Group Theory - WordPress.com€¦ · Web viewGroup Theory Relevancy to Chemistry 4/15/2009 MTH 4110 Joel Guttormson Introduction Group theory is more applicable than one may realize

Group TheoryRelevancy to Chemistry

4/15/2009MTH 4110Joel Guttormson

Joel GuttormsonTerm PaperMTH 41104/15/2009

I ntroduction

Group theory is more applicable than one may realize. In fact, it is integral in the

understanding of chemistry. Specifically, this paper will focus on the group theory of chemical

bonds. Group theory greatly reduces the level of mathematical difficulty and furthers an

understanding of the complex interactions between atomic orbitals that form molecular bonds.

This paper will contain a brief discussion on the construction of character tables, which are

directly related to group operation tables to enable a more coherent explanation of the relevancy

of group theory to chemical bonds. The objective of this paper will be to show the deep

interconnectedness of group theory and chemical bonding in a way that mathematicians as well

as chemists can understand it.

Symmetry and Point Groups

Though most mathematicians know what symmetry is, it is useful to see how chemists

define symmetry. Symmetry is, “that property of a body (or pattern) by which the body (or

pattern) can be brought from an initial spatial position to another, indistinguishable position by

means of a certain operation, known as a symmetry operation.” (Ladd, 1998, p. 5) Though this

definition is worded quite differently from our mathematical notions of symmetry, it is

nonetheless equivalent. Another useful definition is that of something called a point group. A

point group, the use of the word group is justified as will be shown later, is “a set of symmetry

operations, the action of which leaves at least one point of the body invariant, or unmoved”

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(Ladd, 1998, p. 46). In this discussion, it will be useful to focus our examination on a simple and

well-known molecule, water. It is common knowledge that water is comprised of two hydrogen

atoms and one oxygen atom, forming a molecule. At first, you may be thinking, as I did, that the

water molecule follows the same symmetry pattern as that of the group or of the symmetry

properties of an equilateral triangle. However, the symmetry of the water molecule is more

complex than that due to something called an orbital. An orbital is “a specific region of

an atom” that “can contain two electrons with paired spins.” (Anne Marie Helmenstine,

2009)Orbitals, in diagrams, show the charge associated with the atom in the molecule (see Figure

1 below, (Kettle, 2007, p. 20)). These orbitals can have a significant effect on the group

structure. Whereas an equilateral triangle can be “flipped” about a vertical axis, call it z,

and that symmetry operation would be considered an identity operation, this is not the case with

the water molecule. Not only do orbitals complicate the matter of symmetry, rather it is further

complicated by the undeniable fact that, while an equilateral triangle is a two-dimensional object,

a molecule is, at least, a three-dimensional object. However, due to this complication of three

dimensions we will not consider auxiliary axes such as the lines . Thus, we will

only considered what can be called the “normal” axes of x, y and z with their standard positions

and directions intact.

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Figure 1

The Water Molecule: A Symmetry Group

We know water is comprised of three atoms; two hydrogen and one oxygen. The

arrangement, (see Figure 2 below), allows us to visually examine the symmetry group properties

of the water molecule.

Figure 2

As stated previously, the identity element, E, is not the rotation about the z-axis, but a

rotation about the z-axis. There are of course, other symmetry operations. is the operation

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that rotates the water molecule about the z-axis. is an operation that requires a more

sophisticated definition. “The operation…leaves the phases unchanged (although the ‘front’

and ‘back’ of each lobe are interchanged.” (Kettle, 2007, p. 20) It has the effect of putting a

“mirror plane” through the molecule, hence the subscript v for vertical. For further visual

reference, it “has the effect of interchanging the two hydrogen atoms.” (Kettle, 2007, p. 15) The

operation, like the operation, reverses the phases but does so by reversing the lobe of just

the oxygen atom (see Figure 3 below).

Figure 3

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Now that we have these operations, we can develop a multiplication table to see how the

different operations interact with each other.

