386
J. Chem. Inf. Comput. Sci. 2000, 40, 386-398
Regarding Enumeration of Molecular Isomers Ivan Baraldi* and Davide Vanossi Dipartimento di Chimica, Universita´ di Modena e Reggio Emilia, Via Campi 183, 41100 Modena, Italy Received June 22, 1999
In this paper, a simple theoretical approach to counting of substitution isomers is described. It is based on Po´lya’s theorem and on point groups as recently described by us [Baraldi, I.; Fiori, C.; Vanossi, D. J. Math. Chem. 1999, 25, 23-30] and extended in this paper. Several applications are developed that range from molecules without symmetry to molecules with icosahedral symmetry (Ih). The problem of the appearance of stereoisomers is also analyzed. INTRODUCTION
In a recent work1 we were able to express the cycle index of all point groups as a function of a limited number of initial geometrical parameters. Such parameters are the number m of elements composing the domain D of substitution sites (points, elements) of the molecule having G symmetry and the numbers (n(Cn), n(σ), ...) that identify how many points of set D lie on symmetry elements (Cn, σ, ...) of the molecule. One of the primary utilities of the results of this work resides in the fact that the chemical applications of the Po´lyaRedfield theorem, such as the counting of isomers, for which several methods have been developed,2-11 can be made much more understandable to the chemical community. In this paper, the logical development of the previous work is presented. Precisely, after introducing the cycle index of a domain D given by the union of substitution domains corresponding to the different molecular orbits determined by G symmetry on D, significant molecular applications are discussed. Such applications include the development of some series counting isomers, the corresponding formulas for the total number of isomers, and examples of enumeration of molecular isomers. In particular, where possible, we pay attention to the problem of stereoisomerism, and the counting of the isomers is made both with the extended molecular point group (achiral point group) and with the proper rotational group (chiral point group). The problem of the substitutions in the fullerene C60, a topic of recent interest,12 is also analyzed. MATHEMATICAL DEVELOPMENT
We consider a molecule (M) having m atoms with n distinct elements and molecular formula
Am(1)1 Am(2)2 ...Am(n)n
n
m ) ∑ mi
(2)
i)1
G ) {gˆ i, i ) 1, 2, ..., g} denotes the molecular point group of finite order g of M. By action of G on M, the m atoms are split up in no(G) distinct orbits, or better G-orbits, and such a number is given by the formula
no ) no(G) ) no(G,M) )
1
∑ n(gˆ i)
g gˆ i∈G
(3)
where n(gˆ i) is the number of atoms of the molecule that are stationary during the molecular motion connected to the symmetry operation gˆ i. Equation 3 expresses the CauchyFrobenius lemma also called Burnside’s lemma. Each G-orbit is formed of a certain number of equivalent atoms or equivalent molecular groups. With ni we indicate such number, which is a divisor of g, and we have no
m ) ∑ni
(4)
i)1
As a consequence, the number of equivalent atoms is a divisor of g. We now develop the procedure to obtain the enumeration of molecular isomers. Because the elements (points) of each G-orbit are disjoint to the elements of other G-orbits, the G-orbits can be considered independent of each other. Evidently such a procedure can be used also in the case in which two or more G-orbits are considered as a whole. The domain D of substitution sites of M, which include m points, is given by the union of the domains Di (i ) 1, 2, ..., no) of the elements of different G-orbits, no
D ) ∪ Di
(5)
i)1
where mi is the number of atoms of the ith element A(i) of M. It is evident that * To whom correspondence should be addressed. Telephone: 059378457. Fax: 059-373543. E-mail:
[email protected].
where Di indicates the domain of the ith G-orbit of length ni; i.e. Di ) {dij, j ) 1, 2, ..., ni} contains ni ) |Di| elements. It is known that if |St(dij)| indicates the dimension of the stabilizer of the jth element of subset Di, then
10.1021/ci990056g CCC: $19.00 © 2000 American Chemical Society Published on Web 02/09/2000
REGARDING ENUMERATION
OF
MOLECULAR ISOMERS
J. Chem. Inf. Comput. Sci., Vol. 40, No. 2, 2000 387
ni|St(dij)| ) g
no Ri
(6)
To each subset Di we combine a group of distinct permutations given by the permutations of the ni elements of Di, when Di undergoes the symmetry operations belonging to G. Let Pi ) Pi(Hi) ) P(Hi,Di) be such a permutation group, where Hi is one of the subgroups of G (Hi e G), including the banal subgroups C1 and G, and such that Pi is isomorphic to Hi. (Pi is a transitive permutation group on Di.) The cycle index of such a permutation group is
∑
hiP∈Pi
xei11P
ij
(13)
ni ) ∑kij
(14)
i)1 j)1
with Rj
j)1
and then no Rj
Z(Pi,ni) ) Z(Pi(Hi),ni) ) Z(xi1,xi2, ..., xini) ) 1
∏∏wkij
m ) ∑∑kij eniP xei22P...xin i
(7)
where the symbols have the common meaning (hi is the order of Hi, xij the jth indeterminate of Zi, ejP the number of cycles of degree j in the permutation P ∈ Pi). With each G-orbit being treated in an independent way with respect to the substitution, the cycle index of the total permutation group no
P(G) ) P(G,D) ) ∑Pi
(8)
(15)
i)1 j)1
The coefficient of this term, in the expanded form, is the number of isomers of molecular formula (1R1) (21) Ak(12) ...Ak1R Ak21 ...Ak(ij) ...Ak(nnoRRi) Ak(11) 11 12 1 ij
(16)
o i
where the correspondence Ak(ij) ) wkijij has been made, and ij each element has been identified by two indices, i and j. kij gives the numbers of elements of j type in the ith G-orbit. To simplify the notation, in what follows the molecular formula of isomers will be indicated as ...WiXjYk....
i)1
APPLICATIONS
is no
no
i)1
i)1
Z(P(G),m) ) ∏Z(Pi,ni) ) ∏Z(xi1,xi2, ..., xini) ) no
1
∏h ∑ xei1 i)1 iP∈Pi
1P
eniP xei22P...xin (9) i
With each Z(Pi,ni) being written as a function of the initial parameters according to ref 1, also Z(P(G),m) will be a function of such parameters. The counting series of molecular isomers is obtained from eq 9, making the substitution k k k + wi2 + ... + wiR xik f wi1 i
(10)
where Ri indicates the number of different elements or molecular groups, i.e., the distinct substitutes, that can be attached to the ni points of subset Di and wij is the weight function of the jth substitute of subset Di. The formula for the series counting isomers is then no
CI(P(G)m) ) ∏Z(wi1 + wi2 + ... + wiRi, wi12 + wi22 + i)1
ni ni ni ... + wiRi2, ..., wi1 + wi2 + ... + wiR ) (11) i
m
Z(P(C1),m) ) ∏x1i1
(17)
i)1
m
CI(P(C1),m) ) ∏(wi1 + wi2 + ... + wiRi)
(18)
i)1
m
NI(P(C1),m) ) ∏Ri
(19)
i)1
In the case in which R1 ) R2 ) ... ) Rm ) R and wij ) wj (j ) 1, 2, ..., R), then
and that for the total number of isomers is no
NI(P(G),m) ) ∏Zi(Ri,Ri,...,Ri)
The application of the previously developed theory begins with molecules without geometrical symmetry, C1 point group, and then continues with molecules of increasing symmetry, to conclude with molecules of icosahedral symmetry. To be more exact, the molecular systems considered are as follows: oxiran (C2V), molecules with molecular formula AB3 (C3V), adamantane-2,6-dione (D2d), dibenzodioxine (D2h), coronene (D6h), molecules with molecular formula AB4 (Td), molecules with molecular formula AB6 (Oh), cubane (Oh), borane anion (B12H122-, Ih), and fullerene C60 (Ih). Use also of the molecular rotational subgroups (chiral groups) to see when stereoisomers appear completes the analysis of molecular isomers. (1) Molecules without Symmetry. The molecular point group is G ≡ C1 ) {Eˆ }, and the number of G-orbits is equal to the number of atoms of M, no(C1) ) m. We have
(12)
i)1
Expanding eq 11, we obtain a combination of terms of the general form
CI(P(C1),m) ) (w1 + w2 + ... + wR)m ) m! ∑r1!r2!...rR!wr11wr22...wrRR (20) where the sum is taken over all R-tuples of nonnegative
388 J. Chem. Inf. Comput. Sci., Vol. 40, No. 2, 2000
BARALDI
Table 1 XiYj R
m
NI
i
j
isomer no.
