J. Phys. Chem. 1993,97, 8136-8144
8736
Nuclear Spin Statistics of Larger Fullerene Cages (CSO,C~O, C7,5, and C,) K. Balasubramanian Department of Chemistry and Biochemistry, Arizona State University, Tempe, Arizona 85287-1 604 Received: March 15, 1993; In Final Form: May 7, 1993
The nuclear spin multiplets and nuclear spin statistical weights of the rovibronic levels of fullerene cages CSO, C70,c76, and Cs4 are obtained using computerized group-theoretical techniques. For some of the larger cages, nuclear spin statistics of isomeric forms are considered and it is shown that their spin statistics are different.
I. Introduction Following the remarkable discovery of the C a buckminsterfullerene cluster in 1985,numerous papers have appeared which have focused on several carbon cages.'* Now the entire area of fullerene chemistry is a very active topic of research. The earlier mass spectroscopic study of carbon has revealed magic numbers for both n = 50 and 60. The larger fullerene clusters have received more attention quite recently. Several alternative structures based on the isolated-pentagon rule have been proposed.30 For example, there are 24 possible isomers for C84 all of which satisfy the isolated-pentagon rule.30 Likewise, there are 46 isomers for CSOwhich satisfy the isolated-pentagon rule.30 Quantum mechanical studies whether based on ab initio or semiempirical schemes do not always lead to unambiguous determination of the ground states of these clusters since the ordering of these structures changes depending on the level of theory employed. Consequently,alternative schemesaredesirable to discriminate closely-spaced isomers. Spectroscopic characterization of fullerenes is important to establish the structure of a given fullerene cage. In fact, the experimental confirmation of the proposed spheroidal structure for c 6 0 was made possible by 13C NMR. The far-infrared and ultraviolet absorption spectra of carbon dust was observed by Krltschemer et al.1° The infrared emission spectrum of gasphase c 6 0 as well as the UV-visible spectra of C a and C70 have been recorded.1' The 13CNMR spectroscopy of viable isomers of (278 has been considered.26 Interestingly, even the forbidden transitions in the absorption spectra of c60 have been observed.24 The UV photoelectron spectra of c76 and K,C76 have been observed by Hino et aLZs The third and most abundant fullerene in the carbon arc is believed to be Cs4 (C70 being the second most abundant). It is interesting that c84 is probably found as a mixture of isomers. Of the 24 possible isomers which satisfy the isolated-pentagon rule, four possible candidates were considered as most probable by Saito et alez7Among the two possible Dz structures (round and flat) the round D2 structure is considered to be the most favorable candidate for Cs4. Group-theoretical analysis and the nuclear spin statistics of the fullerene clusters are quite important for the analysis and assignment of both optical and NMR spectra. In fact when other procedures fail to differentiate closely-related isomers, the nuclear spin populations are found to be different. The nuclear spin species play an important role in the multiple-quanta and other NMR spectra.4s The spin statistical weights of the rovibronic levels are important for the interpretation of the observed intensity patterns. For these reasons the nuclear spin statistical weights have been studied for several fullerene c a g e ~ . ~ ~ ~With ~ Z ~ the 4 3 exception of
0022-3654f 93f 2Q91-8736$Q4.0O fQ
Cw and C78, nuclear spin multiplets and nuclear spin statistical weights are not readily available for larger fullerene cages. The objective of the present study is the systematic generation of nuclear spin species and statistical weights for larger fullerene cages of experimental interest. We employ computerized combinatorial and grouptheoretical techniques in conjunction with the quadrupole precision arithmetic to generate the nuclear spin species and spin statistical weights of larger fullerene cages. 11. Method of Computation
In a previous investigation on the nuclear spin statistics of C2rC40 fullerene cages we have described the method of computation of nuclear spin statistical weights.51 Consequently we shall be very brief in the present study. The readers may obtain additional details from refs 46-5 1. All of the computations require the generalized character cycle indices (GCCIs) of the irreducible representations of the point groups of the parent cages. The GCCI which corresponds to the character x:g x(g) of an irreducible representation r is defined as
-
where x , b l x2b2 ... x,$ is a permutation cycle representation of g E G if it generates 61 cycles of length 1,bz cycles of length 2, ...b, cycles of length n upon its action on the vertices of the parent carbon cage. For each fullerene isomer with a given point group symmetrythe GCCIs would be different. In fact for all 5 favorable cages of C78 (satisfying the isolated-pentagon rule), the GCCIs were found to differ.42 For the larger fullerene cages considered here, generation of the GCCIs, especially, and the permutation representations for the point group operations required imagination and visualization of the cages in three orthogonal projections. The structures of different cages considered in this study are shown in Figure 1. We now illustrate the generation of GCCIs using Cad which satisfy theisolated-pentagonrule. While Fowler% proposed a rather flat Dz-symmetry structure for (284, recent studies of Saito et al.27as well as Raghavachari and R ~ h l f i n g ~ ~ rule out the flat D2 structure. The flat structure also shows poor agreement with the observed photoelectron spectra.27 Saito et al.Z7 consider two possibilities, namely, a rather round structure with D2 symmetry proposed by Wakabayashi et al.32and another one with Td symmetry to be present in the graphite-arc soot. A Dah isomer is also likely to be present in the carbon soot. Let us consider the Td symmetry cage for C Sto~ illustrate the generation of GCCIs (see Figure 1 for its structure). It can be shown that all operations in a given conjugacy class generate the 0 1993 American Chemical Society
Nuclear Spin Statistics of Larger Fullerene Cages
The Journal of Physical Chemistry, Vol. 97, No. 34, 1993 8737
same permutational-cycle representation while the converse is not true. Consequently, it suffices to consider a representative from each conjugacyclass for the generation of the GCCI. Eight C3 operations in the Td group generate the xjZ8cycle representations. Three C2 operations generate x242. Six S4 operations generate xdZ1while the six IJd planes yield ~ 1 ~ x COnSeqUently 2 ~ ~ . we obtain the following GCCIs for the Td cage.
