Planar trans- (difluorodiazene, - = - with the two fluorines on opposite sides) has…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry

Invariance, Ch Representations, Schur's lemma and characters

Problem

Planar trans-

\mathrm{N_2F_2}

(difluorodiazene,

\mathrm{F}

\mathrm{N}

\mathrm{N}

\mathrm{F}

with the two fluorines on opposite sides) has point group

C_{2h} = \{E, C_2, i, \sigma_h\}

, order

4

and abelian, with

\sigma_h

the molecular plane itself,

C_2

the axis through the midpoint of the

\mathrm{N}

\mathrm{N}

bond perpendicular to that plane, and

i

the inversion centre at that midpoint. ((a)) Label the two nitrogen atoms

\mathrm{N}_1

\mathrm{N}_2

. Since the molecule is planar, every atom is fixed by

\sigma_h

; the

C_2

rotation and the inversion

i

each exchange

\mathrm{N}_1 \leftrightarrow \mathrm{N}_2

. Write the

2\times2

permutation matrices

D(C_2)

and

D(i)

for this labelling, and verify that

D(C_2)D(i)

equals

D(\sigma_h)

, consistent with

C_2\cdot i = \sigma_h

in the group's multiplication table. ((b)) Using the fact that

C_{2h}

is abelian with

4

classes and the great orthogonality relation

\sum_i d_i^2 = |G|

, find the only possible list of dimensions of its irreducible representations.