15064
J. Phys. Chem. 1996, 100, 15064-15067
Enumeration and Symmetry of Substitution Isomers Yihan Shao, Jian Wu, and Yuansheng Jiang* Department of Chemistry, Nanjing UniVersity, Nanjing 210093, PR China ReceiVed: March 12, 1996X
A simple group-theoretical approach, which is based on the subgroup decomposition of the isomer permutation representation, is applied to enumerate the substitution isomers of three fullerene cages, D2-C76, D2-C84, and D2d-C84. Several rules governing the symmetry distribution of substitution isomers are extracted from the enumeration results. These rules are also applicable to other substitution frameworks.
1. Introduction For decades, chemists have shown interest in the enumeration of substitution isomers,1-17 which can be divided into three subproblems, that is, how to compute the total number of isomers, how to compute the number of chiral isomers, and how to find the symmetry distribution of the isomers. So far, these problems have never been solved by using Po´lya’s theorem,1-6 double-coset method,7,8 the table of marks,9-12 etc. In this work, we employ an enumeration method, which is based on the subgroup decomposition of the isomer permutation representation, to enumerate the substitution isomers of three fullerene cages, D2-C76, D2-C84, and D2d-C84. Several rules governing the symmetry distribution of substitution isomers are extracted from the enumeration results. These rules are also applicable to other substitution frameworks. 2. Enumeration Method Suppose a substitution framework possesses point group symmetry G and n substitution sites. If m indistinct substituents are to be introduced into this framework and each substitution site can have only one substituent, there will be a total number n of m isomeric configurations. These configurations span a representation of group G termed the isomer permutation representation.14 In most cases, the isomer permutation representation (Γ) is a reducible one, and thus can be expressed as follows:
()
Γ ) ∑cγΓG(γ)
(1)
γ
where ΓG(γ) denotes the γth irreducible representation of point group G, and cγ its multiplicity. For a specific framework and a fixed number of m, the multiplicities can easily be obtained.17 For an isomeric configuration possessing point group symmetry H (which must be a subgroup of G), there are |G|/|H| equivalents under the symmetry operations of G. All of these |G|/|H| configurations correspond to a single isomer, and thus in most cases, the number of isomers is much smaller than n m . Let Γ(G/H) be the representation of G spanned by the |G|/ |H| configurations. Then one has the following relation:
()
Γ ) ∑ nH Γ(G/H)
(2)
TABLE 1: Number of C76Xm, Substitution Isomers of D2-C76 m
C1
C2
D2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
19 684 17 575 320 226 4 618 710 54 648 408 546 547 350 4 713 915 618 35 616 668 975 238 631 305 676 1 431 790 092 795 7 755 527 599 280 38 181 069 141 200 171 814 801 670 208 710 167 886 026 320 2 707 515 028 799 664 9 555 935 525 207 100 31 322 232 988 142 320 95 615 237 915 961 100 272 503 427 705 944 632 726 675 808 161 304 360 1 816 689 519 500 769 184 4 265 271 047 903 308 200 9 419 140 228 755 879 480 19 591 811 680 035 862 332 38 430 092 137 547 594 160 71 166 837 299 275 568 300 124 541 965 266 480 104 144 206 138 425 280 660 266 800 322 950 199 594 764 286 400 479 216 425 222 352 147 120 673 898 097 952 252 763 856 898 530 797 291 910 275 850 1 136 377 184 788 771 256 040 1 363 652 621 772 428 536 290 1 553 048 819 215 637 934 284 1 678 971 696 476 563 663 300 1 723 155 162 146 806 390 800
0 57 0 1026 0 12 654 0 110 466 0 752 913 0 4 139 568 0 18 930 384 0 73 349 424 0 244 517 460 0 709 083 192 0 1 804 983 432 0 4 061 172 024 0 8 122 425 444 0 14 504 255 568 0 23 206 929 840 0 33 359 848 272 0 43 171 715 070 0 50 366 862 348 0 53 017 895 700
0 0 0 19 0 0 0 171 0 0 0 969 0 0 0 3876 0 0 0 11 628 0 0 0 27 132 0 0 0 50 388 0 0 0 75 582 0 0 0 92 378 0 0
Γ(G/H), the representation of G spanned by all the symmetrically equivalent configurations corresponding to an isomer of symmetry H, is structured as (γ) (γ) Γ(G/H) ) ∑a0,H ΓG
(3)
γ
H
where the summation runs over all the subgroups of G, and nH is the number of isomers possessing symmetry H. X
Abstract published in AdVance ACS Abstracts, August 15, 1996.