+
PGA1= 24[x184 1
+ 3 ~ , +4 6x:’ ~ +6 ~ , ~ x , 4 ~ ]
1 [ 2 ~ 1-~8 ~~ 24
PGE = -
+
3 ’ ~6
~,4~]
+
PGTI = -[3x184 1 - 3 ~ , 4 6~ ~ : ~6xl2x,4’] 24 1 POT’ = - [ 3 ~ 1 ~ 3~ . ~ ; ~- 6 ~ : ’ + 6 ~ 1 ~ ~ , 4 ~ ] 24
As another example let us consider the CWfullerene cage. There are 46 isomers for CWwhich satisfy the isolated-pentagon rule.3o At the present time there is less information available on CWcompared to C84 and thus the structure is not established, and in any case CW would most likely be present as a mixture of isomers. Liu et al.bohave considered three candidates for CWas being favorable. We consider the high-symmetry DV cage among these to illustrate the GCCI construction. All GCCIs of the CW cage with the D5h group are shown below: PoA” = L[.X190 20
+ 4X518+ 5X,45 +
+ 4X;X108 +
X110X240
5x14x,43]
TABLE I: Nuclear Spin Statistics of C a w Frequencies of the Irreducible Representations: A1’(56295090972480), A2’(56295007086400), El’(11259OO98058336) A1”(56294916489408),A1’(56294966821056), El”(112589883309984) Spin Multiplets and Their Frequencies
2s +I 1 7 13 19 25 31 37 43 49
2s
freauencv
+1 -
243096166434 1071120386462 6 17621029641 133643710976 11673136065 388380937 4201569 10733 3
3 9 15 21 27 33 39 45 51
2s
frequency
+1 -
AI’ 675272061262 1009911760666 4130308686931 65649389759 4202423067 98437045 690199 942
243098662578 1071117563442 617621438236 133642831295 11673111271 388331349 4200327 10435
1 7 13 19 25 31 37 43 49
486194828880 2142237949904 1235242467877 267286542335 23346247336 776712261 8401896 21168 6
3 9 15 21 27 33 39 45
1
243095873662 1071118762418 617619398814 133642841809 11672862135 388330093 4196651 10487 1
3 9 15 21 27 33 39 45 51
675271205958 100990998674 410307283819 65648761821 4202256713 98412135 688141 883 A2” 675268917826 1009912274886 410306793505 65649252135 4202239003 98429845 688095 929
L
964671445098 836I65 I42062 246184784784 29149590525 1354106755 21858831 94691 74
1
A,‘ 675269149094 1009912747294 410307134270 65649380145 4202263677 98431985 687945 92 1 0
1 7 13 19 25 31 37 43 49
3 9 15 21 27 33 39 45 51
5
11 17 23 29 35 41 47
frequency
5
11 17 23 29 35 41 47
964673198342 836162853930 24618488573 1 29149177 123 1354092587 21846687 94485 49
El‘
51
1350541210356 2019824502070 820615821201 131298769840 8404686744 196869030 1378150 1863 0
5
11 17 23 29 35 41 47
1929344643440 1672327995882 492369610515 58278767648 2708199367 43705518 189170 123
AI”
5x14x,43]
5x14x,43]
5x14x,43]
PGE” = -[2x190 1 - 2x,‘8 - 2x1IOx:,40 20
+ 2x52xl,8]
Once the cycle indices are constructed the generating functions
for nuclear spin multiplets can be obtained through the following substitution in the cycle index: GFX
P G x [ x k -P ak
+ @]
where a correspondsto spin up while fl corresponds to spin down. The arrow symbol stands for replacing every x k by ak+ f l k . The coefficient of amlflma in the G F gives the number of times the irreducible representation r with character x occurs in the reducible set of nuclear spin functions containing ml a spins and mz fl spins. Once GFx is generated the coefficient in the G F can be sorted in accordance with the total mF nuclear spin quantum number. The computation of G F s can be tediousparticularly for larger fullerene clusters. The subsequent sorting of the coefficients can also be involved for higher cages with higher symmetry. The authoI.48 has developed a Fortran code for this purpose, which was further enhanced to handle the large number of nuclear spin
7 13 19 25 31 37 43 49 1 7 13 19 25 31 37 43 49
243098577818 1071117128038 617621033194 133642639915 11673064029 388326551 4200193 10485 3
3 9 15 21 27 33 39 45
1 7 13 19 25 31 37 43 49
486194451360 2142235890456 1235240432008 267285481780 23345926164 776656624 8396844 20972 4
3 9 15 21 27 33 39 45 51
5
11 17 23 29 35 41 47
964670136302 836163378478 246183643986 29149163595 1354012725 21846919 94009 56
0 5
11 17 23 29 35 41 47
964672840458 836162397850 246184624614 29149096297 1354080023 21846413 94515
56
0
51
El’’ 1350540123784 2019822261740 820614077324 131298013900 8404495716 196841980 1376240 1812
5
11 17 23 29 35 41 47
1929342976760 1672325776228 492368268600 58298259892 2708092768 43693332 188520 112
0
functions for the fullerene cages and was discussed bef0re2~The quadruple precision arithmetic was used uniformly for all the cages. The computation of the frequencies of the nuclear spins can be obtained independently through the following substitution:
“r = P G x ( x k
-
2)
for all spin-l/2 nuclei considered here. All our computations are valid for the spin statistics of 13C, fullerene cages or C,H, fullerenes.
