Ro-vibrational states of triplet H3+ (a3Σu+): The lowest 19 bands

Journal of Molecular Spectroscopy 221 (2003) 163–173 www.elsevier.com/locate/jms 3 þ Ro-vibrational states of triplet Hþ 3 ða Ru Þ: The lowest 19 bands nio J.C. Varandas* Alexander Alijah, Luıs P. Viegas, Mihail Cernei, and Anto Departamento de Quımica, Universidade de Coimbra, 3004-535 Coimbra, Portugal Received 28 March 2003; in revised form 24 July 2003 Abstract We have performed extensive calculations of the ro-vibrational states of triplet Hþ 3 , using the method of hyperspherical harmonics and our recently reported double many-body expansion potential energy surface. The rotational term values of the lowest 19 states are presented here for a total angular momentum of J 6 10. Ó 2003 Elsevier Inc. All rights reserved. 1. Introduction Little is known about the electronically excited states of the hydrogen molecular ion Hþ 3 , despite the fundamental importance of this ion in astrophysics and astrochemistry. Back in 1974, Schaad and Hicks [1] studied a large number of excited electronic states, and their finding that the lowest electronic triplet state is bound has kindled further theoretical studies by Ahlrichs et al. [2], and much later by Wormer and de Groot [3] and by Preiskorn et al. [4]. Only recently complete potential energy surfaces of this state and corresponding vibrational calculations have been reported [5–8], following the desire expressed by Tennyson [9] and by McNab [10] for such work to be performed. In one of our previous publications [5], we also dealt with rotational excitation, but this was considered for the lowest five vibrational states only. We have now extended those calculations significantly and present here the rotational term values for J 6 10 of the 19 lowest bands using our recent double many-body expansion (DMBE) potential energy surface [8]. In his 1995 review article, McNab [10] states that ‘‘no accurate calculations of vibration–rotation levels . . . have been published, and no spectroscopic observations have been reported, which involve the 3 Rþ u state. Such calculation and observations would be extremely interesting.’’ By providing theoretical predic* Corresponding author. Fax: +351-39-827-703. E-mail addresses: alijah@ci.uc.pt (A. Alijah), varandas@qtvs1.qui. uc.pt (A.J.C. Varandas). 0022-2852/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0022-2852(03)00229-7 tions of the ro-vibrational levels, we hope to stimulate experiments aiming at their observation. 2. Symmetry properties of the ro-vibronic states þ In this lowest electronic triplet state, a3 Rþ u , H3 is linear, with three equivalent minima of the potential energy, which correspond to the three possible permutations of the nuclei. The three equivalent nuclear arrangements lead to a triplication of the ro-vibrational levels. Due to the barriers between the equivalent minima on the potential energy surface, each delocated states becomes split into a singly and a twofold degenerate component. This can be understood on the basis of group theory as follows:

Using the wavefunctions jW
I i, jWII i, and jWIII i, which are localised in the minima I, II, and III, respectively, superpositions have to be found that transform as irreducible representations of the three-particle permutation inversion group, S3H or S3  I, which is isomorphic to the molecular symmetry group D3h ðMÞ [11]. As a result we obtain the one-dimensional representation


jW
A i  jWI i þ jWII i þ jWIII i ð1Þ and the two-dimensional representation


2 jW
E;n i  jWI i þ xjWII i þ x jWIII i; ð2Þ


2 jW
E;g i  jWI i þ x jWII i þ xjWIII i ð3Þ with x ¼ e2p i=3 . The two components of the two-dimensional representation are related by complex conjugation. 164 A01 ð0Þ; E0 ð2Þ A002 ð4Þ; E00 ð2Þ A02 ð4Þ; E0 ð2Þ A001 ð0Þ; E00 ð2Þ A002 ð4Þ; E00 ð2Þ A01 ð0Þ; E0 ð2Þ A001 ð0Þ; E00 ð2Þ A02 ð4Þ; E0 ð2Þ Rþ g R g Rþ u R u R g Rþ g R u Rþ u (Pg ; . . .) % R u & Rþ u % R g & Rþ g ð0; 11 ; 0Þþ ð0; 11 ; 0Þ ð0; 11 ; 1Þþ ð0; 11 ; 1Þ ð1ÞN ð1ÞN ð1ÞN ð1ÞN even 6¼ 0 odd odd even 6¼ 0 even odd + ) + ) even odd 0 even + ) + ) ð0; 22 ; 0Þþ ð0; 22 ; 0Þ ð0; 22 ; 1Þþ ð0; 22 ; 1Þ ð1ÞN ð1ÞN ð1ÞN ð1ÞN (Du ; . . .) ð0; 0 ; 0Þ ð0; 00 ; 1Þ (Pu ; . . .) A001 ð0Þ; E00 ð2Þ A02 ð4Þ; E0 ð2Þ A002 ð4Þ; E00 ð2Þ A01 ð0Þ; E0 ð2Þ A02 ð4Þ; E0 ð2Þ A001 ð0Þ; E00 ð2Þ A01 ð0Þ; E0 ð2Þ A002 ð4Þ; E00 ð2Þ R u Rþ u  Rg Rþ g Rþ u R u þ Rg R g N odd N 0 ð1Þ ð1ÞN (Dg ; . . .) % Rþ g & R g þ % Ru & R u A02 ð4Þ; E0 ð2Þ A01 ð0Þ; E0 ð2Þ N even R u R g Rþ g Rþ u Cv ½D1h ðMÞ H vE These characters will be needed for a symmetry classification of the ro-vibronic states. The molecular symmetry group of the symmetric linear Hþ 3 , as long as tunnelling can be neglected, is D1h ðMÞ. In this group the symmetric stretch vibration transforms as Rþ g , the degenerate bending vibrations as Pu , and the antisymmetric stretch vibration as Rþ u . The symmetry of the combination vibrations can be obtained from the