From special unitary groups SU(2I+1) to nuclear spin statistical weights

Journal of Molecular Spectroscopy 219 (2003) 177–185 www.elsevier.com/locate/jms

From special unitary groups SUð2I þ 1Þ to nuclear spin statistical weightsq A. Ruoff *,1 Sektion Schwingungsspektroskopie, Universit€at Ulm, Albert-Einstein-Allee, D-89081 Ulm, Germany Received 23 April 2002; in revised form 22 August 2002 Dedicated to Prof. Dr. Hans B€ urger on the occasion of his 65th birthday

Abstract Following a suggestion of S.R. Polo, Galbraith [J. Chem. Phys. 68 (1978) 1677] was the first to obtain nuclear spin statistical weights for molecules of the type XY4 ðTd Þ, XY5 ðD3h Þ, and XY6 ðOh ) starting from special unitary groups. In the present contribution, an analogous but simpler method is presented allowing the calculation of nuclear spin statistical weights of rigid molecules belonging to all important molecular point groups. The molecules under consideration may contain more than one set of identical nuclei with spins up to 3. The method proposed can easily be extended to nonrigid molecules. Ó 2003 Published by Elsevier Science (USA). Keywords: Nuclear spin statistical weights; Special unitary groups; Group theory

1. Introduction In high-resolution rovibronic spectroscopy of molecular substances in the gas phase, the assignment of the rotational lines is often a difficult task. This is true especially for perpendicular bands of symmetric top molecules. In most cases, this type of band has a very complicated appearance due to its subband structure and to accompanying hot bands. In favorable cases the analysis of a perpendicular band is straightforward if the R Q0 -subband is discernible in the observed spectrum. In this case, the assignment of the subband R Q0 is unambiguous if the nuclear spin statistical weights (henceforth abbreviated as NSW) of the ground state levels K ¼ 0 with J ¼ even and J ¼ odd are known. To a lesser extent, NSW are also useful for the analysis of other bands, especially of hot bands. The usual way of calculating NSW of rovibronic molecular levels consists of four steps. First, one has to q This article was intended for the Special Issue Dedicated to Professor Hans B€ urger on the Occasion of his 65th Birthday. * Fax: +49-0-731-5023187. E-mail address: andreas.ruoff@chemie.uni-ulm.de. 1 Current address: Sektion R€ ontgen- und Elektronenbeugung, Universit€ at Ulm, Albert-Einstein-Allee 11, D-89081 Ulm, Germany.

evaluate the character of the (reducible) representation of the nuclear spin functions jnspini of the molecule under an appropriate spin permutation group, which is usually assumed to be isomorphic with the molecular point group (PG). This may be done by a method given by Landau and Lifshitz [1], as corrected by Jonas [2]. Second, this representation has to be reduced into a sum of irreps. Third, one has to take the direct product of the species of the involved rovibronic levels and of the species of the nuclear spin functions. Finally, one has to see how many times this latter product contains the species of the total internal wavefunction jinti obeying the Pauli exclusion principle. In this approach, the first two steps are often a tedious and time consuming affair. This method has been described in many papers which are summarized in [3,4]. Balasubramanian [4] proposed the application of spin projection operators instead of the first two steps sketched above. For rigid molecules, his method is also based on a group being isomorphic to the group PG. Following a suggestion of S.R. Polo, Galbraith [5] adopted a principally different method for the calculation of the NSW of the molecules XY4 ðTd Þ, XY5 ðD3h Þ, and XY6 ðOh Þ. He reduced the product group SUð2I þ 1Þ  SðN Þ to a group of permutations

0022-2852/03/$ - see front matter Ó 2003 Published by Elsevier Science (USA). doi:10.1016/S0022-2852(03)00002-X

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A. Ruoff / Journal of Molecular Spectroscopy 219 (2003) 177–185