8738 The Journal of Physical Chemistry, Vol. 97, No. 34, 1993
Balasubramanian
TABLE II: Nuclear Spin Statistics of ' C m or C70H70 Frequencies of Irreducible Representations: Alf(59029581133795822848),Azf(59029581047896476928),E1'(118059162181692291456) Ai"(59029580955 125183232). A2"(5902958 1006664790784). Elf'( -. - .118059161961789965952) S&n Multiplets and Their Frequencies I.
2s+ 1
frequency
2s+ 1
frequency
2s+ 1
frequency
1 7 13 19 25 31 37 43 49 55 61 67
155814273700052362 781117300751527006 626760122665526948 235883603974177204 46656465275394000 4920022392396526 269892146978172 7289400928068 88357165170 412114828 559850 138
3 9 15 21 27 33 39 45 51 57 63 69
Aif 442175645944380624 803434936121389814 487729255802082780 147359779709299748 23651971583894940 2014401209800272 87930476177360 1841348184692 16583778990 53388637 43483 4
1 7 13 19 25 31 37 43 49 55 61 67
155814275904013792 781117298329216456 626760123347867948 235883602884912844 46656465305546460 4920022236585442 269892142745680 7289394678220 88357015130 412063524 559389 103
3 9 15 21 27 33 39 45 51 57 63 69
Ai 442 175643481129614 803434937295016334 48772925419 1758020 147359779834116908 2365 1971243803240 2014401203273532 87930454856676 1841347592716 16583465270 53384270 42890 3
5 11 17 23 29 35 41 47 53 59 65 71
659384734220050210 742469623057155090 35 1756250802496712 85847895739263060 11177746511858330 765910097396952 2641 1349179452 423752181524 2719672476 5948503 2618 0
1 7 13 19 25 31 37 43 49 55 61 67
311628549604064418 1562234599080743462 1253520246013394896 47 1767206859091064 933 12930580940460 9840044628981457 539784289723852 14578795606288 176714180349 824178352 1119231 24 1
3 9 15 21 27 33 39 45 51 57 63 69
El' 884351289425510238 1606869873416407667 975458509993840800 294719559543415640 47303942827698 180 4028802413073804 175860931034224 3682695777408 33 167244211 106772907 86373 8
5 11 17 23 29 35 41 47 53 59 65 71
1318769466713777690 14849393682159 18641 703512501274988624 171695792127855840 22355493025259855 1531820256857268 52822700 178292 847505904368 5559374432 11903493 5267 0
1 7 13 19 25 31 37 43 49 55 61 67
155814273521895312 781117299693778396 626760121393771388 235883603055494104 46656464822202300 4920022235751208 269892109326660 7289394622076 88356505992 412068972 558470 114
3 9 15 21 27 33 39 45 51 57 63 69
Ai" 442 175645419718424 803434934910416644 487729254607245660 147359778952606448 2365 1971253416740 2014401108491556 87930454377740 1841345100908 16583482928 53373964 43030 2
5 11 17 23 29 35 41 47 53 59 65 71
659384131671076360 7424696838788 13700 351756249399475612 85847895792058140 11177746281556880 765910096049844 2641 1339194028 423752208652 2179589628 5949582 2566 0
1 7 13 19 25 31 37 43 49 55 61 67
155814275855501532 781117298056159996 626760123031389788 235883602668420664 46656465209275740 4920022207982896 269892137094972 7289394147404 88356980664 4120679 16 559526 114
3 9 15 21 27 33 39 45 51 57 63 69
Af 442175643345401784 803434936984733284 487729253897611020 147359779662241088 23651971 176282540 2014401185625756 87930452162604 1841347316044 16583472016 53384876 43028 4
5 11 17 23 29 35 41 47 53 59 65 71
659384734004682580 742469682732480820 351756250545808492 85847895606936060 11177746466678960 765910087505748 26411347738124 423752126812 2779671468 5949516 2632 0
5 11 17 23 29 35 41 41 53 59 65 71
659384732493727480 742469685158765370 35 1756250472491852 85847896388592780 11177746513401014 165910159460316 26411350999028 423153722844 2779701956 5954982 2649 1
The Journal of Physical Chemistry, Vol. 97, No. 34, 1993 8739
Nuclear Spin Statistics of Larger Fullerene Cages
TABLE II (Continued) 2s+1
frequency
2s+ 1
frequency
2s+ 1
frequency
1 7 13 19 25 31 37 43 49 55 61 67
31 1628549377395148 1562234597749938392 1253520244425161176 471767205723915754 933 1293003147804O 9840044443733614 539784246421632 14578788769480 176713486698 824136888 1117990 228
3 9 15 21 27 33 39 45 51 57 63 69
88435 1288765120208 1606869871895151412 975458508504856680 294719558614846550 47303942429699280 4028802294117312 175860906540520 3682692416952 33 166954902 106758840 86058 6
5 11 17 23 29 35 41 47 53 59 65 71
1318769465675758940 1484939366611293036 703512499945284104 171695791398994200 22355492748236330 1531820183555592 5282268693 1976 847504335464 5559261096 11899104 5198 0
El”
TABLE Itk Nuclear Spin Statistics of C ~ with S 9 Symmetry Frequencies of IrreducibleRepresentations: A1 (1 8889465931684739284992), Bl,Bz,B3( 18889465931409861378048) Spin Multiplets and Their Frequencies
