[CONTRIBUTION FROM
THE
CHEMICAL LABORATORY OF THE UNIVERSITY OF CALIFORNIA]
CALCULATION OF THE NUMBER O F STEREOISOMERS I N CARBON CHAIN COMPOUNDS GERALD E. K. BRANCH
AND
TERRELL L. HILL
Received April 17, 1939 INTRODUCTION
A simple method can be devised for obtaining the number of stereoisomers of any carbon chain compound, whether simple or branched, which can readily be applied to chains containing units of geometric isomerism alone or in combination with asymmetric atoms. This general method consists in working from the periphery of the molecule inward, by treating each region successively in a manner determined by the arrangement of the groups, until finally the complete solution is made about an initially determined part of the molecule. A structural formula only shows which atoms are bonded together, and a single formula may represent two or more compounds. In the same way, parts of a structural formula may be identical as far as the structural formula goes, and yet represent different things. When two structural formulas or two parts of a structural formula are the same, we shall say that the things represented are structurally identical, but when the things represented are the same, we shall use “identical” without any qualifying adverb. We shall follow the same convention with respect to “different”. The things represented by a region of a structural formula we shall call the possibilities of the region, and their number will be expressed by the symbol p. The number of stereoisomers may be obtained from the p-values of all the regions, and the number of possibilities of a compound region from the p-values of the component regions. To do this it is necessary to have certain formulas to express the number of possibilities of a compound region in terms of the p-values of the component regions. These formulas will depend on the procedure adopted for dividing the structural formula into regions and the order of combining the regions, as well as upon the stereochemistry of the elements in the compound. It appears that if we follow certain rules for selecting and combining regions, quite simple formulas are adequate for noncyclic carbon compounds that contain no elements having stereochemical properties other than those characteristic of carbon. 86
CALCULATION OF STEREOISOMERS I N COMPOUNDS
87
OPTICAL ISOMERISM
In the present section we shall restrict ourselves to saturated compounds. For these the fundamental assumption is that a region containing one atom with four different groups attached to it has two possibilities. If thle groups are structurally different such an atom may be recognized from the structural formula, and we shall call such an atom asymmetric. This does not quite conform to common usage. For instance text books refer to compounds of the type a3CCsbCab - . Csb'Cas having either an odd or an even number of asymmetric atoms, while in our use of asymmetrjc atom such a compound cannot have an odd number of such atoms. In using the test of four structurally different groups, a group includes everything that can be reached by passing along bonds, and has an order starting with the bond. Thus
has an asymmetric atom as the groups -CBr2CH2 and -CH&Br2 are not structurally identical. In a saturated noncyclic compound no atom can have four different groups attached to it unless the molecule contains atoms with four structurally different groups. When structurally different regions are combined, the number of possibilities is given by the formula N = ps. pb .pc . . , where N is the number of stereoisomers or the number of possibilities in the compound region according as a b c . is the whole structural formula or part of it. If in a formula, or part of one, there are no structurally identical regions for which p > 1, there are no atoms except asymmetric ones that can have four different groups, and N = 2" where n is the number of asymmetric atoms. N may refer to the whole molecule or to a region connected by bonds. When we speak of structurally identical regions, the identity must exist not only in the contents of the region but also in any groups to which they are bonded. Thus in H3C-CHC1-CH2-CH3 the two methyl groups are not identical regions, but the three hydrogen atoms attached to one carbon atom are. When a formula contains structurally identical regions for which p > 1, we use the following procedure for selecting and combining regions. Working inwards from the periphery we select the first region that includes an asymmetric atom. From this we proceed along the bonds until another region is found containing an asymmetric atom, and these regions are
+ +
88
GERALD E . K. BRANCH AND TERRELL L. HILL
TABLE NUMBEROF INACTIVE FORMS FOR THE VARIOUSTYPES OF STRUCTURE Greek letters represent simple or complex groups of asymmetric or geometric units Equation
Structural type
a d-A-b
Ni = 0
I
C
a--P a
a
\ d / \
a-aor
a
a
/“
a-C-a
\
a
JI
a
“
/ \
6
a /a
\c-c b
’
Ni = 2 b‘
a
\
/a
/c=c\b
a
a
\
/c=c\
a
/a
5
CALCULATION OF STEBEOISOhlEk$ I N COMPOUNDS
89
0:
\
/c=c\
P
a!
