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

Irreducible representation notations

Irreducible representation notations

		Bethe	BSW	EWK
Oh (m3m)
		Γ1		A1g
		Γ1'		A1u
		Γ2		A2g
		Γ2'		A2u
		Γ12		Eg
		Γ12'		Eu
		Γ15'		T1g
		Γ15		T1u
		Γ25'		T2g
		Γ25		T2u
Td (-43m)
		Γ1	Γ1	A1
		Γ2	Γ2	A2
		Γ3	Γ12	E
		Γ4	Γ25	T1
		Γ4	Γ15	T2
D2d (-42m)
		Γ1	A1
		Γ2	A2
		Γ3	B1
		Γ4	B2
		Γ5	E
C2v		Γ1	A1
		Γ3	A2
		Γ2	B1
		Γ4	B2

Abbreviate

Note on abbrievation

To me, there are two cases we have to do abbrievation:
1) when taking notes
2) when writing a computer program

In the 1) case, we want o gain time by abbreviating words, while in 2) case, we want to avoid clutter in our codes. In both cases, the caveat is to make sure that you will be able to recognize these abbreviations afterward without confusion. We should also take advantage of the context.

Here are a list of tricks (some may not apply for computer programmings)

	Original	Abbreviation
Symbols	equal	=
	*	important
	w/	with
Standard	for example	e.g.
1st syllable	dem	democracy
	integer	int
1st syllable+first letter of the second	demo	demonstration
or so much you can recongnize	intro	introduction
Omit vowels	level	lvl
Use the first and last syllables	energy	engy
use the first and last letters	feet	ft

9．Us the first and last syllable

engy = energy