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Appendix A: Exchange reactions

We discuss below the reactions displayed in Table 1.

• ⇌

  ⇌

  These reactions have been studied experimentally by Adams & Smith (1981) at 80 K without differentiating between N¹⁵NH⁺ and ¹⁵NNH⁺. The total rate is 4.6 × 10^-10 cm³ s^-1. We thus take half this value for the forward reaction rate constant, considering the symmetry factor, and we assume no barrier for these rapid reactions. We also introduce the ⇌ reaction for completeness.

• ⇌

  This reaction involves adduct formation. The numerical constant is computed to reproduce the experimental value of Anicich et al. (1977) at room temperature.

• ⇌

  No information is available for this reaction.The bimolecular exit channels ¹²C⁺ + NCN and C₂⁺/N₂ are both endothermic by 170 and 76 kJ/mol, respectively. We performed DFT calculations (at the M06-2X/cc-pVTZ level) to explore the possibility of isotopic exchange. Direct N exchange is impossible because it would require simultaneous bond formation, rearrangement, and bond breaking. However, isotopic exchange could take place through adduct formation. No barrier was found in the entrance channel for NCNC⁺ formation (exothermic by 167 kJ/mol). The most favorable NCNC⁺ isomerization pathway is cyclic. However, the position of the c-NC(N)C⁺ transition state (TS) is highly uncertain and is found to be below the entrance channel at M06-2X/cc-pVQZ level (–14 kJ/mol) but slightly above the entrance level at the RCCSD(T)-F12/aug-cc-pVTZ level (+19 kJ/mol). In any case, the TS position is close to the entrance level so that the NCNC⁺ back dissociation is favored at room temperature. Isotopic exchange may, however, be enhanced at low temperature and may become the main exit channel. Considering a T^-1 temperature dependence of the adduct lifetime, we tentatively suggest k_f = 3.8 × 10^-12 × (T/ 300)^-1 cm³ s^-1.

• ⇌

  Terzieva & Herbst (2000) estimated the value of this rate coefficient from the difference between the Langevin rate and the other measured exothermic reactions reported below: k₁ = 5.0 × 10^-10 cm³ s^-1 and k₂ = 5.0 × 10^-11 cm³ s^-1. No data are available for the two other reactions. We find a barrier for NON⁺ adduct formation (261 kJ/mol at the M06-2X/cc-pVTZ level) and neglect this exchange reaction.

• ⇌

  No information is available for these reactions, which were reported as crucial by Rodgers & Charnley (2008b). We performed DFT calculations (at the M06-2X/cc-pVTZ level) which showed that isotopic exchange through the addition elimination mechanism is impossible as the NN(H)N⁺ and NNNH⁺ ions are metastable, respectively 73 and 332 kJ/mole above the ¹⁴N+ N₂H⁺energy.

  	Fig. A.1 Energy diagram of the NNNH+ system.
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  Figure A.1 displays the various possibilities. In addition, direct nitrogen exchange through NN(H)...N⁺→ N...N(H)N⁺ presents a barrier equal to +87 kJ/mol. We did not find any N₃H⁺ configuration with an energy lower than that of the reactants. We then neglect this reaction.

• ⇌

  Following a similar suggestion for explaining abundance anomalies of the ¹³C species of CCH (Sakai et al. 2010), we consider the above reaction, for which no information is available. We performed various DFT calculations at the M06-2X/cc-pVTZ level. The HNNH⁺ ion is 109 kJ/mole more stable than H + N₂H⁺, but the addition reaction shows a barrier of +12 kJ/mole. The intramolecular isomerization ¹⁵NNH⁺→ N¹⁵NH⁺ also displays a barrier of +172 kJ/mole. Then direct isomerization through tunneling is expected to be very slow, and we neglect that reaction².

• ⇌

  This reaction was also suggested by Terzieva & Herbst (2000) despite the lack of any relevant information. We performed DFT calculations to explore the possibility of isotopic exchange as shown in Fig. A.2.

