Group theory, nuclear spin statistics and tunneling splittings of 1,3,5-triamino-2,4,6-trinitrobenzene

Chemical Physics Letters 398 (2004) 15–21 www.elsevier.com/locate/cplett

Group theory, nuclear spin statistics and tunneling splittings of 1,3,5-triamino-2,4,6-trinitrobenzene Krishnan Balasubramanian

*

Chemistry and Material Science Directorate, Lawrence Livermore National Laboratory, University of California, Livermore, CA 94550, USA Institute of Data Analysis and Visualization, University of California Davis, P.O. Box 808 L-268, Livermore, CA 94550, USA Glenn T Seaborg Center, Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA Received 16 August 2004; in final form 2 September 2004 Available online 28 September 2004

Abstract The symmetry group of the non-rigid 1,3,5-triamino-2,4,6-trinitrobenzene (TATB) is constructed for the internal rotations of the nitro groups that exhibit low rotation barriers of 5 kcal/mole. The permutation group has 48 operations characterized by the wreath product S3[S2]. The nuclear spin statistical weights for TATB with 17O, 14N and 1H species are computed and the tunneling splitting patterns of the rotational and rovibronic levels of TATB are constructed for the first time. It is shown that the ground rovibronic state of TATB is split into A1, A3, T1 and T3 levels.
2004 Elsevier B.V. All rights reserved.

1. Introduction Spectroscopy of 1,3,5-triamino-2,4,6-trinitrobenzene (TATB) is becoming the focus of recent studies [1,2]. General interest on TATB is due to its potential as a high-energy material and its stability under thermal, impact or shock initiation conditions [3–20]. The symmetry aspects of TATB have been explored in the context of unusual second harmonic generation of TATB [10,11] due to the centrosymmetric crystal structure of TATB [8,9]. Some possible explanations for the observed SHG of TATB varied from polymorphism [10], noncentro symmetric stacking of molecular layers [11], to the centrosymmetric deformation of the geometry [12]. There is also considerable interest in the development of graph theoretical techniques for the characterization of related polycyclic aromatics [21,22]. * Address: Institute of Data Analysis and Visualization, University of California Davis, P.O. Box 808 L-268, Livermore, CA 94550, USA. Fax: +1 925 422 6810. E-mail address: [email protected].

0009-2614/$ - see front matter
2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.09.032

Recent temperature-dependent Raman spectra of TATB have revealed [1] that there are more peaks in the spectra than expected or could be attributed to a rigid D3h vibrational mode analysis. As noted by McGrane and Shreve [1], the observed spectra could not be assigned unambiguously due to the presence of more peaks and complex normal modes of the TATB molecule. The temperature dependent spectra of TATB show additional peaks than expected for a rigid TATB. Indeed this appears to be consistent with the group theoretical analysis and predictions in the current work that show splitting of the rovibronic levels of TATB into tunneling levels due to the internal rotation of the nitro groups in TATB. The recent UV-spectra of TATB demonstrate that radicals of TATB can be generated by exposure to irradiation. Hence there is compelling need to understand the symmetry and rovibronic levels. Manna et al. [17] have shown by extensive ab initio computations that the barrier to internal rotation of the nitro groups is only
5.6 kcal/mole suggesting that the TATB molecule would exhibit considerable non-rigidity due to torsion of

16

K. Balasubramanian / Chemical Physics Letters 398 (2004) 15–21

the nitro groups. The barrier for the internal rotation to the amino group is considerably larger (
17 kcal/mole) due to a partial C–N double bond of the anilinic group. Thus the TATB molecule is an interesting case of a nonrigid molecule due to the internal rotations of the nitro groups. The symmetry group of the non-rigid TATB has not been studied up to date. This is especially important for the analysis and assignment of the spectra that exhibit tunneling splittings due to the internal rotation of the nitro groups. Moreover 17O NMR spectroscopy can be a powerful tool for the study of nitro group internal rotations. Such studies would be benefited by the nuclear spin statistical analysis of 17O TATB that we provide here. We have developed the symmetry group of the non-rigid TATB, its nuclear spin statistics and the patterns of rovibronic levels splittings due to the internal rotation of the nitro groups.

