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Enumeration of staggered conformers of alkanes and monocyclic cycloalkanes
Journal of
MOLECULAR STRUCTURE Journal of Molecular
Enumeration
Structure 445
(I 998) 127- 137
of staggered conformers of alkanes and monocyclic cycloalkanes’
S.J. Cyvina’*, B.N. Cyvin”, J. Brunvolla, Jianji Wangb “Deparrment ofPhysical Chemistv, Norwegian University of Science and Technology, N-7034 Trondheim, Norway hDepartment of Chemistry, Inner Mongolia Normal University, Huhehaote 010022, PR China Received 26 August 1997; revised 13 November
1997; accepted
13 November
1997
Abstract Staggered conformers of alkanes are represented by systems which can be embedded in the diamond lattice. Previous enumerations of alkane systems (acyclic) are supplemented. Original contributions are supplied to the numbers and symmetries of monocyclic staggered cycloalkanes, of which cyclohexane is the key structure. These systems are represented by single cycles on the diamond lattice. In particular, the generation of monocyclic staggered cycloalkanes by successive addition of cyclohexane rings is treated in details. 0 1998 Elsevier Science B.V. Keywords:
Alkane; Cycloalkane;
Enumeration
1. Introduction Structural investigations on alkanes, which by definition are acyclic, go back to classical works in gas electron diffraction on ethane [l-5], they include higher molecules with unbranched carbon skeletons [ 1,6- 1 l] and some molecules with branched carbon skeletons [ 1,12- 171. The unbranched alkanes, namely normal paraffins, constitute the prototype of a homologous series in organic chemistry, and it has been investigated by very many researchers. It is impracticable to give a comprehensive survey of all these works here; they involve thermodynamics, spectroscopy and many other areas. We have restricted the list
* Corresponding author. ’Dedicated to the memory of the late Professor Otto Bastiansen.
of references [18-311 to some works where the diamond lattice is mentioned explicitly. This is a crucial concept in the present work inasmuch as all the considered structures should be embeddable in the diamond lattice. The smallest monocyclic cycloalkane within the staggered conformers under consideration is cyclohexane. This molecule was included in the old investigation by Pauling and Brockway [l], but its structure was discussed thoroughly and determined accurately first by Hassel et al. [32-341. Professor Odd Hassel, who became a Nobel laureate, had been the teacher and collaborator of Professor Otto Bastiansen. Cyclohexane became a popular subject for gas electron diffraction investigations also among other researchers [35-391, and now the ‘chair’ configuration (in contrast to ‘boat’) is firmly established.
0022-2860/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved PII SOO22-2860(97)00419-5
128
S.J. Cyvin et al.Nournal
c?fMoleculur
Strucrurr
445 (1998)
127-137
Table I Generating
functions
for the staggered
Symmetry
Function
conformational
Td
x[ 1 + x’b*(x ‘*)I
T
x’[b(x”)
D z<,
x2b*(xh)
DZ
x[b*(x’)
- 1 - .rJb*(x”)]
x’[b(x’)
- b*(d’)]
Sh C,,
xb*(x’)[
&
$r5[b’(x3)
S4
&r[b(x4) - b*(x’)]
stereoisomers
of (acyclic)
alkane systems belonging
to the different symmetry
groups
- b*(x”)]
$x*[b(x’)
- b*(x’)] bb*(.r’)
+ l] -x
+ 2b(x”)]
C?,
G[b*(x’)b*(x’)
C?h C,
b*(x’)
CZ
$x[b’(x’)
C,
x + &b*(.r)b*(x)[xb(x’)
C,
; + ;x2[b2(x2)
- $rzb*(xh) - .r’b*(x”)
+ x[ I - b*(x4)]
- b*(x4)]
- 1 - x’b*@)
&x’[b’(x’)
- b*(x’)b*(x’)] - b*(x’)b*(x*)]
+ .r[b(?)
- b”(x’)]
+ b(x’) - b*(x’)
- x’[b(d’)
- I] + b*(x) - ;b*(x’)[xb*(.x’)
- b(x”)] - ;[b*(x’)
- b*(d’)]
- x’[b(x12)
- b*(x”)]
- .r[b(x”) - b*(x’)] + I] - +,rb*(x”)[xb*(x’)
+ 21 + $2b*(,$‘)
+ x5b*(x”)
-x%*(x6)]
The enumerations of staggered conformers of alkanes are treated in a previous paper 1401, which should be consulted for the definitions of basic concepts and for previous literature. Especially noteworthy is the work of E. Funck [41], which was characterized as ahead of its time and had been undertaken according to an idea suggested by W. Liittke [42]. Later and independently, the same idea was conceived by Balaban [26]. In these works [26,41] the enumeration problem for the unbranched systems was solved completely. The corresponding problem for all acyclic (branched + unbranched) systems turned out to be far more intricate and was originally posed as an ‘open problem’ by one of the present authors [43]. In response, Wang et al. [44] succeeded to solve the crucial part of the problem, while the details were worked out in collaboration between the research groups from Inner Mongolia [44] and Norway [40]. This, we believe, is a genuine collaboration in the spirit of the late Professor 0. Bastiansen. In the present work, these enumerations of staggered alkanes are supplemented. In addition, a contribution to the enumerations of staggered monocyclic cycloalkanes is provided. In precise terms, these systems should contain exactly one cycle each and no side branches. As far as we know, only one enumeration work of such systems exists in the previous literature [45]. Two supplementary references to alkanes and cycloalkanes are of interest [46,47].
2. Staggered
alkanes
2.1. Classi$cation It is reiterated that the alkane systems under consideration should be embeddable in the diamond lattice. Overlapping and non-overlapping systems are distinguished; in the former case at least two carbon atoms occupy the same site. A subclass of the non-overlapping systems, referred to as ‘unstrained’, is defined in a subsequent section. Branched and unbranched systems refer to the branching of the carbon skeleton. The alkane systems are distributed among sixteen symmetry groups: achiral Td, D,,, D,,, S,, C,,, S,, Czv, C2,,, C,, C;; chiral T, D3, D2, C3, CZ, C,. Two systems are said to be isomorphic if they can be brought into each other by translations and symmetry operations; otherwise they are non-isomorphic. In addition to the total number of non-isomorphic systems (spectra1 isomers) for a given carbon content we are also interested in the number of stereoisomers. In the latter case, the chiral systems are counted double in order to account for the enantiomerit pairs. 2.2. Crucial generating functions In order to express the total number of stereoisomers of the staggered alkanes with a given carbon
S.J. Cyvin et al./Journal of Molecular Structure 445 (1998) 127-137 Table 2 Coefficients
in the expansions
”
Table 2 gives numerical values for the coefhcients the expansions of b(x) and b*(x);
of b(x) and b*(x)
b”
b(x)=
2 12 55 273 1428 7752 43 263 246 675 1430715 8414640 50067 108 300 830 572 I 822 766 520 I I 124755664
6
8 9 10 I1 12 13 14 15
2
12 30 55 143 273 728 1428 3876 1752 21318
content and their distribution into the different symmetry groups, a number of auxiliary generating functions were introduced [40,44]. In the present work it was achieved to express all these numbers in terms of one single generating function, namely b(x). This function is determined by b(x) = 1 +xb’(x)
(1)
From the initial condition b(x) = 1 + . ., one successively obtains by iteration: b(x)=l+x+..., b(x)=l+x+3x*+..., b(x)=1+x+3x2+12x”+..., etc. For the total number of stereoisomers (branched + unbranched, overlapping + non-overlapping) the following was arrived at. I(x)=
- ;+
+b4(x)[l -2xb2(x)] +;b(x*),xb(x2)+2]
