Full-dimensional quantum dynamics of CO in collision with H₂

Over many decades much effort has been devoted to theoretical quantum mechanical studies of diatom-diatom inelastic collisions with significant advances resulting from numerical algorithm development and increases in computational processing power. However, full-dimensional (6D) studies of diatom-diatom systems have been limited to H₂–H₂ collisions. These studies have included both time-independent¹ and time-dependent² approaches for solving the Schrödinger equation. Moreover, to make these computations practical, various angular-momentum decoupling approximations have been invoked. To assess the usefulness of such approximations, it is desirable to develop accurate computational methods which are capable of handling large-scale calculations for diatom-diatom (and larger) scattering without resorting to such approximations. Indeed, the first numerically exact 6D close-coupling (CC) computations of rovibrational transitions in the H₂ + H₂ system were reported recently.^3–5 Subsequently, Yang et al.⁶ reported full-dimensional CC calculations of the CO + H₂ system on a 6D potential energy surface (PES) derived from high-level ab initio calculations.

From a theoretical point of view, the development of exact quantum close-coupling methods for molecule-molecule and other polyatomic systems seem viable, but due to the large number of internal states of the molecules involved, their numerical implementation is currently not feasible for systems with more than four atoms. This difficulty has been alleviated to some extent by using angular momentum decoupling schemes, for example, the coupled-states (CS) approximation performed recently by Forrey et al.,^7,8 which has been successfully implemented in H₂ + H₂ and CO + H₂ rovibrational scattering calculations and achieved reasonable agreement with CC results.

Rovibrational collision of CO with H₂ is an important process in a variety of astrophysical environments including interstellar clouds and photodissociation regions. The collisional rate coefficients are crucial to astrophysical modeling, but are difficult to measure, and therefore are mostly obtained from theory. Though great progress has been made in experimental^9–20 and theoretical^6,21–41 studies of CO + H₂ structure and dynamics, there is still a lack of a comprehensive set of state-to-state rovibrational cross sections and rate coefficients for CO + H₂ that are required for astrophysical modeling.

Since the close-coupling calculations of CO + H₂ rate coefficients by Green²¹ using an approximate PES, a number of PESs have been constructed for the CO + H₂ complex.^{6,25,26,32,35–37} Specifically, Jankowski and coworkers reported three accurate four-dimensional (4D) rigid-rotor PESs.^35–37 These 4D PESs were calculated on four-, five-, and six-dimensional (6D) grids using high level electronic structure theory, followed by averaging over the ground intramolecular vibrations of CO and H₂. Comparative studies of collisional cross sections and rate coefficients using the three PESs have been presented in our earlier work^6,40,41 and by other authors.^19,20,33,39

Taking into account the CO vibrational mode, Schinke and co-workers^25,26 extended the rigid-rotor CO + H₂ PES²⁴ to include dependence on CO bond distances and made it available to study vibrationally inelastic processes of CO. This surface was adopted by Bačić et al.^25,26 and Reid et al.³⁰ to investigate the vibrational deactivation of CO (v = 1) by inelastic collisions with H₂. However the dynamical calculations of Bačić et al. and Reid et al. were performed using the infinite order sudden (IOS) and coupled-states (CS) approximations, respectively. The IOS approximation eliminates the rotational channel coupling and neglects the internal rotation. As a result, rotational state-to-state transitions cannot be resolved dynamically. Another 4D ab initio PES including the CO vibrational coordinate dependence was presented by Kobayashi et al.³² Using this potential and the CC method, Flower³⁴ calculated cross sections for the rovibrational excitation of CO in collision with H₂.

