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Comment on nuclear spin weights of 12C60 and 13C60 buckminsterfullerene
Volume 200, number 6
CHEMICAL PHYSICS LETTERS
25 December I992
Comment Comment on nuclear spin weights of 12C&-, and ’3C6Obuckminsterfullerene ILBalasubramanian Department of Chemistry and Biochemistry, Arizona State University,Tempe, AZ 85287-1604, USA Received 12 August 1992; in final form 7 October I992
The nuclear spin statistical weights obtained in a Letter by Hatter and Reimer differ from the values obtained by the author a year earlier. However, the corrected numbers reported in the Erratum by Harter and Reimer agree with our values.
In a recent Letter Harter and Reimer [ 1] have obtained the nuclear spin weights of ‘*C6,,and ‘Q,. However, they appear to be unaware of a paper on the nuclear spin statistical weights of Cdo,CsJIso and ChODeO published a year ago in the same journal [ 2 1. Combinatorial generators based on generalized character cycle indices (GCCI) implemented on a microvax as well as IBM 3090 with quadruple precision arithmetic [ 2 ] yield different weights compared to the values reported in ref. [ 11. Our procedure can also yield the frequencies of the individual nuclear spin multiplets. The total central processing unit time to generate both nuclear spin multiplets and statistical weights for i3Ceois 1.39 s on an IBM 3090. The total nuclear spin statistical weights in ref. [2] for 13Ce0and i2Cb0Hb0are
resentations in Ppln are exactly the total spin statistical weightsfor the irreducible representations of the rotational subgroup I. The exact frequencies computed in the Ih group for the irreducible representations are as follows: 4
96076798852693 12,
TIE
28823036970926496,
T zB 28823036970926496, G,
38430716856193728,
H,
48038396740938240,
AU 9607678793631424, T,”
28823037990981216,
A,(A,)
19215358678900736,
T 2,, 28823037990981216,
T,,(T,,)
57646074961907712,
G,
38439716784610624,
T,,(T,,)
57646074961907712,
HU
48038395577718272.
G,(G,)
76861433640804352,
The frequencies in the rotational subgroup I are as follows:
H,(H,)
96076792318656512.
A,
19215358678900736,
Since for these species F’WIspin could contain either A, or A,, the frequencies of irreducible rep-
T1 57646074961907712,
Correspondence to: K. Balasubramanian, Department of Chemistry and Biochemistry, Arizona State University, Tempe, AZ 85287-1604, USA.
G
76861433640804352 ,
H
96076792318656512.
T2 57646074961907712,
0009-2614/92/$ 05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved.
649
Volume 200, number 6
CHEMICAL PHYSICS LETTERS
As seen from the above, the frequencies for A, and A, add up to that of Ai. The same comment applies to other g and u representations. Since the Pauli exclusion principle places no constraint on parity, both A, and A, may be allowed and thus, the total nuclear spin statistical weights are identical in both the I and I,, groups although the frequencies of the individual spin multiplets differ in the two groups. Our numbers differ from the values reported in table 2 by Hatter and Reimer [1] but agree with the corrected values in ref. [ 3 ] (see the last two paragraphs of this Comment ) . Suppose that the ith irreducible representation occur ni times in Fspin,the irreducible representation of all spin functions. Then C nidim(Ii)=260 i
for ‘3C60,where dim(Fj) is the dimension of the irreducible representation Ii, The least significant digit of 260is easily seen to be 6. It is also readily seen that the least significant digit of Xi N,dim(r,) is 6 for our weights while it is 0 for the numbers in table 2 of ref. [ 11. In quadruple precision we compute 260exactly as 1152921504606846976. It can easily be verified by hand that the weights reported in ref. [ 21 (see above) as well as the frequencies in the total Ih group satisfy
+ 3nTlu+ 3nTiY+4G, + 5H,
=1152921504606846976. This independently establishes that the weights originally reported by Harter and Reimer [ 1 ] are incorrect. The same comment applies to the numbers reported by these authors in table 1 of ref. [ I] which have more than 8 digits relating to the frequencies of A,, T,, T,, G and H species. While the Erratum of Harter and Reimer [ 31 corrects the numbers reported before [ I], it does not compare the results with our previous work. The numbers reported in the Erratum agree with our computed frequency values. The numbers in table 1 of the Erratum of Harter and Reimer [ 31 could also be reproduced by the author. Since 13C nuclei are 650
25 December 1992