S0022-3654(96)00747-2 CCC: $12.00
(γ) is the multiplicity of the γth irreducible representawhere a0,H tion of G. To obtain these multiplicities, one can utilize the following subgroup decomposing formula of each irreducible representation of G:
© 1996 American Chemical Society
Symmetry of Substitution Isomers
J. Phys. Chem., Vol. 100, No. 37, 1996 15065
TABLE 2: Number of C84Xm, Substitution Isomers of D2-C84
Γ(D2/C1) ) A + B1 + B2 + B3
(6a)
m
C1
C2
D2
Γ(D2/C2) ) A + B1
(6b)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
21 840 23 821 481 740 7 718 004 101 611 776 1 132 341 444 10 898 702 556 92 034 196 254 690 255 833 904 4 643 543 538 270 28 248 219 257 468 156 451 697 674 020 793 433 589 398 784 3 702 690 178 285 140 15 967 851 305 335 020 63 871 405 575 418 665 237 743 564 862 972 840 825 846 068 580 413 265 2 683 999 721 782 771 056 8 179 808 679 272 664 720 23 423 997 578 343 118 848 63 142 950 002 448 672 720 160 488 331 247 930 149 296 385 171 995 014 936 903 592 874 044 142 514 756 232 768 1 877 576 306 183 923 196 712 3 822 208 908 977 627 108 208 7 380 817 203 619 560 152 592 13 531 498 206 561 855 959 040 25 570 996 876 075 369 519 568 39 039 463 575 874 948 577 328 61 516 730 483 393 672 723 418 92 275 095 724 899 512 639 760 131 821 565 321 557 870 121 610 179 423 797 242 966 272 749 480 232 766 007 234 462 545 439 960 287 894 798 421 237 014 192 640 339 568 223 779 290 164 130 360 382 014 251 751 316 091 117 448 409 966 514 074 996 661 572 020 419 727 621 552 569 079 156 000
0 63 0 1260 0 17 220 0 167 580 0 1 276 002 0 7 866 684 0 40 467 492 0 177 036 300 0 668 837 715 0 2 207 133 936 0 6 420 842 064 0 16 587 093 936 0 38 278 096 920 0 79 290 169 200 0 148 008 641 424 0 249 764 277 168 0 381 992 890 734 0 530 545 240 680 0 670 162 966 200 0 770 686 882 056 0 807 386 811 660
0 0 0 21 0 0 0 210 0 0 0 1330 0 0 0 5985 0 0 0 20 349 0 0 0 54 264 0 0 0 116 280 0 0 0 203 490 0 0 0 293 930 0 0 0 352 716 0 0
Γ(D2/C2′) ) A + B2
(6c)
Γ(D2/C2′′) ) A + B3
(6d)
Γ(D2/D2) ) A
(6e)
(γ) (η) (γ) (0) ΓG(γ) ) ∑aη,H ΓH ) a0,H ΓH + ...
(4)
η
where ΓH(η) is the ηth irreducible representation of H, and ΓH(0) is the totally symmetric irreducible representation of H. In other words, one can refer to the correlation table of point group G for the multiplicities in eq 3.18 Combining eqs 1, 2, and 3, one has the following set of relations: (γ) nHa0,H ) cγ ∑ H
(5)
ΓG(γ)
in the structure of the where cγ is the coefficent of corresponding isomer permutation representation. As a result, to obtain the number of isomers with specified symmetry, one simply has to carry out the subgroup decomposition of the isomer permutation representation, that is, to solve a linear system governed by eq 5, where the number of unknowns (nH’s) is equal to the number of subgroups of group G, while the number of equations is equal to that of irreducible representations of G. 3. Enumeration Results A. D2-C76 and D2-C84. From the correlation table for the D2 point group, one knows that
{
Thus, the corresponding linear system is
nC1 + nC2 + nC2′ + nC2′′ + nD2 ) cA nC1 + nC2 ) cB1 nC1 + nC2′ ) cB2 nC1 + nC2′′ ) cB3
(7)
where cA, cB1, cB2, and cB3 are the multiplicities in the corresponding isomer permutation representation and can easily be computed.17 Now there are five unknowns but only four equations; thus one of the unknowns should be predetermined. Such a number is easily obtained according to the molecular geometry
{( ) n/4
, m ) 4, 8, 12, ... (8) m/4 0, otherwise Then the linear system is readily solved. The numbers of isomers are collected in Tables 1 and 2, where the numbers of isomers possessing C2, C2′, and C2′′ symmetry have been added up. B. D2d-C84. The linear system for this framework is obtained in a similar way. nD2 )
{
nC1 + nCs + nC2 + nC2′ + nC2V + nD2 + nS4 + nD2d ) cA1 nC1 + nC2 + nS4 ) cA2 (9) nC1 + nC2 + nC2′ + nD2 ) cB1 nC1 + nCs + nC2 + nC2V ) cB2 2nC1 + nCs + nC2′ ) cE
In this case, since there are only five equations, three of the eight unknowns should be predetermined. From the molecular geometry, it is easy to find
{
(
10
)
(m-4)/8 nD2d ) 10 , m/8 0,
{ [(
( ) )
, m ) 4, 12, 20, ... m ) 8, 16, 24, ...