direct products. The molecular symmetry group D1h ðMÞ gives rise to the conventional symmetry classification of the ro-vibronic states of symmetric linear molecules, but for triplet Hþ 3 such symmetry labels are only approximate. The approximate symmetry labels can be related to those of the exact molecular symmetry group of tunnelling Hþ 3, D3h ðMÞ, as described in [5] for the lowest vibrational states. In the present work, we have performed a similar analysis for the general case. The result is shown in a compact form in Table 1. In this table we have listed, as a function of the quantum numbers m2 , l, m3 , and s, the sign of the l-doubling component according to Eq. (4), the vibrational and ro-vibronic symmetry assignments in D1h ðMÞ and the corresponding ro-vibronic symmetry assignments in the exact group D3h ðMÞ. The symmetry classification of the states depends on the parities ð1ÞN þl and ð1Þlþv3 . The symmetry of the rovibronic states can thus be expressed in an even more compact though less obvious way, as shown in Table 2 for D1h ðMÞ. In this form it is used in our computer programme. Example ðv1 ; vl2 ; v3 Þ
ð6Þ s : v3 H Table 1 Symmetry classification of the ro-vibronic states in D1h ðMÞ and D3h ðMÞ N þl vE ðjW
iÞ ¼
ð1Þ N odd for l ¼ 0. In the above equations v1 , v2 , and v3 are the quantum numbers of the symmetric stretching vibration m1 , the double degenerate bending vibration m2 , and the antisymmetric stretching vibration m3 . An excitation of the bending vibration m2 leads to the vibrational angular momentum l, which takes the values l ¼ v2 ; v2  2; . . . ; v2 . In the above equations we have neglected the elec~ ¼~ tronic spin and use N , where N J ~ S , and its external projection m in the rotational part. The internal projection has to be identical to the vibrational angular momentum l. The two states with l 6¼ 0 are split in energy, an effect known as l-type doubling. These two states have opposite parity, i.e., opposite characters [11] with respect to the operator for the inversion of the spatial coordinate system, EH , N even ð5Þ l 1 jWi ¼ pffiffi jWþ i 2 Crve ½D3h ðMÞ for l 6¼ 0 and Crve ½D1h ðMÞ ð4Þ v2 1 jlj jW
i ¼ pffiffi jv1 v2 v3 iðjNlmi
jN  lmiÞ 2 Rþ u Rþ g Each of the localized functions jW
i, dropping now the localization index I, II or III, can be expanded approximately in terms of linear molecule basis functions as A001 ð0Þ; E00 ð2Þ A002 ð4Þ; E00 ð2Þ A. Alijah et al. / Journal of Molecular Spectroscopy 221 (2003) 163–173 A. Alijah et al. / Journal of Molecular Spectroscopy 221 (2003) 163–173 Table 2 Summary of the symmetry classification of the ro-vibronic states in D1h ðMÞ ð1ÞN þl ¼ lþv3 ð1Þ ¼1 ð1Þlþv3 ¼ 1 + Component ) Component 1 )1 1 )1 Rþ u Rþ g R u R g R u R g Rþ u Rþ g Table 1 is rather complex, and therefore we will give two explicit examples to illustrate its use: The first example will be the vibrational ground state, ðv1 ; vl2 ; v3 Þ ¼ ð0; 00 ; 0Þ, which, in D1h ðMÞ, has a vibrational symmetry of Rþ g . Since the electronic state at the equilibrium linear geometry is a 3 Rþ u state, the vibronic symmetry of the vibrational ground state is þ þ Rþ g  Ru ¼ Ru . The same result is obtained for non-zero but even values of the angular momentum N , while for odd values the parity with respect to an inversion of the laboratory coordinate system, EH , is reversed. This H parity, defined by the character vE listed in the fifth column of the table, is indicated by the
superscript of the symmetry labels. Consequently, the ro-vibronic symmetry of the vibrational ground state becomes R u for odd values of N . The two last columns in Table 1 finally contain the symmetry classification in D3h ðMÞ, which is obtained from the reverse correlation table between D1h ðMÞ and D3h ðMÞ, Table 3, which corresponds to Table VI of [5]. Note that in D3h ðMÞ the parity is indicated by ‘‘prime’’ and ‘‘double prime.’’ As a second example we consider the state ð0; 11 ; 1Þ. Since this is a combination of ð0; 00 ; 1Þ and ð0; 11 ; 0Þ, its vibrational symmetry is determined by the direct product Rþ u  Pu ¼ Pg . Of course, since the vibrational angular momentum is l ¼ 1 and thus the angular momentum N must be at least one, this vibrational symmetry label is hypothetic. Indeed, the degeneracy is lifted upon rotational excitation, and the hypothetical Pg state splits into one component of Rþ g symmetry and one of R The