Most recently, Yang et al.⁶ presented a high-level 6D interaction potential surface based on nearly 400 000 geometries fitted with an invariant polynomial approach.⁴² The 6D PES is a significant step forward in quantitative modeling of the CO + H₂ complex. Using this PES, the first essentially exact full-dimensional dynamics computations for inelastic rovibrational quenching of CO from v₁ = 1, j₁ = 0 due to H₂ impact were performed. These dynamics calculations were performed within a full angular-momentum-coupling formulation, representing one of the largest such computations performed to date. Agreement with experimentally determined integral cross sections and rate coefficients was obtained for both rotational and vibrational transitions, confirming the accuracy of the 6D PES and scattering calculations. In this paper, we extend our previous v₁ = 1, j₁ = 0 rovibrational scattering calculations to higher CO rotational states and to v₁ = 2. Further details on the 6D PES are also presented.

The rest of the paper is organized as follows. Sec. II briefly describes the theoretical methods. The results are presented and discussed in Sec. III, and Section IV summarizes the results, discusses the astrophysical applications, and presents an outlook on future work.

In this section, we will briefly describe the methods we used in the potential energy surface calculation and fit and rovibrational inelastic scattering calculations. The reader is referred to Refs. 3, 6, and 42 for more detailed discussions of the methodology. Atomic units are used throughout unless otherwise noted.

The electronic ground state interaction potential of CO + H₂ was computed on a six-dimensional (6D) grid using Jacobi coordinates (R, r₁, r₂, θ₁, θ₂, ϕ) as shown in Fig. 1. R is the center-of-mass distance between CO and H₂. r₁ and r₂ represent the bond lengths describing the vibration of CO and H₂, respectively. θ₁ (θ₂) is the in-plane orientation angle between $r→1$ (⁠$r→2$⁠) and $R→$⁠, and ϕ the out-of-plane dihedral angle. In the computations of the potential energy surface, R spans from 4.0 to 18.0 a₀ and the bond distances are taken over the ranges 1.7359 ≤ r₁ ≤ 2.5359 a₀ and 1.01 ≤ r₂ ≤ 1.81 a₀, which support vibrational states of CO up to v₁ = 5 and H₂ v₂ = 0 in the scattering calculations. All angular coordinates were computed with 0° ≤ θ₁ ≤ 360° and 0° ≤ θ₂, ϕ ≤ 180°.

The electronic structure calculations were performed with MOLPRO2010.1.⁴³ Potential energies were computed using the explicitly correlated coupled-cluster (CCSD(T)-F12B) method^44,45 and cc-pcvqz-f12 orbital basis sets⁴⁶ at all of the configurations. Benchmark calculations at the CCSD(T)-F12b/cc-pcvqz-f12 level were carried out on selected molecular configurations and results were compared with those from the conventional CCSD(T) method using aug-cc-pV5Z and aug-cc-pV6Z basis sets. The counter-poise (CP)⁴⁷ corrected interaction energy agrees closely with those derived from CCSD(T)/aug-cc-pV6Z. The interaction potential energy surface was corrected for basis set superposition error (BSSE).⁴⁸ 

To fit the CO–H₂ interaction potential in 6D (referred to as V6D), an invariant polynomial method⁴² was used. The potential was expressed in terms of Morse-type variables, y_i = e^−0.5d_i,

where d_i are the internuclear distances between two atoms, d₁ = d_HH′, d₂ = d_OH′, d₃ = d_CH′, d₄ = d_CH, d₅ = d_OH, and d₆ = d_CO. The order of the polynomials, n_i, i = 1 − 6 ranges from 0-6 with the total power of the polynomial, $N=∑i=16ni$⁠, restricted to a maximum value of 6. Expansion coefficients c_n1⋯n6 were determined by a weighted linear least-squares fit to the ab initio electronic potential energies up to a maximum of 10 000 cm⁻¹.

Some details of the fitted interaction PES were illustrated in Ref. 6. Fig. 2 shows the V6D interaction PES near the global minimum, which corresponds to the collinear arrangement H–H–C–O (θ₁ = 0, θ₂ = 0, ϕ = 0) with a depth of −85.937 cm⁻¹ at R = 8.0 a₀. Fig. 3 depicts the anisotropy of the V6D and V12 PESs in θ₁, θ₂, and ϕ, respectively. θ₁ describes the orientation of CO, and the anisotropy with respect to θ₁ is the largest because of the larger CO bond length. V6D and V12 PESs show similar behavior; however, some differences can be seen which may partly be attributed to the fact that V6D is a full-dimensional surface, while V12 is a vibrationally averaged 4D surface.

