Question

Using pictures, show that the d(x^2-y^2) orbital has B1 symmetry in the C4v point group

Answer 1

Correlation Table:

C4v	C4	C2v (σv)	C2v (σd)	C2	Cs (σv)	Cs (σd)
A1	A	A1	A1	A	A’	A’
A2	A	A2	A2	A	A’’	A’’
B1	B	A1	A2	A	A’	A’’
B2	B	A2	A1	A	A’’	A’
E	E	B1+B2	B1+B2	2B	A’+A’’	A’+A’’

Representation Products:

⊗	A1	A2	B1	B2	E
A1	A1	A2	B1	B2	E
A2	A2	A1	B2	B1	E
B1	B1	B2	A1	A2	E
B2	B2	B1	A2	A1	E
E	E	E	E	E	A1+A2+B1+B2

Transition Products:

For the C_4v point group, the irreducible representation of the dipole operator is A₁+E. Transitions that are dipole forbidden are indicated by parentheses.

⊗(A1+E)⊗	A1	A2	B1	B2	E
A1	A1+E	(A2+E)	(B1+E)	(B2+E)	A1+A2+B1+B2+E
A2	(A2+E)	A1+E	(B2+E)	(B1+E)	A1+A2+B1+B2+E
B1	(B1+E)	(B2+E)	A1+E	(A2+E)	A1+A2+B1+B2+E
B2	(B2+E)	(B1+E)	(A2+E)	A1+E	A1+A2+B1+B2+E
E	A1+A2+B1+B2+E	A1+A2+B1+B2+E	A1+A2+B1+B2+E	A1+A2+B1+B2+E	A1+A2+B1+B2+4E

Character Table:

C4v	E	C2	2C4	2σv	2σd	Linear	Quadratic	Cubic
A1	1	1	1	1	1	z	x2+y2, z2	z3, z(x2+y2)
A2	1	1	1	-1	-1	Iz
B1	1	1	-1	1	-1		x2-y2	z(x2−y2)
B2	1	1	-1	-1	1		xy	xyz
E	2	-2	0	0	0	(x, y), (Ix, Iy)	(xz, yz)	(xz2, yz2) (xy2, x2y) (x3, y3)

Stereograph	3D object	Molecule SF5 Cl (Sulphur chloride pentafluoride)