labels are determined by the g symmetry. H value of vE for the hypothetic N ¼ 0. We find that the component with s ¼ þ1, ð0; 11 ; 1Þþ, has a character of )1, thus giving rise a R g state. In contrast, the ð0; 11 ; 1Þ component has a character of þ1 and gives rise to a Rþ g state. These characters are listed in the seventh column of the table, denoted Cv ½D1h ðMÞ. Multiplying with the electronic symmetry, Rþ u , we obtain the ro-vibronic symmetry labels of the two l-type douTable 3 Reverse correlation table between D1h ðMÞ and D3h ðMÞ with the spin statistical weights for 1 H3þ given in parentheses D1h ðMÞ R g ð6Þ Rþ g ð2Þ D3h ðMÞ A002 ð4Þ  E00 ð2Þ A01 ð0Þ  E0 ð2Þ D1h ðMÞ R u ð2Þ Rþ u ð6Þ D3h ðMÞ A001 ð0Þ  E00 ð2Þ A02 ð4Þ  E0 ð2Þ 165 bling components for even values of N . For odd values of N , the parity is reversed, and we find the ro-vibronic symmetry labels as listed in the ‘‘N odd’’ column. From here it is straightforward to arrive at the symmetry labels of tunnelling Hþ 3 , using once again the reverse correlation between D1h ðMÞ and D3h ðMÞ, Table 3. In the last columns of Table 1 we have also indicated the spin statistical weight of each state for 1 Hþ 3 . In this molecule, the nuclear spins of the three protons can be combined to yield a quartet state, which is totally symmetric in the three-particle permutation inversion group and thus in the isomorphic molecular symmetry group D3h ðMÞ, and two doublet states, which span a degenerate representation. To yield a total wavefunction which is antisymmetric with respect to an odd permutation of two protons, the quartet spin state has to be combined with a spatial function of A02 or A002 symmetry, while the doublet spin state goes with the degenerate spatial functions. The statistical weights of the ro-vibronic wavefunctions of A1 , A2 or E symmetry, either ‘‘prime’’ or ‘‘double prime,’’ are thus 0, 4, and 2, respectively. 3. Calculation of the ro-vibrational states and their assignments The ro-vibrational states have been calculated using the method of hyperspherical harmonics [12]. To make full use of the exact permutation inversion symmetry, the primitive hyperspherical harmonics are symmetrized such as to transform as the irreducible representations of S3  I. Thus, for each value of the angular momentum N , six independent calculations are performed, yielding the ro-vibronic states of A01 , A02 , E0 , and of A001 , A002 , and E00 symmetry. Note that the latter three irreducible representations cannot be realized for N ¼ 0, so that only three independent calculations need to be performed for this case. To improve the efficiency of the computational procedure, the symmetrized hyperspherical harmonics, typically around 1500 functions, are contracted at some suitable value of the hyperradius q, and 300 contracted functions are retained and used as angular basis. The resulting set of coupled hyperradial equations is integrated numerically within 2:0a0 6 q 6 12a0 with a hyperradius step size of Dq ¼ 0:05a0 . The calculation of the ro-vibrational states of such a floppy ion is a non-trivial task. The results obtained here with our extended basis should nevertheless, for the present surface, be accurate to at least 0.1–0.3 cm1 , depending on the energy. We did not take into account the geometrical phase effect since we have shown in a previous publication [5] that its inclusion leads to negligible differences even for the lowest vibrational states. Furthermore, we neglected any effects of the electronic spin, as the spin–orbit coupling should be rather small. 166 A. Alijah et al. / Journal of Molecular Spectroscopy 221 (2003) 163–173 Table 4 Ro-vibronic term values of the states of group I ðv1 ; vl2 ; v3 Þ N Cð1Þ rve nð1Þ ð2Þ Crve ð0; 00 ; 0Þ 0 1 2 3 4 5 6 7 8 9 10 A02 A001 A02 A001 A02 A001 A02 A001 A02 A001 A02 0 0 0 0 0 0 0 0 0 0 0 E0 E00 E0 E00 E0 E00 E0 E00 E0 E00 E0 0 0 0 0 0 0 0 0 0 0 0 0.00 9.51 28.51 56.99 94.92 142.27 198.99 265.03 340.33 424.81 518.40 ð1; 00 ; 0Þ 0 1 2 3 4 5 6 7 8 9 10 A02 A001 A02 A001 A02 A001 A02 A001 A02 A001 A02 1 1 1 1 1 1 1 1 1 1 1 E0 E00 E0 E00 E0 E00 E0 E00 E0 E00 E0 2 3 3 3 3 3 3 3 3 3 3 ð0; 11 ; 1Þ 1 2 3 4 5 6 7 8 9 10 A001 A02 A001 A02 A001 A02 A001 A02 A001 A02 2 2 2 2 2 2 2 2 2 2 E00 E0 E00 E0 E00 E0 E00 E0 E00 E0 ð0; 20 ; 0Þ 0 1 2 3 4 5 6 7 8 9 10 A02 A001 A02 A001 A02 A001 A02 A001 A02 A001 A02 2 3 3 3 3 3 3 3 3 3 3 ð0; 22 ; 0Þþ 2 3 4 5 6 7 8 9 10 A02 A001 A02 A001 A02 A001 A02 A001 A02 0 1 2 3 4 5 6 7 A02 A001 A02 A001 A02 A001 A02 A001 ð0; 00 ; 2Þ nð2Þ Eð1Þ Eð2Þ ð1Þ ð2Þ Erel Erel D 0.00 9.51 28.51 56.99 94.92 142.29 199.02 265.06 340.36 424.84 518.43 0.00 9.51 28.51 56.99 94.92 142.27 198.99 265.03 340.33 424.81 518.40 0.00 9.51 28.51 56.99 94.92 142.29 199.02 265.06 340.36 424.84 518.43 0.00 0.00 0.00 0.00 0.00 )0.02 )0.03 )0.02 )0.02 )0.02 )0.03 975.50 984.12 1001.35 1027.13 1061.40 1104.09 1155.07 1214.22 1281.38 1356.38 1439.03 975.50 984.12 1001.35 1027.13 1061.41 1104.20 1155.20 1214.36 1281.54 1356.56 1439.21 0.00 8.63 25.85 51.63 85.90 128.59 179.57 238.72 305.88 380.88 463.53 0.00 8.62 25.85 51.63 85.91 128.70 179.70 238.87 306.04 381.06 463.72 0.00 0.00 0.00 0.00 )0.01 )0.11 )0.13 )0.14 )0.15 )0.18 )0.19 4 4 4 4 4 4 4 4 4 4 1271.60 1281.67 1299.68 1326.22 1361.33 1405.13 1457.67 1518.86 1588.81 1667.29 1271.65 1281.62 1299.71 1326.21 1361.37 1405.20 1458.07 1519.45 1589.66 1668.42 0.00 10.07 28.08 54.62 89.73 133.53 186.07 247.26 317.21 395.69 0.05 10.02 28.12 54.62 89.77 133.60 186.47 247.85 318.06 