2. Wreath product group approach to non-rigid TATB The concept of symmetry group of non-rigid molecules as a permutation-inversion group was formulated first by Longuet-Higgins [23] although there have been earlier works that suggested the need for such a framework. Balasubramanian [24] showed that these groups could be cast into elegant algebraic structures known as wreath product groups. The character theory of wreath product groups has been considered and discussed [25,26]. In particular the development of generators for the characters of wreath product groups has provided a general algorithm for the construction for the character tables of wreath product groups [25]. We show here that the wreath product formalism leads to a permutation group of order 48 for the symmetry group of non-rigid TATB. Other applications of wreath products can be seen in [26–31]. The molecular framework of TATB is shown in Fig. 1. The rigid TATB molecule has D3, D3h and even lower symmetries at various levels of theories. Our recent work shows a D3 to D3h equilibrium structure at higher levels. In the present work the non-rigid group of TATB is formulated by allowing for the internal rotations of the nitro groups. It is assumed here that the amino groups would not exhibit large amplitude motions due to a much larger barrier for this process [17]. We first develop the group of all permutation operations and then introduce the inversion operation since the total group is a direct product of the permutation and inversion groups. Let all the permutations of the rigid framework with D3 symmetry be collected into a group which we call G, and as a set of permutations, G is isomorphic to the S3 permutation group in the mathematical literature [32]. In general the Sn group consists of n! permutations of n objects in a set X. Thus

Fig. 1. Geometry of TATB.

the permutations of the nuclei of the rigid TATB framework would consist of six permutations isomorphic to both S3 permutation group and the D3 point group. The internal rotations of the nitro group switch the oxygen atoms. Let the permutations of the oxygen atoms of each nitro group be the group H, which in this case, is the S2 group of permutations. The overall group of TATB then becomes the wreath product of G with H, denoted by G[H]. Specifically for TATB the wreath product group becomes S3[S2]. The group G[H] consists of permutations fðg; pÞ j p mapping of X into H; g 2 Gg; with the composition law ðg; pÞðg; p0 Þ ¼ ðgg0 ; pp0g Þ; where pg ðiÞ ¼ pðg1 iÞ; 8i 2 X; pp0 ðiÞ ¼ pðiÞp0 ðiÞ; 8i 2 X: Since the non-rigidity of TATB arises from the internal rotations of the nitro groups, for simplicity, we can define the set X as the nitrogen nuclei of the nitro groups of TATB. Then all the symmetry operations of non-rigid TATB can be represented as permutations of the oxygen nuclei. The order of the group G[H] is jGjjHjn. For the TATB this means the order of the permutation group is 3

j S3 ½S2  j¼ 3!ð2!Þ ¼ 48: The conjugacy classes of TATB group are characterized by matrix types defined as follows. Let a permutation g 2 G give rise to a1 cycles of length 1, a2 cycles of length 2, . . ., an cycles of length n upon its action on the set X. The cycle type of G is then denoted by Tg = (a1, a2, . . ., an). To illustrate a permutation (1)(23) of the nitrogen nuclei of all three nitro groups would

K. Balasubramanian / Chemical Physics Letters 398 (2004) 15–21

correspond to the cycle type (1,1,0). Let C1, C2, . . ., Cs be the conjugacy classes of the group H. Since G is a complete Sn group of n! permutations we can cast the cycle type of an element in the wreath product G[H] by a s · n matrix, T(g;p) called the cycle type of (g;p) defined as follows. If aik of the cycle products of G belong to the conjugacy class Ci we can define the cycle type of (g;p), which represents the conjugacy class of Sn[H] as Tðg; pÞ ¼ aik

ð1 6 i P s; 1 6 k P nÞ:

Table 1 shows all possible matrix types that represent the conjugacy classes of the group S3[S2] of TATB. The number of conjugacy classes and the number of elements in each conjugacy class are obtained using the following technique. Let P(m) denote the number of partitions of integer m with the convention that P(0) = 1. Let n be partitioned into ordered s-tuples (s = no of conjugacy P classes of H), (n) = (n1, n2, . . ., ns) such that i ni ¼ n. The number of conjugacy classes of Sn[H] is given by X Pðn1 ÞPðn2 Þ    Pðns Þ: n