2
4
16
4 3 2
0 1 2
2 8 6
3
27
Z(P(C2V),7) ) 1/8x111(x221 + x122)(x431 + 3x232)
(31)
NI(P(C2V),7) ) 1/8R1(R22 + R2)(R43 + 3R23)
(32)
1 2 2 ) + (w221 + w222 + ... + w2R )][(w131 + w132 + ... + w2R 2 2
i
j
k
isomer no.
3 2 1
0 1 1
0 0 1
3 18 6
NI(P(C1),m) ) Rm
1 4 2 2 ) + 3(w231 + w232 + ... + w3R ) ] (33) ... + w3R 3 3
In Table 2 we report some results of the total number of isomers (NI) calculated from eq 32. In the case of R1 ) R2 ) R3 ) 2 eq 33 is reduced to
integers (r1, r2, ..., rR) such that r1 + r2 + ... + rR ) m, and
CI(P(C2V),7) ) 1/8(w111 + w112)[(w121 + w122)2 + (w221 + w222)][(w131 + w132)4 + 3(w231 + w232)2] (33′)
(21)
For R ) 2, the last two formulas are reduced to
(mk )w w m NI(P(C ),m) ) 2 ) ∑ ( ) k m
CI(P(C1),m) ) (w1 + w2)m ) ∑
VANOSSI
1 )[(w121 + w122 + CI(P(C2V),7) ) 1/8(w111 + w112 + ... + w1R 1
XiYjZk 3
AND
k
1
m-k
2
(22)
and the enumeration of different isomers is given in Table 3. Using C2 symmetry, no(C2) ) 4, and the domain is thus decomposed
k)0 m
m
1
(23)
k)0
Equation 20 expresses a generalization of the binomial theorem given by eq 22 and is known as the multinomial theorem. Table 1 gives some examples of isomer counting and their enumeration within C1 symmetry. (2) Molecule with C2W Symmetry.
ˆ 2,σˆ xz,σˆ yz} C2V ) {Eˆ ,C
The permutation group of different domains is
P1 ) P(C2,D1) ) P(C1,D1) Pi ) P(C2,D2)
(35)
i ) 2, 3, 4
(36)
and the corresponding cycle index
C2 ) {Eˆ ,C ˆ 2}
Z(P1,1) ) x111
Oxiran (C2H4O). The spatial distribution of the atoms in this molecule and the C2V symmetry generates three distinct G-orbits, no(C2V) ) 3, of dimensions one, two, and four, respectively. The domain D is thus decomposed:
(37)
Z(Pi,2) ) 1/2(x2i1 + x1i2)
i ) 2, 3, 4
(38)
In all we have 4
Z(P(C2),7) ) 1/8x111∏(x2i1 + x2i2)
(39)
i)2 4
NI(P(C2),7) ) 1/8R1∏(R2i + R1i )
(40)
i)2
The permutation group of different domains is
4
P1 ) P(C2V,D1) ) P(C1,D1)
(25)
P2 ) P(C2V,D2) ) P(C2,D2)
(26)
P3 ) P(C2V,D3)
(27)
Z(P1,1) ) x111 Z(P2,2) ) Z(P3,4) ) In all, we have
1
/4(x431
+ ... +
1 2 wiR ) i
+
+ ... + 2 (wi1
(28)
+
x122)
(29)
+
3x232)
(30)
/2(x221
1 wi2
+
w112
+
1 w1R ) 1
2 wi2
∏[(wi11 + i)2
2 + ... + wiR )] (41) i
Table 4 reports some values of the total number of isomers (NI) obtained with eq 40. In the particular case of R1 ) R2 ) R3 ) R4 ) 2, the cycle index is
and the cycle index is
1
CI(P(C2),7) )
/8(w111
1
4
CI(P(C2),7) )
1
/8(w111
+
w112)
∏[(wi11 + wi21)2 + i)2
2 2 + wi2 )] (41′) (wi1
and the repartition of different isomers is given in Table 5.
REGARDING ENUMERATION
OF
MOLECULAR ISOMERS
J. Chem. Inf. Comput. Sci., Vol. 40, No. 2, 2000 389 Table 5a
Table 2 R1
R2
R3
NI(P(C2V),7)
1 2 1 1 2 1 2 2 3 1 1 3 ‚‚‚ 3 ‚‚‚ 3 ‚‚‚
1 1 2 1 2 2 1 2 1 3 1 2 ‚‚‚ 2 ‚‚‚ 3 ‚‚‚
1 1 1 2 1 2 2 2 1 1 3 1 ‚‚‚ 2 ‚‚‚ 3 ‚‚‚
1 2 3 7 6 21 14 42 3 6 27 9 ‚‚‚ 63 ‚‚‚ 486 ‚‚‚
TiUjVkWlXmYnZo i
j
k
l
m
n
o
isomer no.
0 0 0 1
0 0 1 1
0 1 1 1
1 1 1 1
2 1 1 1
2 2 1 1
2 2 2 1
16 24 12 2
a
R1 ) R2 ) R3 ) R4 ) 2.
G-orbit has one element and the other three elements. The decomposition of the domain D is
D)
D2 D1 ∪ one-dimensional three-dimensional (n1 ) |D1| ) 1, n2 ) |D2| ) 3) (42)
The permutation groups of both point groups are Table 3a ViWjXkYlZm
a
P1 ) P(C3V,D1) ) P(C3,D1) ) P(C1,D1)
(43)
i
j
k
l
m
isomer no.
P2(C3V) ) P(C3V,D2)
(44)
0 0 0 0 1 1
0 1 1 1 1 1
1 1 1 2 1 1
2 1 2 2 1 2
4 4 3 2 3 2
8 4 8 12 4 6
P2(C3) ) P(C3,D2)
(44′)
R1 ) R2 ) R3 ) 2.
From ref 1, the general expressions of the cycle index for a domain D of m elements submitted to symmetries C3V and C3 are, respectively, 3) (m-n(C3))/3 x3 + Z(C3V,m) ) 1/6(xm1 + 2xn(C 1
Table 4 R1
R2
R3
R4
1 2 1 1 1 1 2 2 2 ‚‚‚ 2 ‚‚‚ 2 3 1 ‚‚‚ 3 ‚‚‚ 3 ‚‚‚ 3 ‚‚‚
1 1 2 1 1 1 2 1 1 ‚‚‚ 2 ‚‚‚ 2 1 3 ‚‚‚ 2 ‚‚‚ 2 ‚‚‚ 3 ‚‚‚
1 1 1 2 1 2 1 2 1 ‚‚‚ 2 ‚‚‚ 2 1 1 ‚‚‚ 2 ‚‚‚ 2 ‚‚‚ 3 ‚‚‚
1 1 1 1 2 2 1 1 2 ‚‚‚ 1 ‚‚‚ 2 1 1 ‚‚‚ 1 ‚‚‚ 2 ‚‚‚ 3 ‚‚‚
NI(P(C2),7) 1 2 3 3 3 9 6 6 6 18 ‚‚‚ 54 3 6 ‚‚‚ 27 ‚‚‚ 81 ‚‚‚ 648 ‚‚‚
The 12 stereoisomers present are eight of TV2XY3 type and four of TUVXY3 type. (3) Molecules with C3W Symmetry.
