The square-planar molecule (five atoms) belongs to

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

The square-planar molecule

\mathrm{XeF_4}

(five atoms) belongs to

D_{4h}

. Its nine vibrations reduce to

\Gamma_{\rm vib}=A_{1g}+B_{1g}+B_{2g}+A_{2u}+B_{2u}+2E_u

. In

D_{4h}

the linear functions

(x,y)

transform as

E_u

and

z

A_{2u}

; the quadratics

x^2+y^2,\,z^2

transform as

A_{1g}

x^2-y^2

B_{1g}

, and

xy

B_{2g}

; no quadratic transforms as

B_{2u}

. ((a)) Predict how many distinct infrared bands and how many distinct Raman bands appear, and name any vibration that is silent in both. ((b)) Suppose the molecule sits at a site in a crystal whose surroundings above the plane differ from those below, so that the site keeps the four-fold axis, the vertical mirrors

\sigma_v

and the diagonal mirrors

\sigma_d

but loses

\sigma_h

i

and

S_4

: the symmetry falls from

D_{4h}

C_{4v}

. In

C_{4v}

, with classes

(E,\,2C_4,\,C_2,\,2\sigma_v,\,2\sigma_d)

, the two-dimensional species

E

has characters

(2,0,-2,0,0)

, and the other four species are one-dimensional. Restrict

E_u

C_{4v}

, decompose it, and say whether the two

E_u

vibrations stay doubly degenerate.

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