Double group

Double groups

When analyzing splitting of energy levels in external potential using group theory, one may have the problem of half integer angular moment, because the wavefunctions change with rotation of 360 degree. Here we should use so called double group.

The double group consists two the original group G and its product with operation R (rotation with 2\pi degree), which is why it is called double group.

Here are my collections

C_3v' (C_3v)

	E	R	C32 C3R	C3 C32R	σv1 σv2 σv3	σv1R σv2R σv3R	Pseudo vector	polar vector
A1	1	1	1	1	1	1		z
A2	1	1	1	1	-1	-1	Sz
E3/2+	1	-1	1	-1	i	-i
E3/2-	1	-1	1	-1	-i	i
E	2	2	-1	-1	0	0	(Sx,Sy)	(x,y)
E1/2	2	-2	-1	1	0	0

IR product table

	A1	A2	E3/2+	E3/2-	E	E1/2
A1
A2		A1	E3/2-	E3/2+	E	E1/2
E3/2+		E3/2-	A2	A1	E1/2	E
E3/2-		E3/2+	A1	A2	E1/2	E
E		E	E1/2	E1/2	A1+A2+E	E3/2++E3/2-+E1/2
E1/2		E1/2	E	E	E3/2++E3/2-+E1/2	A1+A2+E

Note that it is the direct product instead of Γ^†Γ, which is why E_3/2⁺E_3/2⁺=A₂ instead of A₁.

D_2h' (D_2h)

	E	R	C2x C2xR	C2y C2yR	C2z C2zR	i	i R	σh σhR	σvx σvxR	σvy σvyR	Pseudo vector	polar vector
Γ1+	1	1	1	1	1	1	1	1	1	1
Γ2+	1	1	-1	-1	1	1	1	1	-1	-1	Sz
Γ3+	1	1	1	-1	-1	1	1	-1	1	-1	Sx
Γ4+	1	1	-1	1	-1	1	1	-1	-1	1	Sy
Γ1-	1	1	1	1	1	-1	-1	-1	-1	-1
Γ2-	1	1	-1	-1	1	-1	-1	-1	1	1		z
Γ3-	1	1	1	-1	-1	-1	-1	1	-1	1		x
Γ4-	1	1	-1	1	-1	-1	-1	1	1	-1		y
Γ5+	2	-2	0	0	0	2	-2	0	0	0
Γ5-	2	-2	0	0	0	-2	2	0	0	0

Product table

	Γ1+	Γ2+	Γ3+	Γ4+	Γ1-	Γ2-	Γ3-	Γ4-	Γ5+	Γ5-
Γ1+
Γ2+
Γ3+
Γ4+
Γ1-
Γ2-
Γ3-
Γ4-
Γ5+
Γ5-									Γ1++Γ2+ +Γ3++Γ4+	Γ1+

D₄' (D₄)

D4' (16)		E	R	C4 C43R	C43 C4R	C2 C2R	2C2' 2C2'R	2C2'' 2C2''R
Γ1	A1'	1	1	1	1	1	1	1
Γ2	A2'	1	1	1	1	1	-1	-1
Γ3	B1'	1	1	-1	-1	1	1	-1
Γ4	B2'	1	1	-1	-1	1	-1	1
Γ5	E1'	2	2	0	0	-2	0	0
Γ6	E2'	2	-2	√2	-√2	0	0	0
Γ7	E3'	2	-2	-√2	√2	0	0	0

O' (O_h)

O' (48)		E	R	4C3 4C32R	4C32 4C3R	3C2 3C2R	3C4 3C43R	3C43 3C4R	6C2' 6C2'R
Γ1	A1'	1	1	1	1	1	1	1	1
Γ2	A2'	1	1	1	1	1	-1	-1	-1
Γ3	E1'	2	2	-1	-1	2	0	0	0
Γ4	T1'	3	3	0	0	-1	1	1	-1
Γ5	T2'	3	3	0	0	-1	-1	-1	1
Γ6	E2'	2	-2	1	-1	0	√2	-√2	0
Γ7	E3'	2	-2	1	-1	0	-√2	√2	0
Γ8	G'	4	-4	-1	1	0	0	0	0