geometry in chemistry
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Geometry in Chemistry
Chemistry is the science behind the structure and reactivity of the most basic (and for the
most part, ubiquitous) building blocks of all matter: the elements. A look at the combination of
any number of elements shows a variety of accessible geometries and structures. Chemistry has
advanced to the point where structures once proposed as conjecture can be analyzed and
confirmed with high levels of confidence. Much of the studies done by medicinal chemistry
groups in pharmaceutical companies look at structure-activity relationships, seeking to find the
ideal structural conformation and biochemical activity related to that structure. So much of
chemistry is based on knowing the 3-D structure of molecules and in order to appreciate the
beautiful presence of geometry in Chemistry, one needs to take a look at one of the most
fundamental starting points for molecular structures: valence-shell-electron-pair-repulsion theory
(VSEPR theory).
VSEPR theory is one of the simpler geometric models used to predict the shape of
molecules based on electron pair repulsions. As the name states, only valence (on the outermost
region) electrons are included, since they are the electrons most involved in bonding interactions.
The theory states that “electron pairs repel each other whether they are in chemical bonds (bond
pairs) or unshared (lone pairs). Electron pairs assume orientations about an atom to minimize
repulsion.”1 Chemical bonds are described as the means of connectivity between atoms or
molecules. Most bonds involve the interaction of electrons. The use of VSEPR as a predictive
tool for molecular geometry depends a proposed Lewis dot structure. Using VSEPR to predict
molecular geometries from Lewis dot structures requires counting lone electron pairs and
molecular groups as generic “groups” both occupying the same amount of space. With two
Figure 1 - Tetrahedral, trigonal bipyramidal and octahedral molecular geometries
1 Petrucci, R. H.; Harwood, W. S.; Herring, F. G. General Chemistry UBC Chemistry 121/123 Edition, Volume A; Prentice-Hall: New Jersey, 2002; Chapter 11.
“groups” around an atomic centre, a linear geometry is predicted; with three “groups”, a trigonal
planar geometry; with four, a tetrahedral geometry; with five, a trigonal bipyramidal geometry;
with six, an octahedral geometry.
For a tetrahedral geometry, if given a bond length of r, then the distance between two
adjacent atoms2 (bound to the same central atom) is r322 . The angle formed (going through the
central atom) between these two adjacent atoms is calculated to be ideally 109.5°.3
For a trigonal bipyramidal geometry, the angle formed between axial and equatorial
groups is 90°. The angle formed between two equatorial groups is 120°. If given a bond length of
r, then the distance between an axial and equatorial atom is r2 . The distance between two
equatorial atoms is r3 .
For an octahedral geometry, the angle formed between any atom and the central atom is
90°. The angles formed between any of the non-central atoms are either 45° or 60°. If given a
bond length of r, then the distance between any two non-central atoms is r2 .
Depending on how many of the “groups” around the centre are lone pairs, the remaining
“groups” are positioned in such as way as to minimize the lone pair-lone pair interactions. Lone
pair-lone pair interactions are greatest when they are 90° apart. In a very simplistic approach to
Chemistry, these would be sufficient, but it is hardly the case. VSEPR theory is a useful
geometric structure predictive tool (but not always accurate) for the main group elements (i.e. the
elements in Groups 13 to 18, the p-block). The problem is, the main group elements only account
for approximately one-fourth of all elements. There are other bonding theories that can explain
more complex examples that are not explained with the simpler bonding theories and one of the
most commonly accepted and highly descriptive in terms of bonding is molecular orbital (MO)
theory.