The quantum CC formalism for molecular collisions was first developed in 1960 by Arthurs and Dalgarno⁴⁹ for a rigid-rotor scattered by a spherical atom and subsequently extended to full vibrational motion to study diatom-diatom collisions by several authors.^50–53 The resulting coupled-channel equations, based on the time-independent Schrödinger equation, are solved numerically. In the 6D Jacobi coordinates shown in Fig. 1, the Hamiltonian of CO + H₂ can be written as

where the radial kinetic energy term $T(R→)$ describes the center-of-mass motion of the scattering system, $T(r→1)$ and $T(r→2)$ are the kinetic energy terms for CO and H₂, respectively. $U(r→1,r→2,R→)$ denotes the total potential energy surface and is given by

where $V(r→1,r→2,R→)$ describes the interaction potential between CO and H₂ which vanishes when the two molecules are far apart.

To facilitate the scattering computations, V(R, r₁, r₂, θ₁, θ₂, ϕ) is expressed as

with the bi-spherical harmonic function expressed as

where 0 ≤ λ₁ ≤ 8, 0 ≤ λ₂ ≤ 4 was adopted for the potential expansion in the scattering calculations. Only even values of λ₂ contribute due to the symmetry of H₂.

We use combined molecular state (CMS) notation, (v₁j₁v₂j₂), to describe a combination of rovibrational states for CO (v₁j₁) and H₂ (v₂j₂). v and j are the vibrational and rotational quantum numbers. A CMS represents a unique quantum state of the diatom–diatom system before or after a collision. The rovibrational state-to-state cross section as a function of collision energy E is given by

where (v₁j₁v₂j₂) and (⁠$v1′j1′v2′j2′$⁠) are, respectively, the initial and final CMSs, the wave vector k² = 2μE/ħ², and S is the scattering matrix. l is the orbital angular momentum and J the total collision system angular momentum, where $J→=l→+j→12$ and $j→12=j→1+j→2$⁠.

The total quenching cross section of CO from initial state $(v1j1v2j2)→(v1′;v2′j2′)$ was obtained by summing the state-to-state quenching cross sections over the final rotational state $j1′$ of CO in vibrational state $v1′$⁠,

Full-dimensional rovibrational scattering calculations were performed using the TwoBC code.⁵⁴ The CC equations were propagated for each value of R from 4 to 18.0 a₀ with step-size of ΔR = 0.05 a₀ using the log-derivative matrix propagation method of Johnson⁵⁵ and Manolopoulos.⁵⁶ As discussed in Ref. 6, at least 13-15 rotational states have to be included in the v₁ = 1 basis set to ensure the convergence of the v₁ = 1 → 0 vibrational quenching cross section. The cross sections are converged down to the lowest collision energy of 0.1 cm⁻¹ based on tests with respect to the adopted maximum internuclear distance R_max for the long range part of the V6D PES. We further tested the effect of truncating R_max by performing pure rotational excitation and deexcitation calculations within the rigid-rotor approximation using the 4D V12 PES. No discernible difference was found for R_max = 18 compared to R_max = 75.

The number of discrete variable representation points N_r1 and N_r2; the number of points in θ₁ and θ₂ for Gauss-Legendre quadrature, N_θ1 and N_θ2; and the number of points in ϕ for Chebyshev quadrature, N_ϕ, adopted to project out the expansion coefficients of the interaction potential are listed in Table I. The intramolecular potentials used are V₁(r₁) for the CO monomer presented by Huxley and Murrell⁵⁷ and V₂(r₂) for the H₂ monomer adapted from the work of Schwenke.⁵⁸ 

