Nuclear spin selection rules in chemical reactions by angular momentum algebra

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1 Journal of Molecular Spectroscopy 228 (2004) Nuclear spin selection rules in chemical reactions by angular momentum algebra Takeshi Oka * Department of hemistry and Department of Astronomy and Astrophysics, The Enrico Fermi Institute, The University of hicago, hicago, IL 60637, USA Received 24 January 2004; in revised form 0 August 2004 Available online 27 September 2004 Abstract The detailed selection rules for reactive collisions reported by Quack using molecular symmetry group are derived by using angular momentum algebra. Instead of the representations of the permutation inversion group for both nuclear spin and rovibronic coordinate wavefunctions, those of the rotation group for nuclear spin wavefunction only are used. The method allows more straightforward derivation of QuackÕs results and further extension of the calculation for separating elementary reactions and application to higher proton systems. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Selection rules; Nuclear spin modifications; Angular momentum algebra; Ion-neutral reactions; Molecular ions; H þ 3. Introduction The total nuclear spin quantum number I corresponding to the vector sum of kinetically equivalent identical nuclei, I = P ii i is a robust quantum number which does not change easily because of the weakness of the nuclear magnetic interaction. It is well established that transitions between ortho (I = ) and para (I = 0) spin modifications of two proton molecules such as H 2,H 2 O, and H 2 O are almost rigorously forbidden both in spectroscopy [] and in collision [2]. onversion between the two spin species is so slow that, for many applications, the two spin species may be regarded as independent molecules in the process of thermalization. This applies also to the ortho (I = 3/2) and para (I = /2) species of three proton molecules such as NH 3,H 3 F, etc., and the ortho (I = ), meta (I = 2), and para (I = 0) species of four proton molecules such as H 4. * Fax: address: t-oka@uchicago.edu. The names ortho and para were initially introduced for the He spectrum empirically (parhelium and orthohelium) [3,4] and theoretically explained by Heisenberg [5] for the electron spins in the atom. Heisenberg later used them for the nuclear spins in the hydrogen molecule [6]. Maue [7] introduced ortho, meta, and para for H 4 in the order of statistical weight. In chemical reactions A + fi + D, protons are scrambled and nuclear spin species change. Nevertheless since the overall nuclear spin angular momentum of the reactants and the products conserves nearly rigorously, that is, I A + I = I + I D, there exist nearly rigorous branching ratios between different sets of (I A,I ) fi (I-,I D ). The theory of the relations was first formulated by Quack [8] based on the Hougen [9] Longuet-Higgins [0] molecular symmetry group and using WatsonÕs correlations of symmetry classifications []. Experimental supports of QuackÕs theory have been reported for photodissociation of H 2 O [2] and ion-neutral reactions involving H þ 3 [3,4]. A more thorough discussion of selection rules in general and the literature can be found /$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:0.06/j.jms

