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The full non-rigid group of hexamethylbenzene using wreath product
Chemical Physics Letters 421 (2006) 566–570 www.elsevier.com/locate/cplett
The full non-rigid group of hexamethylbenzene using wreath product Mohammad Reza Darafsheh a
a,*
, Ali Reza Ashrafi b, Arash Darafsheh
c
Department of Mathematics, Statistics and Computer Science, Faculty of Science, University of Tehran, Tehran 14174, Iran b Department of Mathematics, Faculty of Science, University of Kashan, Kashan, Iran c Faculty of Science, Shahid Beheshti University, Tehran, Iran Received 1 February 2006 Available online 3 March 2006
Abstract The character table of the non-rigid hexamethylbenzene, C6(CH3)6, is derived for the first time. The group of all feasible permutations is the wreath product group D6[Z3] = Z3 D6 and it consists of 8748 operations divided into 174 conjugacy classes and irreducible representations. We have shown that the full character table can be constructed using elegant matrix type generator algebra. The group can also be useful for weakly-bound (NH3)6. 2006 Elsevier B.V. All rights reserved.
1. Introduction The group theory for non-rigid molecules finds numerous applications ranging from the rovibronic spectroscopy of molecules exhibiting large amplitude motions, chemical reactions, dynamic stereochemistry to weakly-bound van der Waals complexes. Following the pioneering works of Longuet-Higgins [1], the symmetry group of a non-rigid molecule group consists of all permutations and permutation-inversion operations, which become feasible as molecule tunnel through a number of potential energy maxima separated by multiple minima. Subsequently, several other workers [2–25] have formulated different ways of characterizing non-rigid groups (NRG). For example, Smeyers [2], Altmann [3] and others have defined the NRG group as the complete set of the molecule conversion operations, which commute with a given nuclear Hamiltonian operator, limited to large amplitude motions. In addition, these molecular conversion operations will be expressed in terms of physical operations, such as rotations, internal rotations, inversions, similarly as in the Altmann’s theory, rather than in terms of permutations and
*
Corresponding author. Fax: +98 21 6412178. E-mail address: [email protected] (M.R. Darafsheh).
0009-2614/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2006.02.008
permutations-inversions. This way of expressing the nonrigid operations is indeed more descriptive and flexible [3]. The complete set of molecular conversion operations which commute with the nuclear motion operator contains overall rotation operations, describing the molecule rotating as a whole, and the non-rigid tunneling motion operations, describing molecular moieties moving with respect to the rest of the molecule. Such a set forms a group, which we call the full non-rigid group (f-NRG). Longuet-Higgins [1] investigated the symmetry groups of non-rigid molecules, where changes from one conformation to another can occur easily. In many cases, these dynamical symmetry groups are not isomorphic with any of the familiar point groups of rigid molecules and their character tables are not known. It is therefore of considerable interest and importance to develop simple methods to calculate these character tables, which are needed for the classification of wave functions, correlation of the rovibronic states, determination of selection rules, hyperfine splittings of non-rigid species, and so on. We describe here a general method that is appropriate for molecules which consist of a number of XH3 or XO2 groups attached to a rigid framework. Examples of such molecules are hexamethylbenzene, which is considered here in some detail. For the linear framework, a similar problem has been studied by Bunker [4].
M.R. Darafsheh et al. / Chemical Physics Letters 421 (2006) 566–570
Lomont [5] has given two methods for calculating character tables. These methods are satisfactory for small groups, but both of them require knowledge of the class structure and hence of the group multiplication table and they become very massive as the order of the group becomes even moderately large. For non-rigid molecules, whose symmetry groups may have several thousand elements, they are usually quite impracticable. Group theory for non-rigid molecules is becoming more and more relevant and numerous applications to large amplitude vibrational spectroscopy of small organic molecules have appeared in the literature [5–7]. Our approach here is first to specify the algebraic structure of the full non-rigid group of molecules that contain 6 groups of non-rigid internal rotors such as hexamethylbenzene and a weakly-bound ammonia hexamer (NH3)6. With a geometric consideration of dynamic symmetries of the molecules, we will show that the f-NRG of this molecule can be specified by wreath product of some known groups. Then based on the structure of the group we apply GAP [8] as a useful package for computing the character tables and even the group structure to compute the character table of the f-NRG of this molecule, see Fig. 1. We use [9] for the standard notations and terminology of character theory. In this Letter, we use the Balasubramanian’s method [10–20], for computing the non-rigid group of hexamethylbenzene molecule by wreath product formalism. It is shown that this is a group of order 8748 with 174 conjugacy classes. The motivation for this study is outlined also in [21–25] and the reader is encouraged to consult these Letters for background material as well as basic computational techniques. The developed technique is not only applicable for the hexamethylbenzene molecule but also for ammonia hexamer. As seen from the literature [18,26–29] hydrogen-bonded and van der Waals complexes exhibit tunneling motions among different potential minima. Thus, the symmetry groups of these non-rigid complexes can also be expressed as wreath product groups that we have considered here. Computations were carried out with the aid of GAP and this was done by characterizing the algebraic structure of f-NRG as the wreath product of known groups.
Fig. 1. The structure of hexamethylbenzene.