396.82 )0.05 0.04 )0.04 0.01 )0.05 )0.07 )0.40 )0.59 )0.85 )1.13 E0 E00 E0 E00 E0 E00 E0 E00 E0 E00 E0 3 5 5 5 5 5 5 5 5 5 5 1274.43 1285.63 1305.18 1333.81 1371.42 1418.29 1474.18 1539.10 1612.87 1695.27 1786.18 1274.49 1285.57 1305.24 1333.76 1371.48 1418.38 1474.37 1539.42 1613.64 1697.45 1788.41 0.00 11.19 30.75 59.38 96.99 143.86 199.75 264.67 338.44 420.84 511.75 0.06 11.14 30.81 59.33 97.05 143.95 199.93 264.99 339.21 423.02 513.98 )0.06 0.06 )0.06 0.05 )0.06 )0.09 )0.18 )0.32 )0.76 )2.17 )2.22 4 4 4 4 4 4 4 4 4 E0 E00 E0 E00 E0 E00 E0 E00 E0 6 6 6 6 6 6 6 6 7 1342.36 1377.11 1421.20 1474.39 1536.39 1607.12 1686.42 1774.12 1870.15 1342.38 1377.09 1421.23 1474.46 1536.55 1607.74 1687.26 1771.60 1872.98 0.00 34.75 78.84 132.03 194.03 264.76 344.06 431.76 527.79 0.02 34.73 78.87 132.10 194.19 265.38 344.90 429.23 530.62 )0.02 0.02 )0.03 )0.07 )0.16 )0.63 )0.84 2.53 )2.83 3 4 5 5 5 5 5 5 E0 E00 E0 E00 E0 E00 E0 E00 5 8 9 9 9 9 9 9 1574.43 1582.90 1599.41 1624.42 1657.45 1698.67 1747.82 1804.79 1574.54 1582.81 1599.53 1624.46 1657.65 1699.09 1748.37 1805.67 0.00 8.47 24.98 49.99 83.02 124.24 173.39 230.36 0.11 8.38 25.10 50.03 83.22 124.66 173.94 231.24 )0.11 0.10 )0.12 )0.04 )0.20 )0.42 )0.55 )0.88 167 A. Alijah et al. / Journal of Molecular Spectroscopy 221 (2003) 163–173 Table 4 (continued) N Cð1Þ rve Cð2Þ rve nð2Þ 8 9 10 A02 A001 A02 5 5 5 0 E E00 E0 ð1; 11 ; 1Þ 1 2 3 4 5 6 7 8 9 10 A001 A02 A001 A02 A001 A02 A001 A02 A001 A02 5 6 6 6 6 6 6 6 6 6 ð2; 00 ; 0Þ 0 1 2 3 4 5 6 7 8 9 10 A02 A001 A02 A001 A02 A001 A02 A001 A02 A001 A02 ð1; 20 ; 0Þ 0 1 2 3 4 5 6 7 8 9 10 ð1; 22 ; 0Þþ 2 3 4 5 6 7 8 9 10 ðv1 ; vl2 ; v3 Þ nð1Þ ð1Þ ð2Þ Eð1Þ Eð2Þ Erel Erel D 9 9 9 1869.41 1941.25 2020.23 1870.53 1942.97 2022.18 294.98 366.82 445.80 296.10 368.54 447.75 )1.12 )1.72 )1.95 E00 E0 E00 E0 E00 E0 E00 E0 E00 E0 11 13 13 13 12 12 12 12 12 12 1889.57 1900.69 1910.79 1932.94 1958.48 1992.66 2034.50 2080.75 2138.88 2197.69 1892.91 1898.29 1913.31 1932.44 1960.28 1994.12 2036.29 2085.45 2140.03 2204.77 0.00 11.12 21.22 43.37 68.90 103.09 144.93 191.18 249.31 308.12 3.34 8.72 23.74 42.87 70.70 104.55 146.72 195.88 250.46 315.19 )3.34 2.40 )2.52 0.50 )1.80 )1.46 )1.79 )4.70 )1.15 )7.08 4 6 7 7 7 8 8 8 8 8 8 E0 E00 E0 E00 E0 E00 E0 E00 E0 E00 E0 7 12 14 14 14 16 16 16 16 16 16 1923.94 1935.72 1956.27 1982.79 2014.09 2072.37 2113.48 2178.46 2232.50 2311.34 2376.07 1924.50 1935.76 1956.64 1985.91 2017.82 2069.17 2121.49 2174.58 2242.12 2308.16 2389.03 0.00 11.78 32.33 58.85 90.15 148.44 189.55 254.52 308.56 387.40 452.13 0.56 11.83 32.71 61.97 93.88 145.23 197.56 250.65 318.18 384.22 465.10 )0.56 )0.04 )0.37 )3.12 )3.73 3.20 )8.01 3.88 )9.62 3.18 )12.96 A02 A001 A02 A001 A02 A001 A02 A001 A02 A001 A02 5 7 8 8 8 7 7 7 7 7 7 E0 E00 E0 E00 E0 E00 E0 E00 E0 E00 E0 8 13 15 15 15 15 15 15 15 15 14 1942.17 1963.74 1963.98 1999.80 2021.26 2049.95 2091.99 2137.16 2196.65 2249.47 2324.00 1952.62 1952.79 1974.31 1990.15 2028.01 2066.68 2105.16 2153.68 2213.11 2279.32 2326.22 0.00 21.57 21.81 57.62 79.09 107.78 149.81 194.98 254.48 307.30 381.83 10.45 10.62 32.13 47.98 85.84 124.50 162.98 211.51 270.94 337.15 384.05 )10.45 10.95 )10.33 9.64 )6.75 )16.72 )13.17 )16.52 )16.46 )29.86 )2.22 A02 A001 A02 A001 A02 A001 A02 A001 A02 9 9 9 9 9 9 9 9 9 E0 E00 E0 E00 E0 E00 E0 E00 E0 17 18 18 18 18 18 17 17 18 2035.78 2068.42 2091.38 2138.44 2177.59 2237.21 2292.73 2361.96 2434.59 2042.48 2061.95 2097.82 2133.51 2183.99 2235.59 2298.18 2363.99 2455.05 0.00 32.64 55.60 102.66 141.81 201.42 256.94 326.18 398.81 6.70 26.17 62.04 97.73 148.21 199.81 262.40 328.20 419.26 )6.70 6.47 )6.44 4.93 )6.40 1.62 )5.45 )2.03 )20.45 ð1Þ ð2Þ The energies Eð1Þ and Eð2Þ are with respect to the zero-point energy, the energies Erel and Erel are with respect to the bandhead of the A component. Note that the states of A01 and A001 symmetry have a statistical weight of zero. All energies are in units of cm1 . Cð1Þ and Cð2Þ indicate the ro-vibronic symmetry of the states, while nð1Þ and nð2Þ are indices counting states with the same symmetry and angular momentum N . Since, as mentioned before, the three-particle permutation inversion group is isomorphic with the molecular symmetry group D3h ðMÞ, our calculated rovibrational states can be related directly to tunnelling Hþ 3 . To assign spectroscopic quantum numbers to these states, which we will call hyperspherical states, we use the half-automatic method developped in [13], which has þ been applied to Dþ 3 [14], D2 Hþ [15], H2 Dþ [16], and H3 [17,18] in the electronic ground state. This method has now been adapted to the specific situation of tunnelling triplet Hþ 3 . For a given vibronic state, first the one- dimensional component is analysed and the appropriate hyperspherical state with the required symmetry is selected. Then, the matching two-dimensional component is assigned, such as to produce a splitting between the two components as small as possible. This strategy of