Astron. Astrophys. 333, 1007-1015 (1998)

4. Isomerization of H₂ NC (¹ A₁) into HCNH ()

Given the barrier calculated for formation of H₂ NC in its excited ³ B₂ state from C + NH₃ reactants, the only possibility for production of this metastable ion via reaction (1) would appear to lie in the stabilization of ground state H₂ NC , which is formed as a primary product by the reaction occurring on the ground potential surface (Figs. 1 and 4). Although there is formally enough total energy for isomerization of H₂ NC into HCNH (Fig. 4) over the H₂ NC transition state, there are several reasons that isomerization need not be rapid or even occur at all. First, the exothermicity of reaction may go sufficiently into relative translation between H₂ NC and H that there is not enough internal energy left in H₂ NC for it to overcome the transition state barrier. Secondly, there is a large amount of angular momentum in ion-polar molecule collisions and this angular momentum, if converted into rotational angular momentum of the H₂ NC product, rather than into relative angular momentum of the two products, effectively slows the rate of isomerization since only vibrational energy is useful in this process (Herbst 1996). Thirdly, relaxation of the ground state H₂ NC ion product via emission of infrared radiation (and, in the laboratory, by collision) to vibrational-rotational states below the barrier to isomerization may be more rapid than isomerization. For these three reasons, it is necessary to estimate the rate of isomerization and then determine the fraction of product that can remain as H₂ NC .

Our first consideration is to estimate how much energy goes into relative translation between the products. Although both statistical theories and assorted experiments (Illies et al. 1983) show that the energy of exothermic reactions tends to wind up as vibrational energy in polyatomic products, experiments also show that in the case of potential surfaces with barriers, a significant amount of translational energy, with a distribution centered on the potential energy difference between transition state and products, can be released (Jarrold et al. 1986). Reference to Fig. 4 shows that the transition state between the H₃ NC energy minimum and the H₂ NC (¹ A₁) + H products lies at an energy of 21.4 kcal/mol above the reactants. It is most likely that the separating products carry away this much translational energy, leaving the remainder of the reaction exothermicity of H₂ NC (¹ A₁) + H products ( kcal/mol) as vibrational-rotational energy of H₂ NC (¹ A₁). Since the transition state energy for isomerization of this ion into the lower energy HCNH ion is only 17.9 kcal/mol (see Fig. 4), the H₂ NC ion possesses sufficient internal energy to isomerize as long as its vibrational energy exceeds 17.9 kcal/mol. To have less vibrational energy than this, the ion must relax rapidly or have large amounts of angular momentum. Both possibilities are addressed below.

The rate of isomerization over a barrier can be treated by the RRKM statistical method (Herbst 1996). In this formulation, the rate k(s^-1) as a function of vibrational-rotational energy E and angular momentum quantum number J is given by the expression

where h is Planck's constant, N refers to the total number of vibrational states of the transition state from its minimum allowable energy through , and refers to the density of vibrational states of the H₂ NC isomer at energy . Note that kcal/mol is the energy difference between H₂ NC and the transition state. The rotational energy for either the ion or transition state can be approximated as that of a spherical top with one rotational constant B equal to the geometric mean (cube root) of the product of three rotational constants. For both the number of states of the transition state and the density of states of the complex, the semi-classical expression of Whitten and Rabinovitch (1964) can be used.

With our calculated energies as well as the vibrational frequencies (Table 3) and rotational constants of the H₂ NC ion and transition state (see Fig. 3), we can determine k(E,J). Using the assumption that 21.4 kcal/mol of reaction energy goes into relative translational energy and is thus not available for isomerization of H₂ NC , we still determine that isomerization is very rapid for a wide range of angular momenta and vibrational energies. Let us consider two extreme cases: when the ion possesses no angular momentum (J = 0) and when it possesses a large angular momentum, quantum number J = 78, which is the maximum quantum number for which there is sufficient vibrational energy for isomerization to occur. For the former case, we calculate that the isomerization rate for H₂ NC into HCNH before any vibrational relaxation can occur is while the equivalent rate after sufficient vibrational relaxation such that H₂ NC can just overcome the transition state barrier is . For the latter case, we calculate the isomerization rate to be (vibrational relaxation in this case will prevent isomerization). Consideration of tunneling under the transition state barrier (Herbst 1996) will enlarge these already large values. We conclude that isomerization is a rapid process for all allowable J; in particular, it is much more rapid than radiative relaxation, which is known to occur at a rate of only 10-10³ s^-1 for vibrational states (Herbst 1982; Herbst & Dunbar 1991). Thus we are led to the following picture: after the H₂ NC ion is formed in its ground electronic state, it will rapidly isomerize to the HCNH ion for angular momenta from J = 0 to a maximum value, and, as can be shown by calculations similar to the above, this ion will rapidly isomerize in the backwards direction so that an equilibrium will quickly be achieved between the two ions. As the ions slowly relax vibrationally, the equilibrium will reflect the change in energy. The relevant equilibrium for our purposes will be that corresponding to a vibrational energy just equal to the transition state barrier. A similar analysis was performed previously by De Frees et al. (1985) for the CH₃ CNH - CH₃ NCH isomerization.

To determine the equilibrium populations of ground state H₂ NC and HCNH , it is necessary to calculate the so-called equilibrium coefficient K, which is related to the ratio of the forward and backward rates of isomerization. The equilibrium coefficient in the microcanonical formulation is given by the equation

where once again refers to the vibrational density of states, the subscripts L and NL refer to the linear and non-linear isomers, respectively, and refers to the (positive) difference in energy between the two isomers. Part of this expression can be obtained by dividing the RRKM rate equations for the forward process by that for the backward process since the two processes possess the same transition state. But, one must also consider the rotational degeneracies of the two isomers, which are 2J + 1 for the linear ion and (2J + 1)² for the non-linear ion (assumed to be a spherical top). In addition, there is a symmetry factor for the two-fold symmetry of the non-linear ion. Expression (II) should be evaluated for vibrational energies corresponding to the energy of the transition state. Restricting attention to the energy at the barrier height, and using the Whitten-Rabinovitch (1964) density-of-states formulation, we obtain that K(J) = 3168/(2J + 1) for all values of J for which a sufficient amount of vibrational energy can be obtained. For values of , where is determined by the amount of relative translational energy, a sufficient amount of vibrational energy cannot be obtained, and K(J) = 0. For our standard choice of translational energy, .

The rotationally-averaged equilibrium coefficient K is obtained from the formula

where p(J) is the probability that the reaction produces H₂ NC in a specific rotational level of angular momentum quantum number J. This probability can be estimated by the statistical relation (Herbst 1996)

where N is a normalization coefficient. Eq. (IV) holds for values of J from 0 to a maximum defined by the stronger of several constraints posed by the details of the reaction and by conservation of energy. If is sufficiently large, then the averaged equilibrium constant can be small, so that isomerization need not destroy a large amount of H₂ NC . After doing the rotational averaging, however, we find that K = 44 if the relative product translational energy is set at 21.4 kcal/mol (our preferrred value), and K = 39 if the relative translational energy is set at 0. The weak dependence on translational energy means that this parameter is not of the utmost importance. Since the equilibrium coefficient tells us the ratio of the population of HCNH product to H₂ NC product, we conclude that the amount of the latter is only about 2-3% of the total and that HCNH is the overwhelmingly dominant product for reaction (1) when it occurs on the lowest energy potential surface.

© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998

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