= Solution
For the same pair of atoms, the harmonic approximation gives $\widetilde\nu=(2\pi c)^{-1}\sqrt{k/\mu}$, where $k$ is the <bond force constant> and $\mu$ the <reduced mass>. The large shifts mainly reflect changes in $k$ from CO <pi backbonding>; the complexes are not distinguished simply by their total number of valence <electrons>.
\b[$\mathrm{W(CO)_6}$ has stronger net backbonding than $\mathrm{Cr(CO)_6}$.] Both are group-6 M(0), $d^6$, 18-electron complexes. The more spatially extended <tungsten> $5d$ orbitals provide effective overlap with CO $\pi^*$, consistent with the lower stretching frequency. As a simple same-mode force-constant comparison, $(1977/2000)^2=0.977$, about a 2.3% reduction.
\b[$\mathrm{Ni(CO)_3(PMe_3)}$ has appreciable but less C–O weakening than those hexacarbonyls.] Ni(0) is $d^{10}$ and the complex is also 18-electron. The more contracted, lower-energy late-3d occupied orbitals give less effective donation into CO $\pi^*$ than the group-6 carbonyls in this comparison. <Trimethylphosphine> donates <electron> density and increases backbonding relative to an otherwise comparable less-donating <nickel> complex; that effect does not force its frequency below chromium's. Comparing different metals requires both orbital energies and overlap.
\b[$[\mathrm{Cu(CO)_4}]^+$ has very weak backbonding and a frequency above free CO.] Cu(I) is $d^{10}$ and 18-electron here, but positive charge lowers its occupied orbital energies and suppresses backdonation. Sigma donation and the cation's electrostatic polarization can strengthen the internal C–O bond. Consequently the usual free-CO reference, about $2143\ \mathrm{cm^{-1}}$, lies below this cation's band. A blue shift is compatible with <carbon> coordination; it does not imply absence of a metal–CO bond. Coupled normal modes and measurement conditions limit literal extraction of a local bond force constant from one band.
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