C3V ) {Eˆ ,2C ˆ 3,3σˆ V} ˆ 3} C3 ) {Eˆ ,2C The Case of a Trigonal Pyramid (AB3). The symmetry of this type of molecule is C3V and its chiral group is C3. For both symmetries, the atoms of the molecule are decomposed in two G-orbits, no(C3V) ) no(C3) ) 2. One
v) (m-n(σv))/2 x2 ) (45) 3xn(σ 1 3) (m-n(C3))/3 x3 ) Z(C3,m) ) 1/3(xm1 + 2xn(C 1
(45′)
from which
Z(P2(C3V),3) ) 1/6(x321 + 2x123 + 3x121 x122)
(46)
Z(P2(C3),3) ) 1/3(x321 + 2x123)
(46′)
Z(P1,1) ) x111
(47)
Obviously,
and then the total cycle index is
Z(P(C3V),4) ) 1/6x111(x321 + 2x123 + 3x121 x122)
(48)
Z(P(C3),4) ) 1/3x111(x321 + 2x123)
(48′)
As a consequence 1 ){(w121 + w122 + CI(P(C3V),4) ) 1/6(w111 + w112 + ... + w1R 1 1 3 3 ) + 2(w321 + w322 + ... + w2R )+ ... + w2R 2 2 1 2 )(w221 + w222 + ... + w2R )} (49) 3(w121 + w122 + ... + w2R 2 2 1 ){(w121 + w122 + CI(P(C3),4) ) 1/3(w111 + w112 + ... + w1R 1 1 3 3 ) + 2(w321 + w322 + ... + w2R )} (50) ... + w2R 2 2
NI(P(C3v),4) ) 1/6R1R2(2 + 3R2 + R22)
(51)
NI(P(C3),4) ) 1/3R1R2(2 + R22)
(52)
390 J. Chem. Inf. Comput. Sci., Vol. 40, No. 2, 2000
BARALDI
VANOSSI
Table 9a
Table 6 R1
R2
NI(P(C3),4)
NI(P(C3V),4)
1 1 2 2 2 3 3 3 4 4 ‚‚‚
1 2 1 2 3 2 3 4 3 4 ‚‚‚
1 4 2 8 22 12 33 72 44 96 ‚‚‚
1 4 2 8 20 12 30 60 40 80 ‚‚‚
TiUjVkWlXmYnZo i
j
k
l
m
n
o
isomer no.
2 2 2 2 2
2 2 2 2 2
4 4 4 4 4
4 4 4 4 4
4 4 4 4 4
0 1 2 3 4
8 7 6 5 4
2 2 12 14 13
a
R1 ) R2 ) R3 ) R4 ) R5 ) 1, R6 ) 2.
The permutation groups are
Pi ) P(D2d,Di) ) P(C2,Di)
Table 7a WiXjYkZl
a
AND
isomer no.
i
j
k
l
C3V
C3
1 1 1
0 0 1
0 1 1
3 2 1
12 36 12
12 36 24
R1 ) 3, R2 ) 4.
Pj ) P(D2d,Dj)
i ) 1, 2
j ) 3,4,5
(56)
The cycle index of these permutation groups is obtained from Tables 1 and 2 of ref 1 and is 2 1 + xi2 ) Z(Pi,2) ) 1/2(xi1
R2
R3
R4
R5
R6
NI(P(D2d),24)
1 2 1 1 2 2 2 1 2 2 ‚‚‚
1 1 1 1 2 1 1 1 2 2 ‚‚‚
1 1 2 1 1 2 1 1 2 2 ‚‚‚
1 1 1 1 1 1 1 1 1 1 ‚‚‚
1 1 1 1 1 1 1 2 1 1 ‚‚‚
1 1 1 2 1 1 2 2 1 2 ‚‚‚
1 3 6 43 9 18 129 258 54 2322 ‚‚‚
R2 1 3 ) + 2 2 (2R + R3) NI(P(C),4) 2 2 2
and Table 6 indicate that there are stereoisomers for R2 g 3. In Table 7, the enumeration of different isomers in the case of R1 ) 3, R2 ) 4 is reported. The table shows that the 12 stereoisomers are all of WXYZ type. 4. Molecule with D2d Symmetry.
ˆ 2,2C ˆ ′2,2σd} D2d ) {Eˆ ,2Sˆ 4,C ˆ 2x,C ˆ 2y,C ˆ 2z} D2 ) {Eˆ ,C Adamantane-2,6-dione (C10H12O2). The D2d symmetry of this molecule gives six G-orbits, no(D2d) ) 6, two twodimensional generated by the carbonyl groups, three fourdimensional, and the other eight-dimensional. The last is due to eight equivalent hydrogen atoms. The molecular domain D is then decomposed in such a way
i ) 1, 2
(57)
4 2 1 2 1 + 3xj2 + 2xj4 + 2xj1 xj2) Z(Pj,4) ) 1/8(xj1
j ) 3, 4, 5 (58) Z(P6,8) )
/8(x861
1
+
5x462
+ 2x264)
(59)
The total cycle index is
Z(P(D2d),24) )
In Table 6 we have compared some NI values of both symmetries. The ratio
NI(P(C3V),4)
(55)
P6 ) P(D2d,D6)
Table 8 R1
(54)
2
1
5
2 1 4 2 [∏(xi1 + xi2 )][∏(xj1 + 3xj2 +
j)3 2284 i)1 1 2 1 8 2xj4 + 2xj1 xj2)][x61 + 5x462 + 2x264] (60)
and the series counting isomers and the numbers of isomers have the expressions
CI(P(D2d),24) )
1
2
2284
1 1 1 2 {∏[(wi1 + wi2 + ... + wiR ) + i i)1
5
2 (wi1
+
2 wi2
+ ... +
2 wiR )]}{ i
∏[(wj11 + wj21 + ... + wjR1 )4 + j
j)3
2 2 4 4 2 2 4 + wj2 + ... + wjR ) + 2(wj1 + wj2 + ... + wjR )+ 3(wj1 j j 1 2 2 1 1 2 + wj2 + ... + wjR ) (wj1 + wj2 + ... + 2(wj1 j 2 1 8 )]}{(w161 + w162 + ... + w6R ) + 5(w261 + w262 + ... + wjR j 6 2 4 4 2 ) + 2(w461 + w462 + ... + w6R ) } (61) w6R 6 6
NI(P(D2d),24) )
1
2
5
{∏(R2i + Ri)∏(R4j + 3R2j +
j)3 2284 i)1 2R1j + 2R3j )(R86 + 5R46 + 2R26)} (62)
In Table 8 selected total numbers of isomers (NI) are reported, while in Table 9 the explicit enumeration of isomers for R1 ) R2 ) R3 ) R4 ) R5 ) 1 and R6 ) 2 is presented. With the D2 point group we obtain no(D2) ) 8, and the total domain is thus decomposed,
REGARDING ENUMERATION
OF
MOLECULAR ISOMERS
J. Chem. Inf. Comput. Sci., Vol. 40, No. 2, 2000 391 Table 10
The permutation groups are
Pi ) P(D2,Di) ) P(C2,Di) Pj ) P(D2,Dj)
i ) 1, 2, 3, 4
j ) 5, 6, 7, 8
and using the formula
∑
Z(D2,m) ) 1/4(xm1 +
2k) (m-n(C2k))/2 xn(C x2 ) 1
(64)
k)x,y,z
R1
R2
R3
R4
R5
R6
R7
R8
NI(P(D2),24)
1 2 1 3 1 4 1 2 2 1 ‚‚‚ 2 2 ‚‚‚ 2 ‚‚‚
1 1 1 1 1 1 1 2 1 1 ‚‚‚ 2 2 ‚‚‚ 2 ‚‚‚
1 1 1 1 1 1 1 1 1 1 ‚‚‚ 2 1 ‚‚‚ 2 ‚‚‚
1 1 1 1 1 1 1 1 1 1 ‚‚‚ 1 1 ‚‚‚ 1 ‚‚‚
1 1 1 1 1 1 1 1 1 1 ‚‚‚ 1 1 ‚‚‚ 1 ‚‚‚
1 1 1 1 1 1 1 1 1 1 ‚‚‚ 1 2 ‚‚‚ 1 ‚‚‚
1 1 2 1 3 1 4 1 2 2 ‚‚‚ 1 1 ‚‚‚ 2 ‚‚‚
1 1 1 1 1 1 1 1 1 2 ‚‚‚ 1 1 ‚‚‚ 2 ‚‚‚
1 3 9 6 36 10 100 9 27 81 ‚‚‚ 27 81 ‚‚‚ 2187 ‚‚‚
Table 11a QhRiSjTkUlVmWnXoYpZq
one obtain 4 2 1 2 + 2xj1 xj2 + xj2 ) Z(P(D2,Dj),4) ) 1/4(xj1
j ) 5, ..., 8 (65)
Because
h
i
j
k
l
m
n
o
p
q
isomer no.
