Teaching
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- From 1997 to 2000, I taught general organic chemistry seminars and laboratory practice at the
'Departamento de Química Orgánica' of the 'Facultad de Farmacia' ('Universidad de Granada'), Spain
- From 2006 till now, I teach group theory for chemists and different laboratory practice at the
'Departamento de Química Inorgánica' of the 'Facultad de Ciencias' ('Universidad de Granada'), Spain
You can find a complete web page containing constants, unit converter, periodic table, symmetry point
group character tables and more at http://www.webqc.org/symmetry.php
(You can also discover an excellent multilanguage periodic table at http://www.ptable.com/)
Group theory for chemists - An overview
1. Introduction
Harmony,
balance, proprotion and symmetry. The two first concepts are tightly
related and describe more properly the feeling of the observer in
relation with an order, which in turn depends on his/her preferences. Take a look at their definitions: Harmony is 'a
pleasing combination of elements in a whole'. Balance is 'a harmonious
or satisfying arrangement or proportion of parts or elements, as in a
design'.
Thus, there exists an important subjective component. The term
proportion introduces a necessary relationship between the
parts of an object. Proportion is 'a relationship between things or parts of things with respect to comparative magnitude, quantity, or degree'. This characteristic could be also subjective if it is used as a synonym of balanced or harmonious. However, the
subjective component is removed when one use the term symmetry since,
implicitly, there is a mathematically conception and certain rules must be
accomplished. The word symmetry holds into a defined mathematic
rule consisting in the equality of the parts. Symmetry is the 'exact
correspondence of form and constituent configuration on opposite sides
of a dividing line or plane or about a center or an axis'. Although it
is possible to propose different meanings for symmetry we shall keep,
for our purposes, the already mentionned one.
2. Elements and operations of symmetry
A
symmetry element is a geometric entity (point, line or plane) about
which it is established a symmetry operation. The existence of a
symmetry element exhibits a relationship between the parts of an
object. When a point is a symmetry element it is called center of
symmetry or center of inversion. When we have a line as a symmetry
element it will be called axis. For a plane, then we shall have a plane of
symmetry.
In turn, a symmetry operation is the execution of a
geometric change about a symmetry element in such a manner that the
object seems indistinguishable from the starting position. Obviously,
symmetry elements and operations are interdependent concepts since
their existence is interrelated.
Symmetry element: a geometric entity that relates parts of an object
Symmetry operation: the effect of operating with a symmetry element
For
example, imagine a sphere. It can be said that there exist a center of
symmetry (at the sphere center) because when we apply the inversion
center on each point of the sphere we reproduce the sphere again. It is
important to say that the application of a symmetry element on an
object affects to all the points of such an object.
A center of symmetry is designed by the symbol 'i', and the corresponding operation as 'î'.
In order to simplify the notation along this page we talk about
symmetry elements
and operations using the same notation, although we shall be careful to
correctly specify to which we make reference in each case.
An axis could be, in a general way, represented as Cn,
where n is the axis order, and represents the times that an object
results indistinghishable from the starting structure in a turn of 360º
about such an axis. For example, a C2 (two-fold) axis means that the object seems to be twice the same in a complete turn. This also means that a C2
axis offers two symmetry aperations: a 180º turn (C21), and a 360º turn
(C22). A 360º turn coincides with a 0º turn (a 'nothing to do'
operation), and this operation receives a special name: 'E' as the identity
operation. Thus, we shall more properly write that the two operations
included in a C2 axis are: E
and C21 since on turning 360º we keep the object in the exact starting
state, so equivalent to 'nothing to do'. Therefore, a C2
axis repeats an object exactly every 180º. How to relate degrees
and axis order? By the simple following formulae: n = 360º / α, where n is the already mentionned axis order, and α corresponds to the number of degrees that it is necessary to succesively turn the object in order to replicate it. In this case, n = 360º / 180º = 2. Remark that, mathematically, 'E' is also equivalent to another trivial operation: C1. C1
is a 360º rotation of an object following an arbitrary axis. This leads
to a significant conclusion: in a 'real' object, independently of its
symmetry, there will be always the identity symmetry operation 'E',
since all the objects could be turned 360º to become indistinguishable
from the initial ones (it should be noted that I use the term <
'real' objects > as this is not necessarily true for all the
subatomic species but, since we shall move from the macroscopic world
until an atomic scale, this affirmation is rigorously true).
A
plane of symmetry will be showed as sigma. There are three kinds of planes of
symmetry classes: sigmah, sigmav and sigmad.
Independently of the plane of symmetry kind, their
action is always the same: a reflection of a side of the plane to the
other side in order to reproduce the complete object. Sometimes, we can
asign a plane of symmetry to a 'normal', no-symmetry plane. So, be
cautious. For
example, look at a rectangle. Thus, the plane of symmetry denomination
depends on its position with respect to the other symmetry elements, in
particular the principal axis. An object will have a principal axis if
there are other axis with at least the same order. If the plane is
perpendicular to the principal axis is called sigmah and if the plane contains the principal axis (therefore, it is parallel) is denominated sigmav or sigmad depending on the orientation of the plane.
A combined element of symmetry
There
is another kind of element of symmetry that results from the
combination of an axis (always corresponding to a principal axis) and a
perpendicular plane of symmetry: the improper or roto-reflexion axis,
named as Sn (n being the order axis as mentionned above). It
is important to remark that this is a combined symmetry element and the
resulting operation is the combination of the two operations, a
rotation and a reflection, where the reflection must be executed
through a real or a hypothetical plane of symmetry. In other words, the
existence of a Sn does not necessarily implies the existence of the corresponding plane of symmetry (if there exist must be always a sigmah since it must be perpendicular to the principal axis), although the existence of a sigmah and an axis perpendicular to it necessarily implies the existence of the corresponding Sn axis. This could be better seen with a series of examples.
3. Symmetry Point Groups
It is important to remark that symmetry point groups are not crystalline groups (symmetry spatial
groups) since there are no translation operations. Thus, they corresponds to isolated objects in the space. The label 'point'
is due to the fact that in an isolated object all the symmetry elements
cross through the same point, the geometric center of the object, which
coincides with a center of symmetry when the object has one, indeed.
4. Matrix representation of a symmetry operation
A
symmetry operation just describes a change in the spatial position
of the points of an object. These points could then be taken
into account as tri-dimensional vectors.
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