= Solution
The methoxy <substituent> donates <electron> density into the coordinated <arene> by <chemical resonance>. This increases the <electron> density available at <chromium> for <pi backbonding> to CO, weakening C–O and lowering its stretching frequencies. The <ester> <substituent> withdraws <electron> density from the <arene>, reducing <pi backbonding> and raising the carbonyl stretching frequencies. For both observed bands, \b[the methoxy-substituted compound has the lowest carbonyl frequency, the unsubstituted compound is intermediate, and the ester-substituted compound has the highest].
In the <harmonic oscillator> approximation,
$$
\widetilde\nu=\frac{1}{2\pi c}\sqrt{\frac{k}{\mu}},
$$
so the trend reflects an increasing effective C–O <bond force constant> rather than a change in the CO <reduced mass>. For example, comparison of the higher-frequency bands gives $k_{\rm ester}/k_{\rm methoxy}\approx(1992/1983)^2=1.0091$. This is only an approximate local-bond interpretation because the observed <normal modes> couple the three CO oscillators. In the idealized local $C_{3v}$ geometry they span $A_1+E$, giving two infrared-active frequencies. A substituted <arene> can lower the exact <molecular symmetry> and split the $E$ pair, even if that splitting is unresolved experimentally.
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