isomorphic with the molecular point group in order to avoid the reduction of every reducible representation. At this place, the term ‘‘point group’’ deserves the following comment: If this term is used according to Herzberg [6], i.e., as a group consisting of all geometrical transformations of the positions of the nuclei of a molecule, it will be termed a ‘‘geometrical point group.’’ Following Bunker [3], a point group whose operations act only on the vibronic coordinates is called a ‘‘molecular point group.’’ In the present contribution a method will be presented which is analogous to that of Galbraith, but simpler, and which is extended to molecules with more than one set of nuclei belonging to all important molecular point groups. Tables will be given allowing the calculation of the NSW of molecules belonging to Cnv ðn ¼ 2; . . . ; 6Þ, Cnh ðn ¼ 2; . . . ; 6Þ, Dnh ðn ¼ 2; . . . ; 6Þ, Dnd ðn ¼ 2; . . . ; 6Þ, D1h , Td , and Oh , with one or more sets of up to eight equivalent nuclei and nuclear spins up to 3. 2. Theory 2.1. General In the literature, two alternative symmetry classifications of energy levels of rigid molecules are given. They are reviewed in detail by Ezra [7]. (1) Berger [8], Harter et al. [9], and Moret-Bailly [10] use for nonplanar and nonlinear molecules the following chain: Y SðN Þi ¼ GCNP GP PG: ð1Þ Here, GCNP denotes the complete nuclear permutation group of all i sets of N identical nuclei. GP is a subgroup of GCNP which is isomorphic to the PG. This approach is characterized by the fact that the parity group e does not occur. (2) Contrary to this point of view, Longuet-Higgins [11], Hougen [12], and Bunker [3] include the parity group e and advocate the following chain: e  GCNP ¼ GCNPI GMS PG GMSP PGR : ð2Þ MS

Here, G is the group of all feasible permutations P and permutation–inversions P of the molecule. GMSP denotes the permutation subgroup of GMS . PGR stands for the rotational subgroup of the molecular point group. The present contribution is based on chain (2) and the notations are used according to Bunker [3]. For more information on chain (2), the reader is referred to [3]. As pointed out above, the NSW of a rovibronic level jrvei is equal to the number of internal functions jinti of correct symmetry that are obtainable by combining the functions jrvei with the nuclear spin functions jnspini.

In the language of group theory, the NSW is the number of occurrences of Cint of correct symmetry in the direct product Crve  Cnspin . Usually, the wavefunctions jinti, jri, jvi, and jei are classified under the group GMS or a group isomorphic to it. If the corresponding group GMS contains E (and hence P ), two species of jinti are allowed by symmetry. They are designated as Cþ and C as they have positive or negative parity. Unlike the functions discussed, the spin functions have the property of having only positive parity. Thus, the NSW can either be determined under the groups GMS and an isomorphic spin permutation group or under the smaller group GMSP . In the present contribution, the group GMSP is used and, therefore, a considerable simplification is obtained without any loss of information. 2.2. The overall allowed species Cþ Under the group GMSP only one species is permitted for the function jinti. This is the species with positive parity, i.e., Cþ . According to the Pauli exclusion principle [3], the character of an operation P has the value þ1 for permuting a pair of identical boson nuclei but it has the value þ1 or 1 according to whether P permutes an even or an odd number of pairs of identical fermion nuclei. Following Herzberg [6], the determination of Cþ is facilitated by looking at the geometrical structure of the molecule under consideration and its geometrical symmetry elements, whose operations create the geometrical point group. In the first column of Table 1 all point groups discussed in the present contribution are listed. The corresponding rotational subgroups are given in the second column of Table 1. Herzberg [6] pointed out that the nuclei of a molecule can be divided in sets of identical nuclei. The nuclei of one set are transformed into each other by the operations of the geometrical point group. Or, in other words, the positions of all nuclei of a set are fixed by the effect of all operations of the geometrical point group on one representative member of the set. This representative nucleus of the set may have a general position (not on any element of symmetry), or it may be placed on one or more symmetry elements, or it may lie on all symmetry elements. To specify its position, the following abbreviations [6] are used: m ¼ not on any element of symmetry m0 ¼ on all elements of symmetry mab ; mh ; mv ; md , m0 , m00 ¼ only on the mirror plane rab , rh , rv , rd , r0 , r00 but not on an axis mn ¼ on an axis Cn . In Table 1, all possible sets containing identical nuclei of the point groups of column 1 are given in the columns 4 and 5. Every set is designated by the geometrical position of its representative nucleus followed in parenthesis by the number of its members.

A. Ruoff / Journal of Molecular Spectroscopy 219 (2003) 177–185

179

Table 1 MSP of the group GMS (see text) Species Cþ i of a set mi of fermions under the permutational subgroup G PG GMS

PGR GMSP

Ci

Set of type 2

Set of type 1

C3v C3h C5v C5h Td

C3

—

—

—

—

—

—

—

—

—

—

C2v C2h C4v C4h C6v C6h D3h

C2

mv (3), m3 (2), mv (5), m5 (2), m3 (4), m(24) m(4) m2 (2), m(8) m4 (2), m(12) m6 (2), m(12)