2 636 T. Oka / Journal of Molecular Spectroscopy 228 (2004) in [4]. The nuclear spin selection rules were also observed for reactions of H 3 and H 2 radicals in para- H 2 crystals [5]. In this paper, I present a different derivation of QuackÕs results using angular momentum algebra rather than group theory. Unlike Quack who considered both nuclear spin and rovibronic coordinate wavefunctions, we consider only the former since the latter is automatically taken care of by the requirement of the symmetry of the total wavefunctions with respect to permutation. Use of the angular momentum, that is, representations of the rotation group instead of the permutation inversion (PI) group, makes the treatment more general and straightforward. Also it is easier to extend the calculation for separating elementary reactions and to reactions with higher number of protons. 2. Nuclear spin modifications The almighty formula for addition of angular momenta [6] D I D I2 ¼ D I þi 2 D I þi 2 D ji I 2 j; ðþ is the only relation used in this paper. Using this relation repeatedly for I = I 2 = /2 and calculating ðd =2 Þ n (where we use the notation ðd =2 Þ 2 D =2 D =2 ), we obtain representations of nuclear spin modifications in the rotation group and their statistical weights as listed in Table. Representations using the PI group are also given for a comparison. Note that the dimension and the frequency of a representation are swapped between the rotation group and the PI group. Table can be readily extended to molecules with larger number of protons although we run out of the ortho, meta, and para nomenclature for more than five protons. In general, we have ðd =2 Þ n ¼ n=2;ðn Þ=2 X ¼ X0;=2 k¼0 I¼n=2 ðn 2k þ Þn! ðn k þ Þ!k! D n=2 k ð2i þ Þn! ðn=2 IÞ!ðn=2 þ I þ Þ! D I; ð2þ where the first and the second index above P apply for even and odd n, respectively. The first formula is obtained from a straightforward enumeration and generalization, while the second from a more logical derivation [7]. 3. Two examples Like in many applications, a group theoretical enumeration is easier to carry out for specific simple examples rather than for the general case. Here, we first discuss two purely protonic ion-neutral reactions involving H þ 3 both of which play important roles in interstellar chemistry. 3.. Production of H þ 3 The celebrated ion-neutral reaction with a high exothermicity of.7 ev which produces H þ 3 in hydrogen dominated plasmas can be written as a two step process H 2 þ H þ 2!ðHþ 4 Þ! H þ 3 þ H; ð3þ where ðh þ 4 Þ represents the transient activated complex. For brevity, we sometimes use shorthand (2,2) fi (4) * fi (3,) for reaction (3). Like Quack [8], we assume that the four protons on the activated complex scramble completely so that we can regard all of them as equivalent. This assumption is not valid if, for example, we regard (3) as a proton hop reaction from H þ 2 to H 2. Experiments using deuterated species, however, have demonstrated that the four nuclei are completely scrambled [8,9]. Using total nuclear spin angular momenta, reaction (3) can be written as ðd D 0 ÞðD D 0 Þ!D 2 3D 2D 0!ðD 3=2 2D =2 ÞD =2 ; where the notation ðd D 0 ÞðD D 0 ÞðD D 0 Þ ðd D 0 Þ, etc., are used. Finding the relations of nuclear spin species for the first half of the reaction (2,2) fi (4) * is trivial. For example, D D! D 2 D D 0 demonstrates that the reaction of o-h 2 and o-h þ 2, with the statistical weight of 9, produces o-ðh þ 4 Þ (I = ), m-ðh þ 4 Þ (I = 2), and p-ðh þ 4 Þ (I = 0), with a ratio of 3:5:. It is less trivial to find out how each spin species of ðh þ 4 Þ decomposes into spin species of ðh þ 3 Þ in the second half of the reaction (4) * fi (3,). Table Representations and statistical weights of spin modifications n Examples Rotation group PI group ortho meta para Weight 2 H 2,H 2 O D D 0 3A 0 3: 3 H þ 3,H 3 D 3=2 2D =2 4A 2E 3/2 /2 4:4 4 ðh þ 4 Þ,H 4 D 2 3D 2D 0 5A 3F E 2 0 9:5:2 5 ðh þ 5 Þ ; H þ 5 D 5=2 4D 3=2 5D =2 6A 4G 2H 3/2 /2 5/2 6:0:6