Thus for the TATBÕs S3[S2] group the ordered partitions are (3,0), (0,3), (2,1), (1,2) since the S2 group has 2 conjugacy classes. By substituting the values of P(3) = 3, P(2) = 2, P(1) = 1 and P(0) = 1 in the above expression, we obtain the number of conjugacy classes of S3[S2] as 3 + 3 + 2 + 2 = 10. Ten classes thus obtained are shown in Table 1. The number of elements in each conjugacy class of Sn[H] is given by Q

jSn ½Hj a : aik !ðk jHj=jCi jÞ ik

ð1Þ

i;k

Table 1 Conjugacy classes of the non-rigid group of TATB No 1  2 3  

4 5 6

 

7

8  9  10 

Matrix
3 0 0 0
2 0 1 0
1 0 2 0
0 0 3 0
1 1 0 0
0 1 1 0
1 0 0 1
0 0 1 1
0 0 0 0
0 0 0 0



0 0  0 0  0 0  0 0  0 0  0 0  0 0  0 0  1 0  0 1

17

For example, the order of the fifth conjugacy class in Table 1 is given by 6:23 1

1!ð1:2=1Þ 1!ð2:2Þ

1

¼ 6:

The character table of the S3[S2] group of TATB can be generated using powerful combinatorial generating functions that involve the matrix type manipulations. First the irreducible representations of the S3[S2] group are generated using CliffordÕs algebra of induced representations from a smaller group to a larger group [24,32]. The irreducible representations of the Sn group are well known to be characterized by the partitions of n [32]. They are denoted by [n1, n2, . . ., nm], where n1, n2, . . ., nm is a partition of the integer n. Thus the irreducible representations of the S2 group are [2] and [12], while the irreducible representations of the group S3 are [3], [21] and [13]. As reviewed by one of the authors before [24], the irreducible representations of S3[S2] are constructed by first forming the outer tensor (outer direct) products of the representations of S2 three times (sine there are three nitro groups in TATB) and then finding the inertia factor of each such product, and subsequently inducing the representation from the inertia factor group to the whole group. The unique outer products of for S2 · S2 · S2 are given by ½2#½2#½2; ½2#½2#½12 ; ½12 #½12 #½2; and ½12 #½12 #½12 : The inertia factor groups of the above products are S03 ; S02 ; S02 and S03 , respectively. The overall representations of the S3[S2] groups are obtained by multiplying the above outer products with the irreducible representations of the factor group for each product and then inducing the whole product into S3[S2]. That is, the irreducible representations of S3[S2] are given by

Permutation

Order

16

1

A1 ¼ ð½2#½2#½2Þ  ½3 ;

142

3

E1 ¼ ð½2#½2#½2Þ  ½21 ;

1222

3

3

2

1

T2 ¼ ð½2#½2#½12 Þ  ½12  " S3 ½S2 ;

1222

6

T3 ¼ ð½12 #½12 #½2Þ  ½2 " S3 ½S2 ;

23

6

2

14

6

E1 ¼ ð½12 #½12 #½12 Þ  ½210 ;

24

6

A4 ¼ ð½12 #½12 #½12 Þ  ½13 0 ;

32

8

6

8

0

0

0

A2 ¼ ð½2#½2#½2Þ  ½13  ; 0

T1 ¼ ð½2#½2#½12 Þ  ½2 " S3 ½S2 ; 0 0

0

T4 ¼ ð½12 #½12 #½2Þ  ½12  " S3 ½S2 ; 0

A3 ¼ ð½12 #½12 #½12 Þ  ½3 ;

where the › stands for the induced representation from the factor group to the whole group [24]. Consequently there are 10 irreducible representations in the S3[S2] wreath product group.