2 2 2 2 2 2
2 2 2 2 2 2
2 2 2 2 2 2
2 2 2 2 2 2
4 4 4 4 4 4
4 4 4 4 4 4
4 4 4 3 3 2
0 0 0 1 1 2
4 3 2 3 2 2
0 1 2 1 2 2
4 16 12 16 24 9
a
2 1 + xi2 ) Z(P(D2,Di),2) ) 1/2(xi1
i ) 1, ..., 4 (66)
Ri ) 1 (i ) 1, 2, ..., 6); R7 ) R8 ) 2.
molecular domain is thus decomposed
we have
Z(P(D2),24) )
1 1
4
24 44 i)1
NI(P(D2),24) )
1 1 4
2 4 CI(P(D2),24) )
8
∏(xi12 + xi21)∏(xj14 + 2xj12 xj21 + xj22)
1
j)5
4
(67) 8
∏(R2i + Ri)∏(R4j + 2R3j + R2j )
4 i)1
j)5
From Table 1 and 2 of ref 1, the cycle index of the permutation groups
(68)
P1 ) P(D2h,D1) ) P(C2,D1)
4
∏
2444 i)1
[(w1i1
+
w1i2
+ ... +
1 2 wiR ) i
+
8
2 1 4 )]∏[(w1j1 + w1j2 + ... + wjR ) + (w2i1 + w2i2 + ... + wiR i j
Pj ) P(D2h,Dj)
(j ) 2, 3, ..., 6)
(71) (72)
are
j)5
Z(P1,2) ) 1/2(x211 + x112)
1 2 2 ) (w2j1 + w2j2 + ... + wjR )+ 2(w1j1 + w1j2 + ... + wjR j j
4 + 3x2j2) Z(Pj,4) ) 1/4(xj1
2 2 ) ] (69) (w2j1 + w2j2 + ... + wjR j
j ) 2, 3, ..., 6
(73) (74)
Then Table 10 shows some NI values calculated with eq 68, while Table 11 reports the enumeration of different isomers in the case of Ri ) 1 (i ) 1, 2, ..., 6) and R7 ) R8 ) 2, where we have 38 stereoisomers. (5) Molecule with D2h Symmetry.
ˆ 2x,C ˆ 2y,C ˆ 2z,ıˆ,σˆ xy,σˆ xz,σˆ yz} D2h ) {Eˆ ,C ˆ 2x,C ˆ 2y,C ˆ 2z} D2 ) {Eˆ ,C
Z(P(D2h),22) ) CI(P(D2h),22) )
1 1
1 1
6
4 2 (x211 + x112)∏(xj1 + 3xj2 ) (75)
2 45
j)2
1 2 [(w111 + w112 + ... + w1R ) + 1
2 45
6
1 1 2 1 4 )]∏[(wj1 + wj2 + ... + wjR ) + (w211 + w212 + ... + w1R 1 j j)2
Dibenzodioxin (C12H8O2). Using eq 3 and the D2h point group, for this compound we obtain six G-orbits, no(D2h) ) 6, thus composed: the two oxygens form one G-orbit, the twelve carbon atoms three G-orbits of length four, and the eight hydrogen two G-orbits of four elements. The total
2 2 2 2 + wj2 + ... + wjR ) ] (76) 3(wj1 j
NI(P(D2h),22) )
1 1 2 45
6
[(R21 + R11)∏(R4j + 3R2j )] j)2
(77)
392 J. Chem. Inf. Comput. Sci., Vol. 40, No. 2, 2000
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Table 12 R1
R2
R3
R4
R5
R6
NI(P(D2h),22)
1 2 1 2 1 2 1 2 1 2 1 2 3 1 4 1 ‚‚‚
1 1 2 2 2 2 2 2 2 2 2 2 1 3 1 4 ‚‚‚
1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 ‚‚‚
1 1 1 1 1 1 2 2 2 2 2 2 1 1 1 1 ‚‚‚
1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 ‚‚‚
1 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 ‚‚‚
1 2 7 21 49 147 343 1029 2401 7203 16807 50421 6 27 10 76 ‚‚‚
Table 13a
Pi ) P(D6h,Di)
i ) 1, 2
(79)
Pj ) P(D6h,Dj)
j ) 3, 4
(80)
are 6 3 2 1 2 2 + 4xi2 + 2xi3 + 2xi6 + 3xi1 xi2) Z(Pi,6) ) 1/12(xi1 i ) 1, 2 (81) 12 6 4 2 + 7xj2 + 2xj3 + 2xj6 ) Z(Pj,12) ) 1/12(xj1
SiTjUkVlWmXnYoZp i
j
k
l
m
n
o
p
isomer no.
2 2 2 2 2 2
4 4 4 4 4 4
4 4 4 4 4 4
4 4 4 4 4 4
0 0 0 1 1 2
4 4 4 3 3 2
0 1 2 1 2 2
4 3 2 3 2 2
4 8 12 4 12 9
a
From Table 2 of ref 1, the cycle index of the permutation groups
j ) 3, 4 (82)
The expression for the total cycle index, the series counting isomers, and the number of isomers is 2
6 3 2 1 + 4xi2 + 2xi3 + 2xi6 + Z(P(D6h),36) ) 1/124∏(xi1 i)1
4
2 2 12 6 4 2 3xi1 xi2)∏(xj1 + 7xj2 + 2xj3 + 2xj6 ) (83)
R1 ) R2 ) R3 ) R4 ) 1; R5 ) R6 ) 2.
j)3
Table 14
2
R1
R2
R3
R4
NI(P(D6h),36)
1 2 1 3 1 4 1 2 2 1 2 1 2 ‚‚‚
1 1 1 1 1 1 1 2 1 1 2 2 2 ‚‚‚
1 1 2 1 3 1 4 1 2 2 2 2 2 ‚‚‚
1 1 1 1 1 1 1 1 1 2 1 2 2 ‚‚‚
1 13 382 92 44727 430 1400536 169 4966 145924 64558 1897012 24661156 ‚‚‚
Some total numbers of isomers, computed using eq 77, are reported in Table 12, while Table 13 gives the enumeration of the different isomers when R1 ) R2 ) R3 ) R4 ) 1 and R5 ) R6 ) 2. With respect to previous results obtained with D2h symmetry on dibenzodioxin, no change is obtained with D2 symmetry. There are no stereoisomers in this structure as is evident from planar geometry. (6) Molecule with D6h Symmetry.
D6h ) {Eˆ ,2C ˆ 6,2C ˆ 3,C ˆ 2,3C ˆ ′2,3C ˆ ′′2,ıˆ,2Sˆ 3,2Sˆ 6,σˆ h,3σˆ d,3σˆ V} D6 ) {Eˆ ,2C ˆ 6,2C ˆ 3,C ˆ 2,3C ˆ ′2,3C ˆ ′′2} C ˆ oronene (C24H12). The D6h symmetry of this molecule gives four G-orbits, no(D6h) ) 4. The twelve hydrogen atoms give one G-orbit, while the twenty-four carbon atoms give two G-orbits of six elements and one of twelve elements. The molecular domain D is then decomposed in such a way
1 6 ) + CI(P(D6h),36) ) 1/124∏[(w1i1 + w1i2 + ... + wiR i
4(w2i1
+
w2i2
+ ... +
i)1 2 3 wiR ) i
3 2 + 2(w3i1 + w3i2 + ... + wiR ) + 1
6 1 2 ) + 3(w1i1 + w1i2 + ... + wiR ) 2(w6i1 + w6i2 + ... + wiR i i 4
2 2 1 12 (w2i1 + w2i2 + ... + wiR ) ] ∏[(w1j1 + w1j2 + ... + wjR ) + i j j)3
2 6 3 4 ) + 2(w3j1 + w3j2 + ... + wjR ) + 7(w2j1 + w2j2 + ... + wjR j j 6 2 ) ] (84) 2(w6j1 + w6j2 + ... + wjR j 2
NI(P(D6h),36) ) 1/124∏(R6i + 4R3i + 2R2i + 2R1i + i)1
4
6 4 2 3R4i )∏(R12 j + 7Rj + 2Rj + 2Rj ) (85) j)3
In Table 14 are reported examples of total numbers of isomers (NI) calculated with eq 85, while Table 15 reports the number of different isomers obtained using eq 84 with Rk ) 1 (k ) 1, 2, 3) and R4 ) 4. With respect to previous results obtained with D6h symmetry on coronene, no change is obtained with D6 symmetry. There are no stereoisomers in this structure as is evident from planar geometry. (7) Molecules Having Tetrahedral Symmetry.
