Read an Excerpt
Introduction to Stereochemistry
By Kurt Mislow Dover Publications, Inc.
Copyright © 2002 Kurt Mislow
All rights reserved.
ISBN: 978-0-486-15115-1
CHAPTER 1
STRUCTURE AND SYMMETRY
1–1 Introduction
MOLECULAR structure is a description of the arrangement or distribution of particles in a molecule. We shall elaborate this definition by explaining our use of the terms description and particles.
A description may take many forms. It may be verbal, or it may be visual—as, for example, through the use of molecular models. The most accurate descriptions are also the most complex: These are the equations which describe the motion of the particles as a function of time and spatial position.
There are two kinds of submolecular particles which are of interest to the organic chemist: the electrons and the nuclei. Wave-mechanical arguments lead to a description of electronic structure as a probability distribution of negative charge, i.e., as a smeared-out charge density cloud. In contrast, the distribution of nuclei in space may be discussed more nearly in classical terms, i.e., we may think of nuclei as particles which vibrate with a very small amplitude around well-defined average positions in space. These vibrations are completely analogous to the quivers executed by two weights connected by a spring: the coulombic and exchange forces in the molecule simply take the place of the mechanical restoring force in the spring.
Compared to organic molecules, the hydrogen molecule is an extraordinarily simple one (in fact, except for its positive ion [H 1/2], it is the simplest polyatomic molecule conceivable). It might therefore be appropriate to begin our discussion of structure and symmetry by analyzing the case of the hydrogen molecule. In the formation of that molecule, two protons and two electrons have been brought together. The electrons distribute themselves so that the position of the protons assumes an equilibrium value. This distribution of protons and electrons is indicated in Figure 1–1, which shows a cross section of the molecule made by a plane which contains the nuclei. This plane shows contour lines of equal electronic charge density, and the three-dimensional contour surfaces may be developed by rotation around the internuclear line x. Note that well-defined positions are assigned to the protons, whereas the electrons can only be described in terms of the over-all charge density. The molecule may be pictured as a roughly cigar-shaped region of electron density within which the two protons are buried. As shown by the contour lines, the density is highest at points on the x-axis corresponding to the positions of the nuclei, falls off at all distances, but remains high in the region between the nuclei. This region of electron localization between the nuclei coincides in direction with the internuclear line. The outermost contour line shown in diagrams such as these is conventionally chosen to enclose in the neighborhood of 95% of the electron cloud, that is: the probability of finding the electron or electrons within the outermost contour is about 95%.
This description of the distribution of two electrons in the field of the two protons corresponds to the bonding molecular orbital (MO) of hydrogen molecule.
The protons jiggle about but maintain an equilibrium distance r, the bond length. The bond length (10.74 Å in this case) refers to the molecular configuration of the vibrating system at a potential energy minimum. The potential energy of the molecule is raised whenever the bond is stretched (say to 10.75 Å) or compressed (say to 10.73 Å), very much as in the analogous case of the two vibrating weights connected by a spring.
We shall now consider symmetry in simple molecules. Every molecule may be classified according to its symmetry, and each particular symmetry class is characterized by the number and type of symmetry elements present in such a molecule. For our immediate purposes, the most important such elements are planes of symmetry, symbolized by σ, and simple (or proper) axes of rotation of order n (n-fold axes of symmetry), symbolized by Cn. A σ plane is defined as a mirror plane which bisects a geometric figure so that the half of the figure on one side of the plane is exactly mirrored by the half on the other side. Objects like idealized forks, spoons, hammers, and cups have just one plane of symmetry, and so do molecules like nitrosyl chloride (Figure 1–2), bromocyclopropane and vinyl chloride. A Cn axis is defined as an axis which passes through the molecule so that, by a rotation of 360°/n around this axis, a three-dimensional arrangement is obtained which is indistinguishable from the original. For example, water has a twofold axis of symmetry and ammonia a threefold axis, as indicated in Figure 1–2. It is important to note that planes and axes of symmetry are often both encountered in the same molecule. For example, while nitrosyl chloride does not have an axis of symmetry (the trivial one-fold axis C1 is never considered), water and ammonia have, respectively, two and three planes of symmetry (Figure 1–2) which intersect at the Cn axis. The angles between the planes, the dihedral angles, have the values 90° and 60° for water and ammonia, respectively.
Let us now apply a similar analysis to the case of the hydrogen molecule. It is readily seen (Figure 1–3A) that there exists an infinite number of σ planes which are identical with the cross section shown in Figure 1–1 and which intersect at the internuclear axis. This axis, which is called the cylindrical axis, is also C∞ because an infinitesimal rotation suffices to transform the new position into one indistinguishable from the original. In addition to the above-mentioned planes there also exists a plane which is perpendicular to the internuclear axis, which bisects the molecule, and which contains an infinite number of C2 axes. Molecules such as oxygen and carbon dioxide have that kind of symmetry which is called cylindrical symmetry. Objects such as idealized hourglasses, footballs (American), and doughnuts also have cylindrical symmetry. Conical symmetry is a closely related kind of axial symmetry. In molecules possessing conical symmetry, e.g., hydrogen chloride, there also exists an infinite number of σ planes which intersect at the internuclear axis C∞. However, we now have no σ plane and no C2 axes which are perpendicular to the C∞ axis (Figure 1–3B). Objects like idealized funnels, saucers, soda bottles, pins, and eggs have conical symmetry.