relating the degenerate state to the non-degenerate is useful because of the high density of degenerate states as compared to the non-degenerate states for a given symmetry. Note that there are four one-dimensional representations, but only two two-dimensional ones. Having identified the N ¼ 0 levels, the programme loops 168 A. Alijah et al. / Journal of Molecular Spectroscopy 221 (2003) 163–173 Table 5 Ro-vibronic term values of the states of group II ðv1 ; vl2 ; v3 Þ N Cð1Þ rve nð1Þ ð2Þ Crve ð0; 11 ; 0Þ 1 2 3 4 5 6 7 8 9 10 A002 A01 A002 A01 A002 A01 A002 A01 A002 A01 0 0 0 0 0 0 0 0 0 0 E00 E0 E00 E0 E00 E0 E00 E0 E00 E0 ð0; 00 ; 1Þ 0 1 2 3 4 5 6 7 8 9 10 A01 A002 A01 A002 A01 A002 A01 A002 A01 A002 A01 0 1 1 1 1 1 1 1 1 1 1 ð1; 00 ; 1Þ 0 1 2 3 4 5 6 7 8 9 10 A01 A002 A01 A002 A01 A002 A01 A002 A01 A002 A01 ð1; 11 ; 0Þ 1 2 3 4 5 6 7 8 9 10 ð0; 20 ; 1Þ ð0; 22 ; 1Þþ nð2Þ ð1Þ ð2Þ Eð1Þ Eð2Þ Erel Erel D 1 1 1 1 1 1 1 1 1 1 665.89 681.77 705.75 737.96 778.50 827.43 884.78 950.54 1024.70 1107.17 665.89 681.77 705.76 737.97 778.53 827.47 884.84 950.67 1024.82 1107.31 0.00 15.88 39.87 72.07 112.61 161.54 218.89 284.65 358.81 441.28 0.00 15.88 39.87 72.08 112.64 161.58 218.95 284.78 358.93 441.42 0.00 0.00 0.00 0.00 )0.04 )0.04 )0.06 )0.12 )0.12 )0.14 E0 E00 E0 E00 E0 E00 E0 E00 E0 E00 E0 1 2 2 2 2 2 2 2 2 2 2 739.47 749.71 770.10 800.48 840.65 890.44 949.67 1018.18 1095.84 1182.51 1278.06 739.47 749.72 770.10 800.48 840.66 890.46 949.72 1018.36 1096.47 1183.10 1278.63 0.00 10.24 30.63 61.01 101.18 150.97 210.20 278.71 356.37 443.04 538.59 0.00 10.24 30.63 61.01 101.19 150.99 210.25 278.89 357.00 443.63 539.16 0.00 0.00 0.00 0.00 )0.01 )0.02 )0.05 )0.17 )0.62 )0.59 )0.57 1 2 2 2 2 2 2 2 2 2 2 E0 E00 E0 E00 E0 E00 E0 E00 E0 E00 E0 4 6 7 7 7 7 7 7 7 7 6 1475.37 1480.94 1492.78 1510.92 1536.20 1568.40 1608.09 1654.82 1709.03 1770.23 1838.62 1475.29 1481.02 1492.71 1511.05 1536.23 1568.65 1608.23 1655.34 1710.33 1775.86 1839.86 0.00 5.58 17.41 35.56 60.84 93.03 132.72 179.46 233.66 294.86 363.26 )0.08 5.66 17.35 35.69 60.87 93.29 132.86 179.98 234.96 300.50 364.49 0.08 )0.08 0.06 )0.13 )0.03 )0.26 )0.14 )0.52 )1.30 )5.63 )1.23 A002 A01 A002 A01 A002 A01 A002 A01 A002 A01 3 3 3 3 3 3 3 3 3 3 E00 E0 E00 E0 E00 E0 E00 E0 E00 E0 7 8 8 8 8 8 8 8 8 8 1545.10 1563.95 1592.09 1627.95 1672.08 1723.22 1781.81 1847.10 1919.00 1997.67 1544.89 1564.15 1591.92 1628.19 1672.13 1723.69 1782.39 1848.18 1920.30 1998.72 0.00 18.85 47.00 82.85 126.98 178.12 236.71 302.01 373.90 452.57 )0.20 19.06 46.82 83.09 127.03 178.59 237.30 303.09 375.20 453.62 0.20 )0.21 0.18 )0.24 )0.05 )0.46 )0.59 )1.08 )1.30 )1.05 0 1 2 3 4 5 6 7 8 9 10 A01 A002 A01 A002 A01 A002 A01 A002 A01 A002 A01 2 4 4 4 4 4 4 4 4 4 4 E0 E00 E0 E00 E0 E00 E0 E00 E0 E00 E0 6 9 10 10 10 10 10 10 10 10 10 1731.74 1736.89 1754.69 1775.81 1808.87 1846.74 1895.13 1950.16 2013.80 2085.67 2164.06 1729.80 1738.74 1752.98 1777.34 1807.74 1847.80 1895.09 1950.96 2016.02 2087.65 2167.34 0.00 5.16 22.96 44.08 77.13 115.00 163.39 218.42 282.06 353.93 432.32 )1.94 7.00 21.25 45.60 76.00 116.06 163.35 219.22 284.28 355.92 435.60 1.94 )1.85 1.71 )1.53 1.13 )1.06 0.04 )0.80 )2.22 )1.99 )3.28 2 3 4 5 6 7 8 A01 A002 A01 A002 A01 A002 A01 5 5 5 5 5 5 5 E0 E00 E0 E00 E0 E00 E0 11 11 11 11 11 11 11 1799.13 1820.26 1850.82 1888.26 1934.70 1988.98 2051.96 1798.34 1820.87 1850.43 1888.61 1934.71 1990.08 2053.36 0.00 21.13 51.69 89.13 135.56 189.85 252.83 )0.80 21.74 51.30 89.48 135.58 190.95 254.23 0.80 )0.61 0.39 )0.35 )0.02 )1.10 )1.40 169 A. Alijah et al. / Journal of Molecular Spectroscopy 221 (2003) 163–173 Table 5 (continued) N Cð1Þ rve 9 10 A002 A01 ð0; 31 ; 0Þ 1 2 3 4 5 6 7 8 9 10 ð0; 33 ; 0Þ ð0; 00 ; 3Þ ðv1 ; vl2 ; v3 Þ nð1Þ Cð2Þ rve ð1Þ ð2Þ nð2Þ Eð1Þ Eð2Þ Erel Erel D 5 5 00 E E0 11 11 2123.01 2202.56 2124.78 2203.96 323.88 403.43 325.65 404.83 )1.77 )1.40 A002 A01 A002 A01 A002 A01 A002 A01 A002 A01 5 6 6 6 6 6 6 6 6 6 E00 E0 E00 E0 E00 E0 E00 E0 E00 E0 10 12 12 12 13 13 13 13 13 13 1838.01 1854.67 1885.35 1921.42 1968.17 2022.59 2081.25 2149.24 2209.94 2280.78 1835.91 1856.43 1883.96 1922.28 1968.53 2022.78 2085.77 2149.93 2220.26 2286.54 0.00 16.65 47.34 83.40 130.16 184.57 243.24 311.23 371.93 442.77 )2.10 18.42 45.94 84.27 130.52 184.77 247.75 311.92 382.24 448.53 2.10 )1.77 1.40 )0.87 )0.36 )0.19 )4.51 )0.69 )10.32 )5.76 3 4 5 6 7 8 9 10 A002 A01 A002 A01 A002 A01 A002 A01 7 8 8 8 8 8 8 8 E00 E0 E00 E0 E00 E0 E00 E0 16 17 17 17 17 18 18 17 2004.28 2046.27 2098.97 2159.97 2230.99 2304.85 2372.77 2432.72 2004.02 2046.80 2098.78 2160.79 2232.54 2314.32 2385.36 2437.18 0.00 41.99 94.69 155.69 226.71 300.57 368.49 428.44 )0.26 42.52 94.50 156.51 228.26 310.04 381.08 432.90 0.26 )0.53 0.19 )0.82 )1.56 )9.47 )12.59 )4.46 0 1 2 3 4 5 6 7 8 9 10 A01 A002 A01 A002 A01 A002 A01 A002 A01 A002 A01 3 6 7 8 7 7 7 7 7 7 7 E0 E00 E0 E00 E0 E00 E0 E00 E0 E00 E0 9 14 16 17 16 14 14 14 14 14 15 1974.79 1977.03 1991.19 2005.43 2031.61 2058.83 2098.50 2141.95 