IR Raman Hyper-Raman Modes

Information of the Point Group C_6h (6/m)

Character Table¹

C_6h(6/m)	#	1	6⁺	3⁺	2	3^-	6^-	-1	-6⁺	-3⁺	m	-3^-	-6^-	functions
A_g	Γ₁⁺	1	1	1	1	1	1	1	1	1	1	1	1	x²+y²,z²,J_z
A_u	Γ₁^-	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	z
B_g	Γ₄⁺	1	-1	1	-1	1	-1	1	-1	1	-1	1	-1	·
B_u	Γ₄^-	1	-1	1	-1	1	-1	-1	1	-1	1	-1	1	·
¹E_2g ²E_2g	Γ₃⁺ Γ₂⁺	1 1	w w²	w² w	1 1	w w²	w² w	1 1	w w²	w² w	1 1	w w²	w² w	(x²-y²,xy)
¹E_2u ²E_2u	Γ₃^- Γ₂^-	1 1	w w²	w² w	1 1	w w²	w² w	-1 -1	-w -w²	-w² -w	-1 -1	-w -w²	-w² -w	·
²E_1g ¹E_1g	Γ₅⁺ Γ₆⁺	1 1	-w² -w	w w²	-1 -1	w² w	-w -w²	1 1	-w² -w	w w²	-1 -1	w² w	-w -w²	(xz,yz),(J_x,J_y)
²E_1u ¹E_1u	Γ₅^- Γ₆^-	1 1	-w² -w	w w²	-1 -1	w² w	-w -w²	-1 -1	w² w	-w -w²	1 1	-w² -w	w w²	(x,y)

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_1g

²E_1g

¹E_2g

²E_2g

-d

-e

-f

-d

A_u(z)

B_u

¹E_1u(x)

²E_1u(y)

E_2u

-c

-d

-3g

-g

-h

-i

-d

-c

-g

-h

-i

To transform the tensors please click here.

Information of the Space Group P6₃/m (No. 176)

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 = A_u + ¹E_1u + ²E_1u )

A_g

A_u

B_g

B_u

²E_2g

²E_2u

¹E_1g

¹E_1u

¹E_2g

¹E_2u

²E_1g

²E_1u

12i

Raman Active Modes

A_g

A_u

B_g

B_u

²E_2g

²E_2u

¹E_1g

¹E_1u

¹E_2g

¹E_2u

²E_1g

²E_1u

12i

Hyper-Raman Active Modes

A_g

A_u

B_g

B_u

²E_2g

²E_2u

¹E_1g

¹E_1u

¹E_2g

¹E_2u

²E_1g

²E_1u

12i

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

B_g

B_u

²E_2g

²E_2u

¹E_1g

¹E_1u

¹E_2g

¹E_2u

²E_1g

²E_1u

Infrared

Raman

Hyper-Raman

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 C_6h (6/m)

Character Table¹

Raman and Hyper-Raman Tensors

Information of the Space Group P6₃/m (No. 176)

Selection Rules

Modes Activity

Permutational Representation

Mechanical Representation

Irreps decompositions

Second Order IR Activity

Second Order Raman Activity

Information of the Point Group C6h (6/m)

Character Table1

Raman and Hyper-Raman Tensors

Information of the Space Group P63/m (No. 176)

Selection Rules

Modes Activity

Permutational Representation

Mechanical Representation

Irreps decompositions

Second Order IR Activity

Second Order Raman Activity

Information of the Point Group C_6h (6/m)

Character Table¹

Information of the Space Group P6₃/m (No. 176)