Cylindrically and conically symmetrical objects have one C, axis. The only object possessing more than one such axis is the sphere, which has an infinite number of C∞axes intersecting at the center (spherical symmetry).
We shall elaborate on the subject of symmetry elements in Section 1–4.
1–2 Bonding Geometries in Carbon Compounds
Although molecules containing carbon are far more complex than the hydrogen molecule, a description of bonding in organic molecules may be approached in fundamentally the same manner. A carbon nucleus, one or more other nuclei, and the appropriate number of electrons are brought together, and the electrons are allowed to distribute themselves in a fashion which stabilizes the equilibrium configuration of all the nuclei. As in hydrogen molecules, there exist regions of high electron density between the nuclei which generally coincide in direction with the internuclear lines. The atoms thus bonded to the central carbon atom are called ligand atoms. The nonbonding electrons, e.g., the three pairs of nonbonding electrons in chloromethane or the lone pair in trimethylamine, give rise to additional regions of high electron density in the molecule.
The number of ligand atoms defines the coordination number. Carbon in its various stable combinations may exhibit coordination numbers varying from one to four. We shall discuss the geometry of bonding separately for each coordination number.
COORDINATION NUMBER ONE This coordination number is exhibited by cyanide ion and carbon monoxide. These molecules have conical symmetry and in that respect resemble hydrogen chloride.
COORDINATION NUMBER TWO In triatomic molecules—and carbon with a coordination number of two implies a minimum of three atoms in the molecule—our description of bonding geometry must take into account two factors. First, the ligand atoms may be equivalent or they may be nonequivalent. Second, the molecular array may be linear or nonlinear. As it happens, the latter question rarely arises in practice, for stable compounds of carbon in which carbon exhibits a coordination number of two are linear. The angle between the bonds is 180°.
Carbon dioxide is a linear molecule in which the ligand atoms are equivalent and in which the bond lengths are equal (1.16Å). Carbonyl sulfide, hydrogen cyanide, and acetylene are linear molecules in which the ligand atoms are not equivalent and in which the two bond lengths of carbon must therefore be different. Carbon suboxide contains one carbon atom with two equivalent ligands and two carbon atoms with different ligands (see Formula I).
It may be noted that carbon dioxide, carbon suboxide, and acetylene have the cylindrical symmetry of hydrogen molecule, while hydrogen cyanide and carbonyl sulfide have the conical symmetry of hydrogen chloride.
COORDINATION NUMBER THREE Since four atoms need not lie in one plane, combinations of carbon with three ligand atoms raises for the first time the question of nonplanarity. However, with the exception of carbanions (and possibly radicals), in which the unshared electrons may be regarded as occupying an additional position on the coordination sphere, the most stable groupings containing carbon bonded to three atoms are planar, and our discussion is therefore considerably simplified.
Groupings of this type in which carbon is attached to three identical ligands (as in graphite) are quite exceptional. Other than carbonate ion, the only common members in this group of compounds are the carbonium ions R3C+, of which the methyl cation (CH3+) is the simplest representative. Methyl cation has trigonal symmetry, the chief attributes of which are summarized in Figure 1–4. The three ligands are completely equivalent in space and it is easily seen that the bond angles of such arrays must be 120°. However, in the vast majority of compounds containing tricoordinate carbon, the carbon atom is attached to noneVCquivalent ligands. Hence, the totality of the interactions of the three ligands (including nuclei, bonding, and nonbonding electrons) with each other no longer has regular trigonal symmetry (Figure 1–5). Because the distribution of electrons around the central carbon atom now does not have regular trigonal symmetry, the three bond angles cannot be 120°. In other words, even if the bond angles in such cases were experimentally found to be 120°, such an observation would merely indicate that the differences in the interactions of the three ligands are very slight, and therefore that the differences in the bond angles are extremely small. It cannot be doubted that such differences, though they might perhaps be too small to be observable with currently available measuring techniques, would nevertheless be finite and could in principle be detected by more sensitive devices.