2196.88 2265.48 2345.57 1972.83 1980.40 1992.02 2011.92 2036.93 2055.21 2097.72 2147.16 2199.24 2263.82 2357.85 0.00 2.23 16.40 30.64 56.81 84.04 123.70 167.15 222.09 290.69 370.78 )1.96 5.61 17.22 37.13 62.13 80.42 122.93 172.37 224.44 289.03 383.05 1.96 )3.38 )0.83 )6.49 )5.32 3.62 0.78 )5.21 )2.35 1.66 )12.28 Entries as in Table 4. over the desired values of N and determines for each value of N the symmetry of the one-dimensional component according to Table 1 or Table 2. A matching hyperspherical state is selected, usually the one with the lowest energy which has not yet been used, but such that the energy increases with N . The corresponding E component is then determined as described above. As long as the rotational stacks of the vibronic states do not overlap seriously, this method permits an automatic assignment. If the density of states becomes too high for the automatic algorithm to produce unique assignments, we support this algorithm by comparing the rotational patterns of two or more vibrational states that belong to the same family, i.e., differ only in their degree of excitation of the symmetrical stretching mode. Normally no fitting of data to a phenomenological Hamiltonian is necessary. This is a useful feature of our algorithm, as for floppy molecules such as Hþ 3 the energy expressions derived from a phenomenological Hamiltonian are expected to converge only slowly. For the present system, triplet Hþ 3 , the assignment of the highly excited states is hampered by the fact that the potential energy surface becomes very flat in the high energy region, which results in an increased density of states. Furthermore, in the numerical calculation of the eigenvalues the high energy ones converge only slowly. In the present analysis we have considered those states with band origins up to about 2000 cm1 with respect to the vibrational ground state. The even higher states will be subject to a future publication. Our present analysis includes 19 states. Only the lowest four of these have band origins below the tunnelling barrier. 4. Ro-vibrational term values The rotational term values of the 19 vibrational states have been collected in four tables according to their symmetry. Table 4 contains the term values of those states that transform as ðA02 ; E0 Þ for even values of N , while the states tabulated in Tables 5–7 transform as ðA01 ; E0 Þ, ðA001 ; E00 Þ, and ðA002 ; E00 Þ, respectively. The states listed in Tables 4 and 5 can never interact with those listed in Tables 6 and 7, because they have different parity. Between Tables 4 and 5 or Tables 6 and 7, respectively, the one-dimensional states are separated 170 A. Alijah et al. / Journal of Molecular Spectroscopy 221 (2003) 163–173 Table 6 Ro-vibronic term values of the states of group III ð1Þ ð2Þ ðv1 ; vl2 ; v3 Þ N ð1Þ Crve nð1Þ Cð2Þ rve nð2Þ Eð1Þ Eð2Þ Erel Erel D ð0; 11 ; 1Þþ 1 2 3 4 5 6 7 8 9 10 A02 A001 A02 A001 A02 A001 A02 A001 A02 A001 0 0 0 0 0 0 0 0 0 0 E0 E00 E0 E00 E0 E00 E0 E00 E0 E00 1 1 1 1 1 1 1 1 1 1 1273.94 1285.79 1305.75 1334.10 1371.27 1417.09 1471.73 1534.98 1606.84 1687.14 1273.89 1285.84 1305.70 1334.14 1371.24 1417.14 1471.74 1535.04 1606.88 1687.48 0.00 11.85 31.80 60.16 97.32 143.15 197.79 261.03 332.90 413.19 )0.05 11.89 31.76 60.20 97.29 143.20 197.80 261.09 332.93 413.53 0.05 )0.05 0.04 )0.04 0.03 )0.05 )0.01 )0.06 )0.04 )0.34 ð0; 22 ; 0Þ 2 3 4 5 6 7 8 9 10 A001 A02 A001 A02 A001 A02 A001 A02 A001 1 1 1 1 1 1 1 1 1 E00 E0 E00 E0 E00 E0 E00 E0 E00 2 2 2 2 2 2 2 2 2 1342.44 1377.38 1422.02 1475.98 1539.15 1611.32 1692.43 1782.31 1880.83 1342.43 1377.40 1422.01 1475.99 1539.17 1611.40 1692.51 1782.44 1881.45 0.00 34.94 79.58 133.53 196.70 268.87 349.99 439.86 538.39 )0.02 34.96 79.56 133.55 196.72 268.96 350.06 440.00 539.01 0.02 )0.02 0.02 )0.02 )0.02 )0.08 )0.08 )0.13 )0.62 ð1; 11 ; 1Þþ 1 2 3 4 5 6 7 8 9 10 A02 A001 A02 A001 A02 A001 A02 A001 A02 A001 1 2 2 2 2 2 2 2 2 2 E0 E00 E0 E00 E0 E00 E0 E00 E0 E00 4 6 6 6 6 6 6 5 5 5 1898.20 1906.20 1930.81 1951.98 1989.85 2024.44 2075.48 2123.57 2187.36 2248.73 1894.47 1910.02 1927.21 1956.06 1986.70 2029.04 2082.63 2128.20 2185.57 2255.01 0.00 7.99 32.60 53.78 91.65 126.24 177.27 225.37 289.15 350.52 )3.74 11.82 29.00 57.86 88.50 130.84 184.42 229.99 287.36 356.81 3.74 )3.82 3.60 )4.08 3.15 )4.60 )7.15 )4.63 1.79 )6.29 ð1; 22 ; 0Þ 2 3 4 5 6 7 8 9 10 A001 A02 A001 A02 A001 A02 A001 A02 A001 3 3 3 3 3 3 3 3 3 E00 E0 E00 E0 E00 E0 E00 E0 E00 7 8 8 8 8 8 7 8 7 2044.77 2059.50 2099.27 2129.81 2182.87 2229.22 2293.52 2356.04 2428.84 2037.90 2066.00 2093.08 2135.75 2178.25 2234.47 2291.26 2375.90 2430.96 0.00 14.74 54.50 85.05 138.10 184.46 248.76 311.28 384.07 )6.87 21.23 48.31 90.98 133.49 189.70 246.49 331.14 386.19 6.87 )6.49 6.19 )5.93 4.61 )5.24 2.27 )19.86 )2.12 Entries as in Table 4. strictly by symmetry, while their degenerate counterparts have identical symmetry and can thus interact. Such an arrangement of the ro-vibrational term