- ;xb(x~)[b(x)-3]+
~~*~(x))[h’(~)+h(x’)]
(2)
The expressions for the different symmetry groups (except Cl) are collected in Table 1. The function for C, is obtainable on subtraction from (2). In Table 1, in addition to b(x), the function b*(x) also appears. However, the latter function is given explicitly in terms of b(x) by: b”(x) =
1 1 -xb(x2)
in
b;:
0
4
129
5 b,x”, b*(x)= 5 b;x” ll=ll n=O
(4)
It is observed (Table 2) that b;,, = b,, forp = 0, 1, . . ., 7. We have been able to prove this peculiar phenomenon generally for p = 0, 1, 2, . . . . Modify Eq. (1) to b(x)[l -xb2(x)J = 1
(5)
whereupon
b(x)=
’ 1 -xb*(x)
(6)
emerges. The essence of our proof lies in the similar forms of Eq. (3) and Eq. (6). As a matter of routine, the proof was completed by expansions and multiple summations. 2.3. Computer-generated
supplementary
data
The overlapping alkane systems are not supposed to be realized as chemical counterparts because of severe steric hindrances. Therefore the nonoverlapping systems are more interesting in the chemical context. Since an algebraic solution for their numbers of conformers is not to be expected one has to resort to computer programming [48]. That was also the basis for the previously published numbers of conformers of non-overlapping alkane systems [40]. The list is supplemented in Table 3. The numbers up to n = 14 were produced by an effective exploitation of the constitutional alkane isomers [49]. Furthermore, a coding in analogy with Saunders [45] was employed in the case of the unbranched systems. This method reproduced all the numbers for n 5 14 and made it feasible to extend the list through n = 19 for the unbranched systems. Among the non-overlapping alkane systems there are many members in which the shortest non-bonded carbon-carbon distance is the smallest possible, namely equal to the bond distance (corresponding to the unit distance of the diamond lattice). Because of the hydrogen atoms, severe steric hindrances are also expected in these cases. But there are still more
130
S.J. Cyvin et al.Nournul
of Molecular
Structure
445 (1998)
127-137
of Molecular
S.J. Cyvin et ul.Nournal
Table 4 Numbers of unstrained
5
6
7
x
9
I0 II 12
alkane staggered
conformers
Structure
445 (1998)
(acyclic)
unhranched
0
0
0
0
I
0
0
I
0
0
branched
I
0
0
0
0
0
0
0
I
0
unhranched
0
0
0
0
0
I
0
3
0
I
2
7
I2
hranchrd
0
0
0
0
0
I
0
I
3
0
5
IO
I6
unhranched
0
0
0
0
I
0
0
3
I
0
x
I3
24
branched
0
0
0
0
I
0
I
2
4
0
25
33
61
unhranchrd
0
0
0
0
0
I
0
8
0
3
I9
31
58
branched
0
I
0
0
0
I
I
7
8
3
95
I I6
219
unhranched
0
0
0
0
I
0
0
8
3
0
54
66
I28
branched
0
0
I
1
2
0
0
9
0
363
387
759
unhranched
0
0
0
0
0
I
0
20
0
branched
0
0
0
0
0
3
2
32
21
II
8 I4
unhranched
0
0
0
0
I
0
0
I9
8
0
branched
0
0
0
0
4
0
I
42
37
0
unhranohed
0
0
0
0
0
I
0
49
0
20
branched
0
0
0
0
0
5
0
I30
63
58
unhranchrd
0
0
0
0
I
0
0
47
I9
0
branched
0
0
I
I
7
0
6
I61
127
0
unhranched
0
0
0
0
0
I
Ill
0
49
branched
0
0
0
0
0
8
507
197
226
IS
unhranched
0
0
0
0
I
0
II3
46
0
16
unhranched
0
0
0
0
0
I
283
0
I I9
I7
unhranched
0
0
0
0
I
0
273
106
0
IX
unhranched
0
0
0
0
0
I
677
I9
unhranched
0
0
0
0
I
0
648
13
14
127-137
systems which we wish to eliminate on chemical grounds. Consider two incident edges in the diamond lattice, which are defined by one central and two end vertices. If the two end vertices (at the next-to shortest distance in the diamond lattice) are occupied by carbon atoms, but not the central vertex, then two carbon-hydrogen bonds will point along the two edges in question, and again a substantial (but not so severe) steric hindrance will occur. Systems where the above described situations are not present, are referred to as ‘unstrained’, and the pertinent numbers, which again were obtained by computer programming, are collected in Table 4. Again, the numbers for n 5 14 were obtained for the totals (branched + unbranched systems), while the unbranched systems were enumerated up to n = 19.
0
I59
309
I370
2702
334 4715 804 16832 I984 60281 4777 214939
288
247
I30 I298
0
362
715
4799
9557 I727
x74
3405
17088 205
I
4082
60584
121032
4944
9838 431 329
215880
II615
I I 775
21957
28 360
23 503 56600
67 472
67 852
I35 597
162 276
I63 242
326
390 546
391442
782 636
195
3. Monocyclic staggered cycloalkanes 3.1. Classijcation The systems to be enumerated are single cycles on the diamond lattice. The smallest of these systems corresponds to the ‘chair’ configuration of cyclohexane. Notice that the ‘boat’ form is not embeddable in the diamond lattice. The concepts of overlapping, non-overlapping and unstrained (see above) are immediately applicable to the monocyclic cycloalkanes under consideration. 3.2. Application Saunders
of Saunders’ integer code
[45]
has
identified
each
monocyclic
132
Table
of MoleculurStructure
S.J. Cyvin et aUJournul
445 (1998)
127-137
5
Numbers
of non-overlapping
n
D w
&d
monocyclic D1
Cl,.
staggered Dz
cycloalkancs S,
Cz,
(unbranched) CZh
C?
C?
C,
Spectral
Cl
C,
isomers 6
I
0
0
0
0
0
0
0
0
0
0
0
8
0
I
0
0
0
0
0
0
0
0
0
0
0
IO
0
0
0
0
0
0
0
2
0
I
0
0
0
I2
0
0
0
I
0
0
I
0
0
2
I
0
2
7
I4
0
0
0
0
0
0
0
5
0
6
5
3
I6
35
16
0
2
0
0
0
I
I
0
0
20
I3
0
I8
3
0
I
I
0
0
0
IO
I
48
31
31
20
0
0
0
0
0
0
5
0
0
126
96
22
0
0
0
0
0
0
0
37
0
307
243
24
0
6
0
5
9
8
IO
0
907
617
II
Number n
0
3
II3
150
750
876
4809
226
5036
31263
32 076
204 670
0
206 243
come in between those of our Tables 5 and 6. According to our computations they should be: 1, 1, 1, 4, 13, 51, 239, 1204. Saunders [45] has not given enough details of his results to make it possible to pinpoint errors therein, but we must also stress the reservation about our correct interpretation of Saunders’ definitions.
staggered cycloalkane with n carbon atoms by means of a string of n integers, which indicate the directions in which the carbon-carbon bonds turn around the ring. We have adopted this idea in our lirst computer-generated numbers of non-isomorphic monocyclic staggered cycloalkanes. Table 5 shows the results for the pertinent non-overlapping systems, while the numbers of unstrained systems are listed in Table 6. The distributions into symmetry groups are included. Both agreements and discrepancies were found on comparing our results with those of Saunders [45]. The numbers of spectral isomers of the unstrained systems (Table 6) agree perfectly with the Saunders numbers. However, this author has also reported a set of numbers for n = 6, 8, 10, 12, 14, 16, 18, 20 as 1, 1, 1, 4, 13, 56, 238, 1202, respectively. If we have interpreted these numbers correctly, they include systems with the less severe steric hindrances according to the description above (Section 2.3) while the severe steric hindrances (closest possible proximity of non-bonded carboncarbon atoms) are eliminated. Thus these numbers Table
0
4. Addition of six-membered 4. I. Introductory
rings
remarks
Any polycyclic staggered cycloalkane system consisting of exclusively six-membered (cyclohexane) rings is generated by successive additions of cyclohexane rings within the framework of the diamond lattice. This consideration is very natural in view of the theory of benzenoid hydrocarbons [50]. It is well known that the benzenoids are generated by successive additions of hexagons on the hexagonal lattice, and this property has been successfully exploited in
6 of unstrained
monocyclic
DW
D2d
staggered
cycloalkanes
Ci,.
