed a monotonically decreasing behavior as a function of collision energy, while the excitation functions for higher $n′$ levels rose from their respective energy thresholds and then decreased at higher collision energies. This is the behavior for indirectly scattered CH from CH–H₂ collisions, as illustrated in Fig. 2.

Finally, rate constants for rotational transitions in CH(⁠$X2Π$⁠) in collisions with H₂$(j=0,1)$ were computed by averaging the state-to-state cross sections over the thermal distribution of collision energies $Ec$⁠,

In Eq. (9), μ is the collision reduced mass and $kB$ is the Boltzmann constant. The integral in Eq. (9) was evaluated out to $Ec=2000$ cm⁻¹; this upper limit is sufficient for temperatures up to 300 K. The high-energy tail of the cross sections was obtained by exponential extrapolation beyond the total energy $E=1500$ cm⁻¹.

Thermal rate constants for transitions between pairs of CH levels induced by collisions with both H₂$(j=0)$ and H₂$(j=1)$ were computed from the calculated energy-dependent cross sections for temperatures from 10 to 300 K for collisions of the 16 CH(X²Π) levels associated with the rotational manifolds $n=1$–4 in collisions with H₂$(j=0)$ and H₂$(j=1)$⁠. De-excitation rate constants were computed using Eq. (9), while excitation rate constants were obtained by detailed balance. These, as well as energy-dependent cross sections, are reported in the supplementary material.

In Sec. IV, we noted that there is some disagreement between the experimentally measured and our calculated relative cross sections for CH–D₂ collisions. We assumed in our calculations that the spin-orbit splitting does not vary with nuclear geometry; it is possible that this is invalid when the collision complex is formed at small R. In our calculation of the CH–H₂ potential,¹³ we fixed the CH and H₂ internuclear separations, it may be necessary to allow these bond lengths to relax to describe the potential more accurately, particularly in the region of the deep well. Our neglect of the CD–HD channel should not have affected our calculation of the CH product distribution since the CH products can be formed only in direct collisions.

In this paper, we have reported close-coupled quantum statistical scattering calculations of cross sections of rate constants for collisions of CH(⁠$X2Π$⁠) with molecular hydrogen. Our primary motivation for these calculations is to provide these rate constants in modeling the CH internal state distributions in interstellar media, so that the observed intensities of CH spectroscopic transitions can be used to determine CH column densities and concentrations. Since CH–He rate constants are used as surrogates for CH–H₂ rate constants, it is interesting to compare the magnitudes of the rate constants for these two systems.

Figure 8 compares de-excitation rate constants for temperatures from 10 to 300 K for transitions from the four $n=2$ fine-structure levels to the lowest, $n=1$ $F2e$⁠, level for collisions of CH with H₂$(j=1)$⁠, H₂$(j=0)$⁠, and He. The cross sections for CH–He collisions were taken from the work of Marinakis et al.¹¹ We see that there are dramatic differences in both the magnitude and temperature dependence of the CH–H₂$(j=0,1)$⁠, He rate constants. In particular, the rate constants for CH–H₂$(j=0,1)$ collisions are significantly larger than for collisions of CH with He. Also, the rate constants for CH–He collisions increase, in one case (the $n=2$ $F2f$ initial level) dramatically, with increasing temperature. By contrast, the CH–H₂ rate constants either vary modestly with temperature and reach plateau values by 100 K or are approximately independent of temperature. The large CH–H₂ rate constants at low temperatures reflect the important rotational transitions mediated by the formation and decay of the collision complex at low collision energies, as illustrated by the large contribution of indirect scattering seen in the energy dependent cross sections displayed in Fig. 2. As with the energy-dependent cross sections displayed in Fig. 5, the rate constants are slightly larger for collisions of CH with H₂$(j=1)$ than with H₂$(j=0)$⁠.

To our knowledge, this is the first report of rate constants for rotationally inelastic collisions of CH with molecular hydrogen. These rate constants can be employed to assist in astrophysical modeling, through the estimation of CH concentrations from the intensities of observed CH spectroscopic intensities. It is expected that concentration of interstellasr CH will be quite different if the CH–H₂ rate constants presented in this study are employed, as opposed to CH–He rate constants employed as surrogates. Interstellar CH concentrations are quite important for the chemistry in this environment since CH is a reactive species and is a doorway to the production of more complicated carbon-containing molecules.⁶ It will also be useful to extend these calculations to the computation of cross sections for hyperfine-resolved transitions since the CH lines are often fully resolved in astronomical observations. In future work, we plan to compute rate constants for rotationally inelastic transitions in collisions of CH with atomic hydrogen. The latter species has a significant interstellar concentration, and no information is presently available on rate constants for CH–H collisions.

See supplementary material for a table of cross sections as a function of the total energy and a table of rate constants for collision-induced rotational transitions between the 16 CH(⁠$X2Π$⁠) levels associated with the rotational manifolds $n=1–4$ in collisions with H₂$(j=0)$ and H₂$(j=1)$ for temperatures between 10 and 300 K in steps of 10 K.

The author appreciates the encouragement of Millard Alexander. He is also grateful for support from the U.S. National Science Foundation (Grant No. CHE-1565872). The author also thanks François Lique for providing computed rate constants for CH–He, computed in the study reported in Ref. 11. The scattering calculations were performed on the Maryland Advanced Research Computing Cluster, which was funded by the state of Maryland and is jointly managed by Johns Hopkins University and the University of Maryland College Park.