WEB TUTORIAL - Group Theory

[Re₂Cl₈]^2- Worked Example

D4h	E	2C4 (z)	C2	2C'2	2C''2	i	2S4	h	2v	2d	Linear Functions, Rotations	Quadratic Functions	Cubic Functions
A1g	+1	+1	+1	+1	+1	+1	+1	+1	+1	+1	-	x2+y2, z2	-
A2g	+1	+1	+1	-1	-1	+1	+1	+1	-1	-1	Rz	-	-
B1g	+1	-1	+1	+1	-1	+1	-1	+1	+1	-1	-	x2-y2	-
B2g	+1	-1	+1	-1	+1	+1	-1	+1	-1	+1	-	xy	-
Eg	+2	0	-2	0	0	+2	0	-2	0	0	(Rx, Ry)	(xz, yz)	-
A1u	+1	+1	+1	+1	+1	-1	-1	-1	-1	-1	-	-	-
A2u	+1	+1	+1	-1	-1	-1	-1	-1	+1	+1	z	-	z3, z(x2+y2)
B1u	+1	-1	+1	+1	-1	-1	+1	-1	-1	+1	-	-	xyz
B2u	+1	-1	+1	-1	+1	-1	+1	-1	+1	-1	-	-	z(x2-y2)
Eu	+2	0	-2	0	0	-2	0	+2	0	0	(x, y)	-	(xz2, yz2) (xy2, x2y), (x3, y3)

h = 16

Consider each symmetry element in turn, and answer how many unchanged bonds there are in each case after the symmetry element has been completed. This example is quite tricky but it will help to make a model, or to use the visualiser on Otterbein Website.

Help

One way to help work out the number of unchanged bonds is to label them. The bonds have been labelled in [Re₂Cl₈]^2- for the C₂' rotation.

Some labels are shown on the atoms to make them easier to track.

The same principle applies to i, S₄, and _h symmetry elements, which at first appear to change the position of the 'middle' bond E, but infact this stays as an unchanged bond.

The first number indicates the number of unchanged bonds for E, the second number
indicates the number of unchanged bonds for the C₄ rotation and so forth.

E 2C₄ C₂ 2C'₂ 2C"₂ i 2S_{4 h} 2_v 2_d
• 9 1 1 1 1 1 1 1 5 1
• 9 1 1 0 0 0 0 0 5 1
• 9 1 1 0 0 1 1 0 5 1