WEB TUTORIAL - Group Theory
H2O Worked Example
The Number of Unchanged Bonds ( 
R)
R)| C2v | E | C2 (z) |  v(xz) | v(yz) | Linear functions, rotations | Quadratic functions | Cubic functions |
| A1 | +1 | +1 | +1 | +1 | z | x2, y2, z2 | z3, x2z, y2z |
| A2 | +1 | +1 | -1 | -1 | Rz | xy | xyz |
| B1 | +1 | -1 | +1 | -1 | x, Ry | xz | xz2, x3, xy2 |
| B2 | +1 | -1 | -1 | +1 | y, Rx | yz | yz2, y3, x2y |
R Values are from the unchanged bonds, not the moved bonds.
Consider the C2 rotation:
It is clear from this that there are two moved bonds and therefore zero unchanged bonds since neither is in its original place.
Consider each symmetry element in turn, and answer how many unchanged bonds there are in each case after the symmetry element has been completed and pick one of the following answers.
The first number indicates the number of unchanged bonds for E, the second number
indicates the number of unchanged bonds for the C2 rotation and so forth.