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2. Full non-rigid group of hexamethylbenzene In this section we first describe some notation, which will be kept throughout. Let G be a group and N be a subgroup of G. N is called a normal subgroup of G, if for any g 2 G and x 2 N, g1 · g 2 N. Moreover, if H is another subgroup of G such that H \ N = {e} and G = HN = {xy j x 2 H, y 2 N}, then we say that G is a semidirect product of H by N denoted by H N. Suppose X is a set. The set of all permutations on X, denoted by SX, is a group which is called the symmetric group on X. In the case that X = {1, 2, . . . , n}, we denote SX by Sn or Sym(n). Let H be a permutation group on X, a subgroup of SX, and let G be a group. The set of all mappings X ! G is denoted by GX, i.e. GX = {f j f: X ! G}. It is clear that jGXj = jGjjXj. We put H[G] = G H = GX · H = {(f; p) j f 2 GX, p 2 H}. For f 2 GX and p 2 H, we define fp 2 GX by fp = fop1, where ‘o’ denotes the composition of functions. It is easy to check that the following law of composition: ðf ; pÞðf 0 ; p0 Þ ¼ ðff 0p ; p p0 Þ, makes G H into a group. This group is called the wreath product of G by H and sometimes is denoted by H[G] as well. Before going into the details of the computations of hexamethylbenzene we should mention that we consider the speed of rotations of the methyl groups sufficiently high so that the mean time dynamical symmetry of the molecules makes sense. In order to characterize the f-NRG of hexamethylbenzene, we first note that each dynamic symmetry operation of this molecule, considering the rotations of CH3 groups, is composed of two sequential physical operations. We first have a physical symmetry of the hexagonal framework (as we have to map the CH3 groups on CH3 groups which are on vertices of the hexagonal framework). Such operations are exactly the symmetry operations of a hexagon and, as is well known, such operations form the group D6 of order 12. After accomplishing the first framework symmetry operation we have to map each of the six CH3 group on itself which forms the three elements group C3, also denoted by Z3. The number of all such operations is 36 · 12 = 8748. The composition of such dynamic symmetry elements are described as follows. Let us use numbers {1, 2, 3, 4, 5, 6} to indicate the carbon atoms of hexagonal framework. Now from the symmetry point of view the 1, 2, 3, 4, 5, 6 carbons (corners of the hexagonal framework) are important and the remaining carbons follow the motions of 1, 2, 3, 4, 5, 6 carbons. This means that for computing the f-NRG it is enough to consider the hexagon (1, 2, 3, 4, 5, 6) with rotating CH3 on its corners. We can describe such configuration with a 3 · 6 matrix [ij]3·6 where ij means the ith hydrogen on the jth carbon corner. Now the dynamic symmetry operation described above has the form (a1, a2, a3, a4, a5, a6; r), where r is a symmetry of the hexagonal framework (the first physical operation as above) and if h : {1, 2, 3, 4, 5, 6} ! {e, f, g} is a function into the cyclic group {e, f, g} of order three with identity element e then we write ai = h(i),
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M.R. Darafsheh et al. / Chemical Physics Letters 421 (2006) 566–570
Table 1 The representatives of the conjugacy classes of the wreath product, D6[Z3], the non-rigid group of hexamethylbenzene containing 8748 operations No.
Representatives
Size
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 42 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
(e,e,e,e,e,e,()) (g,g,f,g,f,e,(1,6,5,4,3,2)) (e,f,e,f,g,e,(1,2,3,4,5,6)) (g,f,e,e,e,f,(1,5,3)(2,6,4)) (g,g,g,f,e,f,(1,3,5)(2,4,6)) (g,g,g,g,g,g,()) (f,f,f,f,f,f,()) (e,f,g,g,f,e,(1,4)(2,5)(3,6)) (f,g,e,e,g,f,(1,4)(2,5)(3,6)) (g,e,f,f,e,g,(1,4)(2,5)(3,6)) (f,e,e,f,e,f,(1,5,3)(2,6,4)) (f,g,f,g,f,g,()) (e,f,e,f,e,f,()) (e,g,e,g,e,g,()) (f,f,e,g,g,g,(1,3,5)(2,4,6)) (f,e,g,f,e,e,(1,5,3)(2,6,4)) (g,g,e,f,g,f,(1,3)(4,6)) (e,f,f,g,f,g,(1,3)(4,6)) (g,f,g,g,f,g,()) (f,g,f,f,g,f,()) (f,e,g,e,e,e,(1,3)(4,6)) (g,e,g,f,f,f,()) (f,e,f,g,g,g,()) (e,g,f,e,g,e,(1,3)(4,6)) (g,f,e,e,f,e,(1,3)(4,6)) (f,f,f,e,f,e,()) (g,g,g,e,g,e,()) (g,e,g,g,f,g,()) (f,e,f,f,g,f,()) (g,g,f,f,g,g,()) (f,f,g,g,f,f,()) (f,e,f,e,g,g,(1,6)(2,5)(3,4)) (f,f,e,g,e,g,(1,6)(2,5)(3,4)) (e,g,f,f,g,e,()) (e,f,g,g,f,e,()) (f,g,g,f,f,g,(1,6)(2,5)(3,4)) (f,g,e,g,f,g,(1,6)(2,5)(3,4)) (f,g,f,e,f,g,(1,6)(2,5)(3,4)) (e,e,g,g,e,e,()) (e,e,f,f,e,e,()) (e,g,g,g,g,e,()) (e,f,f,f,f,e,()) (f,e,g,f,f,g,()) (g,e,f,g,g,f,()) (f,g,g,g,g,g,()) (g,f,f,f,f,f,()) (g,g,e,g,f,g,()) (f,f,e,f,g,f,()) (e,g,f,g,e,f,()) (e,f,g,f,e,g,()) (f,g,f,g,e,g,()) (g,f,g,f,e,f,()) (g,f,e,e,e,f,()) (f,g,e,e,e,g,()) (f,g,e,g,e,e,(2,6)(3,5)) (g,e,e,f,e,f,(2,6)(3,5)) (g,g,e,f,e,g,()) (f,f,e,g,e,f,()) (e,g,f,e,f,e,(2,6)(3,5)) (e,e,g,e,g,f,(2,6)(3,5)) (e,g,g,e,g,g,()) (e,f,f,e,f,f,()) (e,e,f,f,e,g,(1,4)(2,5)(3,6))
1 486 486 162 162 1 1 27 27 27 324 2 2 2 324 324 27 27 3 3 27 6 6 54 54 6 6 6 6 6 6 162 162 6 6 81 162 162 6 6 6 6 12 12 6 6 12 12 6 6 6 6 6 6 54 54 6 6 27 27 3 3 81
Table 1 (continued) No.