'y
\
/
c=c
\
P
ff
ff
'\
B
6
a.
\
/
c=c /" \
B
P
ff. \ / ff
c=c
\ CY
combined, this process being carried on until a region is obtained which is bonded directly to a region structurally identical with it, or bonded to the same atom as one or more regions identical with it. These regions are then combined. When structurally identical regions are attached to the same atom, this atom and the structurally identical regions are combined to give R1. One then starts at another part of the periphery and proceeds as before until a region, Rz, attached to R1, or to the same atom as RI is obtained. R1 and Rz are then combined. Constant repetition of this procedure eventually leads to the inclusion of the whole formula. We shall show the procedure by applying it to
in which Greek letters indicate regions with asymmetric atoms. Working from the periphery we find the first simple region a. This is then combined with the next such region to give the compound region a+. This is structurally identical with another a-/3 region attached to the same regions and the atom to which carbon atom. We combine the two they are attached to form (a-@)&c. This region is structurally identical
90
GERALD E. K. BRANCH AND TERRELL L. HILL
with and bonded t o the other (a-j3)ICa-, and these are combined, the whole formula being then included. Had the group “a” contained asymmetric atoms it would have been necessary to start a t the periphery and calculate p for a and combine a and (a-j3),C before the final combination of (a--@)Ce with (CY-~~)~C,. In the above example the p value of a+ is pa. pa or 2”. But multiplication cannot be used to obtain the p values for other compound regions, 8
as all the combinations of the possibilities of a+
and a-@
or of -C(aP)2
8
and -C(a@)Z do not represent different things. Also the atom represented by C can have four different groups according to the possibilities chosen for the two a@-groupseven though C is not an asymmetric atom by our definition. Although this atom does not have four structurally different groups it has no structurally identical groups that do not contain asymmetric atoms. We shall call this type of atom quasiasymmetric. When we include unsaturated compounds it will be necessary to modify slightly the above definition. The central atoms in C,,, C,,,, CaZ8t,,Ca202 are quasiasymmetric. In the first two of these, if CY has only two possibilities, the quasiasymmetry will not contribute to the stereoisomerism. In combining identical regions about a central atom, we divide the groups of this atom into two parts, A and B. One of these, A, contains all structurally identical groups, the other, what is left. The central atom will produce an extra possibility when all four groups are different. This number is SA’SB, in which SA is the number of ways in which the groups comprising A may be chosen from their possibilities so that none are identical. S B has the same meaning for B. If B is composed of one group or of structurally different groups, it cannot introduce any identity in the groups attached to the central carbon atom, and the number of possibilities of the region A plus the central atom can be taken as p+4 SA, and we may proceed to the next region. When B is composed of structurally identical groups, the value of SB is zero, if the groups have only one possibility, but SB has a finite value if the groups have more than one ~. possibility. In this case the whole formula is ACB or C Y ~ C PThe number of possibilities for j3 must be calculated by starting at the peripheral point. The total number of isomers is then K = PA.PB SA.SB. The essence of the method of selecting and combining regions is that we never combine structurally identical regions that are neither attached to the same atom nor form two identical halves of the molecule. In this way we combine structurally identical regions having the same equivalence as that between identical groups attached to a carbon atom. This we shall call complete equivalence. It is defined by the condition that if the structurally identical regions, 81, a2 . . . are replaced by a set of non-
+
+
91
CALCULATION O F STEREOISOMERS IN COMPOUNDS
identical regions, m, n . , the structural formula obtained is independent of the order of the changes. That is, if in the formula AalaZa8, all a2, and a3 are completely equivalent, then Am,,o,A m o n , A n m o , A n o m , A o m n , and A,,, are aJll the same formula, even though more than one substance may be represented by this formula. We shall call the number of completely equivalent regions the degree of equivalence and represent it by the symbol q. The number of possibilities obtainable from the combination of q completely equivalent regions is a function of q and p for one of the component regions, and we shall represent this function by the symbol 1 When the regions are attached to a quasiasymmetric atom, S, extra forms are introduced when the quasiasymmetric atom is included. That is N = 1; S. It is obvious that 1; is the number of combinations of p dissimilar things taken q at a time, when repetitions of the p elements are allowed. q - 1 dissimilar This is equal to the number of combinations of p things taken q at a time. S is the number of combinations of p dissimilar things taken q at a time.
x.
