View source for Symmetry Workshop 1:background

<h4>Background</h4>

<p>
'''''A symmetry operation is defined as an operation which, when performed on a molecule, leaves the molecule in an indistinguishable state.''''' <br/>
Thus a <math> \ C_2 \ </math> (180˚ rotation) about the <math> \ z \ </math> axis in <math> \ H_2O \ </math> can only be detected if we label specific hydrogen atoms as shown: </p>

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[[Image:photo1.png]]
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The result is indistinguishable, but not identical, as the labels show that the two hydrogen atoms have moved.
<p>
A 180˚ rotation about the <math> \ y \ </math> axis is not a symmetry operation, however:</p>

<p>
[[Image:photo2.png]]</p>

<p>
because the result of the operation is not superimposeable on the original.  
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<p>
In finding symmetry operations one needs to develop some skill with 3-dimensional visualisation. In question 2 you will use the graphics display of '''GaussView''' to manipulate molecular models and identify symmetry operations.

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<p>
'''''The complete set of symmetry operations of a molecule form a group.''''' <br/>
What this means is that performing any two operations in succession is equivalent to performing a third operation, which is also a symmetry operation. </p>

<p>
[[Image:photo3.png]] 




'''<math> \qquad \sigma_v^2 \quad = \quad C_3 \circ \sigma_v^3 \quad </math>'''

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<p>
(Notice the convention whereby the molecule moves but the symmetry elements are fixed in space).<br/>
Because symmetry operations can be combined as shown above, in order to identify the group of all symmetry operations of a molecule, one need only identify a few operations called ''generators''. Thus the group of a trigonal planar <math> \ AB_3 \ </math> molecule is <math> \ D_{3h} \ </math> and is made up of the following symmetry operations: </p>


<p>

'''<math> \qquad  D_{3h} \quad = \quad \{E \quad  2C_3 \quad  3C_2 \quad  \sigma_h \quad  3S_3 \quad  3\sigma_v \}  \qquad </math>'''

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<p>
where the symbol '''<math> \ 3C_2 \ </math>''' is used to denote the fact that there are 3 equivalent '''<math> \ C_2 \ </math>''' rotation axes (about each AB bond).<br/>
A set of generators is <br/>

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<p>

'''<math> \qquad  D_{3h} \quad generators \quad \{C_3 \quad  \to \quad  S_3 \quad \to \quad C_2 \quad \to \quad \sigma_h \}  \qquad </math>'''

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<h4>Flowchart</h4>
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All the symmetry operations of <math> \ D_{3h} \ </math> are combinations of the above sub-set of generators. The process of finding generators can be made systematic using a decision table or flowchart, such as the one given below.
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<p>
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'''NB''' <br/>
''The flowchart also indicates that there are relationships between point groups; they do not exist in isolation. This idea is explored further in questions 3, 4 and 6.''
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[[Image:graph_group.png]] <br/>

<!-- The flowchart is also available in a .doc file : [[Media:flowchart.doc]]. -->


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