Carbon tetrachloride,

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 tetrachloride, CCl4\mathrm{CCl_4}, is tetrahedral, point group TdT_d. Use the TdT_d character table (classes EE, 8C38C_3, 3C23C_2, 6S46S_4, 6σd6\sigma_d; dipole (x,y,z)→T2(x,y,z)\to T_2; quadratics x2+y2+z2→A1x^2+y^2+z^2\to A_1, (2z2−x2−y2,x2−y2)→E(2z^2-x^2-y^2,x^2-y^2)\to E, (xy,xz,yz)→T2(xy,xz,yz)\to T_2). ((a)) State Γvib\Gamma_{\rm vib} for this AX4\mathrm{AX_4} molecule (the character of Γ3N\Gamma_{3N} has the same values as for any tetrahedral AX4\mathrm{AX_4}). ((b)) Decompose the direct product E⊗T2E\otimes T_2. ((c)) Using the vanishing integral theorem, decide whether the doubly-degenerate (EE) bending mode is infrared active. ((d)) Decide whether the same mode is Raman active.

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