Representatives
Size
64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
(g,e,f,e,e,g,(1,4)(2,5)(3,6)) (f,e,e,f,e,e,()) (g,e,e,g,e,e,()) (g,e,g,e,f,f,(1,4)(2,3)(5,6)) (e,f,e,f,g,g,(1,4)(2,3)(5,6)) (e,f,f,e,g,g,()) (f,g,e,f,f,g,(1,4)(2,3)(5,6)) (g,e,f,g,f,g,(1,4)(2,3)(5,6)) (f,f,e,e,g,e,(1,4)(2,3)(5,6)) (e,e,g,g,e,f,(1,4)(2,3)(5,6)) (g,e,f,g,f,g,()) (f,e,g,f,g,f,()) (e,g,e,f,f,g,()) (e,f,e,g,g,f,()) (f,f,g,f,g,g,(1,3)(4,6)) (e,g,e,e,f,e,()) (e,e,e,f,g,f,(2,6)(3,5)) (e,g,f,g,e,e,(2,6)(3,5)) (e,e,g,g,f,f,(2,6)(3,5)) (e,g,g,f,f,e,(2,6)(3,5)) (g,g,g,e,f,e,(2,6)(3,5)) (f,e,g,e,f,f,(2,6)(3,5)) (e,e,g,f,e,e,(2,6)(3,5)) (e,e,e,g,f,e,(2,6)(3,5)) (e,e,g,g,g,e,()) (e,e,f,f,f,e,()) (g,e,g,f,e,g,(2,6)(3,5)) (f,f,e,g,f,e,(2,6)(3,5)) (g,e,e,f,f,e,()) (f,e,e,g,g,e,()) (g,e,e,g,e,g,()) (f,e,e,f,e,f,()) (f,e,e,e,g,e,()) (f,e,g,e,g,e,()) (g,e,f,e,f,e,()) (g,e,g,g,f,e,()) (f,e,f,f,g,e,()) (e,g,e,g,e,e,()) (e,f,e,f,e,e,()) (g,g,g,e,e,e,(1,5)(2,4)) (e,e,f,f,f,e,(1,5)(2,4)) (g,e,e,f,e,e,(1,5)(2,4)) (g,f,e,f,g,e,()) (f,g,f,g,f,f,(1,3)(4,6)) (g,f,g,g,g,f,(1,3)(4,6)) (g,f,g,e,g,e,()) (f,g,f,e,f,e,()) (e,g,g,g,g,f,(1,3)(4,6)) (f,f,e,g,f,f,(1,3)(4,6)) (g,e,g,f,g,e,(1,3)(4,6)) (f,e,f,e,f,g,(1,3)(4,6)) (f,e,f,f,f,f,()) (g,e,g,g,g,g,()) (g,f,f,g,f,f,(1,3)(4,6)) (g,g,f,g,g,f,(1,3)(4,6)) (e,e,e,f,e,f,(1,4)(2,5)(3,6)) (g,e,g,e,e,e,(1,4)(2,5)(3,6)) (f,e,g,f,e,g,()) (e,g,g,g,f,g,(1,4)(2,5)(3,6)) (e,e,f,e,f,g,()) (e,e,g,e,g,f,()) (g,f,f,f,g,f,()) (f,g,g,g,f,g,()) (f,e,g,g,e,e,(1,3)(4,6)) (f,e,g,e,e,f,(1,3)(4,6))
81 3 3 81 81 6 162 162 162 162 12 12 12 12 54 6 54 54 54 54 54 54 54 54 6 6 54 54 12 12 12 12 12 6 6 12 12 6 6 54 54 54 6 54 54 6 6 54 54 54 54 6 6 27 27 81 81 6 162 12 12 6 6 54 54
M.R. Darafsheh et al. / Chemical Physics Letters 421 (2006) 566–570 Table 1 (continued) No.
Representatives
Size
129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174
(e,f,e,f,e,e,(2,6)(3,5)) (e,e,e,g,e,g,(2,6)(3,5)) (f,g,f,e,e,e,(1,4)(2,5)(3,6)) (e,e,e,g,f,g,(1,4)(2,5)(3,6)) (f,g,e,f,g,e,(1,3)(4,6)) (e,f,g,e,f,g,(1,3)(4,6)) (e,e,f,g,g,g,()) (e,e,g,f,f,f,()) (e,e,f,g,e,f,()) (e,e,g,f,e,g,()) (e,e,e,e,e,g,()) (e,e,e,e,e,f,()) (f,e,f,e,e,e,(1,6)(2,5)(3,4)) (e,e,e,g,e,g,(1,6)(2,5)(3,4)) (f,e,f,e,f,e,(1,3,5)(2,4,6)) (g,f,e,f,g,e,(1,4)(2,3)(5,6)) (e,f,e,g,e,f,(1,4)(2,3)(5,6)) (f,e,g,e,g,e,(1,4)(2,3)(5,6)) (f,f,g,g,f,g,()) (f,g,f,e,e,f,()) (g,f,g,e,e,g,()) (f,g,g,e,e,g,(2,6)(3,5)) (g,f,e,e,f,f,(2,6)(3,5)) (f,e,f,f,g,g,()) (g,e,g,g,f,f,()) (e,e,f,g,f,e,(1,6)(2,5)(3,4)) (e,g,f,g,e,e,(1,6)(2,5)(3,4)) (g,g,g,f,g,e,(2,6)(3,5)) (f,f,f,e,g,g,()) (g,g,g,e,f,f,()) (e,g,g,e,f,f,(1,2,3,4,5,6)) (g,f,e,g,g,g,()) (f,g,e,f,f,f,()) (g,f,f,f,g,g,()) (f,e,e,e,g,g,()) (g,e,e,e,f,f,()) (g,f,g,g,f,e,(1,5)(2,4)) (g,f,f,g,f,e,(1,5)(2,4)) (g,f,e,e,f,f,(1,3)(4,6)) (e,g,f,g,g,e,(1,3)(4,6)) (g,e,g,f,e,g,()) (f,e,f,g,e,f,()) (f,f,e,e,g,g,()) (g,g,e,f,g,g,(2,6)(3,5)) (f,g,e,e,e,e,()) (g,f,e,e,g,f,())
54 54 81 81 27 27 12 12 12 12 6 6 81 81 162 162 81 81 12 12 12 54 54 12 12 81 81 54 12 12 486 12 12 6 12 12 54 54 54 54 12 12 12 54 12 12
1 6 ai 6 6. The group V of the symmetries of the molecule acts on the 18 entries of [ij]3·6 as follows. Consider (h;r) and [ij], we first do h on the ith hydrogen by the rule h(r(j))(i), and then we do r on the jth corner carbon to obtain [h(r(j))(i)r(j)] which shows the new position of molecule. This composition rule is the wreath product composition. Therefore the f-NRG group G of hexamethylbenzene is isomorphic to Z3 D6. We now apply GAP to obtain the conjugacy classes and character table of the group. Simple counting shows that the number of elements of wreath product D6[Z3] group is 12 · 36 = 8748. Table 1 shows the conjugacy classes, a representative from each class and the order of the conjugacy class. Since the character table of the group, Table 2, is very large in size, it
