Phosphorus trichloride,

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

\mathrm{PCl_3}

, is pyramidal with point group

C_{3v}

, P on the

C_3

axis and the three Cl atoms forming the base. Use the

C_{3v}

character table (classes

E

2C_3

3\sigma_v

; basis functions

z\to A_1

(x,y)\to E

R_z\to A_2

(R_x,R_y)\to E

). ((a)) Find

\chi_{3N}

and decompose it. ((b)) Find

\Gamma_{\rm vib}

and the number of distinct vibrational frequencies. ((c)) Decompose the direct product

A_2\otimes E

. ((d)) Using the vanishing integral theorem, decide whether a transition from the totally symmetric ground state (

A_1

) to an excited state of species

A_2

is allowed by electric-dipole absorption.

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