Isotope effect in the formation of carbon monoxide by radiative association

MNRAS 430, 946 950 (2013) doi:10.1093/mnras/sts615 Isotope effect in the formation of carbon monoxide by radiative association Sergey V. Antipov, Magnus Gustafsson and Gunnar Nyman Department of Chemistry and Molecular Biology, University of Gothenburg, SE-41296 Gothenburg, Sweden Accepted 2012 December 12. Received 2012 December 10; in original form 2012 October 1 1 INTRODUCTION Radiative association of carbon and oxygen atoms in their ground electronic states has been discussed as a possible source of carbon monoxide in the metal-rich ejecta of supernova 1987A (Petuchowski et al. 1989; Gearhart, Wheeler & Swartz 1999). The rate coefficients for that process are required at a variety of temperatures to model the formation of CO in different astrophysical environments (see e.g. Glover et al. 2010). Collisions of C( 3 P)andO( 3 P) atoms can lead to the formation of CO in 81 different molecular electronic states. Dalgarno, Du & You (1990) estimated the role of those states in radiative association and concluded that the main contribution comes from the two lowest-lying electronic states X 1 + and A 1. Radiative association involving those two states can occur through four different transitions: A 1 X 1 +, A 1 A 1, X 1 + X 1 +, X 1 + A 1. Our previous study on the subject (Franz, Gustafsson & Nyman 2011) shows that the rate coefficients for forming CO in the first excited A 1 state are several orders of magnitude smaller than for forming CO in the ground X 1 + state. Therefore, the two most important formation channels are C( 3 P ) + O( 3 P ) CO(A 1 ) CO(X 1 + ) + hν (1) and C( 3 P ) + O( 3 P ) CO(X 1 + ) CO(X 1 + ) + hν. (2) The potential energy curve of the A 1 electronic state has a barrier of 636 cm 1 79 mev (Franz et al. 2011). It was shown that due to the barrier the rate coefficient for reaction (1) at temperatures lower than 600 K is dominated by the contribution from resonances. The resonance contribution arises from the quantum mechanical E-mail: sergey.antipov@chem.gu.se Researcher ID: A-3907-2012. ABSTRACT Rate coefficients for the formation of 12 CO and 13 CO isotopologues of carbon monoxide by radiative association for T = 10 20 000 K are calculated using a quantum mechanical approach. It is shown that the presence of the potential barrier on the A 1 electronic state of CO leads to different formation channels for the isotopologues at low temperatures. The corresponding rate coefficients are fitted to an analytic formula. Key words: astrochemistry molecular data molecular processes methods: numerical ISM: molecules ISM: supernova remnants. tunnelling through the barrier, in which case the colliding atoms form a quasi-bound state. The latter can either tunnel back out through the barrier to reform reactants or undergo a radiative decay to a bound state leading to molecule formation. The tunnelling properties of the system depend strongly on the reduced mass. Since the tunnelling plays an important role in the formation of CO according to reaction (1), a pronounced isotope effect could be expected in the corresponding rate coefficients. In this study, we investigate the formation of 12 CO and 13 CO through the radiative association of carbon and oxygen atoms. The quantum mechanical cross-sections and the rate coefficients for reactions (1) and (2) are calculated. The focus is on the relative importance of those reactions in the formation of CO at low temperatures. 2 THEORY The thermal rate coefficient for a collisional process, such as radiative association, can be written as ( ) 8 1/2 ( ) 1 3/2 k (T ) = πμ k B T 0 Eσ (E)e E/k BT de, (3) where E is the kinetic energy of the colliding particles, σ (E) is the cross-section for the process, k B is the Boltzmann constant, μ is the reduced mass and / are the projections of the electronic orbital angular momentum of the molecule in the initial and final states on the internuclear axis. Since only singlet electronic states of CO are considered we drop the spin multiplicity index to simplify the notation. The total cross-section σ (E) for radiative association can be calculated in different ways. Here we briefly describe two of them, which are relevant for this study: the perturbation theory (PT) and C 2013 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society