By visual inspection of Table 1 below, it is clear that it represents a group, by the

informal definition. That is, all elements are represented in the table, no element appears twice

in the same row or column, and there is an identity element. However, this is not sufficient and

thus we must formally prove that this set, call it is a group. For to be a

group it must be nonempty with a single binary operation and “satisfy the following axioms: 1.

Closure: .; 2. Associativity:

.; 3. There is an element

.; 4. For each there is an element such

that ” (Hungerford, 1997, p. 163)

Table 1

E

E E

E

E

E

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Proof: Define . Then is nonempty. The binary operation of is

“symmetry multiplication”, that is, the consecutive application of symmetry operations. is

closed by observing that in multiplication table an element in the table that is not in .

Symmetric operations are known to be associative. By observing the table, each element is its

own inverse, that is, . There is an identity

element, . Again by observing the table, the first leftmost column and the first row show

that is the identity since all elements times are equal to themselves. Thus, is a group.

From now on, we will denote the group as . We now have a fascinating result at hand.

The water molecule, because of its symmetry, forms a group1. Not only is it a group, but it is

also a point group, by definition, since for any rotation we leave the oxygen atom fixed. Note

that the interchange of the lobes of the oxygen atom does not negate or contradict the assertion

that is a point group or that it fits the definition of such. Now it is useful to expand upon the

aforementioned in the direction of the chemistry in order to show more correlation between it

and the mathematics.

1 Interestingly, is isomorphic to (the set of units in ). Since this result has no bearing on the remainder of the material, the proof of this result is left to the reader.

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Character Tables

Mathematicians would find the above satisfactory in the description of the superficial group

structure of mentioned and developed above. However, chemists require more. Recall the

elements of . In our development, it was left out that these operations have

what are called multipliers associated with them. These multipliers deal with whether or not the

phases are changed. For , it can be easily observed that the phase does not change

under those operations. These operations are called “symmetric”. An operation that is

“symmetric with respect to rotation by ” is given the multiplier value of 1. Those that are

not symmetric in this way, i.e. the phases change due to application of the symmetry operation,

are called “antisymmetric” and are given the multiplier value of -1. (Cotton, 1990, pp. 90-91)

These notions of “symmetric” and “antisymmetric” require us to shift our perspective and also

consider more advanced view of the phase patterns of the water molecule because as defined

they are not complete and do not really tell the whole story. Figure 4 below2 shows the

orientation we will be considering. It is good also to note that we are talking merely about the

phases of the oxygen atom in the molecule. The additional complexity of the consideration of

the hydrogen atoms of the molecule is outside the scope of this paper.

2 Figure 4 shows a shift in orientation with the main axis being the x-axis rather than the z-axis.

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Figure 4

At first glance, one will notice what appear to be two “extra” phases or lobes. For simplicity

only realize that our initial view left out that inside the oxygen molecule there are protons (and

neutrons of course, but they do not affect the charge of the atom) that may be configured in

different ways as to produce several configurations of the oxygen atom’s phases. The

configuration, shown above in Figure 4, is used for ease of understanding though it is

acknowledged that other configurations do exist. Below, in Figure 5, we see the symmetry

operations at work, however this time we get a clearer picture of what is happening. In E, we see

that not much has changed from our original look at it. The lobes do not change. However, in

we see that the two “smaller” lobes (phases) do in fact change. Thus, the above

description of what is going, as said, is not complete or for that matter, entirely correct because

we had, essentially, an incomplete picture. Then, our delegation of the numbers 1 and -1 to

certain elements of (i.e. symmetry operations) has, using the definitions of “symmetric” and

“antisymmetric”, E and as symmetric operations and as antisymmetric operations.

Why is this so? The answer lies within examination of Figure 5 below.

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Figure 5

If we examine the new, “smaller” lobes, we can see that in the charge of the lobes is

opposite that of the molecule before the operation. Whereas, in E and the lobes not only

remain unchanged but are virtually, but not identically, equal to each other. Now, we can begin

translating Table 1 into what is called a “character table”. A character table is a simple way to

tabulate the four representations of the ways that functions may transform in . (Ladd, 1998, p.

