Carbon disulfide,

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

Carbon disulfide,

\mathrm{CS_2}

, is a linear, centrosymmetric triatomic,

\mathrm{S{=}C{=}S}

. As with

\mathrm{CO_2}

, its full symmetry is treated in the finite subgroup

D_{4h}

: for this molecule

\Gamma_{3N}=A_{1g}+E_g+2A_{2u}+2E_u

, with basis functions

z\to A_{2u}

(x,y)\to E_u

, quadratics

x^2+y^2,\,z^2\to A_{1g}

xy\to B_{2g}

x^2-y^2\to B_{1g}

(xz,yz)\to E_g

. ((a)) Remove translations (

A_{2u}+E_u

) and rotations (

E_g

) from

\Gamma_{3N}

to find

\Gamma_{\rm vib}

, and identify the symmetric stretch, the bend and the antisymmetric stretch. ((b)) Decompose the direct product

A_{2u}\otimes E_u

with the characters of the

D_{4h}

table, and say whether it contains

A_{1g}

. ((c)) Using the vanishing integral theorem and the quadratic functions listed, decide whether the antisymmetric stretch (

A_{2u}

) is Raman active. ((d)) Say whether either stretch is active in both the infrared and the Raman spectrum, and name the principle that decides it.

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