the semi-classical (SC) approaches. For more detailed overviews of different methods see Babb & Kirby (1998) and Gustafsson et al. (2012). 2.1 SC approach In SC theory, the cross-section is given by (Bates 1951; Zygelman & Dalgarno 1988) ( μ ) 1/2 σ (E) = 4π P 2E A (R)dRdb b, (4) 0 R c (1 V (R)/E b 2 /R 2 ) 1/2 where A (R) is the transition rate, R is the internuclear distance, R c is the classical turning point (distance of closest approach) for the corresponding value of the impact parameter (b) andp is the statistical weight factor for approaching along a particular molecular potential curve V (R). For the A 1 X 1 + transition P = 2/81 and for the X 1 + X 1 + transition P = 1/81 (Dalgarno et al. 1990; Franz et al. 2011). The transition rate is conventionally defined as A (R) = 64 π 4 ( ) 2 δ0, + D(R) 2 3 (4πɛ 0 )h 2 δ 0, λ 3, (5) (R) where D(R) is the transition dipole moment, λ is the optimal transition wavelength: 1 λ (R) = max ( 0, V (R) V (R) hc ). However, in order to ensure that equation (4) describes the formation of a molecule and not a radiative quenching to the continuum of the final electronic state, the restricted transition probability should be used (Gustafsson et al. 2012): A (R) V Eb2 (R) + < 0 R 2 A Eb (R) = E<V (R) V (R) (6) 0 otherwise, where Eb2 is the classical rotational energy of the molecule. R 2 The presented SC implementation has a major limitation: it is applicable only to the radiative association processes that involve transition between different electronic states of the molecule. Thus, it can be applied to study the formation of CO through reaction (1), but not through reaction (2). 2.2 PT approach The quantum mechanical PT treatment of the radiative association process leads to the Fermi Golden rule-like formula for the crosssection (Babb & Kirby 1998): σ (E) = Jv J 64 3 π 5 4πɛ 0 P k 2 S J J λ 3 E v J M EJ v J 2, (7) where the sum runs over all initial rotational (J) and final vibrational (v ) and rotational (J ) quantum numbers. In equation (7) λ E v J is the wavelength of the emitted photon, k = 2μE is the wavenumber and the Hönl London factors (oscillator strength) S J J for the corresponding transitions are given in Table 1. The summation over parity is not done explicitly in equation (7), but it is instead taken Isotope effect in CO radiative association 947 Table 1. Hönl London factors, S J J. (Hansson & Watson 2005; Watson 2008). For the 1 1 + transition Hönl London factors averaged over parity components of the 1 state are given. J 1 1 + 1 + 1 + J 1 (J + 1)/2 J + 1 J (2J + 1)/2 0 J + 1 J/2 J care of by the factor P. Thus, for the 1 1 + transition the parity averaged Hönl London factors are used. The transition dipole matrix element is defined as M EJ v J = ψ EJ (R) D(R) ψ v J (R), (8) where D(R) is the matrix element of the dipole moment operator between the corresponding molecular electronic wavefunctions. Here ψ EJ (R) is the radial part of the continuum energy-normalized wavefunction of the initial state and ψ v J (R) is the radial part of the final rovibrational wavefunction, normalized to unity. 2.3 Resonance contribution to the rate coefficient A typical radiative association cross-section shows the presence of many exceedingly narrow resonances, see e.g. Bennett et al. (2003) and Franz et al. (2011). Such complicated resonance structures make it practically impossible to calculate the rate coefficient by a numerical integration of the cross-section according to equation (3). Moreover, it is known that the widely used PT approach overestimates the heights of the resonances, for which the tunnelling width is smaller than or comparable to the radiative width (Bennett et al. 2003). Thus, a special treatment of the resonances is required. The established approach to the problem is to use the Breit Wigner theory (Breit & Wigner 1936; Bain & Bardsley 1972). The total rate coefficient is divided into a sum of two terms: k (T ) = k dir (T ) + kres (T ), where k res (T ) is the resonance contribution and kdir (T )isthe direct contribution, which comes from the baseline of the crosssection. According to the Breit Wigner theory the resonance contribution to the rate coefficient can be calculated as ( ) 2π 3/2 k res (T ) = 2 P μk B T 2J + 1 1/Ɣ tun vj vj + e E vj /k BT, (9) 1/Ɣrad vj where (v, J) are vibrational and rotational quantum numbers assigned to quasi-bound states associated with resonances, E vj is the energy of the quasi-bound state (corresponds to the energy at the peak of the resonance), Ɣ vj tun is the width associated with tunnelling and Ɣ vj rad is the (cumulative) width due to the radiative decay to all bound levels of the final electronic state. 