C5 T

C4 B C6

D3 A2

D3d D5h D5 D5d Oh

O

D2h D4 D2d D4h

D4d D6h D6 D6d D1h

D1

B1 B2 B3 B1 A2 B1 B2 A2 A2 B2 B1 B1 B

From the Pauli exclusion principle, it is inherent that only sets containing fermion nuclei have to be taken into account in the determination of the species Cþ . Every þ such set gives a contribution Cþ i to C resulting in Y Cþ ¼ Cþ ð3Þ i : Determining the effect of the permutations P on the nuclei of a set reveals that the sets can be divided into two types. Some sets taken alone yield the totally symmetric species A1 as Cþ i . Sets of this type 1 are collected together in the last column of Table 1. Clearly, all sets m0 ð1Þ also belong to this type. For the sake of brevity, the latter are not listed in Table 1. All other sets (designated as type 2) are summarized in the fourth column of Table 1. Taken alone they yield nondegenerate but nontotally symmetric species Cþ i as given in column 3. The multiplication rules of species have been published by some authors [6,13] and in a very compact form by Salthouse and Ware [14]. The rules relevant to the species Cþ i of Table 1 are given in Table 2 and those for the groups D3 and D6 are collected in Table 3. As an illustration of applying Table 1, two examples will be given:

mxz (2), myz (2) mh (2) mv (4), md (4) mh (4) mv (6), md (6) mh (6) m3 (2), m2 (3), mh (6), mv (6) m6 (2), md (6) m5 (2), mh (10), mv (10) m5 (2), md (10) m4 (6), m2 (12) m2z (2) m2y (2) m2x (2) m4 (2), m2 (4) m4 (2) m002 (4) m02 (4) m8 (2), m2 (8) m6 (2) m00 (6) m0 (6) m12 (2), m2 (12) m1 (2)

m(6) mh (3), m(6) m(10) mh (5), m(10) m2 (6), md (12),

m(4) m(8) m(12)

m2 (6), m(12) m2 (5), m(20) m2 (10), m(20) m3 (8), md (24), mh (24), m(48) myz (4), mxz (4), mxy (4), m(8) md (4), m(8) mh (8), mv (8), md (8), m(16) md (8), m(16) mh (12), m00v (12), m0v (12), m(24) md (12), m(24) —

Table 2 Multiplication rules for nondegenerate species of the groups Cn ðn ¼ 2; 4; 6Þ, Dn (n ¼ 2, 3, 4, 5, 6), D1 , T, and O [13] General rules AA¼BB¼A AB¼B Exception: For D2 : B  B ¼ B if the subscript numbers are not the same. ÔÕ ¼ ‘‘’’ ¼ Õ Ô’’ ¼ ‘‘ Subscripts on A or B: 11¼22¼1 12¼2 Exception: For D2 : 12¼3 23¼1 13¼2

Example 1. The rigid molecule PF5 belongs to the molecular point group D3h , the rotational subgroup of which is D3 . In this molecule, m0 equals 1 (the phosphorus nucleus), m3 equals 2 (the two axial fluorine

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A. Ruoff / Journal of Molecular Spectroscopy 219 (2003) 177–185

Table 3 Multiplication rules for the groups D3 and D6 (see also Table 2) D3

E  E ¼ A1 þ ðA2 Þ þ E

D6

B  E1 ¼ E2 B  E2 ¼ E1 E1  E1 ¼ E2  E2 ¼ A1 þ ðA2 Þ þ E2 E1  E2 ¼ B1 þ B2 þ E1

Table 4 Dimension k of the species of SUðnÞ  Sð8Þ; n stands for (2I þ 1), b stands for ð8!Þ 1

Species in parentheses are the antisymmetric part.

nuclei), and m2 equals 3 (the three equatorial fluorine nuclei). Or, briefly: m0 ð1Þ, m3 ð2Þ, and m2 ð3Þ. According to Tables 1 and 2, and (3), Cþ ¼ A1  A2  A2 ¼ A1 :

ð4Þ

Example 2. The molecule hexadeutero-13 C-benzene, C6 D6 , belongs to the molecular point group D6h , whose rotational subgroup is D6 . The molecule contain two sets m02 (6): the six 13 C nuclei and the six deuteron nuclei. The second set does not play a role in the present calculation because the D nucleus is a boson. Hence

13

Cþ ¼ B1 :

ð5Þ

2.3. The nuclear spin functions As is well known [15,16], spin functions of particles with the spin I transform according to the species of the special unitary group SUð2I þ 1Þ. In the past, methods were developed to adapt permutational symmetry to the irreps of SUð2I þ 1Þ. It can be shown that the operations of the group SUð2I þ 1Þ and the permutations of identical spins commute. Thus, the spin functions of N identical particles with spin I have to be classified under the product group SUð2I þ 1Þ  SðN Þ. Graphically, every species of this product group can be characterized by the associated Young diagram consisting of N cells. Kaplan [15] and Hamermesh [16] have given formulae to calculate the dimension k, i.e., the degree of degeneration of these species, in terms of the characteristic parameters of the associated Young diagrams. Galbraith [5] has given the formulae for the cases N ¼ 4, 5, and 6 in a very convenient form. The formulae for the case N ¼ 8 are derived in Table 4. Every formula for k of a species C of SUðnÞ  SðN Þ has the structure 1

kC ¼ dC  f ðnÞðN !Þ ;