(unbranched)
D2
S4
Cl,
CZh
C2
CA
C,
Cl
Spectral isomers
6
I
0
0
0
0
0
0
0
0
0
0
1
14
0
0
0
0
0
0
I
0
0
0
0
I
I6
0
I
0
0
0
0
0
0
0
0
0
I8
1
0
0
0
0
0
3
2
I
2
2
20
0
0
0
0
0
I
0
6
3
0
9
I9
22
0
0
0
0
0
0
I
I3
5
I7
78
120
24
0
2
I
I
2
1
0
25
I8
0
369
419
1 II
S.J. Cwin et ul.Nournal Table 7 Specification A
of added cyclohexane
of Molecular Structure 445 (1998) 127-137
rings in terms of the coordinates B
of the diamond
133
lattice
C
D
1 0 0-I -II
0-I
I 0, 0
computerized generations and enumerations of benzenoid systems [50-521. The perimeter of a polycylic cycloalkane system is a single cycle on the diamond lattice and represents therefore a monocyclic cycloalkane system within the class of interest in the present work. However, a systematic addition of cyclohexane rings is a far more complex matter than the addition of benzenoid rings, and also the application to systematic generations of single cycles in the diamond lattice meets with complications. One must realize that such a cycle does not always define uniquely a polycyclic cyclohexane system, in sharp contrast to the situation with benzenoids, which are defined uniquely by their perimeters on the hexagonal lattice [50].
4.2. Algebra of the addition of cyclohexanes Consider the addition of a cyclohexane to a carbon-carbon bond C , -Cz so that (at least) one of the cyclohexane bonds merges with C i-C?. Six possibilities are distinguished according to the six bonds of the added cyclohexane. The aim is to specify the coordinates of the carbon atoms of the added cyclohexane. A previous work [48] should
-2 0 2
0 -I
-2 2
-3I
-2 02
-II
I21 -I 321
0
I
1
be consulted for the precise definition of the integer coordinates in relation to the diamond lattice. Suppose first that Ci is situated in the origin and consequently has the coordinates (0, 0, 0). Then C i -Cz may point in one of the directions of the C-H bonds of methane with C in the origin (or one of the C-C bonds in neopentane). The following four cases are distinguished for the coordinates of C?:
(4 i 1, 1, 11 (B) ( 1, -1,
-1)
(Cl ( -1,
1, -1)
(D) 1-1,
-1,
1)
In Table 7 the complete cyclohexane rings are specified, six in each of the cases A, B, C and D. It remains to consider the general case when the CI-CZ bond is associated with the coordinates (xi, y I, z I ) - {x2, y2, z?}. This problem is easily handled by means of a simple translation: 1. Since C , -C2 are bonded atoms, one of the coordinate triples must necessarily consist of even integers only and the other of odd integers only [46]. Without loss of generality, assume that {x,. yi, z2}
S.J. Cyvin
134
et al./Journal
of Molecular Structure
445
(1998)
127-137
are even numbers, while (x2, y2, z?] are odd. Compute (x1-x1, yZ - yl, z2 - zI]. This is the set of transformed coordinates for atom Cl, while C , has been transformed to origin, (0, 0, 0). 2. Identify the pertinent column of Table 7, wherein the transformed C i-C2 bond corresponds to the two first columns in each coordinate set for the added cyclohexane ring. 3. The added cyclohexane is translated back to the original position by the opposite transformation: if the set of coordinates from Table 7 is denoted (a;, bi, Ci), then the new coordinates are ( ai +.x1, bi+_Yl,
232101
Ci+Zll.
The above procedure is illustrated by an example in the following. For the sake of simplicity, assume the original system to be a six-membered ring (cyclohexane), e.g.: (1, 3, 3)-
12, 2, 4)-
13, 1, 31
I
I
(0, 2, 2k
(1, 1, 1I-
IT
0, 21
Here the coordinates for Cl and C2 are given in heavy numerals. Then C I{ 2, 2,4] and C2( 3, 1, 3) should be chosen according to the above conventions and consistent with Table 7. The transformed coordinates for C2 come out as ( 1, - 1, - 1 ), which identifies column B in Table 7. Then the six added cyclopropanes with C 1-C 2 in the original position are found to be: 234321 (1)
2
I
i 432123
234543
432345
234543
434565
2
3
4
3
1
, An,=0
(6)
2
2
0
I
2
[ 432123
3
1
, discarded
The above example demonstrates three types of addition of a cyclohexane ring. In (l)-(4) the added ring shares exactly one carbon-carbon bond with the initial system. The corresponding coordinates are typed in bold. A monocyclic cycloalkane emerges on deleting this bond. In case (5) three carbons belonging to two incident bonds are shared, and the pertinent monocyclic cycloalkane emerges on deleting this whole ‘bridge’, namely the two bonds with their central carbon. In the last case, (6), the added cyclohexane coalesces completely with the initial cyclohexane; this does not give a genuine addition and should therefore be discarded. In order to keep track of the algebra in the two cases of interest, assume that the initial polycyclic cyclohexane system with h rings has n, external carbons along the perimeter. Each bridge between two rings introduces an internal carbon, of which the number is denoted by n,. Then an addition symbolized by h + h + 1 induces the following shifts in n, and ni in the two cases: (l)-(4)
Izi -
ni(A~i=O),
(5) ni+ni+l(An;=l),
n, --t II, +4
n,+n,+2
A modification of case (5) occurs when the added cyclohexane shares two incident bonds from neighbouring cyclohexanes of the initial system. Then the bridge consists of three bonds with one carbon in common. A full study of polycyclic cyclohexanes should also include the case of ‘compact’ rings as in
S.J. Cyvin et al./Joumal
of Moleculur Structure 445 (1998) 127- 137
-... _Q_ 135
\
Table 8 Parameter values during the generation of monocyclic staggered cycloalkanes through additions of cyclohexanes and the numbers of non-overlapping systems
h
n,
Number I
I
6
0
2
8
I
IO
0
I
----__-
\
\
2
\
\
3
IO
12 14
5 12
Fig. 2. The two monocyclic staggered cycloalkanes with h = IO and n, = 0 (CZh left; C2 right, where only one member of the enantio-
4
5
I2
2
I4
I7
I6
71
18
133
I4
7
16
55
I8 20 22
368
-
adamantane [29,53-S]. This system is generated from two cyclohexane rings which share two incident bonds. The added cyclohexane shares two bonds from each of the initial cyclohexanes. The net effect is an addition of one single bridge, while the perimeter is not affected. 4.3. Generation of monocyclic staggered cycloalkanes through additions of cyclohexanes
meric pair is shown).
encountered for h values up to five according to the algebra of the addition of cyclohexanes, which is treated above. Cyclohexane itself (Fig. 1) is characterized by (n,, ni) = (6, 0). Herefrom, on addition of another cyclohexane in different ways, the unique (8, 1) system (Fig. 1) and two (IO, 0) systems (Fig. 2) are generated. The five (12, 1) systems (CsV+C2+Cr+2C~) are obtained on additons to (10, 0) and (8, l), while the 12 systems of (14, 0), which are distributed among the symmetry groups as C,, + 3Cz + C, + C, + 6C r, are obtained from the (10,O) systems. The unique (10, 2) system (Fig. 3) is generated from (8, 1). From (IO, 2) one obtains the two (12, 3) systems which are illustrated in Fig. 4. The procedure was continued with the aid of computer programming. The computed numbers of non-overlapping monocyclic staggered cycloalkanes generated in the
It will be instructive to demonstrate some of the first steps in the building-up of the title systems. Table 8 shows the values of n, and ni which are
Fig. I The unique monocyclic staggered cycloalkanes with h = 6
Fig. 3. The unique monocyclic staggered cycloalkane with h = IO
(Dw) and h = 8 (&I).
and ,I, = 2 (CZh).