569
is deposited as a supplementary material to this Letter. In supplementary Table, /X denotes the complex conjugate of X. Also, the values A, B, . . ., O are as follows: A ¼ eip=3 ; B ¼ 2e2ip=3 ; C ¼ 3e2ip=3 ; D ¼ 6e2ip=3 ; p E ¼ 12e2ip=3 ; F ¼ 3; G ¼ 2A A2 ; H ¼ 4A; I ¼ 2F ; J ¼ 3 F ; K ¼ 6 2F ; L ¼ 4F ; M ¼ A2 5A; N ¼ ð9 þ F Þ=2; O ¼ ð3 5F Þ=2 3. Conclusion We have developed the group theory and character table of the non-rigid hexamethylbenzene and ammonia hexamer for the first time as a wreath product group D6[Z3] = Z3 D6 and it consists of 8748 operations divided into 174 conjugacy classes and irreducible representations. The derived character tables would also be valuable in other applications such as in the context of chemical applications of graph theory [30] and aromatic compounds [31]. In the case of chemical applications of graph theory, applications can range from enumeration f isomers to the automorphism groups of chemical graphs. In other fields such as theory of quarks and generalized special unitary groups, such wreath products and their double groups find important applications. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.cplett. 2006.02.008. References [1] H.C. Longuet-Higgins, Mol. Phys. 6 (1963) 445. [2] Y.G. Smeyers, Adv. Quantum Chem. 24 (1992) 1. [3] S.L. Altmann, Induced Representation in Crystal and Molecules, Academic Press, London, 1977. [4] P.R. Bunker, Molecular Symmetry in Spectroscopy, Academic, New York, 1979. [5] J.S. Lomont, Applications of Finite Groups, Academic, New York, 1959. [6] A.J. Stone, J. Chem. Phys. 41 (1964) 1568. [7] Y.G. Smeyers, M. Villa, J. Math. Chem. 28 (2000) 377. [8] M. Scho¨nert, H.U. Besche, Th. Breuer, F. Celler, B. Eick, V. Felsch, A. Hulpke, J. Mnich, W. Nickel, G. Pfeiffer, U. Polis, H. Theißen, A. Niemeyer, GAP, Groups, Algorithms and Programming, Lehrstuhl De fu¨r Mathematik, RWTH, Aachen, 1995.. [9] I.M. Isaacs, Character Theory of Finite Groups, Academic, New York, 1978. [10] K. Balasubramanian, J. Chem. Phys. 72 (1980) 665. [11] K. Balasubramanian, J. Chem. Phys. 73 (1980) 3321. [12] K. Balasubramanian, J. Chem. Phys. 75 (1981) 4572. [13] K. Balasubramanian, Int. J. Quantum. Chem. 22 (1982) 1013. [14] K. Balasubramanian, J. Chem. Phys. 78 (1983) 6358. [15] K. Balasubramanian, J. Chem. Phys. 78 (1983) 6369. [16] K. Balasubramanian, Studies Phys. Theor. Chem. 23 (1983) 149. [17] K. Balasubramanian, Theoretica Chimica Acta 78 (1990) 31. [18] K. Balasubramanian, J. Phys. Chem. 108 (2004) 5527. [19] K. Balasubramanian, Chem. Phys. Lett. 391 (2004) 64.
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[20] K. Balasubramanian, Chem. Phys. Lett. 398 (2004) 15. [21] A.R. Ashrafi, MATCH Commun. Math. Comput. Chem. 53 (2005) 161. [22] M.R. Darafsheh, A.R. Ashrafi, A. Darafsheh, Int. J. Quantum Chem. 105 (2005) 485. [23] M.R. Darafsheh, A.R. Ashrafi, A. Darafsheh, Acta Chim. Slov. 52 (2005) 282. [24] M.R. Darafsheh, Y. Farjami, A.R. Ashrafi, Bull. Chem. Soc. Jpn. 78 (2005) 996. [25] M.R. Darafsheh, Y. Farjami, A.R. Ashrafi, MATCH Commun. Math. Comput. Chem. 54 (2005) 53.