MO theory describes chemical bonds in terms of the mathematical combination of atomic
orbitals—wave functions that have the potential of showing the probability distribution of an
electron around a nucleus—that make up an entire molecule. It takes a look at the basic atomic
orbitals (s-, p-, d- and f-orbitals) and describes their combinations as molecular orbitals with
increased electron density between nuclei (known as bonding orbitals) or molecular orbitals that
2 All of the ideal bond lengths were determined using Pythagoras’ theorem and the cosine law. 3 Gillespie, R. J.; Hargittai, István. The VSEPR Model of Molecular Geometry, Allyn and Bacon: Massachusetts, 1991; Chapters 1-3.
are formed from the destructive interference of the wave functions (known as antibonding
orbitals). When electrons occupy bonding orbitals, the bond between the two atoms increases in
strength (bond order increases). When electrons occupy antibonding orbitals, the bonds between
the two atoms decreases in strength (bond order decreases). MO theory gives a good explanation
of the higher order bonding (e.g. double bonds, triple bonds) in molecules. Not only does MO
theory give a good explanation of the molecule itself, but the molecular orbitals formed from the
atomic orbital combinations (either bonding or antibonding) can be used for interactions with
other molecules, as is one of the most important parts of describing reactivity.
Organometallic chemistry is the chemistry of compounds involving metal-carbon bonds.
The description of bonding in organometallics complexes can be explained through the
geometric arrangement of atomic orbitals on the central metal and the molecular orbitals on the
bonded molecular groups (known as ligands). One example of the geometric beauty of
organometallic chemistry is Zeise’s salt, a platinum complex containing ethylene (C2H4) as a
ligand. The dx2-y2 orbital on platinum can interact with the π (pi bonding) orbital, forming a σ
(sigma) bond with between platinum and ethylene. The dxy orbital on platinum can interact with
the π* (pi antibonding) orbital, causing back-donation due to occupation of an antibonding orbital
on ethylene. The orbital geometry as exemplified in Zeise’s salt helps to explain the interaction
between platinum and ethylene and why the physical characteristics are the way they are.
Without considering the suitable geometric molecular orbital overlap, there would not be any
reason for the increased carbon-carbon bond length.
Another example of perfect geometric orbital overlap is in chromium hexacarbonyl. The
central metal has access to five different d-orbitals for bonding with molecular orbital
combinations of the six carbonyl (CO) ligands. The geometry of the molecular orbitals around
the octahedral centre shows perfect symmetry. The central dz2 orbital interacts simultaneously
with all six σCO molecular orbitals, four with the central “donut” and two with the axial lobes.
The perfectly arranged pi-interactions between the central chromium’s d-orbitals and the pi-
antibonding orbitals of the carbonyl ligands help to explain some of the chemical reactivity
observed in carbonyl complexes: for example, the electrophilic character of carbonyl ligands due
to electronic occupation of their pi-antibonding orbitals.
Putting aside the theoretical discussion of geometry, there are some more visible
geometric structures in chemistry. (C60-Ih)[5,6]fullerene4 is a member of the chemical class of
compounds known as fullerenes, a family of carbon allotropes (molecules consisting of only
elemental carbon but having different structures). Buckminsterfullerene (or “buckyball” or C60 as
sometimes called) has a beautiful symmetry, having the same
symmetry as a soccer ball. Being formed from geometric
arrangements of pentagons among hexagons, C60 can be
geometrically formed from the truncation of the Platonic solid, the
icosahedron, with carbons atoms located at each vertex of the
icosahedron. The polygons in the buckyball C60 are regular polygons
(having each internal angle 108° for the
pentagons and 120° for the hexagons), causing the buckling to form a
sphere-like object. There are other carbon allotropes falling under the
fullerene classification: C20 (taking the form of the dodecahedron, with
carbon atoms at each vertex), C70 (shaped like the buckyball, but slightly
more cylindrical), C76 and C84 (both elongated cylindrical structures). The
molecular amazement of C60 doesn’t just stop at its sheer geometric symmetry; C60 has found its
place in chemical research. For example, C60 coated with lithium has been investigated for its 4 Fullerene. http://en.wikipedia.org/wiki/Buckyball (Accessed on April 2, 2009).
potential use as a stable hydrogen storage media.5 Other uses of C60 include the doping of C60
with cesium to obtain the superconducting Cs3C60 at 40K.6
Another interesting structure (but not a carbon allotrope) is cubane, C8H8. Cubane7 is
called a Platonic hydrocarbon because it is the molecular equivalent of the Platonic solid, the
cube. Geometrically it is composed of faces that are cubes, with carbon atoms at each vertex and
one hydrogen atom bonded to each carbon. Cubane is highly strained (90° bond angles) but is
kinetically stable. As interesting as cubane may be geometrically, on replacing the hydrogen
atoms with nitro (-NO2) groups, a much more interesting compound is
formed. Octanitrocubane, C8(NO2)8, having the same molecular shape
(shape of the Platonic cube), is incredibly different from cubane.