S.J. Cyvin
136
et ul.Nournal
of Molecular
Fig. 4. The two monocyclic staggered cycloalkanes with h = 12 and n, = 3 (CZ, left; C2 right, where only one member of the enantiomerit pair is shown).
indicated way and characterized by (n,, n,) are collected in Table 8. In order to reproduce the numbers of Table 5, the entries for IZ = n, should be assembled from different places in Table 8. This table is complete for n = n, through 14, but n, 2 16 appear also at h > 5. On adding the numbers for 12, = 6, 8, 10, 12 in Table 8, one finds 1, 1, 3, 7, respectively, in agreement with Table 5. This agreement should not be expected to continue necessarily for all h values because the systems pertaining to Table 8 have been checked for non-isomorphism only within each category of (n,, nJ. It is instructive and no real discrepancy that the numbers for PZ,= 14 add up to 36 while one might expect 35 according to Table 5. It was ascertained that two of the generated systems under IZ, = 14 are isomorphic, specifically one with nj = 0 and one with n, = 4. That is a demonstration of two different polycyclic cycloalkane systems with the same perimeter.
References [l] L. Pauling, L.O. Brockway, .I. Am. Chem. Sot. 59 (1937) 1223. [2] A. Almenningen, 0. Bastiansen, Acta Chem. Stand. 9 (1955) 815. [3] L.S. Bartell,
H.K. Higginbotham, J. Chem. Phys. 42 (1965) 851. [4] K. Kuchitsu, J. Chem. Phys. 49 (1968) 4456. [5] T. Iijima, Bull. Chem. Sot. Japan 46 (1973) 231 I.
Structure
445
(1998)
127-137
[6] K. Kuchitsu, Bull. Chem. Sot. Japan 32 (1959) 748. [7] R.A. Bonham, L.S. Bartell, J. Am. Chem. Sot. 81 (1959) 3491. [8] R.A. Bonham, L.S. Bartell, D.A. Kohl, J. Am. Chem. Sot. 81 (1959) 4765. [9] L.S. Bartell, D.A. Kohl, J. Chem. Phys. 39 (1963) 3097. [IO] T. Iijima, Bull. Chem. Sac. Japan 45 (1972) 1291. [I l] W.F. Bradford, S. Fitzwater, L.S. Bartell, J. Mol. Struct. 38 (1977) 185. [ 121 R.L. Livingston, C. Lurie, C.N.R. Rao, Nature 185 (1960)458. [I31 W. Zeil, J. Haase, M. Dakkouri, Z. Naturforsch. 22a (1967) 1644. [I41 B. Beagley, D.P. Brown, J.J. Monaghan, J. Mol. Struct. 4 (1969) 233. [ 151 R.L. Hildebrandt, J.D. Wieser, J. Mol. Struct. 15 (1973) 27. [ 161 L.S. Bartell. T.L. Boates, J. Mol. Struct. 32 (1976) 379. [I71 L.S. Bartell, W.F. Bradford, J. Mol. Struct. 37 (1977) 113. [ 181 R.S. Stein, J. Chem. Phys. 21 (1953) I 193. [19] R.P. Smith, E.M. Mortensen, J. Chem. Phys. 35 (1961) 714. [20] R.P. Smith, J. Chem. Phys. 38 (1963) 1463. [21] R.P. Smith, J. Chem. Phys. 40 (1964) 2693. [22] R.P. Smith; J. Chem. Phys. 42 (1965) 1162. [23] F.T. Wall, F.T. Hioe, J. Phys. Chem. 74 (1970) 4410. [24] F.T. Wall, F.T. Hioe, J. Phys. Chem. 74 (1970) 4416. [25] A.T. Balaban, Commun. Math. Chem. (Match) 2 (1976) 51. [26] A.T. Balaban, Rev. Roumaine Chim. 21 (1976) 1049. [27] A.T. Balaban, P. van RaguC Schleyer, Tetrahedron 34 (1978) 3599. [28] M. RandiC, Int. J. Quant. Chem.: Quantum Biology Symposium 7 (1980) 187. [29] A.T. Balaban, Computers Math. Applic. 17 (1989) 397; reprinted in I. Hargittai (Ed.), Symmetry 2 Unifying Human Understanding, Pergamon Press, Oxford, 1989. [30] M. Randid, B. Jerman-Blaiid, N. Trinajstid, Computers Chem. 217 (1990) 237. [31] A.T. Balaban, D.J. Klein, C.A. Folden, Chem. Phys. Lett. 217 ( 1994) 266. [32] 0. Hassel, H. Viervoll, Acta Chem. Stand. 1 (1947) 149. [33] M.I. Davis, 0. Hassel, Acta Chem. Stand. 17 (1963) 1181. [34] MI. Davis, 0. Hassel, Acta Chem. Stand. 18 (1964) 813. [35] N.V. Alekseev, AI. Kitaigorodskii, Zh. Strukt. Chem. 4 (1963) 163. 1361 H. R. Buys, H. J. Geise, Tetrahedron Lett. (1970) 2991. [37] H.J. Geise, H.R. Buys, F.C. Mijlhoff, J. Mol. Struct. 9 (1971) 447. [38] 0. Bastiansen, L. Femholt, H.M. Seip, H. Kambara, K. Kuchitsu, J. Mol. Struct. 18 (1973) 163. [39] J.D. Ewbank, K. Kirsch, L. Schafer, J. Mol. Struct. 3 I (1976) 39. [40] S.J. Cyvin, J. Wang, J. Brunvoll, S. Cao, Y. Li, B.N. Cyvin, Y. Wang, J. Mol. Struct. 413-414 (1997) 227. [41] E. Funck, Z. Elektrochem. 62 (1958) 901. [42] W. Liittke, Private communication; see also W. Liittke and G.A.A. Nonnenmacher, J. Mol. Struct. 347 (1995) I. [43] S.J. Cyvin, J. Math. Chem. 17 (1995) 291. [44] J. Wang, S. Cao, Y. Li, J. Math. Chem. 20 (1996) 211. [45] M. Saunders, Tetrahedron 23 (1967) 2105.
S.J. Cyvin et al./Journal [46] L. K. Montgomery,
of Molecular
in I. Hargittai, M. Hargittai (Eds.), Stereochemical Applications of Gas-Phase Electron Diffraction, Part B. VCH Publishers, New York, 1988, p. 209. [47] A. T. Balaban, in A.T. Balaban (Ed.), From Chemical Topology to Three-Dimensional Geometry, Plenum Press, New York, p. 1. [48] J. Brunvoll, B.N. Cyvin, E. Brendsdal, S.J. Cyvin, Computers Chem. 19 (1995) 379. [49] N. Trinajstic, S. NikoliC, J.V. Knop, W.R. Miiller, K. Szymanski, Computational Chemical Graph Theory, Ellis Horwood, Chichester, 1991.
Structure
445 (1998)
127-137
137
[50] I. Gutman, S. J. Cyvin, Introduction to the Theory of Benzenoid Hydrocarbons, Springer-Verlag. Berlin, 1989. [51] J. Brunvoll, S.J. Cyvin, B.N. Cyvin, J. Comput. Chem. 8 (1987) 189. [52] J. Bnmvoll, B.N. Cyvin, S.J. Cyvin, J. Chem. Inf. Comput. Sci. 27 (1987) 171. [53] W. Nowacki, K. Hedberg, .I. Am. Chem. Sot. 70 (1948) 1497. [54] I. Hargittai, K. Hedberg, in S.J. Cyvin (Ed.), Molecular Structures and Vibrations, Elsevier, Amsterdam, 1972, p. 340. [55] A.T. Balaban, Commun. Math. Chem. 2 (1976) 5 1.