[26] K. Balasubramanian, T.R. Dyke, J. Phys. Chem. 88 (1984) 4688. [27] J.A. Odutola, T.A. Hu, D. Prinslow, T.R. Dyke, J. Chem. Phys. 88 (1988) 5352. [28] F.N. Keutsh, R.J. Saykally, Proc. Natl. Acad. Sci. USA 98 (2001) 10533. [29] R.J. Saykally, G.A. Blake, Science 259 (1996) 1570. [30] A.T. Balaban (Ed.), Chemical Applications of Graph Theory, Academic Press, New York, 1976. [31] J.R. Dias, Handbook of Polycyclic Aromatic Compounds, Part A (Elsevier, 1987), Part B (Elsevier, 1988).
The full non-rigid group of hexamethylbenzene using wreath product Mohammad Reza Darafsheh a
a,*
, Ali Reza Ashrafi b, Arash Darafsheh
c
Department of Mathematics, Statistics and Computer Science, Faculty of Science, University of Tehran, Tehran 14174, Iran b Department of Mathematics, Faculty of Science, University of Kashan, Kashan, Iran c Faculty of Science, Shahid Beheshti University, Tehran, Iran Received 1 February 2006 Available online 3 March 2006
Abstract The character table of the non-rigid hexamethylbenzene, C6(CH3)6, is derived for the first time. The group of all feasible permutations is the wreath product group D6[Z3] = Z3 D6 and it consists of 8748 operations divided into 174 conjugacy classes and irreducible representations. We have shown that the full character table can be constructed using elegant matrix type generator algebra. The group can also be useful for weakly-bound (NH3)6. 2006 Elsevier B.V. All rights reserved.
1. Introduction The group theory for non-rigid molecules finds numerous applications ranging from the rovibronic spectroscopy of molecules exhibiting large amplitude motions, chemical reactions, dynamic stereochemistry to weakly-bound van der Waals complexes. Following the pioneering works of Longuet-Higgins [1], the symmetry group of a non-rigid molecule group consists of all permutations and permutation-inversion operations, which become feasible as molecule tunnel through a number of potential energy maxima separated by multiple minima. Subsequently, several other workers [2–25] have formulated different ways of characterizing non-rigid groups (NRG). For example, Smeyers [2], Altmann [3] and others have defined the NRG group as the complete set of the molecule conversion operations, which commute with a given nuclear Hamiltonian operator, limited to large amplitude motions. In addition, these molecular conversion operations will be expressed in terms of physical operations, such as rotations, internal rotations, inversions, similarly as in the Altmann’s theory, rather than in terms of permutations and
*
Corresponding author. Fax: +98 21 6412178. E-mail address: [email protected] (M.R. Darafsheh).
0009-2614/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2006.02.008
permutations-inversions. This way of expressing the nonrigid operations is indeed more descriptive and flexible [3]. The complete set of molecular conversion operations which commute with the nuclear motion operator contains overall rotation operations, describing the molecule rotating as a whole, and the non-rigid tunneling motion operations, describing molecular moieties moving with respect to the rest of the molecule. Such a set forms a group, which we call the full non-rigid group (f-NRG). Longuet-Higgins [1] investigated the symmetry groups of non-rigid molecules, where changes from one conformation to another can occur easily. In many cases, these dynamical symmetry groups are not isomorphic with any of the familiar point groups of rigid molecules and their character tables are not known. It is therefore of considerable interest and importance to develop simple methods to calculate these character tables, which are needed for the classification of wave functions, correlation of the rovibronic states, determination of selection rules, hyperfine splittings of non-rigid species, and so on. We describe here a general method that is appropriate for molecules which consist of a number of XH3 or XO2 groups attached to a rigid framework. Examples of such molecules are hexamethylbenzene, which is considered here in some detail. For the linear framework, a similar problem has been studied by Bunker [4].
M.R. Darafsheh et al. / Chemical Physics Letters 421 (2006) 566–570
Lomont [5] has given two methods for calculating character tables. These methods are satisfactory for small groups, but both of them require knowledge of the class structure and hence of the group multiplication table and they become very massive as the order of the group becomes even moderately large. For non-rigid molecules, whose symmetry groups may have several thousand elements, they are usually quite impracticable. Group theory for non-rigid molecules is becoming more and more relevant and numerous applications to large amplitude vibrational spectroscopy of small organic molecules have appeared in the literature [5–7]. Our approach here is first to specify the algebraic structure of the full non-rigid group of molecules that contain 6 groups of non-rigid internal rotors such as hexamethylbenzene and a weakly-bound ammonia hexamer (NH3)6. With a geometric consideration of dynamic symmetries of the molecules, we will show that the f-NRG of this molecule can be specified by wreath product of some known groups. Then based on the structure of the group we apply GAP [8] as a useful package for computing the character tables and even the group structure to compute the character table of the f-NRG of this molecule, see Fig. 1. We use [9] for the standard notations and terminology of character theory. In this Letter, we use the Balasubramanian’s method [10–20], for computing the non-rigid group of hexamethylbenzene molecule by wreath product formalism. It is shown that this is a group of order 8748 with 174 conjugacy classes. The motivation for this study is outlined also in [21–25] and the reader is encouraged to consult these Letters for background material as well as basic computational techniques. The developed technique is not only applicable for the hexamethylbenzene molecule but also for ammonia hexamer. As seen from the literature [18,26–29] hydrogen-bonded and van der Waals complexes exhibit tunneling motions among different potential minima. Thus, the symmetry groups of these non-rigid complexes can also be expressed as wreath product groups that we have considered here. Computations were carried out with the aid of GAP and this was done by characterizing the algebraic structure of f-NRG as the wreath product of known groups.
Fig. 1. The structure of hexamethylbenzene.