18

K. Balasubramanian / Chemical Physics Letters 398 (2004) 15–21

The character table of the S3[S2] group is generated using the generating functions for each of the above 10 irreducible representations. Balasubramanian [25] has developed the generating function algorithm for the characters of the wreath product groups. According to this method let PvG stand for the generalized character cycle index [31] of the irreducible representation v of the factor subgroup G 0 of G as determined by the outer product. PvG is given by 1 X PvG ¼ vðgÞsb11 sb22    sbn n : jGj g2G Let T(M)i be the matrix type of the representation of the inertia factor. Then the generating function for the irreducible representation F* of Sn[H] is obtained by the following replacement: F

TðG½HÞ

¼ PvG ðsi ! TðMÞi Þ:

In the above expression once every si is replaced by the correpsonding matrix type T(M)i, algebraic manipulations are done with the cycle type matrices. To differentiate that these are not the usual additions and multiplications of matrices (as they involve cycle types), we have introduced ¯, ,  operations for additions, multiplications and subtractions. For example, the following examples illustrate how cycle type matrices are multiplied.
3
 1 0 0 3 0 0 ¼ ; 0 0 0 0 0 0





1

0

0

0

0

1 0




0

0

0

0

1

0

0

0

0

2


0

0




¼

0

0

0

1

0

0



1

0 0

1

0 0


¼

 ;

2

0 0

1

0 0

 :

As an illustration let us consider two examples. First consider the A2 irreducible representation given by 0 0 [2]#[2]#[2] [13] . Since [13] is the factor group irreducible representation, its generalized character cycle index is given by  1 ½13  PS3 ¼ s31  3s1 s2 þ 2s3 : 6 The basic matrix types are given as follows for the representation [2] are given by ½2

TðMÞ1 ¼

1 2

½2 TðMÞ2

1 ¼ 2

½2 TðMÞ3

1 ¼ 2









1 0

0 0


0 0 0  0 1 0

0

1

0

0

0

0

0

0

1

0

0

0











0 0

0 0

0

0 1

0

0 0

0

0 0

1

 ;  ;  :

½13 

Replacing every si by T(M)i in the expression for PS3 we obtain " 

3 1 0 0 0 0 0 1 1 GFA2 ¼  6 23 0 0 0 1 0 0 

 1 0 0 0 0 0 1 3 2  0 0 0 1 0 0 2 

 0 1 0 0 0 0   0 0 0 0 1 0 

 0 0 1 0 0 0 1 þ2 1  : 0 0 0 0 0 1 2

By using the matrix manipulations indicated above for the cycle types, we can simplify the above expression into


 3 0 0 2 0 0 1 1 GFA2 ¼ þ 3 6 23 0 0 0 1 0 0


 1 0 0 0 0 0 1 1 0 þ3 þ 6 2 0 0 3 0 0 0 0 0


 0 1 0 1 0 0 0 0 0 6 6 6 1 0 0 0 1 0 1 1 0

 0 0 0 0 0 1 þ8 þ8 : 0 0 0 0 0 1 The coefficients in the above expression with the order of the group (48) factored out give the product of the character of a conjugacy class and the number of elements in the corresponding conjugacy class. As seen from Table 1, ten matrix types in the above expression are the conjugacy classes of the group S3[S2]. The string of coefficients {1, 3, 3, 1, 6, 6, 6, 6, 6, 8, 8} in the above expression generates the character {1, 1, 1, 1, 1, 1, 1, 1, 1, 1} when the number of elements in each conjugacy class is factored out. Table 2 shows the entire character table of the group S3[S2] thus constructed using the matrix generating function method outlined above. As seen from Table 2, the character table consists of 4 one-dimensional irreducible representations, 2 two-dimensional representations and 4 three-dimensional representations satisfying the condition 4.12 + 2.22 + 4.42 = 48, the order of the group S3[S2]. Moreover, using a computer code, we have checked for the orthogonality of every row of the character table with respect to every other row, thus confirming compliance of the great orthogonality theorem. The character table thus generated has all permutations of the non-rigid group of TATB. The inversion operation [23] does not generate any new permutation and it is thus the E* operation and thus the whole permutation-inversion group is the direct product of the S3[S2] group and the inversion group. Thus the character table of PI group of 96 operations is readily generated from Table 2, by replicating blocks of the same table along