MOLECULAR STRUCTURE Journal of Molecular
Enumeration
Structure 445
(I 998) 127- 137
of staggered conformers of alkanes and monocyclic cycloalkanes’
S.J. Cyvina’*, B.N. Cyvin”, J. Brunvolla, Jianji Wangb “Deparrment ofPhysical Chemistv, Norwegian University of Science and Technology, N-7034 Trondheim, Norway hDepartment of Chemistry, Inner Mongolia Normal University, Huhehaote 010022, PR China Received 26 August 1997; revised 13 November
1997; accepted
13 November
1997
Abstract Staggered conformers of alkanes are represented by systems which can be embedded in the diamond lattice. Previous enumerations of alkane systems (acyclic) are supplemented. Original contributions are supplied to the numbers and symmetries of monocyclic staggered cycloalkanes, of which cyclohexane is the key structure. These systems are represented by single cycles on the diamond lattice. In particular, the generation of monocyclic staggered cycloalkanes by successive addition of cyclohexane rings is treated in details. 0 1998 Elsevier Science B.V. Keywords:
Alkane; Cycloalkane;
Enumeration
1. Introduction Structural investigations on alkanes, which by definition are acyclic, go back to classical works in gas electron diffraction on ethane [l-5], they include higher molecules with unbranched carbon skeletons [ 1,6- 1 l] and some molecules with branched carbon skeletons [ 1,12- 171. The unbranched alkanes, namely normal paraffins, constitute the prototype of a homologous series in organic chemistry, and it has been investigated by very many researchers. It is impracticable to give a comprehensive survey of all these works here; they involve thermodynamics, spectroscopy and many other areas. We have restricted the list
* Corresponding author. ’Dedicated to the memory of the late Professor Otto Bastiansen.
of references [18-311 to some works where the diamond lattice is mentioned explicitly. This is a crucial concept in the present work inasmuch as all the considered structures should be embeddable in the diamond lattice. The smallest monocyclic cycloalkane within the staggered conformers under consideration is cyclohexane. This molecule was included in the old investigation by Pauling and Brockway [l], but its structure was discussed thoroughly and determined accurately first by Hassel et al. [32-341. Professor Odd Hassel, who became a Nobel laureate, had been the teacher and collaborator of Professor Otto Bastiansen. Cyclohexane became a popular subject for gas electron diffraction investigations also among other researchers [35-391, and now the ‘chair’ configuration (in contrast to ‘boat’) is firmly established.
0022-2860/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved PII SOO22-2860(97)00419-5
128
S.J. Cyvin et al.Nournal
c?fMoleculur
Strucrurr
445 (1998)
127-137
Table I Generating
functions
for the staggered
Symmetry
Function
conformational
Td
x[ 1 + x’b*(x ‘*)I
T
x’[b(x”)
D z<,
x2b*(xh)
DZ
x[b*(x’)
- 1 - .rJb*(x”)]
x’[b(x’)
- b*(d’)]
Sh C,,
xb*(x’)[
&
$r5[b’(x3)
S4
&r[b(x4) - b*(x’)]
stereoisomers
of (acyclic)
alkane systems belonging
to the different symmetry
groups
- b*(x”)]
$x*[b(x’)
- b*(x’)] bb*(.r’)
+ l] -x
+ 2b(x”)]
C?,
G[b*(x’)b*(x’)
C?h C,
b*(x’)
CZ
$x[b’(x’)
C,
x + &b*(.r)b*(x)[xb(x’)
C,
; + ;x2[b2(x2)
- $rzb*(xh) - .r’b*(x”)
+ x[ I - b*(x4)]
- b*(x4)]
- 1 - x’b*@)
&x’[b’(x’)
- b*(x’)b*(x’)] - b*(x’)b*(x*)]
+ .r[b(?)
- b”(x’)]
+ b(x’) - b*(x’)
- x’[b(d’)
- I] + b*(x) - ;b*(x’)[xb*(.x’)
- b(x”)] - ;[b*(x’)
- b*(d’)]
- x’[b(x12)
- b*(x”)]
- .r[b(x”) - b*(x’)] + I] - +,rb*(x”)[xb*(x’)
+ 21 + $2b*(,$‘)
+ x5b*(x”)
-x%*(x6)]
The enumerations of staggered conformers of alkanes are treated in a previous paper 1401, which should be consulted for the definitions of basic concepts and for previous literature. Especially noteworthy is the work of E. Funck [41], which was characterized as ahead of its time and had been undertaken according to an idea suggested by W. Liittke [42]. Later and independently, the same idea was conceived by Balaban [26]. In these works [26,41] the enumeration problem for the unbranched systems was solved completely. The corresponding problem for all acyclic (branched + unbranched) systems turned out to be far more intricate and was originally posed as an ‘open problem’ by one of the present authors [43]. In response, Wang et al. [44] succeeded to solve the crucial part of the problem, while the details were worked out in collaboration between the research groups from Inner Mongolia [44] and Norway [40]. This, we believe, is a genuine collaboration in the spirit of the late Professor 0. Bastiansen. In the present work, these enumerations of staggered alkanes are supplemented. In addition, a contribution to the enumerations of staggered monocyclic cycloalkanes is provided. In precise terms, these systems should contain exactly one cycle each and no side branches. As far as we know, only one enumeration work of such systems exists in the previous literature [45]. Two supplementary references to alkanes and cycloalkanes are of interest [46,47].
2. Staggered
alkanes
2.1. Classi$cation It is reiterated that the alkane systems under consideration should be embeddable in the diamond lattice. Overlapping and non-overlapping systems are distinguished; in the former case at least two carbon atoms occupy the same site. A subclass of the non-overlapping systems, referred to as ‘unstrained’, is defined in a subsequent section. Branched and unbranched systems refer to the branching of the carbon skeleton. The alkane systems are distributed among sixteen symmetry groups: achiral Td, D,,, D,,, S,, C,,, S,, Czv, C2,,, C,, C;; chiral T, D3, D2, C3, CZ, C,. Two systems are said to be isomorphic if they can be brought into each other by translations and symmetry operations; otherwise they are non-isomorphic. In addition to the total number of non-isomorphic systems (spectra1 isomers) for a given carbon content we are also interested in the number of stereoisomers. In the latter case, the chiral systems are counted double in order to account for the enantiomerit pairs. 2.2. Crucial generating functions In order to express the total number of stereoisomers of the staggered alkanes with a given carbon
S.J. Cyvin et al./Journal of Molecular Structure 445 (1998) 127-137 Table 2 Coefficients
in the expansions
”
Table 2 gives numerical values for the coefhcients the expansions of b(x) and b*(x);
of b(x) and b*(x)
b”
b(x)=
2 12 55 273 1428 7752 43 263 246 675 1430715 8414640 50067 108 300 830 572 I 822 766 520 I I 124755664
6
8 9 10 I1 12 13 14 15
2
12 30 55 143 273 728 1428 3876 1752 21318
content and their distribution into the different symmetry groups, a number of auxiliary generating functions were introduced [40,44]. In the present work it was achieved to express all these numbers in terms of one single generating function, namely b(x). This function is determined by b(x) = 1 +xb’(x)
(1)
From the initial condition b(x) = 1 + . ., one successively obtains by iteration: b(x)=l+x+..., b(x)=l+x+3x*+..., b(x)=1+x+3x2+12x”+..., etc. For the total number of stereoisomers (branched + unbranched, overlapping + non-overlapping) the following was arrived at. I(x)=
- ;+
+b4(x)[l -2xb2(x)] +;b(x*),xb(x2)+2]
- ;xb(x~)[b(x)-3]+
~~*~(x))[h’(~)+h(x’)]
(2)