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2. Full non-rigid group of hexamethylbenzene In this section we first describe some notation, which will be kept throughout. Let G be a group and N be a subgroup of G. N is called a normal subgroup of G, if for any g 2 G and x 2 N, g1 · g 2 N. Moreover, if H is another subgroup of G such that H \ N = {e} and G = HN = {xy j x 2 H, y 2 N}, then we say that G is a semidirect product of H by N denoted by H N. Suppose X is a set. The set of all permutations on X, denoted by SX, is a group which is called the symmetric group on X. In the case that X = {1, 2, . . . , n}, we denote SX by Sn or Sym(n). Let H be a permutation group on X, a subgroup of SX, and let G be a group. The set of all mappings X ! G is denoted by GX, i.e. GX = {f j f: X ! G}. It is clear that jGXj = jGjjXj. We put H[G] = G H = GX · H = {(f; p) j f 2 GX, p 2 H}. For f 2 GX and p 2 H, we define fp 2 GX by fp = fop1, where ‘o’ denotes the composition of functions. It is easy to check that the following law of composition: ðf ; pÞðf 0 ; p0 Þ ¼ ðff 0p ; p p0 Þ, makes G H into a group. This group is called the wreath product of G by H and sometimes is denoted by H[G] as well. Before going into the details of the computations of hexamethylbenzene we should mention that we consider the speed of rotations of the methyl groups sufficiently high so that the mean time dynamical symmetry of the molecules makes sense. In order to characterize the f-NRG of hexamethylbenzene, we first note that each dynamic symmetry operation of this molecule, considering the rotations of CH3 groups, is composed of two sequential physical operations. We first have a physical symmetry of the hexagonal framework (as we have to map the CH3 groups on CH3 groups which are on vertices of the hexagonal framework). Such operations are exactly the symmetry operations of a hexagon and, as is well known, such operations form the group D6 of order 12. After accomplishing the first framework symmetry operation we have to map each of the six CH3 group on itself which forms the three elements group C3, also denoted by Z3. The number of all such operations is 36 · 12 = 8748. The composition of such dynamic symmetry elements are described as follows. Let us use numbers {1, 2, 3, 4, 5, 6} to indicate the carbon atoms of hexagonal framework. Now from the symmetry point of view the 1, 2, 3, 4, 5, 6 carbons (corners of the hexagonal framework) are important and the remaining carbons follow the motions of 1, 2, 3, 4, 5, 6 carbons. This means that for computing the f-NRG it is enough to consider the hexagon (1, 2, 3, 4, 5, 6) with rotating CH3 on its corners. We can describe such configuration with a 3 · 6 matrix [ij]3·6 where ij means the ith hydrogen on the jth carbon corner. Now the dynamic symmetry operation described above has the form (a1, a2, a3, a4, a5, a6; r), where r is a symmetry of the hexagonal framework (the first physical operation as above) and if h : {1, 2, 3, 4, 5, 6} ! {e, f, g} is a function into the cyclic group {e, f, g} of order three with identity element e then we write ai = h(i),
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Table 1 The representatives of the conjugacy classes of the wreath product, D6[Z3], the non-rigid group of hexamethylbenzene containing 8748 operations No.
Representatives
Size
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 42 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
(e,e,e,e,e,e,()) (g,g,f,g,f,e,(1,6,5,4,3,2)) (e,f,e,f,g,e,(1,2,3,4,5,6)) (g,f,e,e,e,f,(1,5,3)(2,6,4)) (g,g,g,f,e,f,(1,3,5)(2,4,6)) (g,g,g,g,g,g,()) (f,f,f,f,f,f,()) (e,f,g,g,f,e,(1,4)(2,5)(3,6)) (f,g,e,e,g,f,(1,4)(2,5)(3,6)) (g,e,f,f,e,g,(1,4)(2,5)(3,6)) (f,e,e,f,e,f,(1,5,3)(2,6,4)) (f,g,f,g,f,g,()) (e,f,e,f,e,f,()) (e,g,e,g,e,g,()) (f,f,e,g,g,g,(1,3,5)(2,4,6)) (f,e,g,f,e,e,(1,5,3)(2,6,4)) (g,g,e,f,g,f,(1,3)(4,6)) (e,f,f,g,f,g,(1,3)(4,6)) (g,f,g,g,f,g,()) (f,g,f,f,g,f,()) (f,e,g,e,e,e,(1,3)(4,6)) (g,e,g,f,f,f,()) (f,e,f,g,g,g,()) (e,g,f,e,g,e,(1,3)(4,6)) (g,f,e,e,f,e,(1,3)(4,6)) (f,f,f,e,f,e,()) (g,g,g,e,g,e,()) (g,e,g,g,f,g,()) (f,e,f,f,g,f,()) (g,g,f,f,g,g,()) (f,f,g,g,f,f,()) (f,e,f,e,g,g,(1,6)(2,5)(3,4)) (f,f,e,g,e,g,(1,6)(2,5)(3,4)) (e,g,f,f,g,e,()) (e,f,g,g,f,e,()) (f,g,g,f,f,g,(1,6)(2,5)(3,4)) (f,g,e,g,f,g,(1,6)(2,5)(3,4)) (f,g,f,e,f,g,(1,6)(2,5)(3,4)) (e,e,g,g,e,e,()) (e,e,f,f,e,e,()) (e,g,g,g,g,e,()) (e,f,f,f,f,e,()) (f,e,g,f,f,g,()) (g,e,f,g,g,f,()) (f,g,g,g,g,g,()) (g,f,f,f,f,f,()) (g,g,e,g,f,g,()) (f,f,e,f,g,f,()) (e,g,f,g,e,f,()) (e,f,g,f,e,g,()) (f,g,f,g,e,g,()) (g,f,g,f,e,f,()) (g,f,e,e,e,f,()) (f,g,e,e,e,g,()) (f,g,e,g,e,e,(2,6)(3,5)) (g,e,e,f,e,f,(2,6)(3,5)) (g,g,e,f,e,g,()) (f,f,e,g,e,f,()) (e,g,f,e,f,e,(2,6)(3,5)) (e,e,g,e,g,f,(2,6)(3,5)) (e,g,g,e,g,g,()) (e,f,f,e,f,f,()) (e,e,f,f,e,g,(1,4)(2,5)(3,6))
1 486 486 162 162 1 1 27 27 27 324 2 2 2 324 324 27 27 3 3 27 6 6 54 54 6 6 6 6 6 6 162 162 6 6 81 162 162 6 6 6 6 12 12 6 6 12 12 6 6 6 6 6 6 54 54 6 6 27 27 3 3 81
Table 1 (continued) No.
