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).