The expressions for the different symmetry groups (except Cl) are collected in Table 1. The function for C, is obtainable on subtraction from (2). In Table 1, in addition to b(x), the function b*(x) also appears. However, the latter function is given explicitly in terms of b(x) by: b”(x) =
1 1 -xb(x2)
in
b;:
0
4
129
5 b,x”, b*(x)= 5 b;x” ll=ll n=O
(4)
It is observed (Table 2) that b;,, = b,, forp = 0, 1, . . ., 7. We have been able to prove this peculiar phenomenon generally for p = 0, 1, 2, . . . . Modify Eq. (1) to b(x)[l -xb2(x)J = 1
(5)
whereupon
b(x)=
’ 1 -xb*(x)
(6)
emerges. The essence of our proof lies in the similar forms of Eq. (3) and Eq. (6). As a matter of routine, the proof was completed by expansions and multiple summations. 2.3. Computer-generated
supplementary
data
The overlapping alkane systems are not supposed to be realized as chemical counterparts because of severe steric hindrances. Therefore the nonoverlapping systems are more interesting in the chemical context. Since an algebraic solution for their numbers of conformers is not to be expected one has to resort to computer programming [48]. That was also the basis for the previously published numbers of conformers of non-overlapping alkane systems [40]. The list is supplemented in Table 3. The numbers up to n = 14 were produced by an effective exploitation of the constitutional alkane isomers [49]. Furthermore, a coding in analogy with Saunders [45] was employed in the case of the unbranched systems. This method reproduced all the numbers for n 5 14 and made it feasible to extend the list through n = 19 for the unbranched systems. Among the non-overlapping alkane systems there are many members in which the shortest non-bonded carbon-carbon distance is the smallest possible, namely equal to the bond distance (corresponding to the unit distance of the diamond lattice). Because of the hydrogen atoms, severe steric hindrances are also expected in these cases. But there are still more
130
S.J. Cyvin et al.Nournul
of Molecular
Structure
445 (1998)
127-137
of Molecular
S.J. Cyvin et ul.Nournal
Table 4 Numbers of unstrained
5
6
7
x
9
I0 II 12
alkane staggered
conformers
Structure
445 (1998)
(acyclic)
unhranched
0
0
0
0
I
0
0
I
0
0
branched
I
0
0
0
0
0
0
0
I
0
unhranched
0
0
0
0
0
I
0
3
0
I
2
7
I2
hranchrd
0
0
0
0
0
I
0
I
3
0
5
IO
I6
unhranched
0
0
0
0
I
0
0
3
I
0
x
I3
24
branched
0
0
0
0
I
0
I
2
4
0
25
33
61
unhranchrd
0
0
0
0
0
I
0
8
0
3
I9
31
58
branched
0
I
0
0
0
I
I
7
8
3
95
I I6
219
unhranched
0
0
0
0
I
0
0
8
3
0
54
66
I28
branched
0
0
I
1
2
0
0
9
0
363
387
759
unhranched
0
0
0
0
0
I
0
20
0
branched
0
0
0
0
0
3
2
32
21
II
8 I4
unhranched
0
0
0
0
I
0
0
I9
8
0
branched
0
0
0
0
4
0
I
42
37
0
unhranohed
0
0
0
0
0
I
0
49
0
20
branched
0
0
0
0
0
5
0
I30
63
58
unhranchrd
0
0
0
0
I
0
0
47
I9
0
branched
0
0
I
I
7
0
6
I61
127
0
unhranched
0
0
0
0
0
I
Ill
0
49
branched
0
0
0
0
0
8
507
197
226
IS
unhranched
0
0
0
0
I
0
II3
46
0
16
unhranched
0
0
0
0
0
I
283
0
I I9
I7
unhranched
0
0
0
0
I
0
273
106
0
IX
unhranched
0
0
0
0
0
I
677
I9
unhranched
0
0
0
0
I
0
648
13
14
127-137
systems which we wish to eliminate on chemical grounds. Consider two incident edges in the diamond lattice, which are defined by one central and two end vertices. If the two end vertices (at the next-to shortest distance in the diamond lattice) are occupied by carbon atoms, but not the central vertex, then two carbon-hydrogen bonds will point along the two edges in question, and again a substantial (but not so severe) steric hindrance will occur. Systems where the above described situations are not present, are referred to as ‘unstrained’, and the pertinent numbers, which again were obtained by computer programming, are collected in Table 4. Again, the numbers for n 5 14 were obtained for the totals (branched + unbranched systems), while the unbranched systems were enumerated up to n = 19.
0
I59
309
I370
2702
334 4715 804 16832 I984 60281 4777 214939
288
247
I30 I298
0
362
715
4799
9557 I727
x74
3405
17088 205
I
4082
60584
121032
4944
9838 431 329
215880
II615
I I 775
21957
28 360
23 503 56600
67 472
67 852
I35 597
162 276
I63 242
326
390 546
391442
782 636
195
3. Monocyclic staggered cycloalkanes 3.1. Classijcation The systems to be enumerated are single cycles on the diamond lattice. The smallest of these systems corresponds to the ‘chair’ configuration of cyclohexane. Notice that the ‘boat’ form is not embeddable in the diamond lattice. The concepts of overlapping, non-overlapping and unstrained (see above) are immediately applicable to the monocyclic cycloalkanes under consideration. 3.2. Application Saunders
of Saunders’ integer code
[45]
has
identified
each
monocyclic
132
Table
of MoleculurStructure
S.J. Cyvin et aUJournul
445 (1998)
127-137
5
Numbers
of non-overlapping
n
D w
&d
monocyclic D1
Cl,.
staggered Dz
cycloalkancs S,
Cz,
(unbranched) CZh
C?
C?
C,
Spectral
Cl
C,
isomers 6
I
0
0
0
0
0
0
0
0
0
0
0
8
0
I
0
0
0
0
0
0
0
0
0
0
0
IO
0
0
0
0
0
0
0
2
0
I
0
0
0
I2
0
0
0
I
0
0
I
0
0
2
I
0
2
7
I4
0
0
0
0
0
0
0
5
0
6
5
3
I6
35
16
0
2
0
0
0
I
I
0
0
20
I3
0
I8
3
0
I
I
0
0
0
IO
I
48
31
31
20
0
0
0
0
0
0
5
0
0
126
96
22
0
0
0
0
0
0
0
37
0
307
243
24
0
6
0
5
9
8
IO
0
907
617
II
Number n
0
3
II3
150
750
876
4809
226
5036
31263
32 076
204 670
0
206 243
come in between those of our Tables 5 and 6. According to our computations they should be: 1, 1, 1, 4, 13, 51, 239, 1204. Saunders [45] has not given enough details of his results to make it possible to pinpoint errors therein, but we must also stress the reservation about our correct interpretation of Saunders’ definitions.
staggered cycloalkane with n carbon atoms by means of a string of n integers, which indicate the directions in which the carbon-carbon bonds turn around the ring. We have adopted this idea in our lirst computer-generated numbers of non-isomorphic monocyclic staggered cycloalkanes. Table 5 shows the results for the pertinent non-overlapping systems, while the numbers of unstrained systems are listed in Table 6. The distributions into symmetry groups are included. Both agreements and discrepancies were found on comparing our results with those of Saunders [45]. The numbers of spectral isomers of the unstrained systems (Table 6) agree perfectly with the Saunders numbers. However, this author has also reported a set of numbers for n = 6, 8, 10, 12, 14, 16, 18, 20 as 1, 1, 1, 4, 13, 56, 238, 1202, respectively. If we have interpreted these numbers correctly, they include systems with the less severe steric hindrances according to the description above (Section 2.3) while the severe steric hindrances (closest possible proximity of non-bonded carboncarbon atoms) are eliminated. Thus these numbers Table
0
4. Addition of six-membered 4. I. Introductory
rings
remarks
Any polycyclic staggered cycloalkane system consisting of exclusively six-membered (cyclohexane) rings is generated by successive additions of cyclohexane rings within the framework of the diamond lattice. This consideration is very natural in view of the theory of benzenoid hydrocarbons [50]. It is well known that the benzenoids are generated by successive additions of hexagons on the hexagonal lattice, and this property has been successfully exploited in
6 of unstrained
monocyclic
DW
D2d
staggered
cycloalkanes
Ci,.