values in four distinctive tables is hoped to facilitate the verification of their assignments. Note that the two components of an l-type doublet will not appear in the same table, as they differ in parity. Instead they appear in Tables 4 and 6 or Tables 5 and 7. Our assignment procedure worked very well when applied to the 19 lowest vibrational states. Only a few manual corrections have been necessary. The first set of corrections has been made for the vibrational states ð2; 00 ; 0Þ and ð1; 20 ; 0Þ for N P 5 (see Table 4), where we made the two rotational stacks cross in order to keep some similarity in the rotational progressions of the states ð1; 00 ; 0Þ and ð2; 00 ; 0Þ, which belong to the same family. The second set of corrections deals with the rotational stacks of ð0; 33 ; 0Þ and ð0; 00 ; 3Þ (see Table 5), which we make cross at N ¼ 4. This set of corrections has been made to keep the splitting between the two ltype components, ð0; 33 ; 0Þþ and ð0; 33 ; 0Þ sufficiently small (see Tables 5 and 7), which appears sensible since in the phenomenological Hamiltonian [19] the interaction terms are between ðl; l
2Þ, and consequently the splitting of the pair ð0; 33 ; 0Þ
is a higher order effect. In order to get an idea about the scale of Coriolis coupling between different ro-vibrational states, we have attempted to express the term values as [19] F ðv; J Þ ¼ mv þ Bv ½N ðN þ 1Þ  l2   Dv ½N ðN þ 1Þ  l2  3 þ Hv ½N ðN þ 1Þ  l2  : 2 ð7Þ 171 A. Alijah et al. / Journal of Molecular Spectroscopy 221 (2003) 163–173 Table 7 Ro-vibronic term values of the states of group IV ð1Þ ð2Þ ðv1 ; vl2 ; v3 Þ N ð1Þ Crve nð1Þ ð2Þ Crve nð2Þ Eð1Þ Eð2Þ Erel Erel D ð0; 11 ; 0Þþ 1 2 3 4 5 6 7 8 9 10 A01 A002 A01 A002 A01 A002 A01 A002 A01 A002 0 0 0 0 0 0 0 0 0 0 E0 E00 E0 E00 E0 E00 E0 E00 E0 E00 0 0 0 0 0 0 0 0 0 0 667.62 686.88 715.73 754.16 802.12 859.57 926.42 1002.62 1088.09 1182.72 667.62 686.88 715.73 754.16 802.13 859.57 926.46 1002.67 1088.13 1182.76 0.00 19.26 48.12 86.55 134.51 191.95 258.80 335.01 420.47 515.10 0.00 19.26 48.12 86.55 134.51 191.95 258.84 335.05 420.51 515.14 0.00 0.00 0.00 0.00 0.00 0.00 )0.04 )0.04 )0.04 )0.04 ð1; 11 ; 0Þþ 1 2 3 4 5 6 7 8 9 10 A01 A002 A01 A002 A01 A002 A01 A002 A01 A002 1 1 1 1 1 1 1 1 1 1 E0 E00 E0 E00 E0 E00 E0 E00 E0 E00 2 3 3 3 3 3 3 3 3 3 1543.26 1559.93 1584.18 1617.02 1657.26 1705.82 1761.55 1825.21 1895.79 1973.81 1543.47 1559.73 1584.39 1616.83 1657.47 1705.67 1761.90 1825.29 1896.22 1974.02 0.00 16.67 40.92 73.75 114.00 162.56 218.29 281.95 352.52 430.54 0.21 16.47 41.12 73.56 114.20 162.41 218.64 282.03 352.96 430.76 )0.21 0.20 )0.20 0.19 )0.20 0.14 )0.35 )0.08 )0.43 )0.22 ð0; 22 ; 1Þ 2 3 4 5 6 7 8 9 10 A002 A01 A002 A01 A002 A01 A002 A01 A002 2 2 2 2 2 2 2 2 2 E00 E0 E00 E0 E00 E0 E00 E0 E00 4 4 4 4 4 4 4 4 4 1798.08 1821.05 1850.31 1888.49 1934.66 1988.72 2052.20 2122.41 2203.01 1798.86 1820.48 1850.65 1888.43 1934.54 1989.20 2051.70 2123.68 2202.49 0.00 22.97 52.24 90.41 136.58 190.65 254.12 324.33 404.93 0.79 22.40 52.57 90.36 136.46 191.12 253.63 325.61 404.41 )0.79 0.57 )0.34 0.06 0.12 )0.47 0.49 )1.27 0.52 ð0; 31 ; 0Þþ 1 2 3 4 5 6 7 8 9 10 A01 A002 A01 A002 A01 A002 A01 A002 A01 A002 2 3 3 3 3 3 3 3 3 3 E0 E00 E0 E00 E0 E00 E0 E00 E0 E00 3 5 5 5 5 5 5 6 6 6 1834.79 1855.98 1881.06 1919.96 1963.57 2019.59 2081.01 2153.20 2230.68 2316.93 1836.87 1853.96 1882.82 1918.37 1964.93 2018.43 2072.79 2153.20 2232.74 2318.45 0.00 21.19 46.27 85.18 128.79 184.81 246.23 318.42 395.89 482.14 2.08 19.17 48.03 83.58 130.15 183.64 238.00 318.42 397.95 483.66 )2.08 2.02 )1.76 1.60 )1.36 1.16 8.22 0.00 )2.06 )1.52 ð0; 33 ; 0Þþ 3 4 5 6 7 8 9 10 A01 A002 A01 A002 A01 A002 A01 A002 4 4 4 4 4 4 4 4 E0 E00 E0 E00 E0 E00 E0 E00 7 7 7 7 7 8 7 8 2003.89 2046.76 2098.20 2160.14 2228.18 2303.37 2358.16 2440.47 2004.34 2046.32 2098.66 2159.76 2229.67 2302.36 2358.62 2441.95 0.00 42.88 94.31 156.25 224.29 299.48 354.28 436.58 0.45 42.44 94.78 155.87 225.78 298.47 354.73 438.06 )0.45 0.44 )0.47 0.38 )1.49 1.01 )0.45 )1.48 Entries as in Table 4. In this equation, v denotes the set of vibrational quantum numbers. The results are presented in Tables 8 and 9. It is noted that only the states ð0; 00 ; 0Þ, ð0; 11 ; 0Þ
, ð0; 00 ; 1Þ, ð1; 00 ; 0Þ, and ð0; 00 ; 2Þ are represented accurately, with a standard deviation of r 6 0:1 cm1 . Within this group of states, small perturbations between ð0; 11 ; 0Þ and ð0; 00 ; 1Þ lead to a decrease of accuracy (in terms of r) by one order of magnitude. The rotational structure of those states with band origins below 1800 cm1 can be represented satisfactorily (r 6 1 cm1 ) by the third-order expression of Eq. (7). For the higher states, a significant loss of accuracy is observed, as expected. To be sure, the accuracy of the fits could have been increased by extending expression (7) to higher orders, but the significance of the high order centrifugal distortion constants and thus the predictive power of the fit would have been rather poor. 172 A. Alijah et al. / Journal of Molecular Spectroscopy 221 (2003) 163–173 Table 8 Rotational parameters of the non-degenerate states ðv1 ; vl2 ; v3 Þ