zs+ 1 1 5 9 13 17 21 25 29 33 37 -. 41 45 49 53 57 61 65 69 73 77
frequency
2s+ 1
Ai 44183465723260623200 189396197493576352720 237846387540032695260 19468 1672763260540590 116811774337310949440 53375127981708841800 18838280465634157272 5153869184913770436 1090013713144448256 176888190499078352 21766297707452680 1997347216126644 133633751459392 6323741647022 203015389614 4167478905 50042352 303696 722 1
3 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75
frequency 125922877210558774290 227275436940485565180 224632699306297588140 156266225604380930880 81596459999675856700 32736745153605548696 1017267 1447218783696 2448581526597555584 454172379746264908 64293004683301320 6848420423054136 538353065425728 30425537401383 1193158034206 30902642720 491886288 4297439 16834 18
III. Results and Discussion Figure 1 shows preferable structures for fullerene cages CSO, C70, c76, and C S considered ~ here. Since for n < 60, there are no structures with isolated pentagons, we show in Figure 1, structures which are considered as likely candidatesfor the ground states. The nuclear spin statistics of the CWbuckminsterfullerene and five structures of C78 with isolated pentagons have been discussed before. Hence we will not consider these in the present study. Since a fullerene cage should contain only pentagons and hexagons, application of the Euler rule leads to the condition that all fullerene cages must contain exactly 12 pentagons. Consequently only the number of hexagons will vary for each fullerene cage. Table I shows the nuclear spin statistics of WSO or CsoHso. The CSOfullerene has Dsh symmetry. The mass spectra exhibit a magic number for abundance at n = 50 among the clusters of smaller sizes. As a result of the DSh symmetry the spin population is distributed among AI’, Az’,El’, El’, AI”, Ai’, El”, and El’’ irreducible representations of the DSh group. Since the GCCIs of both El’ and E*’ (likewise El” and El”) are the same, their nuclear spin multiplets and spin statistical weights are identical. Hence we show in Table I only the results of El’ and El” representations. The overall species should be of Az’ or A2” symmetry for l3Cs0or CwHso in accord with the requirement of the Pauli exclusion principle for fermions. Thus the spin statistical weights of the rovibronic levels can be readily obtained from the frequencies of the irreducible representations listed in Table I
2s+ 1 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77
x+1
frequency
Bi,BzJh
44183465687915359400 189396197459998352110 237846387511251551880 194681672741020566160 116811774321839662880 53375127972039287700 18838280460219206976 5153869182206295288 1090013711941125968 176888190026344596 21766297544441040 1997347167223 152 133633738839136 6323738886341 203014887672 4167405090 50033916 302993 684 0
frequency 125922877244136774900 227275436969266708560 224632699328537562570 156266225619852217440 81596460009345410800 32736745 159020498992 1017267 1449926258844 2448581527800877872 454172380218998664 642930048463 12960 6848420471957628 538353078045984 30425540162064 1193158536148 30902716535 491894724 4298142 16872 19
3 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75
stipulating that
rNeco rs* c A;
or A;
where rrvc and rSPhare the irreducible representations of the rovibronic level and nuclear spin function, respectively. Thus a rovibronic level of AI’ symmetry could couple to nuclear spins of both A i and Az” symmetries. Hence the total nuclear spin statistical weight for the AI’ rovibronic level would be the sum of the frequencies of Az’ and Az“listed in Table I. Consequently once the frequencies are computed then the actual statistical weights are easily found. The C70 fullerene cage has a DSh symmetry. The C70 fullerene is the second most abundant fullerene in the carbon root. Consequently, there are several theoretical and experimental studies on this fullerene. The nuclear spin statistics and spin multiplets of the C70 fullerene have not been considered thus far. Table I1 shows the computed nuclear spin multiplets and nuclear spin statistical weights for the l3C7ofullerene (or C7oH70). Since theGCCIs of the El’and Ez’representations are the same (likewise El” and El”), we only show in Table I1 the results for the El’ and El” representations. Figure 2 shows the nuclear spin spectrum for the AI’ and El’ irreducible representations. It is interesting that the spin population reaches a maximum at the nonet spin multiplicity for all the irreducible representations. It is interesting to compare the frequencies of spin speciesin the high-spin domain. Note that the frequenciesof the 2s 1 = 69 multiplet are 4, 3, 8,2,4,and6 fortheAl’, Az‘,El’, Al”,Az”,andEl”representations, respectively. The frequencies of the 2s 1 = 67 spin multiplet
