Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2021 / Paper 2A / Section B / 23 / b

γ = 0

, each independent diatomic spring has one translation and one stretch. Consequently the angular frequencies are

ω = 0, 0, 2 k / m, 2 k / m .

The two zero modes are independent centre-of-mass translations of the two molecules. There are no transverse coordinates in this model; adding transverse degrees of freedom would be a different problem.

ii

For the collinear centrosymmetric arrangement, the longitudinal coordinates transform in the D infinity h molecular point group. Inversion sends

q_{i} \mapsto - q_{5 - i}

, because displacement is a polar vector. Therefore

Γ_{z} = 2 Σ_{g}^{+} + 2 Σ_{u}^{+} .

Coordinate	Displacement vector	Symmetry	$ω^{2}$ at $γ = 0$	$t$	$(1, 1, 1, 1) /2$	$Σ_{u}^{+}$	$0$	$r$	$(- 1, - 1, 1, 1) /2$	$Σ_{g}^{+}$	$0$	$s$	$(- 1, 1, - 1, 1) /2$	$Σ_{g}^{+}$	$2 k / m$	$a$	$(- 1, 1, 1, - 1) /2$	$Σ_{u}^{+}$	$2 k / m$