r mv Bv Dv  103 Hv  106 ð0; 00 ; 0Þ ð0; 11 ; 0Þþ ð0; 11 ; 0Þ ð0; 00 ; 1Þ ð1; 00 ; 0Þ ð0; 11 ; 1Þþ ð0; 11 ; 1Þ ð0; 20 ; 0Þ ð0; 22 ; 0Þþ ð0; 22 ; 0Þ ð1; 00 ; 1Þ ð1; 11 ; 0Þþ ð1; 11 ; 0Þ ð0; 00 ; 2Þ ð0; 20 ; 1Þ ð0; 22 ; 1Þþ ð0; 22 ; 1Þ ð0; 31 ; 0Þþ ð0; 31 ; 0Þ ð0; 33 ; 0Þþ ð0; 33 ; 0Þ ð1; 11 ; 1Þþ ð1; 11 ; 1Þ ð2; 00 ; 0Þ ð1; 20 ; 0Þ ð1; 22 ; 0Þþ ð1; 22 ; 0Þ ð0; 00 ; 3Þ 0.000 0.000 0.055 0.061 0.001 0.724 0.975 0.588 0.495 0.448 0.313 0.147 0.236 0.051 0.972 0.356 0.450 1.181 2.089 6.076 1.049 2.968 1.444 5.112 6.114 4.225 4.300 1.539 0.000 662.802 661.848 739.537 975.499 1269.701 1267.317 1275.545 1331.517 1331.450 1474.962 1539.240 1540.573 1574.487 1730.741 1791.448 1790.799 1831.079 1831.805 1984.460 1988.502 1893.550 1887.966 1925.290 1948.476 2030.138 2034.460 1973.390 4.754 4.817 3.978 5.104 4.313 3.244 2.912 4.850 5.724 5.767 2.967 4.111 4.729 4.168 3.817 3.644 3.706 4.660 4.871 5.903 5.213 3.132 2.119 4.884 3.724 4.161 3.793 2.761 0.371 0.429 )1.715 2.663 0.894 )9.961 )12.661 3.158 9.912 8.991 )5.613 1.306 7.508 0.913 )2.614 )3.678 )2.250 1.900 4.847 17.405 )4.530 )3.741 )12.653 7.963 8.725 5.531 )0.791 )3.521 0.006 )0.004 )6.742 7.024 )0.074 )42.342 )52.736 11.658 36.462 32.944 )23.099 1.496 23.865 )1.262 )13.626 )13.935 )5.070 0.328 )19.541 33.978 )125.101 )23.349 )55.091 8.331 53.451 21.684 )14.289 19.574 r mv Bv Dv  103 Hv  106 1.655 0.006 0.056 1.711 1.520 0.737 0.974 1.078 1.758 2.002 1.219 0.083 0.289 1.455 1.204 0.969 0.893 3.230 0.832 4.463 3.316 3.262 1.423 3.249 4.982 2.360 6.682 3.900 )8.995 662.805 661.847 729.862 967.306 1269.661 1267.361 1266.282 1360.087 1360.498 1470.341 1539.337 1540.555 1566.619 1723.153 1809.775 1809.970 1832.626 1831.667 2072.922 2070.194 1890.949 1888.809 1914.814 1943.622 2052.648 2052.663 1968.097 3.144 4.816 3.978 3.387 2.859 3.250 2.903 3.231 7.739 7.696 1.791 4.099 4.722 2.760 2.511 5.053 5.037 4.496 4.799 9.120 8.727 3.446 2.138 3.557 2.310 5.633 4.941 2.065 )12.242 0.395 )1.758 )11.724 )10.750 )9.801 )13.010 )10.490 40.274 34.736 )14.780 0.952 6.911 )10.618 )12.139 9.664 10.169 0.042 1.448 92.469 36.345 1.541 )12.284 )1.857 )14.995 29.400 0.770 )2.901 )44.195 )0.224 )7.016 )44.956 )40.636 )41.080 )54.349 )38.039 228.344 183.365 )53.974 )0.626 19.839 )41.020 )43.205 49.329 54.877 )2.194 )39.287 640.689 )152.205 4.267 )49.178 )12.637 )79.059 211.778 )59.893 28.327 Table 9 Rotational parameters of the degenerate states ðv1 ; vl2 ; v3 Þ
0 ð0; 0 ; 0Þ ð0; 11 ; 0Þþ ð0; 11 ; 0Þ ð0; 00 ; 1Þ ð1; 00 ; 0Þ ð0; 11 ; 1Þþ ð0; 11 ; 1Þ ð0; 20 ; 0Þ ð0; 22 ; 0Þþ ð0; 22 ; 0Þ ð1; 00 ; 1Þ ð1; 11 ; 0Þþ ð1; 11 ; 0Þ ð0; 00 ; 2Þ ð0; 20 ; 1Þ ð0; 22 ; 1Þþ ð0; 22 ; 1Þ ð0; 31 ; 0Þþ ð0; 31 ; 0Þ ð0; 33 ; 0Þþ ð0; 33 ; 0Þ ð1; 11 ; 1Þþ ð1; 11 ; 1Þ ð2; 00 ; 0Þ ð1; 20 ; 0Þ ð1; 22 ; 0Þþ ð1; 22 ; 0Þ ð0; 00 ; 3Þ A. Alijah et al. / Journal of Molecular Spectroscopy 221 (2003) 163–173 5. Conclusions With the present article the first data on the rotational energies of a large number of vibrational states of triplet Hþ 3 have become available, 8 years after the publication of the review articles by Tennyson [9] and McNab [10] in which they called for such a work. We hope to extend our present studies to analyse all vibrational states up to the dissociation limit in the near future. Acknowledgment This work has the support of the Fundacß~ ao para a Ci^encia e a Tecnologia, Portugal. References [1] L.J. Schaad, W.V. Hicks, J. Chem. Phys. 61 (1974). [2] R. Ahlrichs, C. Votava, C. Zirc, J. Chem. Phys. 66 (1977) 2771. 173 [3] P.E.S. Wormer, F. de Groot, J. Chem. Phys. 90 (1989) 2344. [4] A. Preiskorn, D. Frye, E. Clementi, J. Chem. Phys. 94 (1991) 7204. [5] O. Friedrich, A. Alijah, Z.R. Xu, A.J.C. Varandas, Phys. Rev. Lett. 86 (2001) 1183. [6] C. Sanz, O. Roncero, C. Tablero, A. Aguado, M. Paniagua, J. Chem. Phys. 114 (2001) 2182. [7] E. Cuervo-Reyes, J. Rubayo-Soneira, A. Aguado, M. Paniagua, C. Tablero, C. Sanz, O. Roncero, Phys. Chem. Chem. Phys. 4 (2002) 6012. [8] M. Cernei, A. Alijah, A.J.C. Varandas, J. Chem. Phys. 118 (2003) 2637. [9] J. Tennyson, Rep. Prog. Phys. 57 (1995) 421. [10] I. McNab, Adv. Chem. Phys. 89 (1995). [11] P.R. Bunker, P. Jensen, Molecular Symmetry and Spectroscopy, second ed., NRC Press, Ottawa, Ontario, Canada, 1998. [12] L. Wolniewicz, J. Chem. Phys. 90 (1988) 371. [13] A. Alijah, Habilitation Thesis, University of Bielefeld, 1996. [14] A. Alijah, L. Wolniewicz, J. Hinze, Mol. Phys. 85 (1995) 1125. [15] A. Alijah, M. Beuger, Mol. Phys. 88 (1996) 497. [16] A. Alijah, J. Hinze, L. Wolniewicz, Mol. Phys. 85 (1995) 1105. [17] P. Schiffels, A. Alijah, J. Hinze, Mol. Phys. 101 (2003) 175. [18] P. Schiffels, A. Alijah, J. Hinze, Mol. Phys. 101 (2003) 189. [19] G. Duxbury, Infrared Vibration–Rotation Spectroscopy: From Free Radicals to the Infrared Sky, Wiley, Chichester, 2000.