In Ref. 6, the pure rotational j₁ = 0 → 1 excitation cross sections for v₁ = 0 of CO in collision with para-H₂ were calculated at low collision energies using the full-dimensional V6D and 4D V12 PESs. As indicated previously, the V12 PES³⁷ was calculated on a six-dimensional grid using high level electronic structure theory, then reduced to 4D by averaging over the ground intramolecular vibrations of CO and H₂. Results from the scattering calculations on the two PESs were convolved with an experimental energy spread function and compared to the measured relative cross sections of Chefdeville et al.¹⁹ Except for the peak near 8 cm⁻¹, our calculation using V6D was in best agreement with the original experiment. More recently however, Chefdeville et al.²⁰ reported new measurements, new experimental analysis, and new calculations for the rotational excitation of CO with normal-H₂ and the rotational excitation from j₁ = 0 and 1 to $j1′=2$ of CO by para-H₂. The j₁ = 0 → 1 measurement was re-analyzed following a procedure outlined in the work of Naulin and Costes⁵⁹ and theoretical cross sections were convolved using an adjusted experimental energy spread function. The j₁ = 0 → 1 excitation cross sections shown in Fig. 3(a) of Ref. 6, which were obtained using the V6D and V12 PESs, are here convolved using the adjusted experimental energy spread. The new convolved theoretical cross sections are compared to the adjusted measurements in Fig. 4. In general, the cross section obtained with V12 shows good agreement with the new measurement. Though the V6D PES displayed better agreement with the original experiment as shown in Ref. 6, the new convolved cross sections obtained with V6D diverge from the adjusted measured data below ∼12 cm⁻¹.

To compare with the additional measurements of Chefdeville et al.²⁰ for CO excitation (j₁ = 0, 1 → 2) due to para-H₂ collisions and CO excitation (j₁ = 0 → 1) by normal H₂, new full-dimensional and rigid-rotor approximation calculations of the state-to-state cross sections have been carried out using the V6D and V12 PESs, respectively. The MOLSCAT code⁶⁰ was used in the rigid-rotor calculations. The pure rotational excitation cross section calculated from the rigid-rotor calculations using a 4D PES, which was obtained by vibrationally averaging the V6D PES over ground vibrational states of CO and H₂, is nearly identical to the full-dimensional results, and therefore not shown. The computed cross sections were convolved with the new experimental energy function of Chefdeville et al.²⁰ In Fig. 5, j₁ = 0, 1 → 2 cross sections for CO due to collisions with para-H₂(j₂ = 0) are shown as a function of collision energy. Figs. 5(a) and 5(b) compare the computed cross sections for j₁ = 0 → 2 and j₁ = 1 → 2 transitions, respectively, using the V6D and V12 PESs. Fig. 5(c) shows the comparison between the total convolved theoretical cross sections and the relative experiment of Chefdeville et al.²⁰ Each total convolved theoretical cross section includes a 90% contribution from j₁ = 0 → 2 and a 10% contribution from j₁ = 1 → 2 transitions.

For the case of CO with normal H₂, the computed j₁ = 0 → 1 rotational excitation cross sections of CO by para-H₂ (j₂ = 0) and ortho-H₂ (j₂ = 1) using the V6D and V12 PESs are compared in Figs. 6(a) and 6(b). The results of CO with para-H₂ are taken from Ref. 6. All the cross sections display strong resonances, particularly at collision energies below 18 cm⁻¹. Different resonance structures are exhibited for the V6D and V12 PESs. The convolved j₁ = 0 → 1 cross section of CO by normal H₂ with 75% relative population of ortho-H₂ and 25% of para-H₂ is compared to the measurements in Fig. 6(c). It can be seen that the V12 PES shows good agreement with measurement over the whole experimental energy range. The V6D cross section is generally in good agreement with experiment, except near 9 cm⁻¹.