  	Fig. A.2 Energy diagram of the HCNNH+ system.
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  Direct N exchange is found to be impossible because it requires simultaneous bond formation, rearrangement, and bond breaking so that isotopic exchange involves adduct formation. Attack by atomic nitrogen on either side of the HCNH⁺ molecular ion leads to a metastable system through high-energy transition states. We did not consider transition states leading to N addition on the C=N bond of HCNH⁺, nor did we examine N insertion into the N-H or C-H bonds, since all these pathways led to species located above the reagent energy level. We did not find any CN₂H₂⁺ species lying below the reagent energy level, and we consider that this reaction cannot take place.

• ⇌

  The N(⁴S^o) + CN () reaction leads to surfaces in C_∞V symmetry and ^3,5A′′ surfaces in Cs symmetry. The quintuplet surface is repulsive at the MRCI+Q/aug-ccpVTZ level, and the ⁵NCN intermediate is above the N + CN level. Considering only the triplet surface, the only barrierless reaction is attack on the carbon atom that leads to the ground state ³NCN intermediate (Daranlot et al. 2012; Ma et al. 2012). The main exit channel is C + N₂ after isomerization of the NCN intermediate through a tight TS, and then back dissociation may be important and isotope exchange possible. In the nominal model, we neglect this reaction, but some tests were performed to estimate its potential role. The upper limit of the isotope exchange rate constant is equal to the capture rate constant minus the N + CN → C + N₂ rate constant. That value is notably lower than the capture one at low temperature. We thus propose the upper limit value of 2.0 × 10^-10× (T/300)^1/6 cm³ s^-1 for the forward rate constant.

• ⇌

  	Fig. A.3 Energy diagram of the CCNN system.
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  The N(⁴S) + C₂N (²Π) reaction leads to ^3,5Π surfaces in C_∞V symmetry and ^3,5A′ + ^3,5A′′ surfaces in Cs symmetry. Figure A.3 displays the positions of the state energies linked to N + C₂N, which are calculated at the MRCI+Q/aug-cc-pVTZ level:10 e- in 10 OM with the geometry fully optimized at the CASCCF level or 14 e- in 14 OM for the CASCCF and 14e- in 12 OM for the MRCI calculation with non relaxed geometry. The triplet surface pathway leads to a NCCN adduct in a triplet state, corresponding to an excited state of the very stable NCCN linear molecule, located –393 kJ/mol below the N(⁴S) + C₂N (²Π) level. No barrier is present in the entrance valley so that this NCCN adduct very likely leads to CN (in a doublet state) + CN (in a doublet state) products. The occurrence of a small exit barrier cannot be excluded, but its energy should be much lower than that of the ¹⁴N + C₂¹⁵N entrance channel. Triplet surfaces thus cannot lead to isotopic exchanges. However, the quintuplet surface deserves specific attention because the NCCN() adduct is found at an energy of –183 kJ/mol below that of the N(⁴S) + C₂N (²Π) level (at the MRCI+Q, RCCSD(T) and DFT level of calculations). However, no exothermic bimolecular exit channel is available on this quintuplet surface. We thus conclude that the quintuplet surface could lead to isotopic exchange and finally consider the two following possibilities for this reaction involving the very reactive C₂N radical (Wang et al. 2005, 2006), i.e. the triplet channel, and the quintuplet channel,

• ⇌

  There is a barrier for NON adduct formation. The isotope exchange rate is calculated to be very low (Gamallo et al. 2010) since the main exit channel is N₂ + O. Moreover, the quintuplet surfaces are repulsive. We neglect this reaction.

• ⇌

  This reaction was first mentioned by Watson et al. (1976) and has been experimentally studied in detail by Smith & Adams (1980) in the 80–500 K temperature range. Their data can be fitted (Liszt 2007). We introduce a new formula allowing us to describe the full temperature range with a single formula.

• ⇌

  This reaction has also been studied experimentally by Smith & Adams (1980). The exothermicity of the reaction has been reported as 9 K by Langer et al. (1984) from theoretical studies, whereas Smith & Adams (1980) proposed a value of 12 ± 5 K. However, a later study by Lohr (1998) leads to a value of 17.4 K. Mladenović & Roueff (2014) have reconsidered this reaction and confirm the value of Lohr (1998). We include this exothermicity value in the present work. Experimental points are fitted through a power law as given in Table 1.