+
+
- (q + p - l)! (p - l)!q!
(p
+ q - l)C,.
The fundamental formulas are: N = pa*pb.Po . = 2", when structurally d8erent regions are combined : N = I = (p q - l)C,, when completely equivalent regions are combined. N = 1; S = (p q - l)C, pC,, when completely equivalent regions and a quasiasymmetric atom are combined: N = PA'PB SASB = I P.1i8 SA.% = (pa l)Cz-(pp 1)C2 p,!C~.ppC2,when two pairs of completely equivalent regions and a quasiasymmetric atom are combined, that is, when the compound belongs to the type Cazf12. Often it is convenient in compounds of the type Casp7 to usle the formula N = p ~ . p B SA-SB in which B is the region f l ~ . SB is then equal to pB, and N = I iu.pB puC2.pB = I 2pu.pp.p7 paC2. Pa*P7. When q = 2 and there is only one center of equivalence the formulas N = I 4" and N = 1: S reduce to the common expressions, N = 2"-' 2n12--1and N = 2n-1, used for the number of stereoisomers of a compound +
+
+
+
+
+
+
+
+
+
+
+
+
+
+
92
GERALD E. K. BRANCH AND TERRELL L. HILL
of the type a3CCabCab ... CabCa8, according as n is even and odd respectively. With our definition of an asymmetric atom, the latter case is one with an even number of asymmetric atoms plus a quasiasymmetric atom. It becomes N = 2", when n is the number of carbon atoms having four structurally different groups. This formula, N = 2")holds whenever completely equivalent regions having p > 1 occur only in pairs attached to the same atom, as in examples V and VI. The formula N = 2n-1 2n'2-1 holds whenever completely equivalent regions having p > 1 occur only in pairs, but finally reduce to one of the types, a-a, a2Ca2 or a2Cp~. Examples I and I1 and IV are cases of this. For the above formulas to hold for complex cases, it is necessary that n is the number of asymmetric atoms by our definition. It should, perhaps, be pointed out that although the method outlined is general and can be applied to any type of non-cyclic stereoisomerism of tetrahedral atoms by the use of a few simple formulas, it is only in certain cases involving equivalence that Senior's (1) formulas are not also applicable. Examples.-The application of the above method can best be shown by considering a variety of examples. In the following examples the Greek letters, a, 8, etc. represent asymmetric units, the letters a, b, etc. represent symmetrical groups, capital letters A, B, etc. stand for regions of the molecule, NA, N B , etc. are the number of possibilities in regions A, B, etc. respectively, and N is the total number of stereoisomers.
+
Example I a ............1............
NA =
NAI =
1 I l e I l 1Y
I A'
=
12
2' = 4.
'
= - - 10.
3.2
B
Example I1 ..............j ..............
C
a
/I\
a
a
A'
NA =
2
NA' = 1 2 f
2.1 s = 3-1 + = 4. 2.1
For the whole molecule, N = I: = 10.
B
Example I11
I A' 4.5.6 -.2+ 3-2.1
4.3.2 --2, N = 48. 3-2.1
P A
\
A'B
/
a
a
NA = NAt = 22 = 4. N = I: = 10. The SS' term is dropped since the two SVmetrical groups are identical.
\ /a C / \
a
P
24 \
Example IV
/ P A'
\I/ C ................I ................ r
Aa
/I\
b
+
6
+
B
a!
I
I/
C
a!/'
NA = N A ~ = 2' = 4, NB = 2' = 8. 4.3 NAAf = 1; s = 10 - = 16. 2.1 Finally, N = pp' = 8.16 = 128.
C
a!
'\
Example V
a a a !
a A'
C
\ ,
//
..........................
Example VI
c........................
P\/ b
P
C
I\
2.1
2
\a!
r B I 6 B
+ 2.1 - = 4. N A A= ~ 1: = 10. NB = Z5 = 32. Then N = pp' + SS' = pp' + Sp' 4.3 10.32 + - .32, and N = 512. 2.1 NA = NAP= Iz + S = 3
=
a!
I
A
CY-&CY
A"'
Example VI1
" I T
2.1.0 = 4. 3.2.1 5.6.7 4.3.2.1 N = I : + s = - 3.2.1 + - = ~ ~4-3.2.1 .
