When can I pull out the identity operator?
Answers
Answered by
0
In group theory, in order to assign a point group, one must first identify the symmetry elements present in the molecule.
To give an example, water has:
a rotational axis, labelled C2 since it is a two fold axis, with a π/2 rotation giving back (what appears to be)the same thing
two mirror planes, labelled σv and σv' (with additional labels xy/xz to define what plane the mirror plane lies in)
Symmetry elements present in water
(Note that for the C2v point group, the axis are defined 'arbitrarily', different tables/books might swap x and y)
From this, it would look like water has three unique symmetry elements, however this is not the case, in fact, water has four due to the presence of an additional symmetry element, E, known as the identity operator (sometimes seen as I in certain textbooks).
This E operator corresponds to 'do nothing' (i.e. leave atoms where they are), which is easily seen in the character table for the C2v point group (to which water belongs) in which the characters under E are all '1':
C2vA1A2B1B2E1111C211−1−1σv(xz)1−11−1σ′v(yz)1−1−11zRzx,Ryy,Rxx2,y2,z2xyxzyzC2vEC2σv(xz)σv′(yz)A11111zx2,y2,z2A211−1−1RzxyB11−11−1x,RyxzB21−1−11y,Rxyz
C2v character table, from orthocresol's Group Theory Tables
The identity, E, consists of doing nothing; the corresponding symmetry element is the entire object. Because every molecule is indistinguishable from itself if nothing is done to it, every object possesses at least the identity element. (Taken from Atkins' Physical Chemistry).
Since transforming a molecule by the identity operator just gives back itself, the use of 'E' in a qualitative sense appears redundant, however it appears in every character table, and as such much have a purpose.
To give an example, water has:
a rotational axis, labelled C2 since it is a two fold axis, with a π/2 rotation giving back (what appears to be)the same thing
two mirror planes, labelled σv and σv' (with additional labels xy/xz to define what plane the mirror plane lies in)
Symmetry elements present in water
(Note that for the C2v point group, the axis are defined 'arbitrarily', different tables/books might swap x and y)
From this, it would look like water has three unique symmetry elements, however this is not the case, in fact, water has four due to the presence of an additional symmetry element, E, known as the identity operator (sometimes seen as I in certain textbooks).
This E operator corresponds to 'do nothing' (i.e. leave atoms where they are), which is easily seen in the character table for the C2v point group (to which water belongs) in which the characters under E are all '1':
C2vA1A2B1B2E1111C211−1−1σv(xz)1−11−1σ′v(yz)1−1−11zRzx,Ryy,Rxx2,y2,z2xyxzyzC2vEC2σv(xz)σv′(yz)A11111zx2,y2,z2A211−1−1RzxyB11−11−1x,RyxzB21−1−11y,Rxyz
C2v character table, from orthocresol's Group Theory Tables
The identity, E, consists of doing nothing; the corresponding symmetry element is the entire object. Because every molecule is indistinguishable from itself if nothing is done to it, every object possesses at least the identity element. (Taken from Atkins' Physical Chemistry).
Since transforming a molecule by the identity operator just gives back itself, the use of 'E' in a qualitative sense appears redundant, however it appears in every character table, and as such much have a purpose.
Answered by
4
You say C2C2 is an element of your group of symmetries. This forces C2⋅C2⋅C2⋅C2C2⋅C2⋅C2⋅C2 to also be a symmetry. Which element is that product? (Using the fourth power is a consequence of an error in your description of C2C2. Actually, C2C2 is rotation by ππ radians. With this correction, we need only ask "which symmetry is given by 'first do C2C2, then do C2C2 again'?")
Similar questions