Octahedral ( ) has

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Atomic, Molecular & Optical Physics · Molecular structure and spectra

Invariance, Ch Character tables at work: vibrations, selection rules and splitting

Problem

Octahedral

\mathrm{SF_6}

(

O_h

) has

\Gamma_{\rm vib}=A_{1g}+E_g+2T_{1u}+T_{2g}+T_{2u}

. Its

O_h

character table gives dipole species

T_{1u}

and quadratic species

A_{1g}+E_g+T_{2g}

. A distortion that stretches one

\mathrm{S{-}F}

bond lowers the symmetry to

C_{4v}

(that axis becomes the new

C_4

), and correlation of the six irreducible pieces gives

\Gamma_{\rm vib}=4A_1+2B_1+B_2+4E

. In this

C_{4v}

table the dipole species are

A_1

(

z

) and

E

(x,y)

; the quadratic species are

A_1

B_1

B_2

and

E

(every species except

A_2

). A further distortion lowers the symmetry to

C_{2v}

, keeping the two mirror planes that each contain four S-F bonds (the two axial bonds and two equatorial ones), and correlation gives

\Gamma_{\rm vib}=6A_1+A_2+4B_1+4B_2

. In this

C_{2v}

table the dipole species are

A_1

(

z

B_1

(

x

) and

B_2

(

y

); every species (

A_1

A_2

B_1

B_2

) appears among the quadratic functions. ((a)) Using the infrared and Raman selection rules, find the number of infrared-active and Raman-active vibrational bands in

O_h

. ((b)) Repeat for the

C_{4v}

-distorted molecule. ((c)) Repeat for the

C_{2v}

-distorted molecule. ((d)) State, with a reason drawn from the character tables, why the number of Raman bands equals the total number of distinct vibrational frequencies at the

C_{2v}

stage but not at the

O_h

stage.

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