87) However, before we can create a full character table for we must first create the most

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important piece of the character table, which is its “irreducible representations”. It is in this

piece that the 1’s and -1’s become useful. Kettle offers the best explaination of how to translate

this notion and incorperate it with our knowledge of Table 1. “Everywhere in [Table 1] that the

operation E is listed, replave it by the number with which it is associated”. (Kettle, 2007, p. 40)

Table 2

E

1 1 1 1

1 1 -1 -1

1 -1 1 -1

1 -1 -1 1

Then, taking analogous information from above, we can complete the table as show above in

Table 2. With further substitution and the addition of which are the irreducible

representations of the individual lobes. (Ladd, 1998, p. 85) In other words, each row is an

irreducible representation of an operation and we represent each as above with the A’s and B’s.

Now it is possible to create our character table, Table 3 (Kettle, 2007, p. 42) from the above,

accumulated information.

Table 3

E

1 1 1 1

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1 1 -1 -1

1 -1 1 -1

1 -1 -1 1

Conclusion

Although the above examination was brief and touches only on a limited range of topics and uses

of group theory in chemistry, it shows that group theory is an integral part of the understanding

and application of chemistry, especially when it comes to understanding the fundamentals of

bonding and interactions between atoms.

Furthermore, the above has shown that one particular aspect of chemistry, namely

character tables, stand alone as one of the most important fundamentals to learn when studying

chemistry and are completely dependent on concepts in group theory. This finding shows that

even the most abstract of algebra can be applicable. Probably the most interesting, though not

entirely practical, result from the above is that the water molecule, with symmetry operations

actually forms a group and that this group is isomorphic to an abstract, seemingly purely

mathematical group, (the units of the group ). Another remarkable result or at least a

consequence is, that the physical properties that chemistry grapples with can severely influence

group structure and behavior. As with the example of the water molecule, the complexity added

by charge and the positions of the phases (lobes), even just those of the oxygen atom, both

greatly focus and expand our purely mathematical notions of groups and group structure. I find

it astonishing that the physical world in which we live can be defined both by the most elegant

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equation or formula and by the most abstract and theoretical of concepts. I can think of no better

example of this than the application of group theory to chemistry because chemistry, more than

most science, really is part of our everyday lives. This, in the end, makes group theory part of

our everyday lives as well.

Figures and Tables

Table 1 (Kettle, 2007, p. 39).........................................................................................................................................2Table 2 ........................................................................................................................................................................ 2Table 3 ........................................................................................................................................................................2

Figure 1 (Kettle, 2007, p. 20)........................................................................................................................................2Figure 2 (Kettle, 2007, p. 14)........................................................................................................................................2Figure 3 .......................................................................................................................................................................2Figure 4 .......................................................................................................................................................................2Figure 5 .......................................................................................................................................................................2

Bibliography Anne Marie Helmenstine, P. (2009). Orbital Definition: Chemistry Glossary Definition of Orbital. Retrieved 4 18, 2009, from About.com: http://chemistry.about.com/od/chemistryglossary/a/orbitaldef.htm

Cotton, F. A. (1990). Chemical Applications of Group Theory. New York: John Wiley & Sons, Inc.

Hungerford, T. W. (1997). Abstract Algebra: An Introduction. Thomson Learning, Inc.

Kettle, S. F. (2007). Symmetry and Structure: Readable Group Theory for Chemists. Chichester: John Wiley & Sons Inc.

Ladd, M. (1998). Symmetry and Group Theory in Chemistry. Chichester: Horwood Publishing.

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Table of ContentsIntroduction......................................................................................................................................1

Symmetry and Point Groups............................................................................................................1

The Water Molecule: A Symmetry Group......................................................................................2

Character Tables..............................................................................................................................2

Conclusion.......................................................................................................................................2

Figures and Tables...........................................................................................................................2

Bibliography....................................................................................................................................2

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Group Theory - WordPress.com€¦ · Web viewGroup Theory Relevancy to Chemistry 4/15/2009 MTH 4110 Joel Guttormson Introduction Group theory is more applicable than one may realize