Nuclear-spin statistics of C60, C60H60 and C60D60

Volume 183, number 3,4

CHEMICAL PHYSICS LETTERS

30 August 1991

Nuclear-spin statistics of CcO,CbOHbO and CbODbO* K. Balasubramanian

’

DepartmenntofChemistry, Arizona State Uwersify, Tempe, AZ 85.X37-1604,USA Received 29 April 1991; in final form 16 May I99 1

The nuclear-spin statistical weights of the rotational levels of ‘3C60,CeOHGD and CaDdo were obtained using the generalizedcharacter-cycle indices of the I,, group. The nuclear-spin statistical weights approximately follow the I : 3: 3: 4: 5 ratio for 4 (A,), T,, (T,,), Tlg (TZu), G, (G,) and H, (H,) symmetries, respectively. The overall nuclear-spin statistical weights of the Jth rotational level of C60H60 and C&De0 are found to vary approximately as (25+ 1 )X I9 2 I5 358 678 900 736 and (2Jt 1)x706519 304 586988 199 I83 738 259, respectively.

1. Introduction Recently, Smalley and co-workers [ I] have produced CbO,C60H36and solvated bucky ions in gram quantities by contact-arc vaporization of graphite. KrSitschmer, Huffman and co-workers [2,3] reported the first macroscopic preparation of C6,,, although the C6,,cluster was first discovered as a peak in the mass spectra of carbon vapor [ 4,5 1. Ever since the appearance of the mass spectral work, there have been numerous publications dealing with .several aspects of C6,, and related carbon clusters [ l- 17 1. Several spectroscopic studies have been carried out on CbO_These include 13C NMR spectra of chromatographically separated Cm, NMR spectra of Cbo [ 41, the infrared and ultraviolet absorption spectra of laboratory-produced carbon dust [ 2 1, the vibrational Raman spectra of purified solid films of C,, and C,O [ 15 1, the infrared emission spectrum of gasphase CbO[ 161, and UV-visible spectra of fullerenes Cho and C,, [ 17 1. As this manuscript was about to be submitted, an interesting manuscript on the cold molecular-beam electronic spectra of ChOand CT0by Haufler et al. [ 181 was received by the author. The analysis of the intensity patterns of bigh-resolution spectra requires information on the nuclearspin statistical weights of rotational (rovibronic)

levels. Yet to date, presumably due to the cornbinatorial complexity of the problem ( 260 or 360possible spin functions), this information has not been obtained for C60H60,C60D60and 13Cb0.Although the nuclear-spin statistical weights are very large, the ratio of the nuclear-spin statistical weights would provide manageable information on the intensity ratios of the observed transitions. It is also interesting to note that an enumeration technique was used by Klein and co-workers [ lo] to count the large number of the KekulC structures of C6@ In this investigation, we show that the generalizedcharacter-cycle indices (GCCI) previously introduced by the author [ 19-221 yield the nuclear-spin statistical weights of rotational levels of 13Cb0,C60H60 and C60D60.It is interesting to note that a special case of the GCCI was used by King [ 231 in the discussion on chirality polynomials. The nuclear-spin statistical weights of rotational levels of ‘3C60, CbOHGO and C60D60are obtained through the use of GCCIs and higher-precision computational techniques. We show that the ratio of the spin statistical weights of 4 (A,), T,, (T,,), Tzg (Tz,), G, (G,) and H, (H,) is approximately given by the 1: 3 : 3 : 4 : 5 ratio for both C60H60and C60D60_Furthermore, the overall ratio of nuclear-spin statistical weights of the rotational levels is shown to vary as W+ 1.

* Dedicated to Professor Klaus Ruedenberg on his 70th birthday. ’ Camille and Henry Dreyfus Teacher-Scholar.

292

0009-2614/91/$ 03.50 0 I991 Elsevier Science Publishers B.V. All rights reserved.

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30 August 199 1

CHEMICAL PHYSICS LETTERS

2. Generalized-character-cycle indices (GCCI) and computational techniques for the I, group In a previous paper, the author [ 191 developed a general tensor algebraic technique to generate both nuclear-spin statistical weights, nuclear-spin species and symmetry-adapted nuclear-spin functions. A special case of the projection operator defined in the tensor product space yields a group-theoretical structure which the author called the generalized-character-cycle index (GCCI). The GCCIs of the point group of the molecule under consideration were shown to yield the nuclear-spin statistical weights of rovibronic levels. The GCCI which corresponds to the character x:g+x(g) of an irreducible representation r in the group G, geG, is given by

sider the weights in the Ih group to establish this. Table 1 shows all the GCCIs of the point group I,,. For example, the GCCI of the T,, representation is PT’~=~(3x~o+12~~2-27~~0+12~~o). Note that although the x(cs) and x(c:) for the T,, and T,, representations are f( l+&) and f ( 1- $), both c5 and c: yield the same cycle representation of x!,~. Thus, the net effect is that fi cancels out in the GCCIs of both Tlg (T,,) and TZg (Tz,) representations. The generating functions for nuclear-spin species are obtained by replacing every & by cyk+bk if CYdenotes spin up and p denotes spin down for C60H60 and ‘3C60. In symbols, GFX=P;(xk-+cXk+pk)