3 T. Oka / Journal of Molecular Spectroscopy 228 (2004) Quack employed WatsonÕs method [] which traces the symmetry relation of the reverse reaction (4) * (3,) and uses FrobeniusÕ reciprocity. We do the same using angular momentum algebra. For (4) * (3,), we have D 2 D D 3=2 D =2 ð4þ 2D 2D 0 2D =2 D =2 : These equations show that D 2 and D 0 are produced only from D 3=2 D =2 and D =2 D =2, respectively. Reversing these, we learn that D 2 and D 0 dissociates only to D 3=2 D =2 and D =2 D =2, respectively. These agree with the intuitive picture that ðh þ 4 Þ with I = 2 where all proton spins are parallel decomposes only to o-h þ 3, and ðh þ 4 Þ with I = 0 where two pairs of spins are antiparallel decomposes only to p-h þ 3. alancing the dimensions, we obtain the first and third equations of Eqs. (5), shown below. Eqs. (4) show that D is produced from D 3=2 D =2 and D =2 D =2 and therefore decompose into both of them. We thus obtain the formula of D in Eqs. (5) from D 2 and D 0 in Eqs. (5) and the identity D 2 3D 2D 0 ¼ðD 3=2 2D =2 ÞD =2. D 2! 5ðD 3=2 D =2 =8Þ D!ðD 3=2 D =2 =8Þ2ðD =2 D =2 =4Þ ð5þ D 0!ðD =2 D =2 =4Þ: In normalized forms ðd 2 =5Þ ½2Š; ðd 3=2 D =2 =8Þ ½3=2; =2Š, etc., Eqs. (5) are written as ½2Š! ½3=2; =2Š; 3½Š! ½3=2; =2Šþ2½=2; =2Š; 2½0Š!2½=2; =2Š; which are more obviously the reciprocals of Eq. (4) in the sense of FrobeniusÕ reciprocity []. Using Eqs. (5), we have spin relations for the reaction (2,2) fi (3,) as D D! D 2 D D 0! 6ðD 3=2 D =2 =8Þ3ðD =2 D =2 =4Þ, corresponding to o-h 2 þ o-h þ 2! o-hþ 3 þ H ð2=3þ! p-h þ 3 þ H ð=3þ; D D 0 ¼ D 0 D! D!ðD 3=2 D =2 =8Þ2ðD =2 D =2 =4Þ, corresponding to o-h 2 þ p-h þ 2 ¼ p-h 2 þ o-h þ 2! o-hþ 3 þ H ð=3þ! p-h þ 3 þ H ð2=3þ; and D 0 D 0! D 0!ðD =2 D =2 =4Þ, corresponding to p-h 2 þ p-h þ 2! p-hþ 3 þ H that have been used in the analysis of our experiment [4]. Those relations are summarized in Table 2a where the more systematic I values rather than ortho and para notations are used to express spin modifications. We will call this relation a branching ratio matrix Reactive collision of H þ 3 with H 2 In H 2 -dominated plasmas such as those in interstellar molecular clouds, an H þ 3 ion collides with H 2 many times during its life undergoing the reaction H þ 3 þ H 2!ðH þ 5 Þ! H þ 3 þ H 2 ð6þ or (3,2) fi (5) * fi (3,2) in shorthand. Although the reactants and the products are the same, this is a chemical reaction in the sense that the five protons are scrambled and the spin species of H þ 3 may change. In fact this reaction is the major mechanism for thermalization of o-h 2 and p-h 2 in molecular clouds. Unlike for the highly exothermic reaction (3), this is a thermoneutral reaction and the activated complex ðh þ 5 Þ has much smaller excess energy than ðh þ 4 Þ of reaction (3). Therefore, the assumption of complete proton scrambling may not be valid. Indeed the experiment by ordonnier et al. [4] has shown that proton hop reactions are faster than proton exchange reactions in their hollow cathode discharge at room temperature. So we need to separate reaction (6) into separate elementary reactions and calculate nuclear spin branching ratios separately for the proton hop reaction and proton exchange reaction. To do this, we first calculate branching ratios assuming complete scrambling of the protons. With the angular momentum algebra, the reaction (6) is written as ðd 3=2 2D =2 ÞðD D 0 Þ!D 5=2 4D 3=2 5D =2 Proceeding as in Section 3., we have D 5=2 D 3=2 D =2 D 3=2 D D 3=2 D 3=2 D 0 2D 3=2 2D =2 2D =2 D and their reciprocals 2D =2 2D =2 D 0 ½5=2Š!