• ⇌

  There are no bimolecular exit channels for this reaction and two different pathways lead to isotopic exchange. In a direct reaction, ¹³C⁺+ NC →¹³CNC⁺→¹³CN + ¹²C⁺, whereas the indirect pathway involves ¹³C⁺+ CN →¹³CCN⁺→¹³CN+ ¹²C⁺. There is no barrier in the entrance valley for both cases at the M06-2X/cc-pVTZ level. Moreover, the cyclic transition state from CCN⁺ and CNC⁺ is located at –436 kJ/mol below the reactant energy, leading to fast isomerization. The corresponding capture rate constant is 3.82 × 10^-9 × (T/ 300)^-0.40.

• ⇌

  In a similar way to the ionic case, no exothermic bimolecular exit channels are available for this reaction and two reaction pathways occur for possible isotopic exchange. ¹³C+ NC →¹³CNC →¹³CN+ ¹²C holds for the direct process, and ¹³C+ CN →¹³CCN → c-¹³CNC →¹³CN + ¹²C describe the indirect exchange mechanism. No barrier is found in the entrance channel, and the transition state from CCN to c-CCN is low (–298 kJ/mol below the reactant energy), in good agreement with Mebel & Kaiser (2002) so that isomerization is expected to be fast. The capture rate is computed as 3.0 × 10^-10 cm³ s^-1.

• ⇌

  There are no exothermic bimolecular exit channels for this reaction and no barrier for HCNC formation, but isotopic exchange requires isomerization through a TS located close to the reactant level, involving a TS located at –16 kJ/mol at the M06-2X/cc-pVQZ level but +34 kJ/mol at the RCCSD(T)-F12-aug-cc-pVQZ level. Calculation of the rate constant for exchange is complex and similar to the ¹⁵N + CNC⁺ case. We neglect this reaction in the nominal model.

• ⇌

  There are no exothermic product channels for this reaction, and there is very likely to be no barrier. The capture rate constant is equal to 3.0 × 10^-10 cm³ s^-1.

• ⇌

  This reaction could be an additional possibility for enhancing ¹³CO because no barrier has been found in the entrance valley (Le Picard & Canosa 1998). The high-pressure CH + CO → HCCO association reaction rate constant is equal to 3 × 10^-11 × (T/ 300)^-0.9 between 53 and 294 K. The exchange rate has been measured to be ~ × 10^-12 (Taatjes 1997) at room temperature. This value represents 1% of the association reaction rate constant at high pressure, which is explained by a transition state localized at 6 kJ/mole above the reactants energy as computed at the M06-2X/cc-pVTZ level, in good agreement with Sattelmeyer (2004). We do not include this reaction in our models given the high TS which should make this exchange process negligible at low temperature.

Appendix B: ZPE values

We revisit the ZPE values in light of several recent studies. We recall in Table B.1 the atomic masses of various isotopes that may be used to derive spectroscopic constants of isotopic molecules, as found in basic molecular spectroscopy textbooks (Herzberg 1945, 1989). We only refer to the first-order expansion terms for the purpose of computing ZPEs.

Appendix B.1: Diatomic molecules

For diatomic molecules, energy levels are expressed as a Dunham expansion or equivalently as a sum of harmonic + anharmonicity correction factors. The following expression is obtained for the ZPE, corresponding to v = 0: (B.1)where ω₀ is the harmonic contribution of the vibrational frequency and ω_ex_e represents the anharmonic contribution. The reduced mass μ dependence of ω₀ and ω_ex_e are 1/ and 1/μ, respectively. The label computed in Table B.2 indicates the use

of this property to compute the spectroscopic constants, in the absence of other information.

Appendix B.2: Polyatomic molecules

For polyatomic molecules, the following expression extends the diatomic formulae where the different vibrational degrees of freedom are included: (B.2)where ω_i refers to the harmonic frequencies, d_i is the corresponding degeneracy, and x_ij stands for the anharmonic terms. The sum is performed over the number of vibrational modes. In the case of polyatomic linear molecules, the number of vibrational modes is 3N-5, whereas it is 3N-6 in the general case where N is the number of nuclei in the molecule.

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