3 COMPUTATIONAL DETAILS The potential energy curves and dipole moment matrix elements have been taken from our previous study (Franz et al. 2011) and are presented in Fig. 1. Molecular data have been calculated with the

948 S. V. Antipov, M. Gustafsson and G. Nyman Figure 1. Potential energy curves of the X 1 + and A 1 states of CO. The radial dependence of the dipole matrix elements D(R) is shown in the inset. multireference configuration interaction method using the aug-ccpv6z basis set. The mass-dependent diagonal Born Oppenheimer correction has been neglected, thus, the same potential energy curves are used for 12 CO and 13 CO. The bound-state wavefunctions (ψ v J ) are obtained by solving the Schrödinger equation with the discrete variable representation method of Colbert & Miller (1992). The continuum wavefunctions (ψ EJ ) are calculated using the Numerov method (Korn & Korn 1968). The resonance parameters needed for calculation of k res (T ) according to equation (9) have been obtained using the LEVEL program of Le Roy (2007). The peak energies (E vj ) and the tunnelling ) are provided explicitly in the output and the radiative widths (Ɣ vj tun widths have been calculated as Ɣ vj rad = A vjv J, (10) v J where A vjv J is the Einstein A-coefficients for spontaneous transition from the initial quasi-bound state (v, J) to the final, lower-lying, bound state (v, J ). The A-coefficients are given by LEVEL directly. 4 RESULTS AND DISCUSSION Fig. 2 shows the radiative association cross-sections for the formation of 12 CO through reactions (1) and (2). The low-energy behaviour of the cross-sections for the A 1 X 1 + transition is determined by the presence of the barrier on the A 1 potential curve. For energies lower than the barrier height (636 cm 1 79 mev) the SC cross-section is zero, while the quantum (PT) cross-section has a finite magnitude, which comes completely from tunnelling through the barrier. At high collision energies (>10 ev) the discrepancy between the SC and PT cross-sections is due to contribution from non-frank Condon transitions, which are not accounted for by the SC theory (Julienne 1978). For the intermediate energies, the SC cross-section and the baseline of the PT cross-section are in good agreement. The baseline of the X 1 + X 1 + cross-section decreases slowly for collision energies up to 0.1 ev, above which it shows fast decay. The radiative association cross-sections for the formation of 13 CO show the same qualitative features as for 12 CO (Fig. 3). The main difference arises in the resonance structure of the quantum crosssection for the A 1 X 1 + transition at collision energies smaller than the barrier height. Since the reduced masses of the isotopo- Figure 2. Radiative association cross-sections for the formation of 12 CO calculated using perturbation theory (PT). For reaction (1) results of the semi-classical approach (SC) are also shown. Figure 3. The same as in Fig. 2 but for the 13 CO isotopologue. logues are different, it affects the vibrational spacing, which determines the number of supported quasi-bound states and, ultimately, the resonance structure of the cross-section. The observed difference in the cross-sections translates to the rate constants. Fig. 4 shows the calculated rate coefficients for the formation of carbon monoxide through the radiative association of C( 3 P)andO( 3 P) atoms. In all cases the resonance contribution is calculated using Breit Wigner theory (equation 9). The direct component of the cross-sections for the A 1 X 1 + transition is constructed in the following way: the SC cross-section is taken up to 4.45 ev and that of PT otherwise. The baselines of the X 1 + X 1 + cross-sections are approximated by a power law (linear in a log log plot) up to 0.1 ev, which