_

The generating function for the deuterated C60D60 is given by * ... is a cycle representation of the perwhere xf1xh2 mutation representation of EGG if it generates 6, cycles of length 1, b, cycles of length 2, etc., upon its application on the set D of all nuclei possessing nonzero nuclear spin in the molecule. Consider the set F of all nuclear-spin functions. There are 260 such spin functions for C60H60if all carbon nuclei are ‘*C. Likewise there are 360nuclearspin functions for CSOD60.Note that in recent infrared and UV absorption spectra of laboratory-produced CbO,Krhschmer et al. [2] have studied the spectra of 13C-enriched Cbousing a “C-graphite rod. The nuclear-spin statistical weights of the rovibronic levels can be obtained once the irreducible representations spanned by 2”’ (or 360for CbODhO) spin functions are known. However, the conventional method of breaking down the reducible representation of 260 spin functions using the Bumsides method would not lead to a ready solution since it requires the character of the representation spanned by 260 ( 360) spin functions. The GCCI technique developed by the author [ 191 provides a nice solution to the nuclear-spin statistics of these species. Although the point group of buckminsterfullerene is I,,, the I subgroup suffices to generate the nuclearspin statistical weights of the rovibronic levels since, as shown here, the nuclear-spin statistical weights are the same for g and u levels. However, we first con-

GFz=P&(xpAk+&

v”) ,

where A,p and v are the three possible z-projections for the deuterium nuclear spin. The number of times the irreducible representation I- with character x:g+x(g) occurs in the set of all protonic or 13Cspin functions is given by

where every xr; is replaced by 2 since there are two possible spin projections for the hydrogen nucleus or the 13Cnucleus. Table I The GCCIs of the Ih point group ‘) r

Cycle type XfO

A*

T,,

1

G;

3 3 4

H, Au T 1” TZ. G. Hu

5 1 3 3 4 5

T

x:’

24 12 I2 -24 0 24 I2 I2 -20 0

Xi”

20 0 0 20 -20 20 0 0 20 -20

X:”

31 -27 -27 4 15 -1 -3 -3 -4 -5

-GO

24 12 I2 -24 0 -24 12 -12 24 0

X:”

20 0 0 20 -20 -20 0 0 -20 20

a) All cycle indices should be divided by 120.

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The number of times the irreducible representation I- occurs in the set of all deuterated spin functions is given by

To illustrate, the number of times the T, representation occurs in the set of all 260 protonic spin functions is directly given by

30 August 1991

Note that the nuclear-spin statistical weights reported in table 2 for CbOH60 are exact to all digits since they were obtained using double-precision arithmetic which suffices to compute these numbers as the total number of all nuclear-spin functions is 260.The computation of the nuclear-spin statistical weights of the deuterated C60D60required quadruple precision since they contain 28 digits. The quadruple precision guarantees that all 28 digits in table 2 are exact for GOD60.

The actual evaluation of nuclear-spin statistical weights and spin species for all irreducible representations requires computational techniques based on multinomial expansion in real higher-precisjon arithmetic since the numbers involved are very large. Such a program was implemented on a microvax using quadruple-precision arithmetic and was used to generate the spin statistical weights for 13Ce0,&H,, and CbODbO.

3. Results and discussion Table 2 shows the final nuclear-spin statistical weights for 13Cb0,C60H60and CbOD6,,.They were obtained by stipulating that the direct product of the rovibronic species and nuclear-spin species be antisymmetric since 13Cand H nuclei are fermions. Since all the operations of the I group yield either even permutations of nuclei or even numbers of odd permutations, the overall species is A, or A,, for the Ih group. The statistical weights in table 2 were obtained by stipulating that Pe@)rop’” c A, or A, .

Although all nuclear-spin statistical weights contain 17 digits for C60H60and 28 digits for Ce0D6,,,the ratio of the nuclear-spin statistical weights of the rovibronic levels of A,, T+ Tzg, G, and H, symmetries is seen to be approximately 1: 3: 3 : 4: 5. Since the intensity ratios will depend on the ratio of the spin statistical weights, it is comforting that in the final ratio, the large nuclear-spin statistical weight of the A, level is approximately factored out. It is also interesting to note that the approximate ratio of the statistical weights varies as the degeneracy of the irreducible representations. A given rotational level with quantum number J transforms as the rotational DC=‘) representation. This will have to be correlated in the I,, point group to find the total spin statistical weight contributing to a given J level. This is done easily following the procedure of Galbraith and co-workers [24] or the standard procedure of subduced representation for finding the irreducible representation spanned by spherical harmonics 1lm,) outlined by Hamermesh [ 251. Such a correlation for the I,, group was done by Balasubramanian et al. [26] for B,,Hti2, and more recently by Harter and Weeks [ 271. Table 3 shows the correlation table for the rota-