Representatives
Size
64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128
(g,e,f,e,e,g,(1,4)(2,5)(3,6)) (f,e,e,f,e,e,()) (g,e,e,g,e,e,()) (g,e,g,e,f,f,(1,4)(2,3)(5,6)) (e,f,e,f,g,g,(1,4)(2,3)(5,6)) (e,f,f,e,g,g,()) (f,g,e,f,f,g,(1,4)(2,3)(5,6)) (g,e,f,g,f,g,(1,4)(2,3)(5,6)) (f,f,e,e,g,e,(1,4)(2,3)(5,6)) (e,e,g,g,e,f,(1,4)(2,3)(5,6)) (g,e,f,g,f,g,()) (f,e,g,f,g,f,()) (e,g,e,f,f,g,()) (e,f,e,g,g,f,()) (f,f,g,f,g,g,(1,3)(4,6)) (e,g,e,e,f,e,()) (e,e,e,f,g,f,(2,6)(3,5)) (e,g,f,g,e,e,(2,6)(3,5)) (e,e,g,g,f,f,(2,6)(3,5)) (e,g,g,f,f,e,(2,6)(3,5)) (g,g,g,e,f,e,(2,6)(3,5)) (f,e,g,e,f,f,(2,6)(3,5)) (e,e,g,f,e,e,(2,6)(3,5)) (e,e,e,g,f,e,(2,6)(3,5)) (e,e,g,g,g,e,()) (e,e,f,f,f,e,()) (g,e,g,f,e,g,(2,6)(3,5)) (f,f,e,g,f,e,(2,6)(3,5)) (g,e,e,f,f,e,()) (f,e,e,g,g,e,()) (g,e,e,g,e,g,()) (f,e,e,f,e,f,()) (f,e,e,e,g,e,()) (f,e,g,e,g,e,()) (g,e,f,e,f,e,()) (g,e,g,g,f,e,()) (f,e,f,f,g,e,()) (e,g,e,g,e,e,()) (e,f,e,f,e,e,()) (g,g,g,e,e,e,(1,5)(2,4)) (e,e,f,f,f,e,(1,5)(2,4)) (g,e,e,f,e,e,(1,5)(2,4)) (g,f,e,f,g,e,()) (f,g,f,g,f,f,(1,3)(4,6)) (g,f,g,g,g,f,(1,3)(4,6)) (g,f,g,e,g,e,()) (f,g,f,e,f,e,()) (e,g,g,g,g,f,(1,3)(4,6)) (f,f,e,g,f,f,(1,3)(4,6)) (g,e,g,f,g,e,(1,3)(4,6)) (f,e,f,e,f,g,(1,3)(4,6)) (f,e,f,f,f,f,()) (g,e,g,g,g,g,()) (g,f,f,g,f,f,(1,3)(4,6)) (g,g,f,g,g,f,(1,3)(4,6)) (e,e,e,f,e,f,(1,4)(2,5)(3,6)) (g,e,g,e,e,e,(1,4)(2,5)(3,6)) (f,e,g,f,e,g,()) (e,g,g,g,f,g,(1,4)(2,5)(3,6)) (e,e,f,e,f,g,()) (e,e,g,e,g,f,()) (g,f,f,f,g,f,()) (f,g,g,g,f,g,()) (f,e,g,g,e,e,(1,3)(4,6)) (f,e,g,e,e,f,(1,3)(4,6))
81 3 3 81 81 6 162 162 162 162 12 12 12 12 54 6 54 54 54 54 54 54 54 54 6 6 54 54 12 12 12 12 12 6 6 12 12 6 6 54 54 54 6 54 54 6 6 54 54 54 54 6 6 27 27 81 81 6 162 12 12 6 6 54 54
M.R. Darafsheh et al. / Chemical Physics Letters 421 (2006) 566–570 Table 1 (continued) No.
