Solution

Take the eight Mo–Cl bond-length changes as a stretching basis. These scalar coordinates have the same permutation representation as the eight sigma donors:

Γ_{stretch} = A_{1 g} + B_{1 g} + E_{g} + A_{2 u} + B_{2 u} + E_{u} .

Infrared spectroscopy detects modes transforming as a dipole component:

z

A_{2 u}

and

(x, y)

E_{u}

. There are therefore two IR fundamentals, one

A_{2 u}

and one doubly degenerate

E_{u}

Raman spectroscopy detects modes transforming as polarizability components. The relevant modes are

A_{1 g}

B_{1 g}

and

E_{g}

, giving three Raman fundamentals, of which

E_{g}

is doubly degenerate. The remaining

B_{2 u}

stretch is silent in both ordinary spectra.

Thus the prediction is

N_{IR} = 2, N_{Raman} = 3, N_{coincident modes} = 0.

The zero follows from infrared and Raman mutual exclusion: the molecule has inversion symmetry, dipole components are ungerade, and polarizability components gerade. This counts resolved fundamental frequencies of the ideal ion, not separate lines for degenerate components, isotope splittings or accidental numerical overlaps between distinct modes.

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