ˆ 2,8C ˆ 3,6Sˆ 4,6σˆ } Td ) {Eˆ ,3C ˆ 3,4C ˆ 23} T ) {Eˆ ,3C ˆ 2,4C Molecules with Molecular Formula AB4 (CH4, SiF4, MnO4-, ...). Using both symmetries one obtains no(Td) )
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MOLECULAR ISOMERS
J. Chem. Inf. Comput. Sci., Vol. 40, No. 2, 2000 393
Table 15a
Table 17a ViWjXkYlZm
a
ViWjXkYlZm
isomer no.
i
j
k
l
m
isomer no.
i
j
k
l
m
T
Td
6 6 6 6 6 6 6
6 6 6 6 6 6 6
6 6 6 6 6 6 6
0 1 2 3 4 5 6
12 11 10 9 8 7 6
2 2 18 38 100 132 90
1 1 1 1 1
4 3 2 2 1
0 1 2 1 1
0 0 0 1 1
0 0 0 0 1
4 12 6 12 2
4 12 6 12 1
a
Rk ) 1 (k ) 1, 2, 3); R4 ) 4.
R1 ) 1; R2 ) 4.
and the formulas for the total number of isomers and the series counting isomers are
Table 16 R1
R2
NI(P(T),5)
NI(P(Td),5)
1 2 1 2 3 1 3 2 3 4 1 4 2 4 3 4 ‚‚‚ 5 ‚‚‚
1 1 2 2 1 3 2 3 3 1 4 2 4 3 4 4 ‚‚‚ 5 ‚‚‚
1 2 5 10 3 15 15 30 45 4 36 20 72 60 108 144 ‚‚‚ 375 ‚‚‚
1 2 5 10 3 15 15 30 45 4 35 20 70 60 105 140 ‚‚‚ 350 ‚‚‚
NI(P(Td),5) ) 1/24R1(R42 + 6R32 + 11R22 + 6R2) NI(P(T),5) ) 1/12R1(R42 + 11R22)
(92) (92′)
1 )[(w121 + w122 + CI(P(Td),5) ) 1/24(w111 + w112 + ... + w1R 1 1 4 2 2 ) + 3(w221 + w222 + ... + w2R ) + ... + w2R 2 2 4 1 ) + 8(w121 + w122 + ... + w2R )× 6(w421 + w422 + ... + w2R 2 2 3 1 2 ) + 6(w121 + w122 + ... + w2R ) × (w321 + w322 + ... + w2R 2 2 2 )] (93) (w221 + w222 + ... + w2R 2 1 CI(P(T),5) ) 1/12(w111 + w112 + ... + w1R )[(w121 + w122 + 1 1 4 2 2 ) + 3(w221 + w222 + ... + w2R ) + ... + w2R 2 2 1 3 )(w321 + w322 + ... + w2R )] (93′) 8(w121 + w122 + ... + w2R 2 2
no(T) ) 2, and thus
From Tables 1 and 2 of ref 1, the cycle index of the permutation groups
P1 ) P(Td,D1) ) P(T,D1) ) P(C1,D1)
(87)
P2(Td) ) P(Td,D2)
(88)
P2(T) ) P(T,D2)
(88′)
Z(P1,1) ) x111
(89)
are
The comparison of some total numbers of isomers (NI) obtained using eqs 92 and 92′ is reported in Table 16. This table and the ratio NI(P(Td),5)/NI(P(T),5) show that stereoisomers begin to appear for R2 g 4. The enumeration of different isomers for R1 ) 1 and R2 ) 4 is given in Table 17 for both symmetries. There is one stereoisomer of VWXYZ type. 8. Molecules Having Octahedral Symmetry.
ˆ 3,6C ˆ 4,3C ˆ 24,6C ˆ 2,ıˆ,8Sˆ 6,6Sˆ 4,3σˆ h,6σˆ d} Oh ) {Eˆ ,8C ˆ 4,3C ˆ 24,6C ˆ 2} O ) {Eˆ ,8C ˆ 3,6C (8.a) Molecules with Molecular Formula AB6 (SF6, ...). Using eq 3, we obtain no(Oh) ) no(O) ) 2, the atom A gives one G-orbit, and the atoms B the other G-orbit of length six. The molecular domain D is thus decomposed
Z(P2(Td),4) ) 1
/24(x421 + 3x222 + 6x124 + 8x121 x123 + 6x221 x122) (90)
Z(P2(T),4) ) 1/12(x421 + 8x121 x123 + 3x222)
(90′)
Because
The cycle indices of the total permutation groups are
Z(P(Td),5) ) 1
/24x111(x421
P1 ) P(Oh,D1) ) P(O,D1) ) P(C1,D1)
(95)
Z(P1,1) ) x111
(96)
then
+
3x222
Z(P(T),5) )
+
6x124
+
x111(1/12)(x421
8x121
+
x123
8x121
+
x123
6x221
+
x122)
3x222)
(91) (91′)
On the other hand,
394 J. Chem. Inf. Comput. Sci., Vol. 40, No. 2, 2000
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P2(Oh) ) P(Oh,D2)
(97)
P2(O) ) P(O,D2)
(98)
Z(P2(O),6) ) /24(x621 + 6x221 x124 + 3x221 x222 + 8x223 + 6x322) (99)
Z(P2(Oh),6) ) 1/48[x621 + 7x322 + 8(x126 + x223) + 6(x221 x124 + x122 x124) + 9x221 x222 + 3x421 x122] (100) then
Z(P(O),7) ) 1 /24x111(x621 + 6x221 x124 + 3x221 x222 + 8x223 + 6x322) (101) 6(x221 x124 + x122 x124) + 9x221 x222 + 3x421 x122] (102) 1 )[(w121 + w122 + CI(P(O),7) ) 1/24(w111 + w112 + ... + w1R 1 1 6 2 3 ) + 6(w221 + w222 + ... + w2R ) + 8(w321 + ... + w2R 2 2 3 2 1 2 4 ) + 6(w121 + w122 + ... + w2R ) (w21 + w322 + ... + w2R 2 2 4 1 2 2 ) + 3(w121 + w122 + ... + w2R ) (w21 + w422 + ... + w2R 2 2 2 2 ) ] (103) w222 + ... + w2R 2 1 CI(P(Oh),7 ) 1/48(w111 + w112 + ... + w1R )[(w121 + w122 + 1
... +
w622
+
+ ... +
7(w221
6 w2R ) 2
+
+
w222
(w321
+ ... + +
w322
2 3 w2R ) 2
+ ... +
+
8((w621
3 2 w2R )) 2
+
+
1 2 4 4 ) (w21 + w422 + ... + w2R )+ 6((w121 + w122 + ... + w2R 2 2 2 4 )(w421 + w422 + ... + w2R )) + (w221 + w222 + ... + w2R 2 2 1 2 2 2 2 ) (w21 + w222 + ... + w2R ) + 9(w121 + w122 + ... + w2R 2 2 1 4 2 2 ) (w21 + w222 + ... + w2R )] 3(w121 + w122 + ... + w2R 2 2
(104) NI(P(O),7) )
/24R1R22[R42
1
+
3R22
+
12R12
+ 8]
R1
R2
NI(P(O),7)
NI(P(Oh),7)
1 1 2 2 3 1 3 2 3 4 1 4 2 4 3 4 ‚‚‚ 4 ‚‚‚
1 2 1 2 1 3 2 3 3 1 4 2 4 3 4 4 ‚‚‚ 6 ‚‚‚
1 10 2 20 3 57 30 114 171 4 240 40 480 228 720 960 ‚‚‚ 8904 ‚‚‚
1 10 2 20 3 56 30 112 168 4 220 40 440 224 660 880 ‚‚‚ 7084 ‚‚‚
Table 19a
Z(P(Oh),7) ) 1/48x111[x621 + 7x322 + 8(x126 + x223) +
1 6 ) w2R 2
VANOSSI
Table 18
and from Tables 1 and 2 of ref 1, it follows
1
AND
ViWjXkYlZm
isomer no.