+
+
8740 The Journal of Physical Chemistry, Vol. 97, No. 34, 1993
Balasubramanian
TWO POSSIBLE ISOMERS FOR C,6 3-fold axis
2-foM axis
%fora a x i s (1)
2-fola a x i s (2)
2-fozd axis (3)
f
1
(d) D2 ( round )
Two Favorable Isomers of c84 Figure 1. Favorable structures for fullerenes c70,(276, Cm, and Car. The structures for C70 and C76 are from ref 52 while the isomers for Cw are from ref 27.
1
1
1
9
19
29
39
49
59
69
1
9
19
29
2s+1 1
2. Computed nuclear spin frequency spectrum for the AI’ and El’irreducible representations of c70.
39
2s+1
49
59
69
Nuclear Spin Statistics of Larger Fullerene Cages
TABLE Tv: Nuclear Spin Statistics of
The Journal of Physical Chemistry, Vol. 97, No. 34, 1993 8741
with Td Symmetry
Frequencies of Irreducible Representations: A1(3148244322497234862080), A2(3148244321397722972160), E(6296488643894890725376) Ti(9444732965155175006208), Tz(944473296625468637 1840) Spin Multiplets and Their Frequencies ~
2s+1
frequency
2s+ 1
frequency
2s+ 1
20987146203156865818 37879239492938450520 37438783221263578632 26044370937492993680 13599410002708831003 5456124194172106317 1695445242337015452 408096921685688244 75695396887631656 10715500882693360 1141403437915834 89725520613086 5070925215282 198860127966 5150513461 81989894 717012 2859 4
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77
A1 7363910954352675800 31566032917806894302 39641064593314803060 32446945464119835588 19468629059362988600 8895854665955305672 3139713412434260067 858978198315633369 181668952590422392 29481365250059808 3627716344807340 332891221268128 22272296767682 1053958008210 33836088312 694606789 8343422 50847 131 1
3 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77
A2 7363910953397398400 31566032913388690138 39641064586696095360 32446945456964832102 19468629053076173720 8895854661280974928 3139713409443071307 858978196656344493 181668951791060360 29481364916181576 3627716224461620 332891184107420 22272287043682 1053955882564 33835708226 694552646 8337562 50385 109 0
3 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75
20987146200362725612 37879239487220432940 37438783214171922348 26044370930633983280 13599409997181759247 5456124190364438565 1695445240069245780 408096920513176484 7569539636 1443780 10715500678407080 1141403369733458 89725501230910 5070920585179 198859215236 5150368979 8 1972202 715451 2769 2
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77
14727821907755274500 63132065831190384140 79282129180010898420 94893890921087936450 38937258112435893560 17791709327236280600 6279426821878412949 1717956394970896287 363337904381482752 58962730166418484 7255432569091860 665782405375548 44544583824014 2107913878124 67671796538 1389159735 16680684 101232 241 0
3 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75
41974292403519591430 75758478980163340860 74877566435431043580 52088741868126976960 27198819999892633225 10912248384534501907 3390890482406261232 816193842199345428 151390793248594736 21431001561100440 2282806807702422 179451021790866 10141845800461 397719345502 10300880140 163962096 1432488 5603 6
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77
frequency Ti 2209 1732843480041000 94698098727790120 162 118923193752316422090 97340836366932819128 58405887 157776424OOO 26687563983682503672 9419140228614009108 2576934590273516772 545006855570881968 88444094846238996 10883148712047660 998673565033160 66816864557568 3161868380832 101507253793 2083675559 25014028 151275 331
2s+ 1
frequency
3 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75
62961438620671271158 113637718481774345490 112316349660722915352 78133112806496603520 40798230001909144328 16368372577606415620 5086335723829231020 1224290763314183056 227086189846399580 32146502321013340 3424210201885688 269176529331904 15212767765496 596578811709 15451285941 2459385 16 2 148281 8391 8
3 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75
62961438623465503742 113637718487492363070 1 12316349667814647218 78133112813355613920 40798230007436266472 16368372581414083372 5086335726097027824 1224290764486694816 227086190372599084 32 146502525299620 34242 10270071940 2691765487 14080 15212772396568 596579724439 15451430594 245956208 2 149861 8481 11
