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Appendix A: Calculation of nuclear spin branching ratios using symmetry rules

The method used in deriving the nuclear spin branching ratios is demonstrated by three examples, all dealing with varieties of the reaction . The method is described in Quack (1977) and in Park & Light (2007b). The collision partners are classified, or labeled, according to the irreducible representations of an appropriate permutation group. These labels tell how the symmetrized nuclear spin functions belonging to this species transform under operations of the group, such as the transposition of two identical nuclei and the permutation of n identical nuclei (e.g., Bunker & Jensen 2006). The branching ratios are obtained from correlations between the group of the intermediate reaction complex and the direct product groups representing the reactants and products (Quack 1977). The statistical weight of each channel corresponds to the number of nuclear spin functions belonging to that symmetry species.

Appendix A.1: Three identical nuclei

As the first example we consider the reaction D₂H⁺ + NH₂D →. The H and D symmetries are treated separately. The same result is obtained by considering the mixed system in appropriate direct product groups. From the hydrogen point of view, H₂ and H form H₃ which dissociates back to H₂ + H. On the deuterium side, the reaction forms D₃ from D₂ + D. In the present model we assume that the branching ratios are equal to the pure nuclear spin symmetry induction and subduction statistical weights, which for these reactions can be read from Tables 3 and 4 of Hugo et al. (2009). We reproduce the appropriate tables in Table A.1.

The numbers in the leftmost columns of Table A.1 are the frequencies of the irreducible representations A₁, A₂, and E, in the representations of the permutation group S₃ generated by the nuclear spin functions of and . This means that the symmetry representations of H₃ and D₃ in S₃ can be reduced into the following direct sums (denoted by ⊕): (A.1)(e.g., Bunker & Jensen 2006; Hugo et al. 2009). According to Mulliken’s notation used here, one-dimensional representations are labeled as either A or B, two-dimensional ones are labeled E, three-dimensional F, four-dimensional G, etc. The total nuclear spin statistical weight of each symmetry species can be read from these decompositions by multiplying the frequency by the dimension of the representation, f(Γ_i) × dim(Γ_i). The Greek appellations, para, meta, and ortho are used to indicate these statistical weights for the simplest molecules, so that “ortho” has the highest statistical weight. The “para” (E, I = 1/2) and “ortho” (A₁, I = 3/2) species of have, however, equal statistical weights. has no nuclear spin functions of species A₂, which is antisymmetric with respect to interchange of two nuclei⁴. H₂ and D₂ can have the symmetries A or B, and the single nuclei H and D have the A symmetry. The number in the bottom righthand corner of each table gives the total number of linearly independent nuclear spin functions of the molecule, (2I + 1)³, where I = 1/2 for H₃ and I = 1 for D₃. The branching ratios in Table A.1 are obtained by multiplying the correlation table between S₃ and the product group S₂ ⊗ S₁ by appropriate frequencies. This correlation table is given by

S 3	S2 ⊗ S1
	A ⊗ A	B ⊗ A
A 1	1	0
A 2	0	1

The correlation table S₃ ↔ S₂ ⊗ S₁ presented above is derived using the standard methods of group theory, i.e., by forming the character table of direct product group S₂ ⊗ S₁, and the character table for the subduced representations of S₃ on S₂ ⊗ S₁, S₃ ↓ S₂ ⊗ S₁. The latter is obtained from that of S₃ by stripping it from operations that are unfeasible in the direct product group S₂ ⊗ S₁, i.e., the class of cyclic permutations of three nuclei, {(123),(132)}. Finally, the orthogonality relation between the characters of irreducible representations is used.

Tables A.1 can be read either from left to right (“subduction”) or top down (“induction”). For example, one can see that the A₁ species of H₃ and D₃ dissociate exclusively to the A symmetry of H₂ or D₂, whereas the E species dissociate to both A and B symmetries (“ortho” and “para”, respectively) of H₂ or D₂ with equal probability. In the reaction oD₂ + D (A ⊗ A), the A₁ (“meta”) and E (“ortho”) species of D₃ are formed with the probabilities 10/18 and 8/18, respectively, whereas in the reaction pD₂ + D (B ⊗ A), the A₂ (“para”) and E (“ortho”) species of D₃ are formed with the probabilities 1/9 and 8/9. These branching ratios determine relative abundances of , , and resulting from the reaction D₂H⁺ + NH₂D.

Appendix A.2: Six identical nuclei

To determine the nuclear spin branching ratios of the reactions and , one needs to determine the nuclear spin symmetry species of H₆ and D₆ in the permutation group S₆. We first calculate the characters, χ(R), of the representations generated by the nuclear spin functions of H₆ and D₆ under the operations R of S₆. Crabtree & McCall (2013) give a convenient formula for this: (A.2)where I is the nuclear spin (1/2 for H and 1 for D), n the number of identical nuclei (here 6), and x the number of transpositions or pairwise permutations, which correspond to the operation R. The number of interchanges corresponding to a cyclic permutation of k elements is k − 1, and so x = k − 1. For example, x = 2 for the ternary permutation (123). The character table of S₆ is easily obtainable using GAP (2015). By applying the orthogonality relation of the characters of irreducible representations, we derive the following decompositions for the symmetry representations of H₆ and D₆: The correlation tables S₆ ↔ S₃ ⊗ S₃ and S₆ ↔ S₄ ⊗ S₂ needed for the induction and subduction statistics are derived as outlined in the previous example. Finally, the branching ratio tables listed in Tables A.2−A.5 are obtained by multiplying the correlation tables by the frequencies of irreducible representations appearing in the decompositions given above. In addition, the columns of the correlation table are multiplied by the dimensions of the subduced representations in order to make the weights correspond to the total number of nuclear spin functions belonging to each representation. In the previous example, all products had one-dimensional representations. The correct normalization of the weights can be checked by inspecting the last rows and columns of the tables. In these tables, species with zero frequencies are not listed.

The use of the branching ratio tables is illustrated in Figs. A.1 and A.2. Figure A.1 gives the outcome of the reaction assuming that one of the molecules is of the A₁ (“ortho”) symmetry and the other one is E (“para”). The branching probabilities indicated next to the arrows are obtained from Tables A.2 (reading it top down) and from Table A.3 (reading it from left to right). After combining the probabilities of pathways leading to different symmetry species of , one finds that the F₁, A₁, and E species are formed in ratios 15:7:2 in this reaction. Figure A.2 describes the branching of the reaction assuming that both species are A₁ (“meta”). The branching ratios are obtained from Tables A.4 and A.5. This reaction produces the F₁, A₁, and E species of in ratios 9:9:2.

© ESO, 2015