PChem Teaching Lab | Group Theory

Now that the Reduction Forumla has given us the A_1g, B_1g, E_g, A_2u, B_2u and A_2g representations, we have to check the Character Table for Normal and Quadratic Terms to determine whether the representations will be IR or Raman active.

D_4h E 2C₄ (z) C₂ 2C'₂ 2C''₂ i 2S₄ _h 2_v 2_d Linear Functions,
Rotations Quadratic
Functions Cubic
Functions

A_1g +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 - x²+y², z² -

A_2g +1 +1 +1 -1 -1 +1 +1 +1 -1 -1 R_z - -

B_1g +1 -1 +1 +1 -1 +1 -1 +1 +1 -1 - x²-y² -

B_2g +1 -1 +1 -1 +1 +1 -1 +1 -1 +1 - xy -

E_g +2 0 -2 0 0 +2 0 -2 0 0 (R_x, R_y) (xz, yz) -

A_1u +1 +1 +1 +1 +1 -1 -1 -1 -1 -1 - - -

A_2u +1 +1 +1 -1 -1 -1 -1 -1 +1 +1 z - z³, z(x²+y²)

B_1u +1 -1 +1 +1 -1 -1 +1 -1 -1 +1 - - xyz

B_2u +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 - - z(x²-y²)

E_u +2 0 -2 0 0 -2 0 +2 0 0 (x, y) - (xz², yz²) (xy², x²y), (x³, y³)

From this we see that they are both IR and Raman active. This is reported as below:

D_4h	E	2C₄ (z)	C₂	2C'₂	2C''₂	i	2S₄	_h	2_v	2_d	Linear Functions, Rotations	Quadratic Functions	Cubic Functions
A_1g	+1	+1	+1	+1	+1	+1	+1	+1	+1	+1	-	x²+y², z²	-
A_2g	+1	+1	+1	-1	-1	+1	+1	+1	-1	-1	R_z	-	-
B_1g	+1	-1	+1	+1	-1	+1	-1	+1	+1	-1	-	x²-y²	-
B_2g	+1	-1	+1	-1	+1	+1	-1	+1	-1	+1	-	xy	-
E_g	+2	0	-2	0	0	+2	0	-2	0	0	(R_x, R_y)	(xz, yz)	-
A_1u	+1	+1	+1	+1	+1	-1	-1	-1	-1	-1	-	-	-
A_2u	+1	+1	+1	-1	-1	-1	-1	-1	+1	+1	z	-	z³, z(x²+y²)
B_1u	+1	-1	+1	+1	-1	-1	+1	-1	-1	+1	-	-	xyz
B_2u	+1	-1	+1	-1	+1	-1	+1	-1	+1	-1	-	-	z(x²-y²)
E_u	+2	0	-2	0	0	-2	0	+2	0	0	(x, y)	-	(xz², yz²) (xy², x²y), (x³, y³)

Ignoring E_g, we will therefore expect to see 2 bands in the IR spectra of [Re₂Cl₈]^2-, and 2 in the Raman spectra. But what about E_g - Should we expect it to result in an additional band in the IR and Raman spectra, or just as an additional band in the IR spectra, or just as an additional band in the Raman spectra?

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