0 Ti 22091732844435318400 94698098732208231948 118923193758935129790 97340836374087747032 58405887 164063238880 26687563988356784028 9419140231605197868 2576934591932778516 545006856370244000 88444095180105600 10883148832393380 998673602189992 66816874281568 3161870505509 101507633879 2083729531 25019888 151718 353 0
E
are 138, 103, 241, 114, and 228 for the AI’, Az‘, El’, AI!‘, Ai‘, and El’’representations, respectively. While the frequency of the El’is twice that of AI’ for the 2s + 1 = 69 spin multiplet, the frequency of El’is much less than twice AI’for 2s + 1 = 67.
But for the 2s + 1 = 67 spin multiplet the frequency of the El’ representation is more than twice the frequency of AI’. Thus we see an alternation in the relative frequencies of AI’and El’.The ratios of the frequencies of the successive spin multiplets drop
Balasubramanian
8742 The Journal of Physical Chemistry, Vol. 97, No. 34, 1993
TABLE V: Nuclear Spin Statistics of
with 4
Symmetry Frequencies of Irreducible Representations: A1(4835703278461815233708032), Bi.BzBa(4835703278457417187196928) Spin Multiplets and Their Frequencies
2s
+1
frcuuencv
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85
9761 107478379804395640 42446027972796614221860 55 12879 1 1874446317 18880 47602231921938948162480 30758365241887832807076 15468466700049343538418 6150681003090304447872 1944632602872994196976 488872 147538097426096 97345381262068624776 1524418890549 1296192 1858153655409512076 ina72159956391890 12265161304092165 636981932192256 23604683587212 5982229 13652 9766528902 93910992 459200 882 1
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85
9761 107477841546521200 42446027972282822614440 55128791186997856408080 47602231921585251041430 30758365241633170879920 15468466699882833816816 6 15068 1002991632020256 19446326028201 3 3967896 4888721475 12578694816 97345381251010507888 15244188901210734816 1858 153653938069103 1738721595 105OOO80 12265 161 186061980 63698 1905213928 23604678341426 598222062984 9766416972 93899512 458339 840 0
2s +1
frwuencv
3 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75 79 83
27952262322909883219800 51673425357382986971520 53342209990965727155870 39546469596276364591116 22477266907268959665432 10039498669365504919104 3558608294562642758904 1003532163630888867024 224683663750419606096 397 189524176847 1 1808 5495808955282739379 588102503048602710 47903554093041360 2909256548418864 128203470548538 3953286428364 81135325908 1030712448 7234983 22918 20
7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75 79 83
27952262323423674827220 51673425357829762282320 533422099913 19424276920 39546469596531026518272 22477266907435469387034 10039498669464177346720 3558608294615502987984 1003532163656407598304 224683663761477722984 39718952421965273184 5495808956754182352 588 102503494494520 4790355421 1071545 2909256575397192 128203475794324 3953287279032 81 135437838 1030723928 7235844 22960 21
from 34.5 to 2.84 in moving from the 2s + 1 = 69 spin multiplet to the singlet spin multiplet, a trend that is consistent with Figure 2. The c 7 6 fullerene cage has been a topic of several studies. Hino et al.25 have recorded the ultraviolet photoelectron spectra of c 7 6 as well as Kxc76with a synchrotron radiation light source. The 13C NMR spectra of c 7 6 obtained by Ettl et al.31 appear to suggest a chiral DZstructure for c 7 6 . Saito et al.z7have made tight-binding electronic structure calculations combined with a model potential study on c76. Two possible isomers actually satisfy the isolated-pentagon rule, one with the D2 symmetry discussed before and the other with Td point group symmetry shown in Figure 1. We consider both of these cages for nuclear spin statistics. Table 111shows the nuclear spin statistics of the chiral D2 c 7 6 cage. The GCCIs of the B1, B2, and B, representations are identical for the 4 group, and thus these three irreducible representationshave identical nuclear spin statistics. In the highspin domain, there are differences in the populations of the A1 representation compared to the (B1, B2, B,) triplet of represen-