i
j
k
l
m
O
Oh
1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 1 0 1
0 0 0 0 1 1 1 2 1
0 1 2 3 1 2 1 2 2
6 5 4 3 4 3 3 2 2
4 12 24 12 24 72 20 24 48
4 12 24 12 24 72 16 20 36
a
R1 ) 1; R2 ) 4.
Table 20 R1
R2
NI(P(Oh),16)
NI(P(O),16)
1 1 1 1 1 1 ‚‚‚ 2 2 2 2 ‚‚‚
1 2 3 4 5 6 ‚‚‚ 2 3 4 5 ‚‚‚
1 22 267 1996 10375 41406 ‚‚‚ 484 5874 43912 228250 ‚‚‚
1 23 333 2916 16725 70911 ‚‚‚ 529 7659 67068 384675 ‚‚‚
(105)
NI(P(Oh),7) ) /48R1[R62 + 3R52 + 9R42 + 13R32 + 14R22 + 8R12] (106)
1
From eqs 105 and 106 one obtains the data reported in Table 18 for selected total numbers of isomer (NI). The table and the ratio NI(P(Oh),7)/NI(P(O),7) show that the stereoisomers appear for R2 g 3. Table 19 gives the enumeration of different isomers in the case of R1 ) 1 and R2 ) 4. The table shows that of twenty stereoisomers four are of VWXYZ3 type, four of VX2Y2Z2 type, and twelve of VWXY2Z2 type. (8.b) Cubane (C8H8). An interesting molecule with Oh symmetry is cubane. From eq 2 we obtain two G-orbits of length six for both point groups, no(Oh) ) no(O) ) 2, the eight hydrogen atoms give one G-orbit, and the eight carbon atoms the other G-orbit. The molecular domain is
Because
Pi(Oh) ) P(Oh,Di) Pi(O) ) P(O,Di)
i ) 1, 2 i ) 1, 2
(108) (108′)
from Tables 1 and 2 of ref 1, it follows 8 4 1 1 2 2 + 13xi2 + 8(xi2 xi6 + xi1 xi3) + Z(Pi(Oh),8) ) 1/48[xi1 2 4 2 12xi4 + 6xi1 xi2]
i ) 1, 2 (109)
8 4 2 2 2 + 9xi2 + 8xi1 xi3 + 6xi4 ] Z(Pi(O),8) ) 1/24[xi1
i ) 1, 2 (109′)
The formulas for the total cycle index, the counting isomers,
REGARDING ENUMERATION
OF
MOLECULAR ISOMERS
J. Chem. Inf. Comput. Sci., Vol. 40, No. 2, 2000 395
Table 21a
Table 23a WiXjYkZl
a
isomer no.
WiXjYkZl
isomer no.
i
j
k
l
Oh
O
i
j
k
l
Ih
I
8 8 8 8 8 8 8 8 8 8
8 7 6 6 5 5 4 4 4 3
0 1 2 1 3 2 4 3 2 3
0 0 0 1 0 1 0 1 2 2
3 6 18 9 18 36 18 60 48 51
3 6 18 9 18 42 21 78 66 72
12 12 12 12 12 12 12
0 0 0 0 0 0 0
12 11 10 3 4 5 6
0 1 2 9 8 7 6
2 2 6 10 20 24 18
2 2 6 10 24 28 24
R1 ) 1; R2 ) 3.
a
R1 ) 1; R2 ) 2′.
Table 24
Table 22 R1
R2
NI(P(I),7)
NI(P(Ih),7)
1 1 2 2 3 ‚‚‚
1 2 2 3 3 ‚‚‚
1 96 9216 873504 82791801 ‚‚‚
1 82 6724 441078 28933641 ‚‚‚
R
NI(P,60)
1(Ih) 2(Ih) 3(Ih) ‚‚‚ 1(I) 2(I) 3(I) ‚‚‚
1 9 607 679 885 269 312 353 259 652 293 727 442 874 919 719 ‚‚‚ 1 19 215 358 678 900 736 706 519 304 586 988 199 183 738 259 ‚‚‚
Table 25
and the number of isomers are
C60-nYn
2
8 4 1 1 2 2 Z(P(Oh),16) ) 1/482∏[xi1 + 13xi2 + 8(xi2 xi6 + xi1 xi3) + i)1
2 4 2 + 6xi1 xi2] (110) 12xi4 2
8 4 2 2 2 + 9xi2 + 8xi1 xi3 + 6xi4 ] (111) Z(P(O),16) ) 1/242∏[xi1 i)1 2
1 8 ) + CI(P(Oh),16) ) 1/482∏[(w1i1 + w1i2 + ... + wiR i i)1
13(w2i1
+
w2i2
2 4 2 + ... + wiR ) + 8((w2i1 + w2i2 + ... + wiR ) i i
6 ) + (w1i1 + w1i2 + ... + (w6i1 + w6i2 + ... + wiR i 1 2 3 3 2 ) (wi1 + w3i2 + ... + wiR ) ) + 12(w4i1 + w4i2 + ... + wiR i i 4 2 1 4 2 ) + 6(w1i1 + w1i2 + ... + wiR ) (wi1 + w2i2 + ... + wiR i i 2 2 ) ] (112) wiR i 2
1 8 ) + CI(P(O),16) ) 1/242∏[(w1i1 + w1i2 + ... + wiR i i)1
9(w2i1
+
w2i2
2 4 1 2 + ... + wiR ) + 8(w1i1 + w1i2 + ... + wiR ) i i
3 2 4 2 (w3i1 + w3i2 + ... + wiR ) + 6(w4i1 + w4i2 + ... + wiR )] i i
(113)
isomer no.
60-n
n
Ih
I
60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 15 26 27 28 29 30
1 1 23 303 4 190 45 718 418 470 3 220 218 21 330 558 123 204 921 628 330 629 2 855 893 755 11 661 527 055 43 057 432 740 144 549 869 700 443 284 859 624 1 246 738 569 480 3 226 849 468 425 7 708 584 971 055 17 040 023 323 785 34 932 048 763 560 66 537 224 405 790 117 952 355 252 550 194 877 787 472 550 300 436 595 453 640 432 628 675 734 195 582 384 767 014 701 733 373 386 161 407 864 332 935 668 892 953 746 664 302 456 985 538 239 868 528
1 1 37 577 8 236 91 030 835 476 6 436 782 42 650 532 246 386 091 1 256 602 779 5 711 668 755 23 322 797 475 86 114 390 460 289 098 819 780 886 568 158 468 2 493 474 394 140 6 453 694 644 705 15 417 163 018 725 34 080 036 632 565 69 864 082 608 210 133 074 428 781 570 235 904 682 814 710 389 755 540 347 810 600 873 146 368 170 865 257 299 572 455 1 164 769 471 671 687 1 466 746 704 458 899 1 728 665 795 116 244 1 907 493 251 046 152 1 971 076 398 255 692
2
NI(P(Oh),16) ) 1/482∏[R8i + 20R2i + 21R4i + 6R6i ] i)1
(114) 2
NI(P(O),16) ) 1/242∏[R8i + 6R2i + 17R4i ] (115) i)1
Some results obtained using eqs 114 and 115 are presented
in Table 20. The table and the ratio NI(P(Oh),16)/NI(P(O),16) show that stereoisomers already begin to appear from R1 or R2 equal to 2. Table 21 indicates the enumeration of isomers of cubane for R1 ) 2 and R2 ) 3. The number and the type of stereoisomers indicated in Table 21 are as follows: 6 W8X5Y2Z, 3 W8X4Y4, 18 W8X4Y3Z, 18 W8X4Y2Z2, 21 W8X3Y3Z2.