(unbranched)
D2
S4
Cl,
CZh
C2
CA
C,
Cl
Spectral isomers
6
I
0
0
0
0
0
0
0
0
0
0
1
14
0
0
0
0
0
0
I
0
0
0
0
I
I6
0
I
0
0
0
0
0
0
0
0
0
I8
1
0
0
0
0
0
3
2
I
2
2
20
0
0
0
0
0
I
0
6
3
0
9
I9
22
0
0
0
0
0
0
I
I3
5
I7
78
120
24
0
2
I
I
2
1
0
25
I8
0
369
419
1 II
S.J. Cwin et ul.Nournal Table 7 Specification A
of added cyclohexane
of Molecular Structure 445 (1998) 127-137
rings in terms of the coordinates B
of the diamond
133
lattice
C
D
1 0 0-I -II
0-I
I 0, 0
computerized generations and enumerations of benzenoid systems [50-521. The perimeter of a polycylic cycloalkane system is a single cycle on the diamond lattice and represents therefore a monocyclic cycloalkane system within the class of interest in the present work. However, a systematic addition of cyclohexane rings is a far more complex matter than the addition of benzenoid rings, and also the application to systematic generations of single cycles in the diamond lattice meets with complications. One must realize that such a cycle does not always define uniquely a polycyclic cyclohexane system, in sharp contrast to the situation with benzenoids, which are defined uniquely by their perimeters on the hexagonal lattice [50].
4.2. Algebra of the addition of cyclohexanes Consider the addition of a cyclohexane to a carbon-carbon bond C , -Cz so that (at least) one of the cyclohexane bonds merges with C i-C?. Six possibilities are distinguished according to the six bonds of the added cyclohexane. The aim is to specify the coordinates of the carbon atoms of the added cyclohexane. A previous work [48] should
-2 0 2
0 -I
-2 2
-3I
-2 02
-II
I21 -I 321
0
I
1
be consulted for the precise definition of the integer coordinates in relation to the diamond lattice. Suppose first that Ci is situated in the origin and consequently has the coordinates (0, 0, 0). Then C i -Cz may point in one of the directions of the C-H bonds of methane with C in the origin (or one of the C-C bonds in neopentane). The following four cases are distinguished for the coordinates of C?:
(4 i 1, 1, 11 (B) ( 1, -1,
-1)
(Cl ( -1,
1, -1)
(D) 1-1,
-1,
1)
In Table 7 the complete cyclohexane rings are specified, six in each of the cases A, B, C and D. It remains to consider the general case when the CI-CZ bond is associated with the coordinates (xi, y I, z I ) - {x2, y2, z?}. This problem is easily handled by means of a simple translation: 1. Since C , -C2 are bonded atoms, one of the coordinate triples must necessarily consist of even integers only and the other of odd integers only [46]. Without loss of generality, assume that {x,. yi, z2}
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et al./Journal
of Molecular Structure
445
(1998)
127-137
are even numbers, while (x2, y2, z?] are odd. Compute (x1-x1, yZ - yl, z2 - zI]. This is the set of transformed coordinates for atom Cl, while C , has been transformed to origin, (0, 0, 0). 2. Identify the pertinent column of Table 7, wherein the transformed C i-C2 bond corresponds to the two first columns in each coordinate set for the added cyclohexane ring. 3. The added cyclohexane is translated back to the original position by the opposite transformation: if the set of coordinates from Table 7 is denoted (a;, bi, Ci), then the new coordinates are ( ai +.x1, bi+_Yl,
232101
Ci+Zll.
The above procedure is illustrated by an example in the following. For the sake of simplicity, assume the original system to be a six-membered ring (cyclohexane), e.g.: (1, 3, 3)-
12, 2, 4)-
13, 1, 31
I
I
(0, 2, 2k
(1, 1, 1I-
IT
0, 21
Here the coordinates for Cl and C2 are given in heavy numerals. Then C I{ 2, 2,4] and C2( 3, 1, 3) should be chosen according to the above conventions and consistent with Table 7. The transformed coordinates for C2 come out as ( 1, - 1, - 1 ), which identifies column B in Table 7. Then the six added cyclopropanes with C 1-C 2 in the original position are found to be: 234321 (1)
2
I
i 432123
234543
432345
234543
434565
2
3
4
3
1
, An,=0
(6)
2
2
0
I
2
[ 432123
3
1
, discarded
The above example demonstrates three types of addition of a cyclohexane ring. In (l)-(4) the added ring shares exactly one carbon-carbon bond with the initial system. The corresponding coordinates are typed in bold. A monocyclic cycloalkane emerges on deleting this bond. In case (5) three carbons belonging to two incident bonds are shared, and the pertinent monocyclic cycloalkane emerges on deleting this whole ‘bridge’, namely the two bonds with their central carbon. In the last case, (6), the added cyclohexane coalesces completely with the initial cyclohexane; this does not give a genuine addition and should therefore be discarded. In order to keep track of the algebra in the two cases of interest, assume that the initial polycyclic cyclohexane system with h rings has n, external carbons along the perimeter. Each bridge between two rings introduces an internal carbon, of which the number is denoted by n,. Then an addition symbolized by h + h + 1 induces the following shifts in n, and ni in the two cases: (l)-(4)
Izi -
ni(A~i=O),
(5) ni+ni+l(An;=l),
n, --t II, +4
n,+n,+2
A modification of case (5) occurs when the added cyclohexane shares two incident bonds from neighbouring cyclohexanes of the initial system. Then the bridge consists of three bonds with one carbon in common. A full study of polycyclic cyclohexanes should also include the case of ‘compact’ rings as in
S.J. Cyvin et al./Joumal
of Moleculur Structure 445 (1998) 127- 137
-... _Q_ 135
\
Table 8 Parameter values during the generation of monocyclic staggered cycloalkanes through additions of cyclohexanes and the numbers of non-overlapping systems
h
n,
Number I
I
6
0
2
8
I
IO
0
I
----__-
\
\
2
\
\
3
IO
12 14
5 12
Fig. 2. The two monocyclic staggered cycloalkanes with h = IO and n, = 0 (CZh left; C2 right, where only one member of the enantio-
4
5
I2
2
I4
I7
I6
71
18
133
I4
7
16
55
I8 20 22
368
-
adamantane [29,53-S]. This system is generated from two cyclohexane rings which share two incident bonds. The added cyclohexane shares two bonds from each of the initial cyclohexanes. The net effect is an addition of one single bridge, while the perimeter is not affected. 4.3. Generation of monocyclic staggered cycloalkanes through additions of cyclohexanes
meric pair is shown).
encountered for h values up to five according to the algebra of the addition of cyclohexanes, which is treated above. Cyclohexane itself (Fig. 1) is characterized by (n,, ni) = (6, 0). Herefrom, on addition of another cyclohexane in different ways, the unique (8, 1) system (Fig. 1) and two (IO, 0) systems (Fig. 2) are generated. The five (12, 1) systems (CsV+C2+Cr+2C~) are obtained on additons to (10, 0) and (8, l), while the 12 systems of (14, 0), which are distributed among the symmetry groups as C,, + 3Cz + C, + C, + 6C r, are obtained from the (10,O) systems. The unique (10, 2) system (Fig. 3) is generated from (8, 1). From (IO, 2) one obtains the two (12, 3) systems which are illustrated in Fig. 4. The procedure was continued with the aid of computer programming. The computed numbers of non-overlapping monocyclic staggered cycloalkanes generated in the
It will be instructive to demonstrate some of the first steps in the building-up of the title systems. Table 8 shows the values of n, and ni which are
Fig. I The unique monocyclic staggered cycloalkanes with h = 6
Fig. 3. The unique monocyclic staggered cycloalkane with h = IO
(Dw) and h = 8 (&I).
and ,I, = 2 (CZh).