Species of S(8)

k

[8] [7, 1] [6, 2] [6, 12 ] [5, 3] [5, 2, 1] [5, 13 ] [42 ] [4, 3, 1] [4, 22 ] [4, 2, 12 ] [32 , 2] [32 ; 12 ] [3, 22 , 1] [24 ] [4, 14 ] [3, 2, 13 ] [23 ; 12 ] [3, 15 ] [22 ; 14 ] [2, 16 ] [18 ]

nðn þ 1Þðn þ 2Þ    ðn þ 7Þb 7ðn 1Þnðn þ 1Þ    ðn þ 6Þb 20ðn 1Þn2 ðn þ 1Þ    ðn þ 5Þb 21ðn 2Þðn 1Þnðn þ 1Þ    ðn þ 5Þb 28ðn 1Þn2 ðn þ 1Þ2 ðn þ 2Þ    ðn þ 4Þb 64ðn 2Þðn 1Þn2 ðn þ 1Þ    ðn þ 4Þb 35ðn 3Þðn 2Þ    n    ðn þ 4Þb 14ðn 1Þn2 ðn þ 1Þ2 ðn þ 2Þ2 ðn þ 3Þb 70ðn 2Þðn 1Þn2 ðn þ 1Þ2 ðn þ 2Þðn þ 3Þb 56ðn 2Þðn 1Þ2 n2 ðn þ 1Þðn þ 2Þðn þ 3Þb 90ðn 3Þðn 2Þðn 1Þn2 ðn þ 1Þðn þ 2Þðn þ 3Þb 42ðn 2Þðn 1Þ2 n2 ðn þ 1Þ2 ðn þ 2Þb 56ðn 3Þðn 2Þðn 1Þn2 ðn þ 1Þ2 ðn þ 2Þb 70ðn 3Þðn 2Þðn 1Þ2 n2 ðn þ 1Þðn þ 2Þb 14ðn 3Þðn 2Þ2 ðn 1Þ2 n2 ðn þ 1Þb 35ðn 4Þðn 3Þ    n    ðn þ 3Þb 64ðn 4Þ    ðn 1Þn2 ðn þ 1Þðn þ 2Þb 28ðn 4Þðn 3Þðn 2Þðn 1Þ2 n2 ðn þ 1Þb 21ðn 5Þðn 4Þ    n    ðn þ 2Þb 20ðn 5Þðn 4Þ    ðn 1Þn2 ðn þ 1Þb 7ðn 6Þðn 5Þ    nðn þ 1Þb ðn 7Þðn 6Þ    ðn 1Þnb

priate group GMSP according to Table 6. Finally, Cnspin is obtained as Y Cnspin ¼ CðnspinÞi : ð7Þ The multiplication rules for the groups D3 and D6 are given in Table 3 as an example. For all other groups, the reader is referred to the standard text books [6,13,14]. In the following, the application of Tables 3, 5, and 6 will be explained for the molecules PF5 and 13 C6 D6 . Example 1. For the molecule PF5 , the sets m2 and m3 produce the following spin representations for I ¼ 1=2: m2 :

CFnspin ¼ 4A1 þ 2E ðunder Sð3ÞÞ

m3 :

CFnspin ¼ 3A þ B ðunder Sð2ÞÞ ¼ 3A1 þ A2 ðunder Sð3Þ D3 Þ:

Applying Eq. (7) yields under S(3) Cnspin ¼ ð4A1 þ 2EÞ  ð3A1 þ A2 Þ ¼ 12A1 þ 4A2 þ 8E:

ð6Þ

where dC denotes the dimension of the species C and f ðnÞ is a polynomial in n of degree N. The numerical results are listed in Table 5. The character tables of S(N) are given in [15,16] for N up to 8. These latter tables have been used to derive the correlations collected in Table 6. The species and the dimensionality of the total nuclear spin function of a molecule can be obtained in the following way: First, find for every set of N identical nuclei the corresponding representation CðnspinÞi , applying Table 5. Then correlate all these representations to the appro-