396 J. Chem. Inf. Comput. Sci., Vol. 40, No. 2, 2000
BARALDI
AND
VANOSSI
Table 26 C60-j-kYjZk
isomer no.
60 - j - k
j
k
Ih
I
60 58 57 56 56 55 55 54 54 54 53 53 53 52 52 52 52 51 51 51 51 50 50 50 50 50 49 49 49 49 49 48 48 48 48 48 ‚‚‚ 47 47 47 47 47 ‚‚‚ 46 46 46 46 46 ‚‚‚ 45 45 45 45 45 ‚‚‚ 44 44 44 44 44 ‚‚‚ 43 43 43 43 43 ‚‚‚ 42 42 42 42 42 ‚‚‚
0 1 2 3 2 4 3 5 4 3 6 5 4 7 6 5 4 8 7 6 5 9 8 7 6 5 10 9 8 7 6 11 10 9 8 7 ‚‚‚ 12 11 10 9 8 ‚‚‚ 13 12 11 10 9 ‚‚‚ 14 13 12 11 10 ‚‚‚ 15 14 13 12 11 ‚‚‚ 16 15 14 13 12 ‚‚‚ 17 16 15 14 13 ‚‚‚
0 1 1 1 2 1 2 1 2 3 1 2 3 1 2 3 4 1 2 3 4 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 ‚‚‚ 1 2 3 4 5 ‚‚‚ 1 2 3 4 5 ‚‚‚ 1 2 3 4 5 ‚‚‚ 1 2 3 4 5 ‚‚‚ 1 2 3 4 5 ‚‚‚ 1 2 3 4 5 ‚‚‚
1 31 871 16 297 24 635 227 794 455 560 2 503 774 6 261 710 8 345 202 22 530 942 67 592 070 112 649 922 170 579 826 597 046 590 1 194 038 370 1 492 581 510 1 108 750 851 4 434 993 576 10 348 257 762 15 522 380 874 6 282 867 981 28 272 997 233 75 394 156 968 131 940 023 670 158 327 652 018 31 414 258 005 157 071 208 125 471 213 051 075 942 426 020 250 1 319 396 199 030 139 936 042 155 769 648 590 165 2 565 492 007 305 5 772 358 314 450 9 235 769 953 410 ‚‚‚ 559 743 873 780 3 358 462 751 280 12 314 359 752 240 30 785 898 643 500 55 414 514 806 460 ‚‚‚ 2 023 688 790 060 13 153 978 191 900 52 615 896 256 560 144 693 719 816 100 289 387 418 354 580 ‚‚‚ 6 649 262 306 220 46 544 833 883 100 201 694 262 591 820 605 082 783 215 820 1 331 182 104 087 240 ‚‚‚ 19 947 785 411 700 149 608 392 875 100 698 172 434 405 460 2 269 060 427 169 900 5 445 744 922 357 740 ‚‚‚ 54 856 407 810 105 438 851 254 192 560 2 194 256 204 656 560 7 679 896 695 577 260 19 967 731 315 672 140 ‚‚‚ 138 754 440 115 335 1 179 412 744 162 455 6 290 201 089 111 320 23 588 254 119 543 300 66 047 111 141 027 940 ‚‚‚
1 59 1711 32 509 48 981 455 126 910 252 5 006 386 12 519 010 16 688 080 45 057 474 135 172 422 225 287 370 341 149 446 1 194 050 466 2 388 046 122 2 985 098 760 2 217 471 399 8 869 885 596 20 696 400 864 31 044 599 586 12 565 671 261 56 545 698 807 150 788 055 132 263 879 452 746 316 654 915 830 62 828 356 305 314 141 781 525 942 425 344 575 1 884 850 689 150 2 638 790 964 810 279 871 768 995 1 539 295 620 135 5 130 982 438 035 11 544 712 697 700 18 471 536 753 670 ‚‚‚ 1 119 487 075 980 6 716 922 455 880 24 628 715 671 560 61 571 789 178 900 110 829 220 522 020 ‚‚‚ 4 047 376 351 620 26 307 949 848 180 105 231 785 142 120 289 387 419 828 780 578 774 818 281 660 ‚‚‚ 13 298 522 298 180 93 089 656 087 260 403 388 509 737 300 1 210 165 529 134 380 2 662 364 164 095 900 ‚‚‚ 39 895 566 894 540 299 216 763 414 900 1 396 344 841 308 900 4 538 120 775 224 400 10 891 489 762 209 420 ‚‚‚ 109 712 808 959 985 877 702 471 679 880 4 388 512 356 399 400 15 359 793 254 397 900 39 935 462 461 434 540 ‚‚‚ 277 508 869 722 315 2 358 825 424 830 765 12 580 402 094 155 800 47 176 507 981 557 900 132 094 221 987 821 940 ‚‚‚
REGARDING ENUMERATION
OF
MOLECULAR ISOMERS
J. Chem. Inf. Comput. Sci., Vol. 40, No. 2, 2000 397
Table 26. Continued C60-j-kYjZk
isomer no.
60 - j - k
j
k
Ih
I
41 41 41 41 41 ‚‚‚ 40 40 40 40 40 ‚‚‚ 35 35 35 35 35 ‚‚‚ 30 30 30 30 30 ‚‚‚
18 17 16 15 14 ‚‚‚ 19 18 17 16 15 ‚‚‚ 24 23 22 21 20 ‚‚‚ 29 28 27 26 25 ‚‚‚
1 2 3 4 5 ‚‚‚ 1 2 3 4 5 ‚‚‚ 1 2 3 4 5 ‚‚‚ 1 2 3 4 5 ‚‚‚
323 760 356 124 975 2 913 843 180 259 935 16 511 777 818 410 105 66 047 111 199 045 900 198 141 333 249 029 940 ‚‚‚ 698 640 762 881 985 6 637 087 247 896 875 39 822 522 912 381 825 169 245 722 439 007 830 541 586 310 578 988 840 ‚‚‚ 10 815 716 292 073 875 129 788 595 268 670 520 995 045 895 055 550 520 5 472 752 421 742 555 950 22 985 560 166 027 497 830 ‚‚‚ 29 566 145 470 111 336 428 709 109 093 298 632 4 001 285 010 715 996 492 27 008 673 821 432 285 976 140 445 103 844 530 719 360 ‚‚‚
647 520 696 018 735 5 827 686 264 168 615 33 023 555 496 955 485 132 094 221 987 821 940 396 282 665 963 465 820 ‚‚‚ 1 397 281 501 935 165 13 274 174 343 496 605 79 645 045 610 304 405 338 491 444 181 800 140 1 083 172 620 300 140 700 ‚‚‚ 21 631 432 489 303 455 259 577 189 871 641 460 1 990 091 789 015 917 860 10 945 504 839 587 548 230 45 971 120 326 267 704 150 ‚‚‚ 59 132 290 782 430 712 857 418 216 926 936 024 8 002 570 019 222 905 544 54 017 347 633 822 290 312 280 890 207 674 702 370 360 ‚‚‚ 2