Octanitrocubane is a highly explosive compound, due to the
geometrically ideal (but highly strained) 90° bond angles and its
decomposition into carbon dioxide and nitrogen gas.8 The
hybridization of each carbon atom is sp3 (meaning its most stable bond
angle is 109.5°) so in cubane or octanitrocubane (and other chemicals with strained bond angles)
energy is released from strained systems. (Going from 90° to 109.5° represents the release of
substantial potential energy stored.)
One final example of the importance of geometry in Chemistry is seen through the
importance of stereochemistry. Stereochemistry describes the unique 3-D arrangement of
molecules around a central point. In pharmaceuticals, stereochemical specificity (known as
enantioselectivity) can mean the difference between a recognized medicine and a useless
molecule. (For example, for the racemic drug zopiclone only the enantiopure eszopiclone is
active.) A more practical result of the stereospecificity in products is polypropylene. Propylene is
known as a pro-chiral substrate, meaning that it has the potential to become chiral9 or develop a
potential for asymmetry with respect to spatial arrangement of molecules around the central
atom. With polypropylene, it is very important that there be control of which isomer is formed
5 Sun, Q.; Jena, P.; Wang, Q.; Marquez, M. J. Am. Chem. Soc. 2006, 128, 9741-9745. 6 Palstra, T. T. M.; Zhou, O.; Iwasa, Y.; Sulewski, P. E.; Fleming, R. M.; Zegarski, B. R. Solid State Communs. 1995, 93, 327-330. 7 Cubane. http://en.wikipedia.org/wiki/Cubane (Accessed on April 4, 2009). 8 Octanitrocubane. http://en.wikipedia.org/wiki/Octanitrocubane (Accessed on April 4, 2009). 9 Chirality is also sometimes known as handedness.
(isotactic, syndiotactic or atactic).10 The atactic isomer—with random orientations of the methyl
groups at the chiral centres—is amorphous, has a low melting point and with these
characteristics, finds practically no use commercially. The syndiotactic isomer—having
stereoregular alternating configurations along the whole polymer—also has a low melting point
and does not find much use commercially as well. The only commercially viable isomer is the
isotactic isomer, having all methyl groups in the same orientation—all pointing in the same
direction. This causes the overall polymer to twist into a helical shape and line up against other
helices, giving the polymer higher strength and crystallinity.11 Stereoregularity of the polymer is
obtained by using specific catalytic conditions: in this case, the Ziegler-Natta catalytic system of
a titanium (IV) chloride and trialkylaluminum. In terms of geometry, it is important in the
plastics industry to synthesize selectively the isotactic form of polypropylene. Without the unique
stereo-control of the polymerization, companies could spend millions of dollars to find that they
invested in amorphous goop.
Geometry is a very beautiful subject. In nature, there are so many examples of patterns
and shapes. From the basic molecular structure theory of VSEPR to the super-explosive
octanitrocubane and the ubiquitous isotactic polypropylene, hints of geometry are visible in
Chemistry. This paper, however, cannot do justice to the presence of geometry in Chemistry. The
importance of geometry in Chemistry sometimes may be overlooked, but hopefully we can learn
to appreciate the beauty within Chemistry.
10 Collman, J. P.; Hegedus, L. S.; Norton, J. R.; Finke, R. G. Principles and Applications of Organotransition Metal Chemistry; University Science Books: Mill Valley, CA, 1987; Chapter 11. 11 Polypropylene. http://en.wikipedia.org/wiki/Polypropylene (Accessed on April 3, 2009).
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