K. Balasubramanian / Chemical Physics Letters 398 (2004) 15–21

19

Table 2 Character table of the non-rigid group of TATB

A1 A2 A3 A4 E1 E2 T1 T2 T3 T4

16

142

1222

23

1222

23

124

24

32

6

1 1 1 1 1 2 2 3 3 3 3

3 1 1 1 1 2 2 1 1 1 1

3 1 1 1 1 2 2 1 1 1 1

1 1 1 1 1 2 2 3 3 3 3

6 1 1 1 1 0 0 1 1 1 1

6 1 1 1 1 0 0 1 1 1 1

6 1 1 1 1 0 0 1 1 1 1

6 1 1 1 1 0 0 1 1 1 1

8 1 1 1 1 1 1 0 0 0 0

8 1 1 1 1 1 1 0 0 0 0

the rows and columns and multiplying the last block by –1. This is analogous to how the character table of the rigid planar structure, D3h is generated from the D3 group since the inversion operation for the rigid structure also corresponds to E*, and is thus a direct product. Since the inversion operation does not generate any new permutation operations, it suffices to work with the permutation group S3[S2], isomorphic with Oh, for all applications. 3. Tunneling splittings of rotational/rovibronic levels and nuclear spin statistics of non-rigid TATB Table 3 shows the tunneling splitting patters of the rotational levels of TATB by correlating the D(J) representation into the D3 (rigid) and S3[S2] groups using the subduced and induced representation techniques. As seen from Table 3, even in the ground rotational state, non-rigid TATB exhibits tunneling splitting into A1 + A3 + T1 + T3 levels. The realistic description involves the direct product of the tunneling, vibrational and electronic irreducible representations. Since the ground electronic state of TATB is a 1A1 state the overall rovibronic representation is the product of the tunneling and vibrational irreducible representations. It is thus evident that the coupling of tunneling levels with the vibrational levels can influence the observed spectral splittings as more spectral lines than expected consistent with the recent experimental study [1]. The coupling be-

tween tunneling levels and vibrational levels would be larger in the excited vibrational levels due to the break down of Born–Oppenheimer approximation due to non-rigidity of TATB. In order to complete the analysis of the TATB rovibronic levels, we have provided the nuclear spin species and spin statistical weights. The nuclear spin species can also be helpful in analyzing the 17O NMR or 17O multiple quantum NMR spectra of TATB. Since 17O NMR can be a powerful tool to confirm the internal rotations of the nitro groups predicted theoretically [17], we have provided the 17O nuclear spin species data in the S3[S2] group. The spin statistics is obtained using a combinatorial multinomial generating function method [27,28,31] using the character table shown in Table 2. However, the 17O nuclear spin generation was quite challenging in that 17O has a spin 5/2 nucleus and thus exhibits six different spin states per nucleus. Such spin analysis with group theoretical induction and subductions have been done extensively [33,34], and also the character-based cycle indices have been used in many applications [31] including chirality [35]. Let us represent the 6 ms spin functions of 17O by a1, a2, . . ., a6, where a1 represent ms = 5/2, a2 represents ms = 3/2, . . ., a6 representing ms = 5/2. To illustrate consider the GCCI of the A4 representation from Table 2 is given by 4 PA G ¼

1 6 s  3s41 s2 þ 3s21 s22  3s32  6s21 s22 þ 6s32 þ 6s21 s4 48 1   6s2 s4 þ 8s23  8s16 :

Table 3 Tunneling splittings of rotational levels non-rigid TATB J

Rigid(D3)

Non-rigid

0 1 2 3 4 n

A1 A2 + E A1 + 2E A1 + 2A2 + 2E 2(A1 + A2 + 2E)+D0 2(A1 + A2 + 2E)+Dn  4

A1 + A3 + T1 + T3 (A2 + A4 + T2 + T4) + (E1 + E2 + T1 + T2 + T3 + T4) (A1 + A3 + T1 + T3) + 2(E1 + E2 + T1 + T2 + T3 + T4) (A1 + A3 + T1 + T3) + 2(A2 + A4 + T2 + T4) + 2(E1 + E2 + T1 + T2 + T3 + T4) 2(A1 + A3 + T1 + T3) + 2(A2 + A4 + T2 + T4) + 2(E1 + E2 + T1 + T2 + T3 + T4) + (A1 + A3 + T1 + T3) 2(A1 + A3 + T1 + T3) + 2(A2 + A4 + T2 + T4) + 2(E1 + E2 + T1 + T2 + T3 + T4)+Dn  4