fermions the total internal wavefunction must be antisymmetric as stipulated by the exclusion principle (see for example refs. [ 4-71 which illustrate the computation of spin statistical weights from the frequencies). This means that the direct product of the rovibronic and nuclear spin functions must be either A, or A, for 13C60.For example, a rovibronic level of A,, symmetry could couple to nuclear spin functions of both $ and A, symmetries since $@Ag= Ag and A,@A,=A,. Therefore the total spin statistical weight of the rovibronic level of A, symmetry should be the sum of the frequencies of nuclear spin multiplets for A, and A, which is 19215358678900736 (same as the frequency of the A, representation in the I group). However, the Pauli principle applies strictly only to nuclear permutations and not to inversion operations. Therefore the differences in the nuclear spin populations of the g and u levels could be observed in high-resolution spectra if high-spin states can be excited, as demonstrated in refs. [ 8,9] for SF6. The relative differences in the g and u nuclear spin multiplets are especially significant for the high-spin nuclear spin multiplets. For example, the frequency of the 2S+ 1 = 57 multiplet of the A,, T,,, T2*, G,, H,, A,, T,,, Tzu, G, and H, representations are 22, 36, 36, 58, 80, 14, 42, 42, 56 and 70, respectively. Likewise while the 2S+ 1 = 55 nuclear spin multiplet has a frequency of 280 for the A, representation, the corresponding frequency for the A, representation is 260. Hence these differences are significant for highspin states. Therefore, the difference in the spin populations of the g and u states can be differentiated if the high-spin states can be excited. It remains to be seen if this is feasible experimentally. References [l] W.G. Harter and T.C. Reimer, Chem. Phys. Letters 194 (1992) 230. [2] K. Balasubramanian, Chem. Phys. Letters 183 (1991) 292. [3] W.G. Harter and T.C. Reimer, Chem. Phys. Letters 198 (1992) 429 (E). [4] P.R. Bunker, Molecular symmetry and spectroscopy (Academic Press, New York, 1979). [ 5] K. Balasubramanian, J. Chem. Phys. 74 ( 1981) 6824. [6] J.T. Hougen, J. Chem. Phys. 37 (1962) 433. [7] K. Balasubramanian and T.R. Dyke, I. Phys. Chem. 88 (1984) 4688. [ 81 W.G. Harter, Phys. Rev. A 24 ( 198 1) 192. [9] J. Borde and Ch. Borde, Chem. Phys. 71 (1982) 417.
CHEMICAL PHYSICS LETTERS
25 December I992
Comment Comment on nuclear spin weights of 12C&-, and ’3C6Obuckminsterfullerene ILBalasubramanian Department of Chemistry and Biochemistry, Arizona State University,Tempe, AZ 85287-1604, USA Received 12 August 1992; in final form 7 October I992
The nuclear spin statistical weights obtained in a Letter by Hatter and Reimer differ from the values obtained by the author a year earlier. However, the corrected numbers reported in the Erratum by Harter and Reimer agree with our values.
In a recent Letter Harter and Reimer [ 1] have obtained the nuclear spin weights of ‘*C6,,and ‘Q,. However, they appear to be unaware of a paper on the nuclear spin statistical weights of Cdo,CsJIso and ChODeO published a year ago in the same journal [ 2 1. Combinatorial generators based on generalized character cycle indices (GCCI) implemented on a microvax as well as IBM 3090 with quadruple precision arithmetic [ 2 ] yield different weights compared to the values reported in ref. [ 11. Our procedure can also yield the frequencies of the individual nuclear spin multiplets. The total central processing unit time to generate both nuclear spin multiplets and statistical weights for i3Ceois 1.39 s on an IBM 3090. The total nuclear spin statistical weights in ref. [2] for 13Ce0and i2Cb0Hb0are
resentations in Ppln are exactly the total spin statistical weightsfor the irreducible representations of the rotational subgroup I. The exact frequencies computed in the Ih group for the irreducible representations are as follows: 4
96076798852693 12,
TIE
28823036970926496,
T zB 28823036970926496, G,
38430716856193728,
H,
48038396740938240,
AU 9607678793631424, T,”
28823037990981216,
A,(A,)
19215358678900736,
T 2,, 28823037990981216,
T,,(T,,)
57646074961907712,
G,
38439716784610624,
T,,(T,,)
57646074961907712,
HU
48038395577718272.
G,(G,)
76861433640804352,
The frequencies in the rotational subgroup I are as follows:
H,(H,)
96076792318656512.
A,
19215358678900736,
Since for these species F’WIspin could contain either A, or A,, the frequencies of irreducible rep-
T1 57646074961907712,
Correspondence to: K. Balasubramanian, Department of Chemistry and Biochemistry, Arizona State University, Tempe, AZ 85287-1604, USA.