Overall, based on the comparison to adjusted experiments of Ref. 20, the low-energy pure rotational excitation cross sections from V12 PES show better agreement. The V6D PES appears to be less accurate for rotationally inelastic cross sections at low energies. Further improvements on the long-range part of the V6D PES may yield better agreement with experiment. As some uncertainty remains in the experimental analysis, perhaps, new measurements/improved analysis schemes would be useful. In particular, absolute cross sections would provide a more stringent discriminant. On the other hand, it is shown that the state-to-state cross sections on both PESs display numerous resonances and these resonances are very sensitive to the details of the PESs. The resonances generally shift by 2-3 cm⁻¹ between V6D and V12 and this may be partially explained by the equilibrium geometries which for the vibrationally averaged surfaces have well depths of −83.82 cm⁻¹ at (R, θ₁, θ₂, ϕ) = (7.926 a₀, 180°, 0°, 0°) and −93.651 cm⁻¹ at (7.911 a₀, 180°, 0°, 0°), respectively, for the V6D and V12 PESs. Furthermore, the differences in the well depths of the two PESs lead to differences in the energy level spectrum of the van der Waals complexes and corresponding resonance structures.

Full-dimensional calculations of the collision energy dependence of state-to-state cross sections were performed for initial CMSs (1j₁0j₂), for para-H₂ (j₂ = 0), and ortho-H₂ (j₂ = 1), j₁ = 1–5. The collision energy ranged from 0.1 to 1000 cm⁻¹. In this work, we consider only the rotation of H₂, and its vibration is fixed in the ground state $v2=v2′=0$⁠.

Examples of the state-to-state cross sections for quenching from initial CMSs (1200) and (1401) into individual final CO rotational levels in v₁ = 0 are shown in Figs. 7 and 8, respectively, where $j1′=0,2,4,…,22$ and $j2=j2′$ (i.e., elastic in H₂) are displayed. The insets show the distributions of final rotational levels of CO in $v1′=0$ at a collision energy of 2 cm⁻¹. As shown in Fig. 7, each of the cross sections decreases initially as the collision energy increases from 0.1 cm⁻¹ exhibiting the threshold behavior predicted by Wigner’s Law.⁶¹ The cross sections display the presence of a number of resonances at energies between 1.0 and 30 cm⁻¹. The resonant relative magnitudes decrease with increasing of $j1′$⁠. Above the van der Waals region, the cross sections increase with increasing collision energy. Furthermore, the cross sections of larger $j1′$ increase faster. At a collision energy of 1000 cm⁻¹, the cross sections to $j1′=12$ and 22 merge and become the largest. An exception to this behavior is the CO rotational elastic transition to $j1′=2$⁠, which has the largest cross section for collision energies below 20 cm⁻¹ with less resonance like behavior between 1 and 30 cm⁻¹, and becomes the second smallest cross section at energies above 100 cm⁻¹. The inset of Fig. 7 shows CO final rotational distribution in $v1′=0$ from quenching of (1200), and it can be seen that the final rotational distribution is broad and dominated by Δj₁ = 0 − 10. The distribution falls off rapidly above $j1′=16$⁠.

For the vibrational quenching cross sections from CMS (1401) shown in Fig. 8, the trends noted for para-H₂ (1200) in Fig. 7 are also observed except for a suppression of resonance magnitudes. Similarly, for collision energies below 100 cm⁻¹, the CO rotational elastic transition dominates. At collision energies higher than 400 cm⁻¹, the cross section to $j1′=22$ gradually becomes the largest indicating that the rotational basis set may not be sufficient at high collision energies. The inset of Fig. 8 shows that the final rotational distribution of CO in $v1′=0$ from quenching of (1401) is broad and dominated by Δj₁ = 0 − 10 at a collision energy of 2 cm⁻¹.