20

K. Balasubramanian / Chemical Physics Letters 398 (2004) 15–21

The 17O nuclear spin generator is obtained by replacing every sk in the above expression by ðak1 þ ak2 þ    þ ak6 Þ. Thus the 17O nuclear spin generator for the A4 representation is given by 1h 6 4 ða1 þ a2 þ    þ a6 Þ  3ða1 þ a2 þ    þ a6 Þ GFA4 ¼ 48  ða21 þ a22 þ    þ a26 Þ  3ða1 þ a2 þ    þ a6 Þ2  ða21 þ a22 þ    þ a26 Þ2 þ 3ða21 þ a22 þ    þ a26 Þ3 2

þ 6ða1 þ a2 þ    þ a6 Þ ða41 þ a42 þ    þ a46 Þ

i 2 þ 8ða31 þ a32 þ    þ a36 Þ  8ða61 þ a62 þ    þ a66 Þ :

m6 The coefficient of a typical term am1 1 am2 2    a6 yields 17 the number of O nuclear spin functions containing m1a1 spins, m2a2 spins, . . ., m6a6 spins that transform as the A4 irreducible representation. The irreducible representations for the nuclear spin functions are sorted according to their total MF nuclear spin quantum numbers, which then enumerate the nuclear spin multiplets for each of the irreducible representations. This involves considerable combinatorics and multinomial expansions and numerous terms that need to be kept track of. Hence a computer code was developed for 17O nuclear spin generator of TATB. The results are shown in Table 4 together with the 14N and 1H species. The 14N and 1H species are generated in the rigid

Table 4 The 17O and

17

A1

A4 E1

E2 T1

T2

T3

T4

N, 1H)

Symmetry

Nuclear spin statistical weights (17O, 14N, 1H)a

Nuclear spin statistical weights (16O, 14N, 1H)b

A1 A2 A3 A4 E1 E2 T1 T2 T3 T4

72 71 26 26 144 52 154 154 110 110

8112 7464 – – 15540 – – – – –

a b

156 870 317 171 026 487 416 212 326 122

672 904 080 280 820 820 780 660 860 740

sum of (stat weights · dimension of reps)=26 · 36 · 66. sum of (stat weights · dimension of reps)=26 · 36.

D3 group and then induced into the non-rigid group since the amino groups are assumed to be rigid due to a large barrier for internal rotation. As seen from Table 4, TATB exhibits rich non-rigid 17O nuclear spin statistics, and thus it would be interesting to look at the 17O NMR of TATB. The overall nuclear spin statistical weights are obtained as the direct product of 17 O, 14N and 1H species. These weights are shown in Table 5 for all the irreducible representations of the non-rigid group of TATB. The rovibronic levels and

N and 2H spin species of non-rigid TATB 3

A3

14

14

Symmetry

A2

Table 5 Nuclear spin statistical weights of TATB (17O,

O Species 5

14

Proton Species

1

A1(5) 3A1(6) 5A1(8) 7A1(6) 9 A1(4) 11A1(1) 13A1(1)

