WEB TUTORIAL - Group Theory

H₂O Worked Example

The Number of Unchanged Bonds ( _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

Number of symmetry elements, h = 4

The planes of symmetry are at right angles to one another as denoted by the term in brackets in the character table and hence have different values for the unchanged bonds when the symmetry element is performed:

The two planes, 'xz' and 'yz' respectively, are shown below to help you visualise the unchanged bonds.

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.

• 1 0 1 0
• 1 1 1 1
• 0 2 0 2
• 2 0 2 0
• 2 0 2 2
• 3 1 3 1