ð8Þ

If the spin of the central phosphorus nucleus has to be taken into account, the result of Eq. (8) has to be multiplied by the factor (2IP þ 1Þ ¼ 2. Example 2. For the molecule C6 D6 with IC ¼ 1=2 and ID ¼ 1, the two sets m02 (6) give rise to CCnspin ¼ 7A1 þ 5H11 þ 3M1 þ H21

ð9aÞ

and CD nspin ¼ 28A1 þ 35H11 þ 27M1 þ 10N1 þ 10H21 þ 8T þ H22 :

ð10aÞ

A. Ruoff / Journal of Molecular Spectroscopy 219 (2003) 177–185

181

Table 5 Dimension k of the species of the groups SUð2I þ 1Þ  SðN Þ for I ¼ 1/2; 1; 3/2; 2; 5/2; and 3 and for N ¼ 1, 2, 3, 4, 5, 6, and 8 S(N)

Species of S(N)

SU(2) ðI ¼ 1=2Þ

S(1)

[1], A

2

3

S(2)

[2], A [12 ], B

3 1

S(3)

[3], A1 [2, 1], E [13 ], A2

4 2

[4], A1 [3, 1], F2 [22 ], E [2, 12 ], F1 [14 ], A2

5 3 1

S(4)

S(5)

S(6)

S(8)

SU(5) ðI ¼ 2Þ

SU(6) ðI ¼ 5=2Þ

4

5

6

7

6 3

10 6

15 10

21 15

28 21

10 8 1

20 20 4

35 40 10

56 70 20

84 112 35

—

15 15 6 3

—

—

35 45 20 15 1

70 105 50 45 5

126 210 105 105 15

210 378 196 210 35

[5], A1 [4, 1], G1 [3, 2], H1 [3, 12 ], I [22 , 1], H2 [2, 13 ], G2 [15 ], A2

6 4 2 —

21 24 15 6 3

—

—

56 84 60 36 20 4

—

—

—

126 224 175 126 75 24 1

252 504 420 336 210 84 6

462 1008 882 756 490 224 21

[6], A1 [5,1], H11 [4,2], M1 [4, 12 ], N1 [32 ], H21 [3, 2, 1], T [23 ], H22 [3, 13 ], N2 [22 , 12 ], M2 [2, 14 ], H12 [16 ], A2

7 5 3

—

28 35 27 10 10 8 1

—

—

—

—

84 140 126 70 50 64 10 10 6

—

—

—

210 420 420 280 175 280 50 70 45 5

—

—

—

—

462 1050 1134 840 490 896 175 280 189 35 1

924 2310 2646 2100 1176 2352 490 840 588 140 7

[8] [7, 1] [6, 2] [6, 12 ] [5, 3] [5, 2, 1] [5, 13 ] [42 ] [4, 3, 1] [4, 22 ] [4, 2, 12 ] [32 , 2] [32 ; 12 ] [3, 22 , 1] [24 ] [4, 14 ] [3, 2, 13 ] [23 , 12 ] [3, 15 ] [22 , 14 ] [2, 16 ] [18 ]

9 7 5

45 63 60 21 42 24

—

—

—

1 —

SU(3) ðI ¼ 1Þ

SU(4) ðI ¼ 3=2Þ

SU(7) ðI ¼ 3Þ

—

3

—

—

—

—

—

—

165 315 360 189 280 256 35 105 175 84 45 45 20 15 1

—

—

—

—

—

—

—

—

—

495 1155 1500 945 1260 1440 315 490 1050 560 450 315 210 175 15 35 40 10

—

—

—

—

—

—

—

—

1287 3465 4950 3465 4410 5760 1575 1764 4410 2520 2430 1470 1176 1050 105 315 384 105 21 15

—

—

—

—

—

3003 9009 13 860 10 395 12 936 18 480 5775 5292 14 700 8820 9450 5292 4704 4410 490 1575 2016 588 189 140 7

—

—

—

—

—

—

—

3 — —

—

1 —

15 15 6

—

—

—

Reduction from the group S(6) to the group D6 yields (Table 6) CCnspin ¼ 13A1 þ A2 þ 7B1 þ 3B2 þ 9E1 þ 11E2

ð9bÞ

and CD nspin ¼ 92A1 þ 38A2 þ 73B1 þ 46B2 þ 116E1 þ 124E2 : ð10bÞ

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A. Ruoff / Journal of Molecular Spectroscopy 219 (2003) 177–185

Table 6 Correlation tables of the chain SðN Þ GMSP for N ¼ 1, 2, 3, 4, 6, and 8 S(1)

All groups

S(2), C2

C6 ; C5 ; C4 ; C3

A

Að1Þ

A B

A A

S(2), C2

D1 ; D6 ; D5 ; D4 ; D3

D2 (z)

D2 (y)