G
76861433640804352 ,
H
96076792318656512.
T2 57646074961907712,
0009-2614/92/$ 05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved.
649
Volume 200, number 6
CHEMICAL PHYSICS LETTERS
As seen from the above, the frequencies for A, and A, add up to that of Ai. The same comment applies to other g and u representations. Since the Pauli exclusion principle places no constraint on parity, both A, and A, may be allowed and thus, the total nuclear spin statistical weights are identical in both the I and I,, groups although the frequencies of the individual spin multiplets differ in the two groups. Our numbers differ from the values reported in table 2 by Hatter and Reimer [1] but agree with the corrected values in ref. [ 3 ] (see the last two paragraphs of this Comment ) . Suppose that the ith irreducible representation occur ni times in Fspin,the irreducible representation of all spin functions. Then C nidim(Ii)=260 i
for ‘3C60,where dim(Fj) is the dimension of the irreducible representation Ii, The least significant digit of 260is easily seen to be 6. It is also readily seen that the least significant digit of Xi N,dim(r,) is 6 for our weights while it is 0 for the numbers in table 2 of ref. [ 11. In quadruple precision we compute 260exactly as 1152921504606846976. It can easily be verified by hand that the weights reported in ref. [ 21 (see above) as well as the frequencies in the total Ih group satisfy
+ 3nTlu+ 3nTiY+4G, + 5H,
=1152921504606846976. This independently establishes that the weights originally reported by Harter and Reimer [ 1 ] are incorrect. The same comment applies to the numbers reported by these authors in table 1 of ref. [ I] which have more than 8 digits relating to the frequencies of A,, T,, T,, G and H species. While the Erratum of Harter and Reimer [ 31 corrects the numbers reported before [ I], it does not compare the results with our previous work. The numbers reported in the Erratum agree with our computed frequency values. The numbers in table 1 of the Erratum of Harter and Reimer [ 31 could also be reproduced by the author. Since 13C nuclei are 650
25 December 1992
fermions the total internal wavefunction must be antisymmetric as stipulated by the exclusion principle (see for example refs. [ 4-71 which illustrate the computation of spin statistical weights from the frequencies). This means that the direct product of the rovibronic and nuclear spin functions must be either A, or A, for 13C60.For example, a rovibronic level of A,, symmetry could couple to nuclear spin functions of both $ and A, symmetries since $@Ag= Ag and A,@A,=A,. Therefore the total spin statistical weight of the rovibronic level of A, symmetry should be the sum of the frequencies of nuclear spin multiplets for A, and A, which is 19215358678900736 (same as the frequency of the A, representation in the I group). However, the Pauli principle applies strictly only to nuclear permutations and not to inversion operations. Therefore the differences in the nuclear spin populations of the g and u levels could be observed in high-resolution spectra if high-spin states can be excited, as demonstrated in refs. [ 8,9] for SF6. The relative differences in the g and u nuclear spin multiplets are especially significant for the high-spin nuclear spin multiplets. For example, the frequency of the 2S+ 1 = 57 multiplet of the A,, T,,, T2*, G,, H,, A,, T,,, Tzu, G, and H, representations are 22, 36, 36, 58, 80, 14, 42, 42, 56 and 70, respectively. Likewise while the 2S+ 1 = 55 nuclear spin multiplet has a frequency of 280 for the A, representation, the corresponding frequency for the A, representation is 260. Hence these differences are significant for highspin states. Therefore, the difference in the spin populations of the g and u states can be differentiated if the high-spin states can be excited. It remains to be seen if this is feasible experimentally. References [l] W.G. Harter and T.C. Reimer, Chem. Phys. Letters 194 (1992) 230. [2] K. Balasubramanian, Chem. Phys. Letters 183 (1991) 292. [3] W.G. Harter and T.C. Reimer, Chem. Phys. Letters 198 (1992) 429 (E). [4] P.R. Bunker, Molecular symmetry and spectroscopy (Academic Press, New York, 1979). [ 5] K. Balasubramanian, J. Chem. Phys. 74 ( 1981) 6824. [6] J.T. Hougen, J. Chem. Phys. 37 (1962) 433. [7] K. Balasubramanian and T.R. Dyke, I. Phys. Chem. 88 (1984) 4688. [ 81 W.G. Harter, Phys. Rev. A 24 ( 198 1) 192. [9] J. Borde and Ch. Borde, Chem. Phys. 71 (1982) 417.