Representatives
Size
129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174
(e,f,e,f,e,e,(2,6)(3,5)) (e,e,e,g,e,g,(2,6)(3,5)) (f,g,f,e,e,e,(1,4)(2,5)(3,6)) (e,e,e,g,f,g,(1,4)(2,5)(3,6)) (f,g,e,f,g,e,(1,3)(4,6)) (e,f,g,e,f,g,(1,3)(4,6)) (e,e,f,g,g,g,()) (e,e,g,f,f,f,()) (e,e,f,g,e,f,()) (e,e,g,f,e,g,()) (e,e,e,e,e,g,()) (e,e,e,e,e,f,()) (f,e,f,e,e,e,(1,6)(2,5)(3,4)) (e,e,e,g,e,g,(1,6)(2,5)(3,4)) (f,e,f,e,f,e,(1,3,5)(2,4,6)) (g,f,e,f,g,e,(1,4)(2,3)(5,6)) (e,f,e,g,e,f,(1,4)(2,3)(5,6)) (f,e,g,e,g,e,(1,4)(2,3)(5,6)) (f,f,g,g,f,g,()) (f,g,f,e,e,f,()) (g,f,g,e,e,g,()) (f,g,g,e,e,g,(2,6)(3,5)) (g,f,e,e,f,f,(2,6)(3,5)) (f,e,f,f,g,g,()) (g,e,g,g,f,f,()) (e,e,f,g,f,e,(1,6)(2,5)(3,4)) (e,g,f,g,e,e,(1,6)(2,5)(3,4)) (g,g,g,f,g,e,(2,6)(3,5)) (f,f,f,e,g,g,()) (g,g,g,e,f,f,()) (e,g,g,e,f,f,(1,2,3,4,5,6)) (g,f,e,g,g,g,()) (f,g,e,f,f,f,()) (g,f,f,f,g,g,()) (f,e,e,e,g,g,()) (g,e,e,e,f,f,()) (g,f,g,g,f,e,(1,5)(2,4)) (g,f,f,g,f,e,(1,5)(2,4)) (g,f,e,e,f,f,(1,3)(4,6)) (e,g,f,g,g,e,(1,3)(4,6)) (g,e,g,f,e,g,()) (f,e,f,g,e,f,()) (f,f,e,e,g,g,()) (g,g,e,f,g,g,(2,6)(3,5)) (f,g,e,e,e,e,()) (g,f,e,e,g,f,())
54 54 81 81 27 27 12 12 12 12 6 6 81 81 162 162 81 81 12 12 12 54 54 12 12 81 81 54 12 12 486 12 12 6 12 12 54 54 54 54 12 12 12 54 12 12
1 6 ai 6 6. The group V of the symmetries of the molecule acts on the 18 entries of [ij]3·6 as follows. Consider (h;r) and [ij], we first do h on the ith hydrogen by the rule h(r(j))(i), and then we do r on the jth corner carbon to obtain [h(r(j))(i)r(j)] which shows the new position of molecule. This composition rule is the wreath product composition. Therefore the f-NRG group G of hexamethylbenzene is isomorphic to Z3 D6. We now apply GAP to obtain the conjugacy classes and character table of the group. Simple counting shows that the number of elements of wreath product D6[Z3] group is 12 · 36 = 8748. Table 1 shows the conjugacy classes, a representative from each class and the order of the conjugacy class. Since the character table of the group, Table 2, is very large in size, it
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is deposited as a supplementary material to this Letter. In supplementary Table, /X denotes the complex conjugate of X. Also, the values A, B, . . ., O are as follows: A ¼ eip=3 ; B ¼ 2e2ip=3 ; C ¼ 3e2ip=3 ; D ¼ 6e2ip=3 ; p E ¼ 12e2ip=3 ; F ¼ 3; G ¼ 2A A2 ; H ¼ 4A; I ¼ 2F ; J ¼ 3 F ; K ¼ 6 2F ; L ¼ 4F ; M ¼ A2 5A; N ¼ ð9 þ F Þ=2; O ¼ ð3 5F Þ=2 3. Conclusion We have developed the group theory and character table of the non-rigid hexamethylbenzene and ammonia hexamer for the first time as a wreath product group D6[Z3] = Z3 D6 and it consists of 8748 operations divided into 174 conjugacy classes and irreducible representations. The derived character tables would also be valuable in other applications such as in the context of chemical applications of graph theory [30] and aromatic compounds [31]. In the case of chemical applications of graph theory, applications can range from enumeration f isomers to the automorphism groups of chemical graphs. In other fields such as theory of quarks and generalized special unitary groups, such wreath products and their double groups find important applications. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.cplett. 2006.02.008. References [1] H.C. Longuet-Higgins, Mol. Phys. 6 (1963) 445. [2] Y.G. Smeyers, Adv. Quantum Chem. 24 (1992) 1. [3] S.L. Altmann, Induced Representation in Crystal and Molecules, Academic Press, London, 1977. [4] P.R. Bunker, Molecular Symmetry in Spectroscopy, Academic, New York, 1979. [5] J.S. Lomont, Applications of Finite Groups, Academic, New York, 1959. [6] A.J. Stone, J. Chem. Phys. 41 (1964) 1568. [7] Y.G. Smeyers, M. Villa, J. Math. Chem. 28 (2000) 377. [8] M. Scho¨nert, H.U. Besche, Th. Breuer, F. Celler, B. Eick, V. Felsch, A. Hulpke, J. Mnich, W. Nickel, G. Pfeiffer, U. Polis, H. Theißen, A. Niemeyer, GAP, Groups, Algorithms and Programming, Lehrstuhl De fu¨r Mathematik, RWTH, Aachen, 1995.. [9] I.M. Isaacs, Character Theory of Finite Groups, Academic, New York, 1978. [10] K. Balasubramanian, J. Chem. Phys. 72 (1980) 665. [11] K. Balasubramanian, J. Chem. Phys. 73 (1980) 3321. [12] K. Balasubramanian, J. Chem. Phys. 75 (1981) 4572. [13] K. Balasubramanian, Int. J. Quantum. Chem. 22 (1982) 1013. [14] K. Balasubramanian, J. Chem. Phys. 78 (1983) 6358. [15] K. Balasubramanian, J. Chem. Phys. 78 (1983) 6369. [16] K. Balasubramanian, Studies Phys. Theor. Chem. 23 (1983) 149. [17] K. Balasubramanian, Theoretica Chimica Acta 78 (1990) 31. [18] K. Balasubramanian, J. Phys. Chem. 108 (2004) 5527. [19] K. Balasubramanian, Chem. Phys. Lett. 391 (2004) 64.
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[20] K. Balasubramanian, Chem. Phys. Lett. 398 (2004) 15. [21] A.R. Ashrafi, MATCH Commun. Math. Comput. Chem. 53 (2005) 161. [22] M.R. Darafsheh, A.R. Ashrafi, A. Darafsheh, Int. J. Quantum Chem. 105 (2005) 485. [23] M.R. Darafsheh, A.R. Ashrafi, A. Darafsheh, Acta Chim. Slov. 52 (2005) 282. [24] M.R. Darafsheh, Y. Farjami, A.R. Ashrafi, Bull. Chem. Soc. Jpn. 78 (2005) 996. [25] M.R. Darafsheh, Y. Farjami, A.R. Ashrafi, MATCH Commun. Math. Comput. Chem. 54 (2005) 53.
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