D2 (x)

A B

A1 A2

A B1

A B2

A B3

S(3), D3

A(3), C3

A1 E A2

A Eþ þ E A

S(4), O

A(4), T

D4

C4

D2

C2 (z)

A1 F2 E F1 A2

A F Eþ þ E F A

A1 B1 þ E A1 þ B2 A2 þ E B2

A B þ Eþ þ E AþB A þ Eþ þ E B

A B1 þ B2 þ B3 2A B1 þ B2 þ B3 A

A A þ 2B 2A A þ 2B A

S(5)

D5

C5

A(5)

A1 ; A2 G1 ; G2 H1 ; H2 I

A1 E1 þ E2 A1 þ E1 þ E2 2A2 þ E1 þ E2

A E1þ þ E1 þ E2þ þ E2 A þ E1þ þ E1 þ E2þ þ E2 2A þ E1þ þ E1 þ E2þ þ E2

A G H Iþ þ I

S(6)

D6

O

D3

A1 H11 M1 N1 H21 T H22 N2 M2 H12 A2

A1 B1 þ E1 þ E2 2A1 þ B2 þ E1 þ 2E2 2A2 þ B1 þ B2 þ 2E1 þ E2 A2 þ 2B1 þ E1 A1 þ A2 þ B1 þ B2 þ 3E1 þ E2 2A1 þ B2 þ E2 A1 þ A2 þ 2B2 þ E1 þ 2E2 A2 þ 2B1 þ 2E1 þ E2 A1 þ E1 þ E2 B1

A1 E þ F1 A1 þ E þ 2F2 A2 þ 2F1 þ F2 2A2 þ F1 2E þ 2F1 þ 2F2 2A1 þ F2 A1 þ F1 þ 2F2 A2 þ E þ 2F1 E þ F2 A2

A1 A2 þ 2E 3A1 þ 3E 3A2 þ A1 þ 3E 3A2 þ E 2A1 þ 2A2 þ 6E 3A1 þ E 3A1 þ A2 þ 3E 3A2 þ 3E A1 þ 2E A2

D6

C6

A1 A2 B1 B2 E1 E2

A A B B E1þ þ E1 E2þ þ E2

S(8)

O 8

[8], [1 ] [7, 1], [2, 16 ] [6, 2], [22 ; 14 ] [6, 12 ], [3, 15 ] [5,3], [23 ; 12 ] [5, 2, 1], [3, 2, 13 ] [5, 13 ], [4, 14 ] [42 ], [24 ] [4, 3, 1], [3, 22 , 1] [4, 22 ], [32 ; 12 ] [4, 2, 12 ] [32 , 2]

A1 A1 þ 2F2 2A1 þ 3E þ F1 þ 3F2 A1 þ E þ 3F1 þ 3F2 2A1 þ E þ 3F1 þ 5F2 2A1 þ 2A2 þ 6E þ 8F1 þ 8F2 A1 þ 4A2 þ 3E þ 5F1 þ 3F2 3A1 þ A2 þ 2E þ F1 þ F2 3A1 þ 3A2 þ 5E þ 8F1 þ 10F2 3A1 þ 3A2 þ 7E þ 6F1 þ 6F2 2A1 þ 4A2 þ 6E þ 14F1 þ 10F2 2A1 þ 2E þ 6F1 þ 6F2

A. Ruoff / Journal of Molecular Spectroscopy 219 (2003) 177–185

183

Table 6 (continued) S(8)

D4 8

[8], [1 ] [7, 1], [2, 16 ] [6, 2], [22 ; 14 ] [6, 12 ], [3, 15 ] [5,3], [23 ; 12 ] [5, 2, 1], [3, 2, 13 ] [5, 13 ], [4, 14 ] [42 ], [24 ] [4, 3, 1], [3, 22 , 1] [4, 22 ], [32 ; 12 ] [4, 2, 12 ] [32 , 2]

A1 A2 þ B1 þ B2 þ 2E 5A1 þ A2 þ 3B1 þ 3B2 þ 4E A1 þ 4A2 þ 2B1 þ 2B2 þ 6E A1 þ 5A2 þ 3B1 þ 3B2 þ 8E 8A1 þ 8A2 þ 8B1 þ 8B2 þ 16E 6A1 þ 3A2 þ 5B1 þ 5B2 þ 8E 6A1 þ 2B1 þ 2B2 þ 2E 7A1 þ 9A2 þ 9B1 þ 9B2 þ 18E 12A1 þ 4A2 þ 8B1 þ 8B2 þ 12E 8A1 þ 14A2 þ 10B1 þ 10B2 þ 24E 2A1 þ 8A2 þ 4B1 þ 4B2 þ 12E