(10a)
otherwise
]
1 21 - nD2d , m ) 4, 8, 12, ... (10b,c) nD2 ) nS4 ) 2 m/4 0, otherwise Then all the other numbers of isomers will be obtained as solutions of eq 9. Table 3 includes the final enumeration results. 4. Symmetry Considerations Now let us take a look at the enumeration results. From Tables 1 and 2, it is obvious that, for these two frameworks of D2 symmetry, C2 is a possible symmetry only for substitution isomers with even numbers of substituents. Similarly, D2 is a
15066 J. Phys. Chem., Vol. 100, No. 37, 1996
Shao et al.
TABLE 3: Number of C85Xm, Substitution Isomers of D2d-C84 m
C1
Cs
C2
C2V
D2
S4
D2d
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
10 410 11 890 240 660 3858 592 50 803 028 566 165 392 5 449 323 348 46 017 047 492 345 127 704 380 2 321 771 394 436 14 124 108 317 620 78 225 846 588 816 396 716 787 955 380 1 851 345 077 901 600 7 983 925 623 161 460 31 935 702 739 935 210 118 871 782 320 015 890 412 923 034 115 034 850 1 341 999 860 523 529 872 4 089 904 339 075 782 656 11 711 998 788 101 426 832 31 571 474 999 644 605 376 80 244 165 621 200 558 992 192 585 997 503 519 124 336 437 022 071 250 998 452 944 938 788 153 083 151 560 176 1 911 104 454 475 598 525 904 3 690 408 601 792 159 999 936 6 765 749 103 256 259 911 376 11 785 498 438 005 968 622 336 19 519 731 787 895 846 909 136 30 758 365 241 645 297 638 356 46 137 547 862 386 090 901 076 65 910 782 660 703 142 820 580 89 711 898 621 394 712 167 960 116 383 003 617 130 216 399 680 143 947 399 210 506 813 352 600 169 784 111 889 522 750 730 080 191 007 125 875 529 597 745 048 204 983 257 037 363 766 317 400 209 863 810 776 149 975 201 768
1 20 41 420 820 5720 10 660 55 860 101 270 425 144 749 398 2 622 228 4 496 388 13 488 024 22 481 940 59 012 100 95 548 245 222 941 060 350 343 565 735 711 312 1 121 099 408 2 140 265 184 3 159 461 968 5 529 031 312 7 898 654 920 12 759 326 880 17 620 076 360 26 430 056 400 35 240 152 720 49 336 136 288 63 432 274 896 83 254 759 056 103 077 446 706 127 330 837 608 151 584 480 450 176 848 413 560 202 112 640 600 223 387 487 440 244 662 670 200 256 895 627 352 269 128 937 220 269 128 752 464
0 31 0 620 0 8600 0 83 690 0 637 906 0 3 932 682 0 20 233 176 0 88 515 180 0 334 416 435 0 1 103 556 816 0 3 210 413 280 0 8 293 519 896 0 19 139 029 080 0 39 645 026 520 0 74 004 281 952 0 124 882 036 944 0 190 996 382 382 0 265 272 473 480 0 335 081 399 120 0 385 343 264 796 0 403 693 313 452
0 1 0 10 0 20 0 100 0 190 0 660 0 1140 0 2970 0 4845 0 10 152 0 15 504 0 27 072 0 38 760 0 58 080 0 77 520 0 101 640 0 125 970 0 146 860 0 167 960 0 176 232 0 184 756
0 0 0 10 0 0 0 100 0 0 0 660 0 0 0 2970 0 0 0 10 152 0 0 0 27 072 0 0 0 58 080 0 0 0 101 640 0 0 0 146 860 0 0 0 176 232 0 0
0 0 0 10 0 0 0 100 0 0 0 660 0 0 0 2970 0 0 0 10 152 0 0 0 27 072 0 0 0 58 080 0 0 0 101 640 0 0 0 146 860 0 0 0 176 232 0 0
0 0 0 1 0 0 0 10 0 0 0 10 0 0 0 45 0 0 0 45 0 0 0 120 0 0 0 120 0 0 0 210 0 0 0 210 0 0 0 252 0 0
TABLE 4 m
G′
2l + 1 4l + 2 4l
C1, Cs C1, C2, Cs, C2V C1, C2, Cs, D2, S4, C2V, D2d