12 6 4 2 Z(P(Ih),24) ) 1/1202∏(xi1 + 16xi2 + 20xi3 + 20xi6 +
9. Molecules with Icosahedral Symmetry.
Ih )
i)1 2 24xi1
{Eˆ ,12C ˆ 5,12C ˆ 25,20C ˆ 3,15C ˆ 2,ıˆ,12Sˆ 10,12Sˆ 210,20Sˆ 6,15σˆ }
2 1 1 4 4 xi5 + 24xi2 xi10 + 15xi1 xi2) (121)
2
ˆ 25,20C ˆ 3,15C ˆ 2} I ) {Eˆ ,12C ˆ 5,12C
12 2 2 4 6 + 24xi1 xi5 + 20xi3 + 15xi2 ) Z(P(I),24) ) 1/602∏(xi1 i)1
(122)
2-
(9.a) Borane Anion (B12H12 ). The application of symmetry operations of Ih and I point group on B12H122- gives
8 6 4 NI(P(Ih),24) ) 1/1202(R12 1 + 15R1 + 16R1 + 44R1 + 8 6 4 2 44R21)(R12 2 + 15R2 + 16R2 + 44R2 + 44R2) (123)
no(Ih) ) no(I) ) 2
6 4 12 6 NI(P(I),24) ) 1/602(R12 1 + 15R1 + 44R1)(R2 + 15R2 +
44R42) (124)
and thus 2
1 12 ) + CI(P(I),24) ) 1/602∏[(w1i1 + w1i2 + ... + wiR i i)1
1 2 5 5 2 ) (wi1 + w5i2 + ... + wiR ) + 24(w1i1 + w1i2 + ... + wiR i i 3 4 2 6 ) + 15(w2i1 + w2i2 + ... + wiR )] 20(w3i1 + w3i2 + ... + wiR i i
(125)
The permutation groups of two domains
Pi(Ih) ) P(Ih,Di) Pi(I) ) P(I,Di)
2
i ) 1, 2
(117)
i ) 1, 2
(118)
1 12 ) + CI(P(Ih),24) ) 1/1202∏[(w1i1 + w1i2 + ... + wiR i i)1
24(w1i1
+
w1i2
1 2 5 5 2 + ... + wiR ) (wi1 + w5i2 + ... + wiR ) + i i
3 4 ) + 16(w2i1 + w2i2 + ... + 20(w3i1 + w3i2 + ... + wiR i
have cycle index
2 6 6 2 ) + 20(w6i1 + w6i2 + ... + wiR ) + 24(w2i1 + w2i2 + wiR i i
12 6 4 2 + 16xi2 + 20xi3 + 20xi6 + Z(Pi(Ih),12) ) 1/120(xi1
10 1 1 2 10 )(w10 ... + wiR i1 + wi2 + ... + wiRi) + 15(wi1 + wi2 + ... + i
i ) 1, 2 (119)
1 4 2 2 4 ) (wi1 + w2i2 + ... + wiR ) ] (126) wiR i i
12 2 2 4 6 + 24xi1 xi5 + 20xi3 + 15xi2 ) Z(Pi(I),12) ) 1/60(xi1 i ) 1, 2 (120)
Table 22 shows selected total numbers of isomers (NI) for borane anion obtained using eqs 123 and 124, while Table 23 shows the enumeration of different isomers that appear for R1 ) 1 and R2 ) 2. The stereoisomers already begin to appear for R1 or R2 equal to 2, and in the example of Table
2 2 1 1 4 4 xi5 + 24xi2 xi10 + 15xi1 xi2) 24xi1
In all we have
398 J. Chem. Inf. Comput. Sci., Vol. 40, No. 2, 2000
BARALDI
23 we found four stereoisomers of W12Y4Z8 type, four of W12Y5Z7 type, and six of W12Y6Z6 type: in all fourteen stereoisomers. (9.b) Fullerene (C60). The importance of having an enumeration of possible heterofullerenes is well-documented.12 Several approaches have been used, and in the case of fullerene C60, to our knowledge, no research has used the cycle index of Ih group. In this section, a comparison of isomers of fullerene C60 using the chiral (I) and achiral (Ih) group of icosahedron is reported. With both symmetries we have only one G-orbit, no(Ih) ) no(I) ) 1, thus one domain
D ) D1
(n1 ) |D1| ) 60)
and the permutation groups
P(Ih) ) P(Ih,D1)
AND
VANOSSI
In Table 24 we have reported some values of NI calculated using eqs 129 and 130. In passing from the Ih point group to the I point group, for R g 2 NI almost doubles, because
(
)
1 1 15R32 + R30 + 20R10 + 24R6 ) ) + NI(P(I),60) 2 2 R60 + 15R30 + 20R20 + 24R12 1 1 + p(R) (133) 2 2
NI(P(Ih),60)
and the fraction p(R) ) 1 for R ) 1 but is negligible with respect to one for R g 2. Table 25 shows the enumeration of C60-nYn type isomers (n ) 0, ..., 30), while Table 26 shows a partial list of the enumeration of C60-j-kYjZk type isomers. The result obtained for C46Y10Z4 can be compared with those reported in ref 12. A small difference is found for the number of isomers corresponding to Ih symmetry.
P(I) ) P(I,D1) REFERENCES AND NOTES
The usual development gives 30 20 12 Z(P(Ih),60) ) 1/120(x60 1 + 16x2 + 20x3 + 24x5 + 6 4 28 20x10 6 + 24x10 + 15x1 x2 ) (127) 30 20 12 Z(P(I),60) ) 1/60(x60 1 + 15x2 + 20x3 + 24x5 )
(128)
NI(P(Ih),60) ) 1/120(R60 + 16R30 + 20R20 + 24R12 + 20R10 + 24R6 + 15R32) (129) NI(P(I),60) ) 1/60(R60 + 15R30 + 20R20 + 24R12) (130) CI(P(I),60) ) 1/60[(w11 + w12 + ... + w1R)60 + 15(w21 + w22 + ... + w2R)30 + 20(w31 + w32 + ... + w3R)20 + 24(w51 + w52 + ... + w5R)12] (131) CI(P(Ih),60) ) 1/120[(w11 + w12 + ... + w1R)60 + 16(w21 + w22 + ... + w2R)30 + 20(w31 + w32 + ... + w3R)20 + 24(w51 + w52 + ... + w5R)12 + 20(w61 + w62 + ... + w6R)10 + 10 10 6 1 1 1 4 24(w10 1 + w2 + ... + wR ) + 15(w1 + w2 + ... + wR)
(w21 + w22 + ... + w2R)28] (132)
(1) Baraldi, I.; Fiori, C.; Vanossi, D. On the cycle index of point groups. J. Math. Chem. 1999, 25, 23-30. (2) Po´lya, G. Kombinatorische Anzahlbestimmungen. fu¨r Gruppen, Graphen und chemische Verbindungen. Acta Math. 1937, 68, 145254. Po´lya, G.; Read, R. C. Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds; Springer: Berlin, 1987. (3) Fowler, P. W. Isomer Counting using Point Group Symmetry. J. Chem. Soc., Faraday Trans. 1995, 91, 2241-2247. (4) Fujita, S. Symmetry and Combinatorial Enumeration in Chemistry; Springer-Verlag: Berlin, 1991. (5) Balaban, A. T. Chemical Applications of Graph Theory; Academic Press: London, 1976. (6) Balasubramanian, K. Applications of Combinatorics and Graph Theory to Spectroscopy and Quantum Chemistry. Chem. ReV. 1985, 85, 599616. (7) Balasubramanian, K. Enumeration of Chiral and Positional Isomers of Substituted Fullerene Cages (C20-C70). J. Phys. Chem. 1993, 97, 6990-6998. (8) Ruch, E.; Ha¨sselbarth, W.; Richter B. Doppelnebenklassen als Klassenbegriff und Nomenklaturprinzip fu¨r Isomere und ihre Abza¨hlung. Theor. Chim. Acta 1970, 19, 288-300. (9) Ha¨sselbarth, W. Substitution symmetry. Theor. Chim. Acta 1985, 67, 339-367. (10) Shao, Y.; Wu, J.; Jiang, Y. Isomer counting and isomer permutation representation. Chem. Phys. Lett. 1996, 248, 366-372. (11) Shao, Y.; Wu, J.; Jiang, Y. Enumeration and Symmetry of Substitution Isomers. J. Phys. Chem. 1996, 100, 15064-15067. (12) Zhang, F.; Li, R.; Lin, G. The enumeration of heterofullerenes. J. Mol. Struct. (THEOCHEM) 1998, 453, 1-6.
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