In Fig. 9, the state-to-state quenching cross sections for CO from v₁ = 1, j₁ = 3 to $v1′=0,j1′$ are compared between para-H₂ (v₂ = 0, j₂ = 0) and ortho-H₂ (v₂ = 0, j₂ = 1). The selected final rotational $j1′$ states are 1, 3, 7, 11, 15, and 19; H₂ rotational transitions are elastic. As shown in Fig. 9, for each individual transition, the state-to-state rovibrational quenching cross sections are of very similar shape and magnitude for the two colliders. The difference between para-H₂ and ortho-H₂ cross sections is somewhat larger at low collision energies, except for $j1′=19$ for which the cross sections are very similar. The inset presents the comparison of the distributions of final rotational levels $j1′$ in $v1′=0$ from quenching of (1300) and (1301) at a collision energy of 2.0 cm⁻¹. For these final $j1′$ levels, similar cross section magnitudes are observed for para-H₂ and ortho-H₂. Other final $j1′$ in $v1′=0$ which are not shown in Fig. 9 result in similar cross section trends.

Using Eq. (7), the state-to-state cross sections from each initial CMS are summed over all final rotational states $j1′$ in $v1′$ to obtain the total vibrational quenching cross sections as a function of collision energy. Figs. 10-12 depict the total quenching cross sections of CO from CMSs (1j₁0j₂), j₁ = 1, 3, and 5 with j₂ = 0. As shown in Table I, the basis sets include rotational states 0, 2, and 4 for para-H₂ and 1 and 3 for ortho-H₂. For all initial CO j₁ states, the upper panels show the CO total quenching cross sections for H₂ rotational transitions $j2=0→j2′=0$⁠, 2, and 4, and the lower panels show the CO total quenching cross sections for H₂ rotational transitions $j2=1→j2′=1$ and 3. Figs. 10-12 show clearly that the cross sections exhibit almost the same trends for each initial CO j₁ state. The total quenching cross sections show qualitatively similar structure with a number of resonances in the energy range between 0.7 and 30 cm⁻¹ induced by the van der Waals interaction, like those shown in Figs. 7 and 8 for the state-to-state cross sections. However the suppression of the resonances can be found with increasing initial CO rotational quantum number j₁. While for energies higher than 70 cm⁻¹ the cross sections increase with increasing collision energy. Further observation shows that for CO with para-H₂, over the whole collision energy range elastic H₂ transitions are dominant over inelastic H₂ transitions with Δj₂ = 4 being the smallest. While for the case of ortho-H₂ below 100 cm⁻¹, the cross sections for elastic and inelastic H₂ transitions are of similar magnitude. We note that this may be an artifact of the small ortho-H₂ basis, but we are currently limited by memory considerations. For collision energies greater than 100 cm⁻¹, the cross section difference between elastic and inelastic H₂ transitions increases with increasing collision energy.

In Fig. 13, the total cross sections for vibrational quenching of CO with para-H₂ from (1j₁00) to (⁠$v1′=0;v2′=0,j2′=0$⁠) are compared for j₁ = 0–5. The result for j₁ = 0, taken from Ref. 6, shows the strongest resonances at relatively low collision energies. The resonances decrease with increasing j₁, with the total cross section of j₁ = 5 having the weakest resonances. As the collision energy increases, the cross section for j₁ = 5 becomes the largest with j₁ = 0 being the smallest. The total cross sections for vibrational quenching of CO with ortho-H₂ from (1j₁01) to (⁠$v1′=0;v2′=0,j2′=1$⁠) for j₁ = 0–5 are displayed in Fig. 14. The comparison shows similar behavior as found for para-H₂ in Fig. 13, except that the resonances are more suppressed.

For each CO initial rotational quantum number j₁ = 0–5 in the v₁ = 1 state, Fig. 15 presents comparisons of the total cross sections of vibrational quenching of CO with para-H₂ and ortho-H₂ from $(1j10j2)→(v1′=0;v2′=0,j2′)$ and H₂ rotational transitions remaining elastic. As shown in Fig. 15, there are small differences between para-H₂ and ortho-H₂ for low collision energies, with the differences becoming smaller with increasing j₁. For collision energies above 10 cm⁻¹, the cross sections of the two colliders are very similar. In Fig. 15(a), the total cross sections of Flower³⁴ for the vibrational de-excitation of CO(v₁ = 1) by para-H₂ and by ortho-H₂ are also presented. The CC method was adopted in Flower’s calculations, but with a 4D PES. There is a two order of magnitude dispersion between the current results and Flower’s. As discussed in Ref. 6, this is due to the difference between 6D and 4D PESs and the fact that Flower likely used an insufficient CO basis in the scattering calculations.