possible symmetry only when m, the number of substituents, is a multiple of 4. In contrast, C1 is always a possible symmetry. In the comparatively complicated case of D2d-C84, the possible symmetries according to the enumeration results have been collected in Table 4. Once again, the substitution isomers display a regular symmetry distribution. From our point of view, a concept of site symmetry may account for the above regularities governing the symmetry distribution of substitution isomers. For a framework of point group symmetry G, the site symmetry of a fixed substitution site is defined as the group made up of the symmetry operations of G which leave the site invariant. In other words, the site symmetry for a site is the symmetry of the framework viewed from that site. By inspection, one easily finds that all the substitution sites (carbon atoms) of D2-C76 and D2-C84 possess C1 site symmetry, while the substitution sites of D2d-C84 fall into two categories: 80 sites possessing C1 symmetry and four sites possessing Cs symmetry. From the enumeration results in the last section, the following rules can be extracted. Rule 1. If all the substitutions sites in a framework possess
site symmetry C1, then G′, the symmetry of the framework having m substituents, satisfies that its order |G′| is a divisor of m. Rule 2. If all the substitution sites in a framework possess site symmetry Cs or C1, then G′, the symmetry of the framework having m substituents, satisfies that (i) if G′ is a group without reflection plane, its order |G′| is a divisor of m; (ii) if G′ is a group with one or more reflection planes, its order |G′| is a divisor of 2m. Rule 3. If the site symmetries of a framework are either S or its subsymmetries, then G′, the symmetry of the framework having m substituents, satisfies that its order |G′| is a divisor of m|G′∩S|. Rule 4. If the substitution sites in a framework possess site symmetries S1, S2, ..., then G′, the symmetry of the framework having m substituents, satisfies that there exists k so that |G′| is a divisor of m|G′∩Sk|. All these rules can be proved to be applicable to frameworks other than the three fullerene cages considered in this work. 5. Conclusions A simple group-theoretical approach, which is based on the subgroup decomposition of the isomer permutation representation, is applied to enumerate the substitution isomers of three fullerene cages, D2-C76, D2-C84, and D2d-C84. Four rules governing the symmetry distribution of substitution isomers are
Symmetry of Substitution Isomers extracted from the enumeration results. These rules are also applicable to other substitution frameworks. References and Notes (1) Po´lya, G. Acta Math. 1937, 68, 145. (2) Balaban, A. T. Chemical Application of Graph Theory; Academic Press: London, 1976. (3) Balasubramanian, K. Chem. ReV. 1985, 85, 599, and references therein. (4) Balasubramanian, K. J. Phys. Chem. 1993, 97, 6990. (5) Balasubramanian, K. Chem. Phys. Lett. 1995, 237, 229. (6) Trinajstic´, N. Chemical Graph Theory; CRC: Boca Raton, 1992. (7) Ruch, E.; Ha¨sselbarth, W.; Ritcher, B. Theor. Chim. Acta 1970, 19, 288. (8) Brocas, J. J. Am. Chem. Soc. 1986, 108, 1135.
J. Phys. Chem., Vol. 100, No. 37, 1996 15067 (9) Ha¨sselbarth, W. Theor. Chim. Acta 1985, 67, 339. (10) Fujita, S. Theor. Chim. Acta 1989, 76, 247. (11) Fujita, S. Bull. Chem. Soc. Jpn. 1990, 63, 2759. (12) Fujita, S. Bull. Chem. Soc. Jpn. 1991, 64, 3215. (13) Mead, C. A. J. Am. Chem. Soc. 1987, 109, 2130. (14) Fowler, P. W. J. Chem. Soc., Faraday Trans. 1995, 91, 2241. (15) Shao, Y. H.; Jiang, Y. S. Chem. Phys. Lett. 1995, 242, 191. (16) Shao, Y. H.; Jiang, Y. S. J. Phys. Chem. 1996, 100, 1554. (17) Shao, Y. H.; Wu, J.; Jiang, Y. S. Chem. Phys. Lett. 1996, 248, 336. (18) Wilson, E. B., Jr.; Decius, J. C.; Cross, P. C. Molecular VibrationsThe Theory of Infrared and Raman Spectra; McGraw-Hill: New York, 1955.
JP9607475