Table 2

The nuclear-spin statistical weights of the rovibronic levels of ‘%Zbo,C60H60and &De0 Rovibronic symmetry YeI

Statistical weight a) C6OH60

C60D60

4th)

19215358678900736 57646074961907712 57646074961907712 76 861433 640 804 352 96076792318656512

706519304586988199183738259 2119557913760758702931804286 2119557913760758702931804286 2826077218347746902115011 104 3 532 596 522 934 735 097 811964 962

TI, (TI.) Tz8 (T,,) Gs (G,) H, (H,) a’ The statistical

microvax.

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weights of g and u levels are the same. All statistical weights were obtained using quadruple-precision arithmetic on a

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Table 3 Nuclear-spin statistical weights of the rotational levels J=O-30 ‘) for CIDHBO and C60D60 J 0

I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

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CHEMICAL PHYSICS LETTERS

Irreducible representations

Statistical weight Xp)

A T, H T2+G GtH T,+T,+H AtT,tGtH T,+T,tGtH T,+G+ZH T,+T2+2GtH AtT,tTZtGt2H 2T,tTz+Gt2H AtTItTjt2Gt2H T,t2T2t2Gt2H T,tT1+2Gt3H At2T,t2T2t2G+2H At2T,tTZt2Gt3H 2T,+2Tz+2G+3H AtT,t2T2t3G+3H 2T,t2T2t3G+3H At2T,t2TZt2G+4H At3T,t2T2t3Gt3H At2T,t2T2t3Gt4H 2T1t3T2t3G+4H At2T,t2TZt4Gt4H A+3T,t3TZt3Gt4H At3T,t2T2t3Gt5H At3T,t3T2t4Gt4H At2T,t3Tzt4G+5H 3T,t3Tzt4G+5H 2At3T,t3T2+4G+5H

1 3 5 7 9 I1 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61

a)The irreducible representations for J>3l are givenby q(At3TIt3T2t4Gt5H)tT(r), whereq is the quotient obtained by dividing J by 30,and r is the remainder. T(r) is the set of irreducible representations spanned by J= r listed in this table (see text for further discussion). Note that since nuclear-spin statistical weights are the same for g and u symmetries,wedo notshowgoru. "f=19215358678900736 for GoH6o; f=706519304586988199183738259forC6r,Ds0.

are given by D’J’~I,,=q(D(30)~l,, -A)+

(D”‘&)

,

where 4 is the quotient obtained by dividing J by 30 while r is the remainder, D’30’~1, are the subduced irreducible representations contained in the J= 30 level while D(‘)& are the subduced irreducible representations contained in the I= r level. As seen from table 3, D(30)&,-A is simply A+3T,+3Tz+4G+ 5H. It is interesting to note that in Dt3”, every irreducible representation is multiplied by its dimension. Hence, using the above formula, the subduced representations for DcJ) for J> 30 can be readily obtained from the results in table 3. The DtJ) correlations that we obtain are identical to that of Balasubramanian et al. [ 26 ] as well as Harter and Weeks [ 271, although in the former work, a correlation table was not explicitly included. For example, the irreducible representations spanned by J=65 and J= 195 rotational levels are shown below: D’65’~l, =2(D’30’& -A)+

(D’5’&,)

=3A+7T, +7Tz +SG+H , D”95)llh=6(At3T, +(A+2T,

+3T,t4Gt5H)

+2Tz +2G+2H)

=7A+2OT, t20T2+26G+32H. Since the nuclear-spin statistical weights of the irreducible representations of the I,, group are approximately in the ratio of their dimensions (degeneracy), the total nuclear-spin statistical weights of the J levels should be approximately proportional to the total degeneracy of D V) which is 2J+ 1. This result is verified in table 3 for all J. Therefore, we conclude that the ratios of all statistical weights are approximately proportional to 2J+ 1,

4. Conclusion tional levels with J=O-30 in the I point group as well as the nuclear-spin statistical weights of the rotational levels in an approximate factored form. Since the statistical weights are invariant to g and u symmetries, it is sufficient to find the correlation in the I group (see ref. [ 26 ] ), We show in table 3 the unique results only up to J=30 since it can be readily seen that the subduced representations spanned by DtJ)

We obtain the nuclear-spin statistical weights of the rovibronic levels of 13Cbo,CsOH60and C60Ds0_ We showed that the nuclear-spin statistical weights of the rotational levels of C60H60and C60D60with quantum numbers J approximately varied respectively as (25+1)~19215358678900736 and (2J+1)x706519304586988199183738259. 295

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Acknowledgement This research was supported in part by the US National Science Foundation under Grant CHE88 18869.

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