3

1

1

N Species

7

9

11

13

A1(13) A1(9) A1(22) A1(15) A1(21) A1(15) 15 A1(17) 17A1(9) 19A1(11) 21A1(5) 23A1(5) 25A1(2) 27 A1(2) 31A1(1) 1 A2(7) 3A2(5) 5A2(14) 7A2(13) 9A2(19) 11A2(13) 13 A2(17) 15A2(10) 17A2(10) 19A2(6) 21A2(5) 23A2(1) 25 A2(2) 1 A3(7) 3A3(2) 5A3(11) 7A3(7) 9A3(13) 11A3(6) 13 A3(10) 15A3(4) 17A3(5) 19A3(2) 21A3(2) 25A3(1) 3 A4(7) 5A4(4) 7A4(11) 9A4(6) 11A4(8) 13A4(5) 15 A4(5) 17A4(1) 19A4(2) 1 E1(4) 3E1(18) 5E1(26) 7E1(32) 9E1(34) 11E1(36) 13 E1(30) 15E1(27) 17E1(21) 19E1(15) 21E1(10) 23 E1(7) 25E1(3) 27E1(2) 29E1(1) 1 E2(4) 3E2(9) 5E2(17) 7E2(16) 9E2(19) 11E2(16) 13 E2(13) 15E2(9) 17E2(7) 19E2(3) 21E2(2) 23E2(1) 1 T1(13) 3T1(16) 5T1(38) 7T1(35) 9T1(47) 11T1(37) 13 T1(40) 15T1(27) 17T1(25) 19T1(14) 21T1(12) 23 T1(5) 25T1(4) 27T1(1) 29T1(1) 1 T2(5) 3T2(23) 5T2(29) 7T2(40) 9T2(39) 11T2(41) 13 T2(33) 15T2(29) 17T2(20) 19T2(15) 21T2(8) 23T2(5) 25 T2(2) 27T2(1) 1 T3(2) 3T3(23) 5T3(23) 7T3(37) 9T3(31) 11 T3(36)13T3(25) 15T3(24) 17T3(14)19T3(12) 21T3(5) 23 T3(4) 25T3(1) 27T3(1) 1 T4(7) 3T4(16) 5T4(27) 7T4(30) 9T4(33) 11T2(29) 13 T2(26) 15T2(19) 17T2(14) 19T2(8) 21T2(5) 23 T2(2)25T2(1)

A2(2) 3A2(6) 5A2(4) 7A2(5) A2(1) 11A2(0) 13A2(0)

A1(1) 7A1(1)

A2(3) 5A2(1)

9

None

None

None

None

1

1

E1(4) 3E1(12) 5E1(14) 7E1(9) E1(5) 11E1(2)

E1(1), 3E1(3)5E1(2)

9

None

None

None

None

None

None

None

None

None

None

K. Balasubramanian / Chemical Physics Letters 398 (2004) 15–21

statistical weights of the rigid D3 structure into the non-rigid TATB are correlated by the use of induced representations as follows: A1 ðD3 : 362867472Þ " S3 ½S2  ¼ A1 ð72156672Þ þ A3 ð26317080Þ þ T1 ð154416780Þ þ T3 ð110326860Þ; A2 ðD3 : 362727504Þ " S3 ½S2  ¼ A2 ð71870904Þ þ A4 ð26171280Þ þ T2 ð154212660Þ þ T4 ð110122740Þ; EðD3 : 725593680Þ " S3 ½S2  ¼ E1 ð144026820Þ þ E2 ð52487820Þ þ T1 ð154416780Þ þ T2 ð154212660Þ þ T3 ð110326860Þ þ T4 ð110122740Þ:

4. Conclusion The symmetry group of the non-rigid TATB which exhibits internal rotations of the nitro groups was shown to be the wreath product group S3[S2] which contains 48 permutations. The total PI group of TATB is a direct product of this group with the inversion group and contains 96 operations. The character table of the group was obtained using powerful combinatorial generators using matrix type manipulations and was utilized to establish tunneling splittings of the rovibronic levels of TATB. It was shown that even in the ground state of TATB the rovibronic level is split into A1, A3, T1 and T3 tunneling levels. The nuclear spin species for the 17O, 14N and 1H nuclei of TATB were obtained using multinomial generating functions. The 17O nuclear spin statistics was found to be quite rich suggesting that 17O NMR should be a powerful technique to probe into the predicted internal rotations of the nitro groups of TATB. We have also obtained the total nuclear spin statistical weights of the rovibronic levels of TATB. Our prediction that the rovibronic levels of TATB should undergo tunneling splitting is consistent with the recently observed temperature-depended Raman spectra [1], which exhibit splitting of the signals that do not correspond to the vibrational normal modes alone. Since interaction with crystal environment could also cause such splittings, it would be necessary to obtain the gas-phase spectra to establish the predicted tunneling splittings.

Acknowledgements The Research at UC Davis was supported by the National Science Foundation under Grant No. CHE-

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0236434. The work at LLNL was performed in part under the auspices of the US Department of Energy by the University of California, LLNL under contract number W-7405-Eng-48.

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