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### Group Theory

For our purposes we consider a set of operations (rotations, reflections..) A, B, C,.. such that
(i) the set is closed under multiplication ie AB=C, with C being in the set
(ii) It is associative ie (AB)C=A(BC). This always holds for our cases.
(iii) There is an identity E in the set such that AE=EA=A
(iv) Each operation A has an inverse A-1 such that AA-1=A-1A=E
Such a set of operations forms a group.
The mathematicians spend a lot of time playing with abstract groups (eg the numbers 1,-1,i,-i under multiplication or eg the set of all nn matrices with nonzero determinant) but we shall only be concerned with those groups constructed from symmetry operations. Let us consider H2O again.
The set of symmetry operations which leave the molecule in an indistinguishable state are
E, the identity, do nothing, operation
C2, a rotation of  about the principal axis
, relection in the plane perpendicular to the molecular plane, and containing the symmetry axis
, reflection in the plane of the molecule
Remember that a molecule is a three-dimensional object and therefore all these operations are different. We can construct a multiplication table for this group, which shows the result of multiplying one operation (along the top) by another (down the side)
 E C2 E E C2 C2 C2 E E C2 C2 E
This table may be generated by seeing what happens to an arbitrary point (x,y,z) under the operations. By inspection we observe that the sets of operations {E,C2},{E,},{E, } each obey the group axioms, and are therefore called subgroups. A simple way of determining the number of symmetry operations in the group is find out how many points have the same electron density in the molecule. (eg in benzene, there are 24).
The different types of symmetry operation which are met are:
E.....The identity
Cn....A rotation of  about an axis. Axis of highest n is called the principal axis. eg in NH3, there are two C3 operations (one clockwise, one anticlockwise).
....Reflection in a mirror plane. There are three types
... reflection in plane perpendicular to the principal axis
... reflection in plane containing the principal axis
... reflection in plane containing the principal axis, bisecting two 2-fold axes perpendicular to the principal axis.
...eg  in BF3 (plane of molecule)

...eg  (plane of molecule),  (perpendicular plane), in H2O

...eg  in allene C3H4

Sn....A rotation of  followed by a reflection. eg S4 in allene.

i....Inversion. eg in CO2.

Hence if we consider BF3 (a planar molecule), there are 12 equivalent density points (six above and six below the plane):

The 12 group operations are E, 2C3, 3C2, 2S3, 3. One may then use the `find the group chart' to show that the name of this group is D3h.
Similarly we may show that the group for H2O is C2v

We may look up the character table:

 C2v E C2 A1 1 1 1 1 z A2 1 1 -1 -1 B1 1 -1 1 -1 x B2 1 -1 -1 1 y

This character table tells us immediately that the possible symmetries of the symmetry orbitals are A1, A2, B1, B2. We can identify the symmetries of the combinations (1s1sB) by writing in what happens to these functions under the group operations. We use the convention that the atomic functions move, but that the atoms remain fixed. We enter  according as the function is changed or not. Thus

 C2v E C2 1sA+1sB 1 1 1 1 1sA-1sB 1 -1 -1 1

We see that the symmetry of 1s1sB are A1,B2

Next: To Determine Symmetry Orbitals Up: Contents - Previous: Symmetry

Nicholas Handy

1998-09-23