In addition to rovibrational quenching calculations from CO v₁ = 1, we also carried out quenching calculations from the CO v₁ = 2 state, with CMS (2000). The total cross sections of CO for vibrational quenching $v1=2→v1′=1$ and $v1=2→v1′=0$ are presented in Fig. 16. For each $Δv1=v1′−v1$ quenching, H₂ rotational transitions are $j2→j2′=0$⁠, 2, and 4. As can be seen in Fig. 16, the total quenching cross sections for both Δv₁ = − 1 and −2 show similar behavior, display resonances at low collision energies, reach minima near 50 cm⁻¹, and then increase with increasing collision energy. The cross sections with H₂ rotationally elastic are the largest. It can also be seen from this figure that the cross section of Δv₁ = − 1 quenching is about two to three orders of magnitude larger than those of Δv₁ = − 2 quenching. This indicates that for a high initial vibrational state, the Δv₁ = − 1 transition dominates the vibrational quenching process as expected due to the highly harmonic character of the CO potential energy. In Fig. 17 the total cross sections of CO quenching from $v1=2→v1′=1$ are compared with the results of CO vibrational quenching $v1=1→v1′=0$⁠, the initial CMSs are (2000) and (1000), and H₂ final rotational states are $j2′=0$ and 2. As can be seen the cross sections for quenching $v1=1→v1′=0$ show significant resonance structure with some sharp features at certain collision energies. In contrast to the quenching of $v1=1→v1′=0$⁠, the cross sections for quenching $v1=2→v1′=1$ show different resonance structure with less sharp features. However, the magnitude of the cross sections is observed to be approximately three times larger than those of $v1=1→v1′=0$⁠.

Additionally, we computed the CO pure rotational quenching cross sections within the vibrational state v₁ = 0 and 1. In each vibrational state, rotational quenching was performed for initial CO rotational states j₁ = 1–5 with the H₂ rotational transition elastic (para-H₂,  j₂ = 0 and ortho-H₂,  j₂ = 1). As examples, Fig. 18 shows the pure rotational quenching in CO v₁ = 1. Cross sections are from j₁ = 1, 3, and 5 to all possible lower levels for CO in collisions with para-H₂ and ortho-H₂, respectively. In contrast to the vibrational quenching cross sections, the pure rotational cross sections are significantly larger. In the case of CO with para-H₂, the quenching cross sections of j₁ = 3 are dominated by the 3 → 2 transition, the smallest cross section is for j₁ = 3 → 0 transition. For initial j₁ = 5, the smallest cross section is still for final $j1′=0$⁠; however, the dominant transition varies with collision energy. When the collision energy is below 10 cm⁻¹, the cross section for the j₁ = 5 → 3 transition is the largest. Above 10 cm⁻¹, the transition j₁ = 5 → 4 dominates. In the case of pure rotational quenching of CO by ortho-H₂, the trends found for para-H₂ are also evident as displayed in Fig. 18. Given a CO rotational quenching $j1→j1′$⁠, the ratios of the cross sections of CO in collisions with para-H₂ to the cross sections of ortho-H₂ are presented in Fig. 19. The initial CO rotational state is j₁ = 1, 3, and 5. It can be seen from the figure that rotational state-to-state cross sections show differences at low collision energies, where the ratios show large fluctuations due to resonances. Starting from roughly 10 cm⁻¹, the cross sections of para-H₂ and ortho-H₂ are of similar magnitude with ratios typically ≲1.5.