tations. For example, the frequency of the 2S + 1 = 75 spin multiplet is 18 for the Al representation while it is 19 for the Bl (B2, B3) representations. The frequency of 2 s 1 = 73 spin multiplet is 722 for A1 while it is 604 for BI (B2, B3). Hence there is an alternation of spin population frequencies between A1 and the rest (B1, Bz, B3). The maximum population is reached at the nonet nuclear spin multiplicity. The nuclear spin statistics of the higher-symmetry (Td) c 7 6 cage is substantially different from the chiralD2 cage. First, the spin population is more distributed in the case of Td fullerene. Table IV shows the nuclear spin statistics of the c 7 6 fullerene cage with the Td point group symmetry. In particular note that the spin populations of the totally symmetric A1 representation are smaller for the Td cage compared to the chiral DZcage. The 1 = 75 spin multiplet of the A1 frequency of the 2S representation of the Tdcage is 4 compared to that of the DZcage which is 18. Likewise, the frequency of the 2S 1 = 73 spin multiplet is 72 for the 4 cage compared to 13 1 for the Td cage (Al representation). It is thus evident that the totally symmetric representation of the Td cage has a smaller population compared to the D2 cage. The frequencies for the 2 s 1 = 75 spin multiplet are 4, 2, 6, 8, and 11 for the Al, A2, El, TI, and T2 representations, respectively. Likewise, the frequencies for the= 1 = 73 nuclear spin multiplet are 131, 109,241,331, and 353 for the AI, Az, El, T1, and T2 representations. Again the alternation in the frequencies seen for other cages is found for the Td c 7 6 cage too. The frequencies of the low-spin multiplets are found to vary approximately as the dimensions of the irreducible representations. We conclude that the nuclear spin statistics of the Td and chiral 0 2 cages are substantially different. The Cs4 fullerene is the third most abundant fullerene to be found in the carbon soot (C, and C70 being the first two). For this reason the Cs4 fullerene has been the subject of several studies. Liu et find 24 isomers for c84satisfying the isolated-pentagon rule. The Hiickel method of Fowler28predicted a rather flat D2 structure for Cs4 as the most stable cage. Saito et al.27 showed that this structure is rather unfavorableand proposed a round D2 structure for Cs4. In addition cages with Td and D6h symmetries are likely to be present in the carbon soot. Liu et a1.4 state that c 8 4 fullerene is most likely to be present as a mixture of isomers. We consider two forms of the c 8 4 fullerene, one with D2symmetry considered to be favorable by Saito et al.2’ and the other with Td symmetry. The computed nuclear spin statistics of the two cages are shown in Tables V and VI, respectively. Again since the cycle indices of the B1, B2, and B3 representations are the same in the D2 group, the corresponding spin statistics are identical for the D2 cage. The frequencies of the 2s + 1 = 83, 81, and 79 multiplets are (20,21), (882,840), (22 918,22 960) for the (Al, B1) representations,respectively. Clearly there is an alternation is the frequencies. The maximal value of frequency is reached at the nonet spin multiplicity. The tetrahedral C84 cage has considerably more distributed nuclear spin statistics compared to the chiral D2 cage. The frequenciesof the 2 s 1 = 83 spin multiplet are 3,3,7, 10, and 11, for the AI, A2, E, TI, and T2, representations, respectively. The frequencies of the 2 s + 1 = 81 and 79 multiplets are (1 57,137,294,410,430) and (3839,38 19,7630,11470,11490), respectively for the (AI, A2, E, TI, T2) set of representations. The highest frequency is reached at the nonet spin multiplicity similar to the D2 cage. The total frequencies vary approximately as the dimensiions of the irreducible representations. The overall species should have A1 or A2 symmetry since the Pauli exclusion principle places no constraint on the composite permutation-inversion operation. The CZaxes of the Td group generate the x242cycle representation and are thus even permutations. Consequently
+
+
+
+
+
+
The Journal of Physical Chemistry, Vol. 97, No. 34,1993 8743
Nuclear Spin Statistics of Larger Fullerene Cages
TABLE Vk N U CSpln ~ StPtiStic~Of 'Tu rritb Td Symmetry Frequencies of IrreducibleRepreclentatiom:
A1(805950546412501652209664), A2(805950546408103604649984)
E( 1611901092820604988424192) T~(2411851639226509510867200),T2(2417851639230901616329728) Spin Multiplets and Their Frequencies 2S+l
1 5 9 13 17 21 25 29 33 31 41 45 49 53 57 61 65 69 13 71 81 85
frequency
2S+1 AI 1626851246396641438140 3 7074337995412206544524 7 9188131864584742793880 11 7933705320335800281185 15 5126394206993425068222 19 2578011183351468600228 23 1025113500522103146226 21 324105433816568445356 31 81418691258719926376 35 16224230211530618635 39 2540698151424411992 43 309692216094219394 41 28918693389889125 51 2044193568911340 55 106163659861764 59 3934114875101 63 99703979882 61 1621171522 71 15654418 15 16124 19 151 83 1
frequency
2s+ 1