S.J. Cyvin
136
et ul.Nournal
of Molecular
Fig. 4. The two monocyclic staggered cycloalkanes with h = 12 and n, = 3 (CZ, left; C2 right, where only one member of the enantiomerit pair is shown).
indicated way and characterized by (n,, n,) are collected in Table 8. In order to reproduce the numbers of Table 5, the entries for IZ = n, should be assembled from different places in Table 8. This table is complete for n = n, through 14, but n, 2 16 appear also at h > 5. On adding the numbers for 12, = 6, 8, 10, 12 in Table 8, one finds 1, 1, 3, 7, respectively, in agreement with Table 5. This agreement should not be expected to continue necessarily for all h values because the systems pertaining to Table 8 have been checked for non-isomorphism only within each category of (n,, nJ. It is instructive and no real discrepancy that the numbers for PZ,= 14 add up to 36 while one might expect 35 according to Table 5. It was ascertained that two of the generated systems under IZ, = 14 are isomorphic, specifically one with nj = 0 and one with n, = 4. That is a demonstration of two different polycyclic cycloalkane systems with the same perimeter.
References [l] L. Pauling, L.O. Brockway, .I. Am. Chem. Sot. 59 (1937) 1223. [2] A. Almenningen, 0. Bastiansen, Acta Chem. Stand. 9 (1955) 815. [3] L.S. Bartell,
H.K. Higginbotham, J. Chem. Phys. 42 (1965) 851. [4] K. Kuchitsu, J. Chem. Phys. 49 (1968) 4456. [5] T. Iijima, Bull. Chem. Sot. Japan 46 (1973) 231 I.
Structure
445
(1998)
127-137
[6] K. Kuchitsu, Bull. Chem. Sot. Japan 32 (1959) 748. [7] R.A. Bonham, L.S. Bartell, J. Am. Chem. Sot. 81 (1959) 3491. [8] R.A. Bonham, L.S. Bartell, D.A. Kohl, J. Am. Chem. Sot. 81 (1959) 4765. [9] L.S. Bartell, D.A. Kohl, J. Chem. Phys. 39 (1963) 3097. [IO] T. Iijima, Bull. Chem. Sac. Japan 45 (1972) 1291. [I l] W.F. Bradford, S. Fitzwater, L.S. Bartell, J. Mol. Struct. 38 (1977) 185. [ 121 R.L. Livingston, C. Lurie, C.N.R. Rao, Nature 185 (1960)458. [I31 W. Zeil, J. Haase, M. Dakkouri, Z. Naturforsch. 22a (1967) 1644. [I41 B. Beagley, D.P. Brown, J.J. Monaghan, J. Mol. Struct. 4 (1969) 233. [ 151 R.L. Hildebrandt, J.D. Wieser, J. Mol. Struct. 15 (1973) 27. [ 161 L.S. Bartell. T.L. Boates, J. Mol. Struct. 32 (1976) 379. [I71 L.S. Bartell, W.F. Bradford, J. Mol. Struct. 37 (1977) 113. [ 181 R.S. Stein, J. Chem. Phys. 21 (1953) I 193. [19] R.P. Smith, E.M. Mortensen, J. Chem. Phys. 35 (1961) 714. [20] R.P. Smith, J. Chem. Phys. 38 (1963) 1463. [21] R.P. Smith, J. Chem. Phys. 40 (1964) 2693. [22] R.P. Smith; J. Chem. Phys. 42 (1965) 1162. [23] F.T. Wall, F.T. Hioe, J. Phys. Chem. 74 (1970) 4410. [24] F.T. Wall, F.T. Hioe, J. Phys. Chem. 74 (1970) 4416. [25] A.T. Balaban, Commun. Math. Chem. (Match) 2 (1976) 51. [26] A.T. Balaban, Rev. Roumaine Chim. 21 (1976) 1049. [27] A.T. Balaban, P. van RaguC Schleyer, Tetrahedron 34 (1978) 3599. [28] M. RandiC, Int. J. Quant. Chem.: Quantum Biology Symposium 7 (1980) 187. [29] A.T. Balaban, Computers Math. Applic. 17 (1989) 397; reprinted in I. Hargittai (Ed.), Symmetry 2 Unifying Human Understanding, Pergamon Press, Oxford, 1989. [30] M. Randid, B. Jerman-Blaiid, N. Trinajstid, Computers Chem. 217 (1990) 237. [31] A.T. Balaban, D.J. Klein, C.A. Folden, Chem. Phys. Lett. 217 ( 1994) 266. [32] 0. Hassel, H. Viervoll, Acta Chem. Stand. 1 (1947) 149. [33] M.I. Davis, 0. Hassel, Acta Chem. Stand. 17 (1963) 1181. [34] MI. Davis, 0. Hassel, Acta Chem. Stand. 18 (1964) 813. [35] N.V. Alekseev, AI. Kitaigorodskii, Zh. Strukt. Chem. 4 (1963) 163. 1361 H. R. Buys, H. J. Geise, Tetrahedron Lett. (1970) 2991. [37] H.J. Geise, H.R. Buys, F.C. Mijlhoff, J. Mol. Struct. 9 (1971) 447. [38] 0. Bastiansen, L. Femholt, H.M. Seip, H. Kambara, K. Kuchitsu, J. Mol. Struct. 18 (1973) 163. [39] J.D. Ewbank, K. Kirsch, L. Schafer, J. Mol. Struct. 3 I (1976) 39. [40] S.J. Cyvin, J. Wang, J. Brunvoll, S. Cao, Y. Li, B.N. Cyvin, Y. Wang, J. Mol. Struct. 413-414 (1997) 227. [41] E. Funck, Z. Elektrochem. 62 (1958) 901. [42] W. Liittke, Private communication; see also W. Liittke and G.A.A. Nonnenmacher, J. Mol. Struct. 347 (1995) I. [43] S.J. Cyvin, J. Math. Chem. 17 (1995) 291. [44] J. Wang, S. Cao, Y. Li, J. Math. Chem. 20 (1996) 211. [45] M. Saunders, Tetrahedron 23 (1967) 2105.
S.J. Cyvin et al./Journal [46] L. K. Montgomery,
of Molecular
in I. Hargittai, M. Hargittai (Eds.), Stereochemical Applications of Gas-Phase Electron Diffraction, Part B. VCH Publishers, New York, 1988, p. 209. [47] A. T. Balaban, in A.T. Balaban (Ed.), From Chemical Topology to Three-Dimensional Geometry, Plenum Press, New York, p. 1. [48] J. Brunvoll, B.N. Cyvin, E. Brendsdal, S.J. Cyvin, Computers Chem. 19 (1995) 379. [49] N. Trinajstic, S. NikoliC, J.V. Knop, W.R. Miiller, K. Szymanski, Computational Chemical Graph Theory, Ellis Horwood, Chichester, 1991.
Structure
445 (1998)
127-137
137
[50] I. Gutman, S. J. Cyvin, Introduction to the Theory of Benzenoid Hydrocarbons, Springer-Verlag. Berlin, 1989. [51] J. Brunvoll, S.J. Cyvin, B.N. Cyvin, J. Comput. Chem. 8 (1987) 189. [52] J. Bnmvoll, B.N. Cyvin, S.J. Cyvin, J. Chem. Inf. Comput. Sci. 27 (1987) 171. [53] W. Nowacki, K. Hedberg, .I. Am. Chem. Sot. 70 (1948) 1497. [54] I. Hargittai, K. Hedberg, in S.J. Cyvin (Ed.), Molecular Structures and Vibrations, Elsevier, Amsterdam, 1972, p. 340. [55] A.T. Balaban, Commun. Math. Chem. 2 (1976) 5 1.