I a-c-c-&a!
NA=Ii+S=4+-
(Y-&a!
I
A"
A'
a!
A C Y
I
a-&CY
I
Example VI11 NA = NA' = I:
CY
............ / 1.... 1 6
B
(3
.*.'
LY
NAAt =
A'
I:
I: = 3, N = pp' 4.3 2.1 -.= 36. 2.1 2.1
= 10, NB =
SS' = 10.3 93
+ S = 4 + 32.1.0 - = 4. -2.1
+
+
94
GERALD E. K, BRANCH AND TERRELL L. HILL GEOMETRIC ISOMERISM
A pair of doubly bonded carbon atoms having different groups attached to each carbon atom allows two possibilities for a region including the pair of carbon atoms. If both the differences are structural they can be recognized from the structural formula, and the double bonded system may be called a geometric unit. But the isomerism exists whether the differences are structural or not. Hence it is possible to have a quasigeometric unit analogous to the quasiasymmetric atom. Further, since two structurally identical geometric units can exist in different forms, it is possible to have four different groups attached to an atom that is not asymmetric without having an asymmetric atom in the molecule. It is therefore advisable to modify the definition of quasiasymmetry. A quasiasymmetric atom is one whose four groups are not all structurally different, but has no structurally identical groups that contain neither an asymmetric atom nor a geometric unit. Similarly, a quasigeometric unit is a pair of doubly bonded carbon atoms that has two structurally identical groups attached to a t least one of the atoms, but in which neither carbon atom has structurally identicaf groups that contain neither an asymmetric atom nor a geometric unit. Thus the central carbon atom in a
I
is quasiasymmetric, and the central pair of doubly bonded carbon atoms in
a-C-C-
/ \
II
C
C
a
C-a
I
II
ba
/I-\
C
/ \
ba
b
and the pair of doubly bonded carbon atoms in U
\ /" C
I
C
/ \
a
b
are quasigeometric units. When both the carbon atoms of a quasigeometric unit are different an extra possibility or stereoisomer can exist.
CALCULATION OF STEREOISOMERS IN COMPOUNDS
95
When a structural formula or part of one shows no structurally identical regions for which p exceeds unity no quasiasymmetric atom nor quasigeometric unit is present. For such a formula or part of one, N = pa.pb-pc . . and N = 2”, where n is the sum of asymmetric atoms and geometric units, When structurally identical regions having p > 1 are present, quasiasymmetric atoms and quasigeometric units may exist, and all combinations of the possibilities of component regions are not different. Hence N need no longer be equal to the product of the numbers of possibilities of component regions. Structurally identical geometric units attached to the same carbon atom or to themselves, or two structurally identical regions attached to a doubly bonded carbon atom, have the same complete equivalence that exists in structurally identical regions in a saturated chain compound. Hence, except when the double bond introduces a new kind of equivalence, the method of selecting and combining regions described in the previous section and the formulas used will enable us to calculate the number of stereoisomers for a given structural formula whether it contains asymmetric atoms, geometric units, or both. In most cases doubly bonded carbon atoms introduce no new equivalences or methods of combination. Thus for
-
a(;
/I
(2
a
/ \
has two possibilities and there are two of these groups joined by
b
a quasigeometric unit, so N = a
I
b-C\
/
C
+
+
+ S = 3Cz + 2G = 4,just as for a
I
bd& b
a
\
C
N =: I S = 3C2 2Cz = 4. The following examples show fundamental cases of complex quasigeometric units, and geometric units and the corresponding analogous cases of quasiasymmetric and asymmetric atoms. For each member of an analogous pair, the number of stereoisomers is the mme and is also given. In the formulas Greek letters have been used to represent regions resulting from any combination of asymmetry, quasiasymmetry and geometric and quasigeometric isomerism.
96
K. BRANCH AND ‘PERRELL L. HfTLL
GERALD E.
IX o!
\
a
X
7 C
/ \
7
CY
\
I
X’
IX and IX’,
Examples XI1 and XII’
C
/ \
/ \
B
r
Y
XII’
6
XIII’
+ S where S = Paba2- 1) + Paba
= [I2
N
I
C
XI’
Examples X and X’, Examples XI and XI’,
B
N =
C
II
/ \
B
B
/B
*\
C
C
/ \
XI11
/“
\
C
II
a
o!
