IR Raman Hyper-Raman Modes

Information of the Point Group T_h (m-3)

Character Table¹

T_h(m-3)	#	1	3⁺	3^-	2	-1	-3⁺	-3^-	m	functions
Mult.	-	1	4	4	3	1	4	4	3	·
A_g	Γ₁⁺	1	1	1	1	1	1	1	1	x²+y²+z²
¹E_g ²E_g	Γ₂⁺ Γ₃⁺	1 1	w w²	w² w	1 1	1 1	w w²	w² w	1 1	(2z²-x²-y²,x²-y²)
T_g	Γ₄⁺	3	0	0	-1	3	0	0	-1	(xz,yz,xy),(J_x,J_y,J_z)
A_u	Γ₁^-	1	1	1	1	-1	-1	-1	-1	·
¹E_u ²E_u	Γ₂^- Γ₃^-	1 1	w w²	w² w	1 1	-1 -1	-w -w²	-w² -w	-1 -1	·
T_u	Γ₄^-	3	0	0	-1	-3	0	0	1	(x,y,z)

w = exp(2iπ/3)

To get the list of the irreducible representations in their matrix form please click here.

Raman and Hyper-Raman Tensors

Raman Tensors

Hyper-Raman Tensors

A_g

¹E_g

²E_g

T_g

b+3^1/2c

c-3^1/2b

b-3^1/2c

c+3^1/2b

-2b

-2c

T_u(x)

T_u(y)

T_u(z)

A_u

Extra information about the Hyper-Raman tensors
The HRS tensor related to A_u has only one nonzero component β_xyz.
The HRS tensor related to E_u has nonzero components only for the nonsymmetric part of the HRS tensor.

To transform the tensors please click here.

Information of the Space Group Pa-3 (No. 205)

Selection Rules

In the following tables one can find which are the allowed representations for each one of the selected Wyckoff positions for IR, Raman and Hyper-Raman processes.

IR Active Modes: (Γ_acoustic = T_u )

WP	A_g	A_u	¹E_g	¹E_u	²E_g	²E_u	T_u	T_g
4b	·	·	·	·	·	·	3	·

Raman Active Modes

WP	A_g	A_u	¹E_g	¹E_u	²E_g	²E_u	T_u	T_g
4b	·	·	·	·	·	·	·	·

Hyper-Raman Active Modes

WP	A_g	A_u	¹E_g	¹E_u	²E_g	²E_u	T_u	T_g
4b	·	1	·	1	·	1	3	·

The polarization selection rules for Raman and Hyper-Raman processes.

Modes Activity

This table is a summary of the activity of the different modes of the space group.

-	A_g	A_u	¹E_g	¹E_u	²E_g	²E_u	T_u	T_g
Infrared	·	·	·	·	·	·	x	·
Raman	x	·	x	·	x	·	·	x
Hyper-Raman	·	x	·	x	·	x	x	·

Note: x ≡ represents the modes which can be detected.

More information:

Information about second order processes:

¹George F. Koster, John O. Dimmock, Robert G. Wheeler, Hermann Statz (1963). Properties of the thirty-two point groups. Published by the M.I.T. press, Cambridge, Massachusetts.
¹Simon L. Altmann and Peter Herzig (1994). Point-Group Theory Tables. Oxford Science Publications.

Information of the Point Group T_h (m-3)

Character Table¹

Raman and Hyper-Raman Tensors

Information of the Space Group Pa-3 (No. 205)

Selection Rules

Modes Activity

Permutational Representation

Mechanical Representation

Irreps decompositions

Second Order IR Activity

Second Order Raman Activity

T_h (m-3)	A_g	A_u	¹E_g	¹E_u	²E_g	²E_u	T_u	T_g
V	·	·	·	·	·	·	1	·
A	·	·	·	·	·	·	·	1
[V²]	1	·	1	·	1	·	·	1
[V²] x V	·	1	·	1	·	1	5	·

Information of the Point Group Th (m-3)

Character Table1

Raman and Hyper-Raman Tensors

Information of the Space Group Pa-3 (No. 205)

Selection Rules

Modes Activity

Permutational Representation

Mechanical Representation

Irreps decompositions

Second Order IR Activity

Second Order Raman Activity

Information of the Point Group T_h (m-3)

Character Table¹