Actually the differences in the bond angles are usually large enough to be detected, and a few examples can be adduced (Formula II):
[FORMULA NOT REPRODUCIBLE IN ASCII]
One exception to the generalization implicit in Figure 1–5A is the case of a molecule having D6h symmetry (section 1–4), such as benzene (Formula III):
[FORMULA NOT REPRODUCIBLE IN ASCII]
This planar array of twelve atoms has perfect hexagonal symmetry (i.e., it contains a C6 axis perpendicular to the plane of the paper which is also a σ plane) and all the bond angles are precisely 120°. Another example is the inner ring in coronene (Formula IV):
[FORMULA NOT REPRODUCIBLE IN ASCII]
Symmetry arguments such as these are extremely useful in stereochemical discussions. It must be re-emphasized that the nature and magnitude of the electronic interactions are completely immaterial to the validity of such arguments and become important only when we attempt to predict or justify the direction and magnitude of the observed effects.
COORDINATION NUMBER FOUR Groupings of this type, in which carbon is attached to four ligands, are the ones most commonly encountered in organic chemistry. In diamond these ligands are identical. As shown by a great variety of physical measurements, the atoms attached to carbon in compounds of type CX4 are completely equivalent. Examples of such compounds are (Formula V):
[FORMULA NOT REPRODUCIBLE IN ASCII]
Equivalence in three dimensions means that a molecule of the type CX4 has only one X—C—X angle and only one X—X internuclear distance. A square planar array of four X groups with a carbon atom at the center does not meet these specifications. It may be shown that equivalence in three dimensions signifies regular tetrahedral symmetry, an analysis of which is given in Figure 1–6. This figure shows a carbon atom, called the tetrahedral carbon atom, attached to four hydrogen atoms numbered 1 through 4. Inspection of the figure shows that there are four C3 axes, one for each of the C—H bonds. In this respect we are reminded of the case of ammonia (Figure 1–2C). Each of the four C3 axes in methane is the locus of intersection of three σ planes, so that one might expect a total of twelve planes: However, each of the σ planes contains two C3 axes, so that there are actually only six σ planes. These six planes may be ordered into the three pairs shown in Figure 1–6. In each pair the two planes are perpendicular to each other and they intersect to form a C2 axis. The three resulting C2axes are mutually perpendicular, and they bisect the six tetrahedral angles.
The value of the regular tetrahedral angle is arc cos -1/3. Expressed in degrees, this is approximately 109.5°.
As soon as the ligands are no longer equivalent, the regular tetrahedral symmetry vanishes since the various interactions of the ligands around the central carbon are no longer identical. It follows that the bond angles can also no longer be 109.5°. The conclusion of this symmetry argument is borne out by numerous experimental observations. Two examples are given below (Formula VI):
[FORMULA NOT REPRODUCIBLE IN ASCII]
Even in acyclic hydrocarbons, such as propane and n-butane, the C—C—C bond angle is approximately 112°, not 109.5°. The regular tetrahedral angle is therefore the exception rather than the rule in organic chemistry.
1–3 Bonding Orbitals in Carbon Compounds
When atoms are brought together to form a molecule, the electrons and nuclei interact. The electronic distributions in the atoms (i.e., the atomic orbitals or AO's) are perturbed. We say that the AO's overlap. The charge density distribution in the localized bonding MO bears a relationship to the charge density distribution in the original AO's, and it is therefore customary to discuss the localized bonding MO with reference to the AO's which would result at infinite separation of the nuclei. For example, in the case of hydrogen molecule the bonding MO may be mathematically described in terms of the component hydrogen AO's.
A discussion of localized bonding orbitals in carbon compounds is rendered far more complex by the fact that carbon has four low-lying AO's which are available for bonding, whereas hydrogen has only one (i.e., 1s). These four orbitals are the 2s and the three 2p orbitals. In its lowest energy state, the carbon atom has two electrons in the 2s orbital and two electrons in the 2p orbitals, so that the description of the electron configuration is 1s22s2p2. A slightly higher energy state is described by 1s22sp3, where one electron occupies each of the four AO's. Like all s-orbitals, the 2s-orbital is spherically symmetrical and has its highest charge density at the nucleus. Each p-orbital, however, is cylindrically symmetrical, and the charge density at the nucleus is zero. The three p-orbitals differ only in direction in space and the three cylindrical axes define a Cartesian coordinate system (x, y, and z). The σ planes which are perpendicular to the cylindrical axes (i.e., yz, xz, and xy, respectively) define the regions of zero electron density and are the nodal surfaces.Figure 1–7A shows projected contour lines of equal electronic charge density for the 2pz AO. The three-dimensional contour surfaces may be developed by rotation around the C∞ axis z. It is seen that the charge density is concentrated in the lobes, two separated distorted ellipsoids which float above and below the nodal xy-plane.
If these AO's were directly developed into bonding orbitals, tetra-coordinate carbon would have three equivalent bonds at right angles to each other, and a fourth nonequivalent bond. This does not correspond to any experimentally observed geometry of carbon bonding orbitals. It follows, therefore, that the wave function (i.e., the mathematical description of the motion and distribution of an electron in the field of the nucleus) of the bonding AO's must be different from the wave functions of the AO's of unbonded carbon.
(Continues...)
Excerpted from Introduction to Stereochemistry by Kurt Mislow. Copyright © 2002 Kurt Mislow. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