provides the converged direct contributions to the rate coefficients up to 120 K. The rate coefficients for the formation of the considered isotopologues of CO through reaction (1) increase steadily with temperature all the way up to 20 000 K. In the case of 12 CO the total rate coefficient for reaction (1) is higher than that for reaction (2) even at temperatures as low as 10 K, at which they differ roughly by a factor of 2. In contrast, for 13 CO at T = 10 K the rate constant for approaching on the A 1 potential is four orders of magnitude lower than that for approaching on the X 1 + potential and exceeds it for T > 36 K (Fig. 4). The difference is caused by the resonance

Isotope effect in CO radiative association 949 Figure 4. Radiative association rate coefficients for the formation of (a) 12 CO and (b) 13 CO. contribution to the rate constant at low temperatures being substantially smaller in the case of 13 CO compared to 12 CO.Atsuchtemperatures the biggest contribution to the rate coefficient for reaction (1) comes from resonances, which correspond to the quasi-bound states with energies below the barrier height on the A 1 potential. Comparison of the corresponding resonance parameters for both Table 2. Fit parameters for the total rate coefficient curves shown in Fig. 4. isotopologues reveals that, while the radiative widths are comparable in magnitude, the tunnelling widths are generally one two orders of magnitude smaller for 13 CO,leadingtothelowerrate coefficient. As can be seen from Fig. 4, the rate constant for the formation of 12 CO through reaction (2) does not vary much at low temperatures and stays around 1 10 22 cm 3 s 1, which already at T = 100 K is three orders of magnitude smaller than the rate constant for the A 1 X 1 + transition. The same holds true for the 13 CO isotope. Our previous study (Franz et al. 2011) shows that at higher temperatures the rate coefficient for the X 1 + X 1 + transition does not go above 2 10 21 cm 3 s 1 and, thus, gives a negligible contribution to the total rate of radiative association of CO. Each of the total rate coefficient curves shown in Fig. 4 can be approximated using a three-parameter formula (Kooij 1873): ( ) T α k(t ) = A e β/t (11) 300 with the fitting parameters given in Table 2. The rate coefficients are of the interest for modelling of chemistry in the interstellar medium and results will be made available in the data base KIDA (Wakelam et al. 2012). 5 SUMMARY In this study, we have calculated the rate coefficients for the formation of 12 CO and 13 CO through the radiative association of C( 3 P) and O( 3 P) atoms. A quantum mechanical approach to the process has been employed and the Breit Wigner theory has been used to properly account for the resonance contribution to the rate coefficients. The presence of a barrier on the potential energy curve of the A 1 electronic state leads to the different formation channels for the isotopologues of CO at low temperatures. Thus, for all considered temperatures the formation of 12 CO through radiative association is dominated by carbon and oxygen approaching on the A 1 state; contribution from the X 1 + X 1 + transition varies from small to negligible (T > 20 K). In the case of 13 CO for T < 36 K the X 1 + X 1 + transition is the main channel of formation, while the rate coefficient for the A 1 X 1 + transition is substantially smaller due to its smaller resonance contribution. At higher T, the A 1 X 1 + transition dominates the formation of the 13 CO isotopologue also. It should be mentioned that authors are not aware of any experimental observation of the isotope effect in radiative association nor the observed depletion of 13 C in CO formed in supernovae and additional work in that area is desired. Isotopologue Transition T (K) α β (K) A (cm 3 s 1 ) 12 CO A 1 X 1 + 10 100 1.833 42.62 5.152 10 18 A 1 X 1 + 100 1000 0.901 153.0 4.575 10 18 A 1 X 1 + 1000 20000 0.159 1156 2.721 10 17 X 1 + X 1 + 10 120 0.155 1.064 1.848 10 22 A 1 X 1 + 10 27.5 7.320 16.81 1.556 10 16 A 1 X 1 + 27.5 200 0.356 379.0 8.000 10 18 13 CO A 1 X 1 + 200 2000 0.512 435.1 9.647 10 18 A 1 X 1 + 2000 20000 0.103 1499 3.458 10 17 X 1 + X 1 + 10 120 0.167 1.136 1.766 10 22

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