Jose 1. Carlos, Jr.
Areneo d e Manila University Loyolo Heights, Quezon City Philippines
Molecular Symmetry and Optical Inactivity
The criteria which have been used in the past to predict optical activityor inactivity have been insufficient or not easily applicable to complicated molecules'. In this paper, we discuss a criteriou for otpical inactivity which is both sufficient and easily applicable. A mathematical proof of the sufficiency is presented. Also presented and applied for the first time is a way of systematically finding hypothetical molecules which may be used to experimentally test the criterion. Historical Background ( 1, 2, 3)
In 1815, the French physicist Biot discovered the property of some molecules which is called "optical activity." He noted this property, for instance, in quartz crystals which turned the plane of polarized light in some cases clockwise aud in othen, counterclockwise. Quartz had been known to have two kinds of configuration; one looked like the mirror-image of the other. I n 1820, John Herschel1 suggested that there might he a relationship between the kind of quartz crystal and its effect on optical rotation, that one kind of quartz crystal would cause an optical rotation by the same amount but in the opposite direction to the other kind. It was Louis Pasteur who in 1848 investigated the view of Herschel1 and found it to be consistent with fact. The significance of configuration was a disputed question for quite some time. In 1873, Wislicenus suggested that optical activity might he explained by the geometrical arrangements of the various atoms in ordinary three-dimensional space. The following year, van't Hoff and Le Bel published their theory of the tetrahedral carbon atom in the case of CX4 molecules. With this theory they proposed probable arrangements of the atoms in certain organic molecules, as well as introduced the concept of an asymmetric carhon atom in organic chemistry. The theory could not at that time be proven concIusiveIy and thus it led to heated arguments and personal attacks among chemists. About fifty years later, the results of infrared and Ramau spectra proved the theory (4). Such was the significance of the theory of van't Hoff and Le Be1 that structural formulas had to be extended to three-demensional space; thus, it became the basis of all stereochemical discussion. From their theory, van't Hoff and Le Be1 came up with a norm for the presence of optical aetivity, namely, optical activity exists when a molecule contains an asymmetric carbon atom, asymmetric in that the four ligands to the carbon atom are different from one another in composition or structure. Thus the theory waa found applicable in the case of the isomeric molecules (I) and (11), in which 248
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lournol of Chemical Mucotion
R is an unsymmetrical radical such as menthyl D-R x-C-Y
L-R
I
I
Y-C-x
I
I
The asymmetric carbou atom criterion can, however, be shown to be insufficient. Apparently, the uorm works satisfactorily for molecules with a single carbon atom. If several asymmetric carbon atoms are present, it is quite possible that the molecule may no longer he asymmetric. The asymmetric carbon atom criterion, although it is an unsatisfactory norm for optical activity, is still among the criteria usually employed in deciding whether optical activity mill be present. This norm is used not exclusively for carbon: it works for central atoms that can havc three or more ligands, that is, when the stereochemistry of the moleculeis significant. Thus, the asymmetric carbon atom criterion is not acceptable on two counts: first, there exist optically inactive molecules which have asymmetric carbon atoms. A classic example is nzeso-tartaric acid (111), where there are two asymmetric carbou atoms, and yet the molecule is optically inactive. Second, there exist optically active molecules which htlvc no asymmetric carbon atom. Agaiu a classic example is the keto-dilactone of benzophenono2,4,2',4'-tetracarboxylic acid (IV). COOH
I HO-C-H I
HO-y-H
C-0
E
l
o
o
c
~
0-C
COOH
It was Pasteur who suggested that optical inactivity is due to molecular symmetry, that is, the presence of a nonsuperposable mirror image implies optical activity. This norm works for the two examples cited above. Thus it seems that asymmetric carbon atom criterion is only a special case of Pasteur's more general principle of symmetry. Symmetry Considerations
Since Pasteur's principle is based on the symmetry of the molecule, it is plausible that optical inactivity, or activity, can be predicted from the symmetry classification of molecules, that is, the classification according to the operations which can be performed on a
{
~
molecule which result in a configuration equivalent to the original configuration. The symmetry properties of a molecule are described by five basic symmetry operations to which are associated corresponding symmetry elements and their symbols (5-7). These are shown in the table.
that in this particular instance the symmetry operations actually do commute, i.e., uhCn= Cnun. We therefore pick an arbitrary point in space (xo,yo,zo)and perform the operation uhand C , in both ways:
Svmmetrv Pro~erties
Symbol
E
C.
: 8.
Symmetry Element
Symmetry . Oueration .
Leave the molecule unchanged. Rothte it about im axis through z,/n. plane of symmetry Reflect it in the plane. center of symmetry Invert it through the center of symmetry. n-fold alternating Rotate it about an axis through BXIB 2rIn and reflect it in a ulane pebendicnlsr to the ax&.