S(8)

D2

[8], [18 ] [7, 1], [2, 16 ] [6, 2], [22 ; 14 ] [6, 12 ], [3, 15 ] [5,3], [23 ; 12 ] [5, 2, 1], [3, 2, 13 ] [5, 13 ], [4, 14 ] [42 ], [24 ] [4, 3, 1], [3, 22 , 1] [4, 22 ], [32 ; 12 ] [4, 2, 12 ] [32 , 2]

A A þ 2B1 þ 2B2 þ 2B3 8A þ 4B1 þ 4B2 þ 4B3 3A þ 6B1 þ 6B2 þ 6B3 4A þ 8B1 þ 8B2 þ 8B3 16A þ 16B1 þ 16B2 þ 16B3 11A þ 8B1 þ 8B2 þ 8B3 8A þ 2B1 þ 2B2 þ 2B3 16A þ 18B1 þ 18B2 þ 18B3 20A þ 12B1 þ 12B2 þ 12B3 18A þ 24B1 þ 24B2 þ 24B3 6A þ 12B1 þ 12B2 þ 12B3

D4

C4

A1 A2 B1 B2 E

A A B B Eþ þ E

The representation Cnspin , then, is obtained by multiplication of (9b) and (10b) following Table 3 as Cnspin ¼ 4291A1 þ 3535A2 þ 4145B1 þ 3605B2 þ 7735E1 þ 7805E2 :

ð11Þ

2.4. Vibronic functions and NSW The species of jrvei, Crve , is given by Crve ¼ Cr  Cv  Ce :

ð12Þ

For vibrational spectroscopy of molecules with closed electronic shells, Eq. (12) is simplified to Crve ¼ Cr  Cv :

ð13Þ

Analogously, for rotational spectroscopy of molecules in the vibrational ground state, (13) is further simplified to Crve ¼ Cr :

ð14Þ

As is well known (see, e.g., [3]), the group GR is a subgroup of the group of a sphere, K, which is iso-

morphic to the group SO(3). If a state jrvei has an angular momentum F, it belongs to the species DF under K. In the present case F is equal to J. Lowering the spherical symmetry to that of a real molecule produces a splitting of the rotational levels. Group theoretically, this is given by correlating the group K down to the group GR . This effect is summarized in the Tables 7 and 8. If GR is one of the cubic groups (i.e., I, O, T ), a rotational level is characterized by the angular momentum quantum number J and by one of the species of the corresponding group GR . Different to this, the rotational level of a symmetric top molecule is characterized by the quantum number J and either by the quantum number K or by a species of the corresponding group GR . This correspondence of the quantum number K and the species of the D1 and its subgroups is given in Tables 7 and 8. As stated above, the NSW of a molecular level is equal to the number of occurrence of the species Cþ in the representation of the product Crve  Cnspin . Principally, one should, therefore, perform this latter multiplication to obtain the NSW.

184

A. Ruoff / Journal of Molecular Spectroscopy 219 (2003) 177–185

Table 7 Correlation tables of the chains K Oð16Þ, and K D1 ðn ¼ 1; 2; 3; . . .Þ K

O n

ðn þ 1Þ

D12n D12nþ1 D12nþ2 D12nþ3 D12nþ4 D12nþ5 D12nþ6 D12nþ7 D12nþ8 D12nþ9 D12nþ10 D12nþ11

A2 þ 2E þ 3F1 þ 3F2 A1 þ A2 þ 2E þ 2F1 þ 3F2 A1 þ A2 þ E þ 3F1 þ 2F2 A1 þ 2E þ 2F1 þ 2F2 A2 þ E þ 2F1 þ 2F2 A1 þ A2 þ E þ F1 þ 2F2 E þ 2F1 þ F2 A1 þ E þ F1 þ F2 A2 þ F1 þ F2 E þ F2 F1 A1

A1 F1 E þ F2 A2 þ F1 þ F2 A1 þ E þ F1 þ F2 E þ 2F1 þ F2 A1 þ A2 þ E þ F1 þ 2F2 A2 þ E þ 2F1 þ 2F2 A1 þ 2E þ 2F1 þ 2F2 A1 þ A2 þ E þ 3F1 þ 2F2 A1 þ A2 þ 2E þ 2F1 þ 3F2 A2 þ 2E þ 3F1 þ 3F2

K

D1

2n

D D2nþ1 K

A1 þ E1 þ E2 þ    þ E2n A2 þ E1 þ E2 þ    þ E2n þ E2nþ1 0 1 2    2n 2n þ 1

As usual, K is the quantum number of the component of angular momentum in the molecular z-axis.