½3=2; Š 4½3=2Š!½3=2; Š½3=2; 0Š2½=2; Š 5½=2Š!½3=2; Š2½=2; Š2½=2; 0Š which give!ðd 3=2 2D =2 ÞðD D 0 Þ: D 5=2! 6ðD 3=2 D =2Þ D 3=2!ðD 3=2 D =2ÞðD 3=2 D 0 =4Þ 2ðD =2 D =6Þ D =2! 2=5ðD 3=2 D =2Þ4=5ðD =2 D =6Þ 4=5ðD =2 D 0 =2Þ These lead to the branching ratio matrix listed in Table 2 that are also given in [3] and [4]. The complete scrambling of protons considered for reaction (6) contains 5!/3!2! = 0 cases of proton partitions containing one identity, three proton hops that

4 638 T. Oka / Journal of Molecular Spectroscopy 228 (2004) Table 2 ranching ratios of spin modifications in chemical reactions A. (2,2) fi (3,) Ex. H 2 þ H þ 2! Hþ 3 þ H Spin species Weight (3/2,/2) (/2,/2) (,) (,0) 3 2 (0,) 3 2 (0,0) 0. (3,2) fi (3,2) Ex. H þ 3 þ H 2! H þ 3 þ H 2 Spin species Weight (3/2,) (3/2,0) (/2,) (/2,0) (3/2,) 2 37/5 4/5 4/5 (3/2,0) (/2,) 2 4/5 2 28/5 8/5 2(/2,0) 4 4/5 0 8/5 8/5. (3,3) fi (4,2) Ex. H þ 3 þ NH 3! NH þ 4 þ H 2 Spin species Weight (2,) (2,0) (,) (,0) (0,) (0,0) (3/2,3/2) 6 25/3 23/5 2/3 2/5 2(3/2,/2) 6 8/ /3 0 2(/2,3/2) 6 8/ /3 0 4(/2,/2) 6 4/3 0 32/5 4 8/3 8/5 D. (4,3) fi (5,2) Ex. H 4 þ H þ 3! Hþ 5 þ H 2 Spin species Weight (5/2,) (5/2,0) (3/2,) (3/2,0) (/2,) (/2,0) (2,3/2) 20 65/7 40/7 8/7 5/7 5/7 2(2,/2) 20 8/7 2 72/7 6/7 20/7 0 3(,3/2) 36 27/7 3 20/7 24/7 45/7 5/7 6(,/2) 36 2/7 0 72/7 48/7 90/7 30/7 2(0,3/2) 8 4/7 0 6/7 6/7 20/7 0 4(0,/2) /7 0 20/7 20/7 E. (4,4) fi (5,3) Ex. H þ 4 þ H 4! H þ 5 þ H 3 Spin species Weight (5/2,3/2) (5/2,/2) (3/2,3/2) (3/2,/2) (/2,3/2) (/2,/2) (2,2) 25 45/4 5/2 40/7 20/7 25/4 25/4 3(2,) 45 57/4 5/2 4/7 60/7 75/4 45/4 2(2,0) 0 / /2 0 3(,2) 45 57/4 5/2 4/7 60/7 75/4 45/4 9(,) 8 45/4 9/2 08/7 80/7 225/4 225/4 6(,0) 8 9/4 0 8/7 36/7 45/4 45/7 2(0,2) 0 / /2 0 6(0,) 8 9/4 0 8/7 36/7 45/4 45/7 4(0,0) / /7 are equivalent and six proton exchanges that are equivalent. They are expressed by H þ 3 þ H e 2! H þ 3 þ H ~ 2 ; ð7aþ H þ 3 þ H e 2! HH e þ 2 þ H 2; and ð7bþ H þ 3 þ H e þ 2! H 2H e þ HH; e ð7cþ respectively, where H e represents protons that are originally in H 2. The branching ratio matrix for identity (8a) is diagonal and is given by 0 6= = =5 0 A ; ð8aþ =5 where the fractions are results of multiplying the probability of /0 to the identity matrix. The proton hop reaction (7b) is decomposed into a succession of reactions H þ 3! H 2 þ H þ and H þ þ H e 2! HH e þ 2 ; i:e:; ð3þ!ð2; Þ and ð; 2Þ!ð3Þ: Using D 3=2 2D =2 D =2 ðd D 0 Þ, the calculation is straightforward and we obtain the matrix for proton hop reactions as 0 2=5 0 6= =5 6=5 6=5 3=5 3=5 A : ð8bþ 0 0 3=5 3=5 The branching ratio matrix for proton exchange reaction (7c) is obtained by subtracting (8a) and (8b) from the total matrix on Table 2 tobe

5 0 9=5 8=5 4=5 3=5 4=5 8=5 4=5 9=5 A : 4=5 0 3=5 ð8cþ For many applications, the branching ratios of ortho and para H 2 are unimportant since H 2 is more abundant than H þ 3 by many orders of magnitude (the ratio of the H 2 number density to the H þ 3 number density is comparable or greater than 0 6 both in laboratory and space plasmas). In this case we can add elements (3/2,) and (3/2,0), and (/2,) and (/2,0) in the same row of the branching ratio matrix. We thus have ortho para branching ratios of H þ 3 o-h þ 3 þ o-h 2!ð2=3Þo-H þ 3 þð=3þp-hþ 3 for both proton hop and proton exchange reactions, but o-h þ 3 þ p-h 2! p-h þ 3, for proton hop reaction while o-h þ 3 þ p-h 2!