I
identity n-fold =is
A particular set of symmetry operations form a symmetry group. For instance, the symmetry group S4 includes the symmetry operations of E, C2, S4, and 82. We should notice that in some cases a symmetry element can be broken down to simpler elements. When n is odd the presence of S. necessarily implies the presence of both C , and an, (a horizontal mirror plane) whereas such is not the case when n is even. A trivial case of molecules with mirror images that are clearly superposable is any molecule with u ( G 8,) or i (= 8%).Then obviously all molecules with the S, axis where n is odd are optically inactive since S , (n odd) implies the presence of both C , and un. Likewise, we expect optical inactivity for molecules that belong to the symmetry groups St,S6,SIO, etc., since these symmetry groups include i. A mathematical proof showing the equivalence of the mirror image superposability and S , criteria should now be presented. By definition 8, = uhCn. That is, first a rotation of 2s/n is performed and then a r e flection through a plane perpendicular to the C , axis. By the term "superposable mirror image" we mean that the mirror image of the molecule can be translated and rotated in some manner to bring the image into coincidence with the original molecule. The position of the mirror plane is arbitrary to the term "superposable mirror image." We usually think of the process intuitively by choosing any mirror plane external to the molecule. However, there is no reason why the mirror plane should necessarily be outside the molecule in question! Since the mirror plane is arbitrary, we are free to choose it anywhere-even inside. Suppose a molecule possesses an S. axis (the operation S. being equivalent to the successive operations C , followedby U A : S , = unC.). If wechoose the mirror plane to be identical with the plane ah involved in the rotation-reflection operation S , then it turns out that we have only to perform a rotation C , afterwards to bring the image into coincidence with the moleculethat is, to superpose the mirror image. The preceding argument shows that mirror image superposability can be represented as Club while the S , axis criterion for optical inactivity involves the operations in the reverse order, S , = ahC,. Symmetry operations do not necessarily commute. So here, all we need to do to prove the equivalence of the S, axis criterion to mirror image superposability is to prove
Here we have chosen to arrange our coordinate system such that the z-axis is coincident with the C , axis and the xy-plane coincident with uh. We obtain the same result. Therefore ah and C , commute. The S , axis criterion for optical inactivity is then equivalent to the mirror image superposability criterion. Since it has been shown that the presence of a superposable mirror image is equivalent to the presence of an S , axis, we claim that the presenceof 8, guarantees optical inactivity, that is, the presence of S , is sufficient for optical inactivity. We should like to experimentally verify this. The particularly interesting cases are the non-trivial ones, that is, those molecules which have neither i nor u but are still necessarily optically inactive due to the presence of S , (S4,S8,SI2, etc., i.e., Sap). Although there are several molecules that belong to the group S4, only a few have yet been synthesized (8-10). I n the case of molecules containing an S I axis, it can be observed that the structure is basically perpendobiplanar, that is, the structure consists of some kind of plane perpendicular to another plane. Such a structure can be expected in quaternary tetrahe dral types of molecules, such as derivatives of methane, neopentane, pentaerythritol, ammonium ion, and others, where the four substituent groups are in some way dissymmetric. The basic perpendobiplanar stmcture may also be expected in quadrisymmetric cyclic molecules, such as the derivatives of cyclobutane and cyclooctane. A third type may be the stericallyhindered biphenyl derivatives. A fourth type may be the spiranes. Finally, we may consider the derivatives of the cuhane-type or octahedral molecules. Shown below are models of molecules belonging to 8,; the configuration of A is the mirror image of A'. H H I I A'=-C-CH, A=-CICHJ I I CaH6
CzHs
Volume 45, Number 4, April 1968
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It is unfortunate that there are so few known molecules that belong to S4,Ss,S12and SO on, in order that we may experimentally test the optical inactivity of such molecules. However, we can suggest a systematic way of finding such molecules. The method is simple: (1) Find molecules of high symmetry containing the desired symmetry element S.. (2) Find ways of reducing the symmetry by proposing a systematic destruction of the unwanted symmetry elements. We shall try this method in the case of the possible molecules belonging to Ss. From the group character tables we find that a molecule having an element Ss will belong to either Can,Dab Dad,or Ss. We can, however, reduce the problem to the symmetry groups Dsh and Dad. Molecules belonging to Can and Daj, have the same basic structure of a planar octagon, or perhaps, octagonal bipyramid; Danhas a higher degree of symmetry than Csh. iV10lecules belonging to Dadand Ss also have the same basic structure of a square antiprism; of the two, Dl,, has the higher symmetry. The use of the basic structures for Dad and Danpoint groups is very helpful in symmetry considerations since only the minimum requirements of such structures are involved. These basic structures are merely examples and other structures might crop up.
Is S, a Necessary Criterion for Optical Inactivity?
The question which now arises is "Is the presence of an 8, axis a necessary criterion for optical inactivity?" I n 1954, Mislow (If) postulated a molecule which belongs to the group Cl and predicted it to be optically inactive: meso-4,4'-di (see-buty1)-2,6,2'GJ-tetramethyl biphenyl (VI). It was pointed out that this asymmetric molecule posesses no center of inversion, plane of symmetry, nor S, axis, even though there 1s restricted rotation between the two benzene rings and free rotation between the benzene rings and the butyl groups. A year later hlislow and Bolstad (IS) synthesized a molecule of such properties: D-klenthyl-Lmenthyl-2,6,2'G'-tetranitro-4,4'-diphenae (VII). This
0-n-menthyl VII
molecule is optically inactive by experiment and is a configurationally pure compound. The molecule may perhaps he regarded like a meso-diastereomer which turns out to he optically inactive. Tho molecule may also be considered as an unresolvable racemic mixture. I n any case, the molecule is unique in that it appears to belong to the point group CI and not to have an S, axis. The absence of an Sasymmetry element was emphasized by filislow (14) since performing the S4 symmetry At present, no molecule belonging to Dgh or can operation would necessarily require an additional has yet been synthesized (11). This can perhaps be rotation by a/2 of one of the menthyl groups in order explained by an apparent overcrowding of ligands to obtain the original conformation. along one plane. For molecules with eight ligands, It should be remcmbered, however, that there is free the more probable structures would be the cube rotation between the menthyl groups and the benzene (Oh), irregular dodecahedron (D2