In practice, this is not necessary because only portions of this product are of interest. This will be demonstrated in the following section. 2.5. Examples 2.5.1. Example 1: The vibrational ground state of the (rigid) molecule PF5 As described above the following results were obtained for PF5 : The GMSP is D3 (M), Cþ is A1 , and the spin functions are represented by 12A1 þ 4A2 þ 8E. Inspection of Tables 7 and 8 reveals that if K ¼ 0 if K ¼ 3p if K ¼ 3p  1

then Cr ¼ A1 ðJ evenÞ or Cr ¼ A2 ðJ oddÞ; then Cr ¼ A1 þ A2 and then Cr ¼ E ðfor p ¼ 1; 2; 3; . . .Þ:

As Tables 2 and 3 demonstrate the species of Cþ , A1 , is only obtained under D3 as A1  A1 and as A2  A2 and is contained once in the product E  E. From these facts, the following result can be deduced easily: K 0 ðJ evenÞ 0 ðJ oddÞ 3p 3p  1

NSW 1  12 ¼ 12 14¼4 12 þ 4 ¼ 16 1  8 ¼ 8:

ð15Þ

In the column NSW the first factor is the number of occurrences of the appropriate species in the representation Cr . The second factor is the number of oc-

Table 8 Correlation tables of the chains D1 D6 ; D5 ; D4 ; D3 , and D2 D1

D6

A1 A2 Eðnþ1Þ Eðnþ2Þ Eðnþ3Þ Eðnþ4Þ Eðnþ5Þ Eðnþ6Þ

A1 A2 E1 E2 B1 þ B2 E2 E1 A1 þ A2

n ¼ 0; 6; 12; . . . D1

D5

A1 A2 Eðnþ1Þ Eðnþ2Þ Eðnþ3Þ Eðnþ4Þ Eðnþ5Þ

A1 A2 E1 E2 E2 E1 A1 þ A2

n ¼ 0; 5; 10; . . . D1

D4

A1 A2 Eðnþ1Þ Eðnþ2Þ Eðnþ3Þ Eðnþ4Þ

A1 A2 E B1 þ B2 E A1 þ A2

n ¼ 0; 4; 8; . . . D1

D3

A1 A2 Eðnþ1Þ Eðnþ2Þ Eðnþ3Þ

A1 A2 E E A1 þ A2

n ¼ 0; 3; 6; . . . D1

D2

A1 A2 Eðnþ1Þ Eðnþ2Þ n ¼ 0; 2; 4; 6; . . .

A B1 B2 þ B3 A þ B1

currence of the appropriate species in the spin representation. 2.5.2. Example 2: The vibronic state B2 of the molecule 13 C6 D6 The following results have been obtained for the molecule 13 C6 D6 : The group GMSP of this molecule is D6 (M), Cþ is of the species B1 and the spin representation is given by Eq. (11). An inspection of Tables 7 and 8 and the use of the multiplication rules of Tables 2 and 3 show that, for a rovibronic state B2 , the following hold:

A. Ruoff / Journal of Molecular Spectroscopy 219 (2003) 177–185

if K ¼ 0 if if if if

K ¼ 6p K ¼ 3p K ¼ 3p  1 K ¼ 6p  1

then Cr Cr then Cr then Cr then Cr then Cr

Acknowledgments

¼ B2 ðJ evenÞ or ¼ B1 ðJ oddÞ; ¼ B1 þ B2 ; ¼ A1 þ A2 ; ¼ E1 and ¼ E2 :

The author thanks Dr. J. Vogt (Sektion Strukturdokumentation, Universit€at Ulm) for a critical reading of the manuscript and Ms. K. Ruoff and Mr. J. Wandel for typing the manuscript.

The species B1 is obtained under D6 as A1  B1 and as A2  B2 and is contained once in the product E1  E2 . These facts imply that K 0 ðJ evenÞ 0 ðJ oddÞ 6p 3p 3p  1 6p  1

NSW 3535 4291 7826 7750 7805 7735

185

ð16Þ

3. Conclusion The present method yields the NSW for rigid molecules containing one or more sets of identical nuclei. It is based on the reduction of the product group SUð2I þ 1Þ  SðN Þ onto the group GMSP . Therefore, it avoids the redundancy which is contained in the use of a spin permutation group being isomorphic with the molecular point group. Clearly, this method may be extended to the nonrigid molecules. The results given in the Eqs. (15) and (16) demonstrate an effect well known to all spectroscopists: The term ‘‘favorable case’’ which has been used in the Section 1 means the case of molecules with small sets of nuclei and/or low nuclear spins I. The theoretical reason for this is given by the results of Table 4.

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