For CO pure rotational quenching cross section from j₁ = 1 to $j1′=0$ in v₁ = 0 and 1, we consider here the transitions (0100) → (0000) and (1100) → (1000) for para-H₂ and (0101) → (0001) and (1101) → (1001) for ortho-H₂. In Fig. 20 the ratios of cross sections within v₁ = 1 to those within v₁ = 0 are displayed for para-H₂ and ortho-H₂, respectively. Despite some differences at low collision energy, where the ratios show large fluctuations in the resonance regions, the $j1=1→j1′=0$ cross sections in v₁ = 1 are almost the same as those for v₁ = 0. In other words, the CO rotational state-to-state cross section in excited vibrational levels can be approximately estimated from the results in v₁ = 0 for collision energies above 10 cm⁻¹, which is likely related to the harmonic nature of the CO potential.

Molecular hydrogen and carbon monoxide are the most abundant molecular species in the majority of interstellar environments. CO has a large dissociation energy and is stable at low temperatures; therefore, it is widespread and a good tracer of molecular gas.⁶² Most studies have focused on pure rotational transitions observed in the far infrared to the radio or electronic absorption in the near ultraviolet. Recently pure rotational transitions in the CO vibrational states v₁ = 1 and 2 were detected in the circumstellar shell of the star IRC +10 216.⁶³ In particular, CO vibrational transitions have been detected; for example, emission due to the fundamental vibrational band near 4.7 μm was observed in star-forming regions in Orion with the Infrared Space Observatory⁶⁴ and from protoplanetary disks (PPDs) of young stellar objects^65–67 with the Gemini Observatory and the Very Large Telescope (VLT). Our current full-dimensional scattering calculation is able to provide accurate rovibrational state-to-state CO–H₂ collisional data for future modeling of protostars, the infrared sources discussed above, and future FIR and submillimeter observations with Herschel and the VLT. Further, CO vibrational bands in the 1-5 μm region will be accessible by the James Webb Space Telescope to be launched in 2018. In PPDs, CO vibrational lines probe the inner warm regions which are exposed to the UV radiation from the protostar.

Full-dimensional dynamics computations for inelastic rovibrational quenching of CO due to H₂ impact have been carried out for initial combined molecular states (1j₁0j₂), j₁ = 1-5, and (2000) with j₂ = 0 and 1 for para-H₂ and ortho-H₂, respectively. The full-dimensional scattering calculations were performed on a 6D CO–H₂ interaction potential surface computed with high-level electronic structure theory and fitted with an invariant polynomial approach. See the supplementary material for a FORTRAN code to compute the V6D PES.⁶⁸ In addition, pure rotational excitation of CO (v₁ = 0) in collision with para-, ortho-, and normal-H₂ was computed. All scattering calculations were done within a full angular-momentum-coupling formulation and compared to available experimental data and prior calculations. In general, good agreement was found between full-dimensional calculations and available measurements, as highlighted in Ref. 6, but additional inelastic experiments which obtain absolute cross sections, particularly for cold collision energies, would be desirable. Both state-to-state and total quenching cross sections from CO vibrational states v₁ = 1 and v₁ = 2 show resonance structures at intermediate energies. The V12 PES appears to be more accurate for pure rotationally inelastic collisions at low energies, while the present V6D PES is appropriate for vibrationally inelastic collisions. To further improve V6D, it will be extended to long-range in the future, but in six-dimensions. The current calculations together with large scale coupled-states (CS) approximation⁸ results will be essential in the construction of a database of CO rovibrational quenching rate coefficients urgently needed for astrophysical modeling including that of the inner zone of protoplanetary disks.

The work at UGA and Emory was supported by NASA Grant No. NNX12AF42G from the Astronomy and Physics Research and Analysis Program, at UNLV by NSF Grant No. PHY-1505557, and at Penn State by NSF Grant No. PHY-1503615. This study was supported in part by resources and technical expertise from the UGA Georgia Advanced Computing Resource Center (GACRC), a partnership between the UGA Office of the Vice President for Research and Office of the Vice President for Information Technology. We thank Shan-Ho Tsai (GACRC), Jeff Deroshia (UGA Department of Physics and Astronomy), and Greg Derda (GACRC) for computational assistance. We thank A. Bergeat for providing us with their experimental data.