ð2=3Þo-H þ 3 þð=3þp-hþ 3 for proton exchange reaction as listed in Table IV of [4]. Since the three proton hop reactions and the six proton exchange reactions are equivalent among themselves, the matrices (8b) and (8c) give nearly rigorous branching ratios of the elementary reactions. For analyzing experimental data, relative rates of the proton hop reaction and the proton exchange reaction should be multiplied to (8b) and (8c) to obtain the overall branching ratio matrix [4]. T. Oka / Journal of Molecular Spectroscopy 228 (2004) the frequency, respectively, of D I in ðd =2 Þ n for combination (I,I 2 ) j. While we can enumerate branching ratios for systems of arbitrary numbers of protons following this procedure, such calculations may be academic for reactions involving a very large number of protons because the assumption of complete proton scrambling is not valid. Perhaps, the assumption is more valid for reactions with higher exothermicity. Separating those reactions into elementary reactions is more complicated than the case of Section 3.2. Extension of this method for reactions with more than two product molecules is straightforward, although I cannot find a good example of such reactions. Extension of this method for nuclear species with spin higher than /2 is also straightforward. Acknowledgments This paper is dedicated to Jon T. Hougen as a sign of our friendship over many years. I wish to thank.j. Mcall, M. Quack, and J.K.G. Watson for critical reading of this paper. This work has been supported by NSF Grant PHY References 4. Extensions The procedure given above can be readily extended to systems with larger numbers of protons if complete scrambling can be assumed. Results for (3,3) fi (6) * fi (4,2), for example, H þ 3 þ NH 3!ðNH þ 6 Þ! NH þ 4 þ H 2, with a large exothermicity of 4.46 ev, (3,4) fi (7)* fi (5,2), for example, H þ 3 þ H 4! ðh þ 7 Þ! H þ 5 þ H 2, with an exothermicity of.33 ev and (4,4) fi (8)* fi (5,3), for example, H þ 4 þ H 4! ð 2 H þ 8 Þ! H þ 5 þ H 3, with an exothermicity of 0.7 ev, are given in Table 2 E. Elements of branching ratio matrices R ij satisfy general sum rules X X R ij ¼ w i ; R ij ¼½ð2I þ Þð2I 2 þ Þf f 2 Š j ; j i X R ij ¼ X w i ¼ X ½ð2I þ Þð2I 2 þ Þf f 2 Š j ¼ 2 n ; ð9þ i;j i j where w i represents the weight of combination i given in the table and (2I + ) and f represent the dimension and [] G. Herzberg, Molecular Spectra and Molecular Structure, vol. 3, Krieger Pub. o. Malabar, Florida (989 99). [2] T. Oka, Adv. At. Mol. Phys. 9 (973) [3]. Runge, F. Paschen, Astrophys. J. 3 (896) [4] A. Fowler, Report on Series in Line Spectra (922). [5] W. Heisenberg, Z. Phys. 38 (926) [6] W. Heisenberg, Z. Phys. 4 (927) [7] A.-W. Maue, Ann. Phys. 30 (937) [8] M. Quack, Mol. Phys. 34 (977) [9] J.T. Hougen, J. hem. Phys. 37 (962) [0] H.. Longuet-Higgins, Mol. Phys. 6 (963) [] J.K.G. Watson, an. J. Phys. 43 (965) [2]. Schramm, D.J. amford,.. Moore, hem. Phys. Lett. 98 (983) [3] D. Uy, M. ordonnier, T. Oka, Phys. Rev. Lett. 78 (997) [4] M. ordonnier, D. Uy, R.M. Dickson, K.E. Kerr, Y. Zhang, T. Oka, J. hem. Phys. 3 (2000) [5] M. Fushitani, T. Momose, J. hem. Phys. 6 (2002) [6] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Pergamon Press (977) 3 and 06. [7] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Pergamon Press (977) 63. [8] J.R. Krenos, K.K. Lehmann, J.. Tully, P.M. Hierl, G.P. Smith, hem Phys. 6 (976) [9] P.M. Hierl, Z. Herman, hem. Phys. 50 (980)

5.61 Physical Chemistry Lecture #36 Page

5.61 Physical Chemistry Lecture #36 Page 1 NUCLEAR MAGNETIC RESONANCE Just as IR spectroscopy is the simplest example of transitions being induced by light s oscillating electric field, so NMR is the simplest