\ /*
C
b
XI1
CY
C
IX’
Examples
XI
= PA’PB
- 1’1 Pa
2
‘
f SA*SB where PA = p a , PB = 12
+
PJPa
2
- I)]
Pa.P7
Examples XI11 and XIII’, N = 2pa.pp. p7.pa. However, geometric isomerism does include a type of equivalence not found in saturated compounds. This is the possibility of the structural identity of the two carbons of the geometric unit. The analogous case is not possible in optical isomerism as the fundamental unit, the asymmetric atom, consists of one atom. Examples are
In the first of these the equivalence is immaterial. The next two are essentially the same, so we shall give the solution for the more complex of the two. Hence we have the two special cases
'
The solutions are: Example XIV.
N = 2 PA 12
Example XV.
N
, where
1 gA+ j s.4, I
=
= pa.pp
PA
1
where PA = pa and SA = pa(pa - 1) 12 2
The following are more specific examples (here a, ,6 etc. are again single, asymmetric units) : Example XVI
D
Y
\
A
= 24 = 16, Ng = 2, NBD = 16.2 = 32, and N* = If2 = 3. Then N = pp' SS' = pp' Sp' = 3.32 2.1 -e32 = 128. 2.1
ND
C=C--cu-B-C-a
/
I
Y
II /\ C
6
B
a
b
+
B
B
d
a
I c=c / I
a D
\ / C /I b CY C--e \ / C / \ C-a a-C II II C C A / \ / \ \
a
Example XVII
N~
=
23 = 8,
h'D
N a = NAf = I:
+
= 1,
+S=
3 1 = 4, and NBD = 8 . 1 = 8. N = pp' SS' = pp' 4.3 = Sp' = Ii.8 -.8 2 10.8 6 - 8 = 128.
+
+
+
+
A'
ff
Lya
B E
1
6
+
+
e
f
1 I c=c--c=c
Example XVIII
N% = 23 = 8, N~ = I; 2.1 ~=3+-=4,and 2.1
+
C---a
II
C
b
\
I
/ \
C---a
d
II
A
C
/ \
d
F
b 97
ThenNBD= 2.64 = 128. Finally, N = pp' SS' = pp' Sp' = 3.128 2.1 -128, N 512. 2.1
+
+
+
9s
GERALD %.
K. BRANCH AND 'PERRELL L. BILL
Groups of the spirane type, a
\
b
/c=c-c*
/" - '="\ ' b
are analogous to geometric units and may be included in this method as geometric units, as long as no distinction between optically active and inactive forms is made. When the number of unsaturated atoms is odd, these units produce optical, and when even, geometric isomers. Isomerism resulting from steric hindrance has not been included in our method. The assumption is made that rotation about a single bond is sufficiently easy to prevent isomerism, and that about a, double bond is sufficiently hindered to produce isomerism. OPTICALLY INACTIVE FORMS
The method of working from the periphery to a chosen final group can be used to obtain the number of meso forms. It is necessary to keep account of both active and inactive possibilities. The number of inactive possibilities in a region, pi, or of inactive isomers in the inolecule Ni can be calculated for different types of structures by the equations given below. The number of active possibilities or active isomers, pa, or N,, can be obtained by difference. The fundamental rules used to obtain these formulas Eire as follows: (a) A geometric unit produces inactive isomers; (b) an asymmetric atom produces active isomers; (c) an asymmetric atom assures optical activity unless its mirror image is also present; (d) even then, compounds of the type a*
\ a/-\
/a
c / \
and
b
s
a
are active; (e) Compounds of the type abC=C. inactive and two active forms.
-
Cab
-
C, == Cab have two
We are indebted to Dr. James K. Senior for some valuable criticism. SUMMARY
A method for calculating the number of stereoisomers of straight or branched carbon chain compounds is described.
CALCULATION OF STEREOISOMERS IN COMPOUNDS
99
It is pointed out that the method is applicable to compounds containing asymmetric carbon atoms, units of geometric isomerism, or both. The method can be used to calculate the number of optically inactive forms, but it is then necessary to introduce a number of additional equations. These additional equations are given, with the type of structures to which they are applicable. BBIRKELEY, CALIF. REFERENCE (1) SENIOR, Ber., BOB, 73 (1927).