1626851246396641438140 7014337995459913234656 9188131864563467119080 1933705320310536054145 5126394206969111551350 2518017783331645912518 1025113500508001085138 324105433807758349036 81418691253919215656 16224230209160995027 2540698150405230712 309692215108891298 28978693262492065 2044193532381195 106163650868988 3934113000947 99703655818 1627132112 15649498 16324 131
2s+ 1
4658110381151163681816 861223155957448115OO40 8890368331840243018615 6591018266058187523622 374621115122141 1677712 1673249778234627809358 593101382431512116504 161255360601580802164 37447271292920349653 6619825403457042608 915968159406868450 98017083905001035 7983925700438640 484876095932292 21361246021013 658881233426 13522518010 111187742 1206020 3839 3
1 5 9 13 17 21 25 29 33 31 41 45 49 53 51 61 65 69 73 77 81 85
TI 4880553738920113260600 21223013986135294828644 27564395593488290696640 23801115960719993554160 15379182620804458681524 7734233349931505666328 3015340501488761979584 972316301405661993928 244436013753858992048 48672690624320469272 7622094450095776168 929076826776350678 86936079691551210 6132580514765910 318490948110576 11802338234298 299110869460 4883185886 46947296 228980 410 0
4658110381145530724124 8612231559553206135240 8890368331814919085505 6591018266033934006150 3146211151201589193552 1673249118220531748210 593101382422702136464 16725536060212OO91444 37447217290550180309 66 19825402437861328 9159681590215OOlO3 98011083777609375 7983925663908480 484876086939516 21367244148183 658880909362 13522532810 171782822 1205641 3819 3
1 5 9 13 11 21 25 29 33 31 41 45 49 53 57 61 65 69 73 77 81 85
4880553138920773260600 21223013986141527785196 27564395593509565111440 23801115960805251487210 15319182620828112198396 7734233349951328150488 3015340501502864040612 972316301414471913968 244436013758119102168 48672690626690038616 1622094451114958048 92901682716 1I 18425 86936019818948870 6132580611296010 318490951103352 11802340107128 299111193524 4883231086 46952216 229359 430
3
freumcy
11 15 19 23 21 31 35 39 43 41 51 55 59 63 67 71 75 79 83
13976131161705720758676 25836712678904243633760 2667 1104995647080024940 19773234798253386500 11238633453107823349692 5019149334725040642816 17193041473033~5a32 501766081825173443792 11234ia31879~~4049688 19859476210473045952 2747904478184397128 294051251683548430 23951777087267700 14546282a320220a 64101736960082 1976643471484 40567696214 515359504 3611122 11470 10
3 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 61 I1 I5 79 83
13976131161717954068544 258361 1267892551a m 5 6 0 26671104995672344251980 19773234798277640017572 11238633453727646037342 5019749334739136703904 1179304147312156542152 501766081830634154512 1123418318819236732% 19859476211492227232 2747904418569785224 294051251810946090 23951717123803845 1454628292194984 64101138834242 i916643801548 40567741624 515364424 3618122 11490 11
I
T2
Az 1 5 9 13 17 21 25 29 33 31 41 45 49 53 57 61 65 69 73 I1 81 85
frequency
3 7 11 15 19 23 21 31 35 39 43 41 51 55 59 63 61 I1 I5 19 83
0
0
E 1 5 9 13 17 21 25 29 33 31 41 45 49 53 57 61 65 69 73 71 81 85
3253102492193254159680 14148675990932211221340 18316263729148210572960 15861410640646305913515 10252188413962618093152 5156155566683114512806 2050221001030091108254 648210867624333701292 162951382512699142032 32448460420688505557 5081396301830826144 619384551803170692 51957386652005050 4088387101396815 212327310730752 7868221855519 199401638916 3255509634 31303538 153016 294
3
I 11 15 19 23 21 31 35 39 43 47 51 55 59 63 61 71 I5 79 83
9317420114303294406600 11224475119121649843120 11780736663655252525875 13182156532092121530312 7492422302422979391084 3346499556455172680738 1186202164854214252968 334510121210293986708 14894554583414238061 13239650805894903936 1831936318427185113 196034161682993150 15961851364347120 969752182773528 42734490189671 1317162142788 21045107544 343570942 2411661 1630 7
0
both A1 and A2 are allowed as overall species. This means a rovibronic level of A1 symmetry could couple to nuclear spin species of both A1 and A2 symmetries. Likewise T1 could couple
to Tzor itself. Using this condition the total nuclear spin statistical weights can be readily obtained from the frequencies in Table
VI.
8744 The Journal of Physical Chemistry, Vol. 97, No. 34, 1993
Balasubramanian
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Aclmow1edg”t. Thisresearch was supported by theNationa1 Science Foundation under Grant CHE 9204999.
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I
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