Allyl Heptacity ($\eta^x$) Determination

The allyl ligand ($C_3H_5$) typically coordinates to a metal center through all three carbon atoms in organometallic complexes. This mode of binding is denoted as $\eta^3$.

• Allyl ligand heptacity: $\eta^3$

Cp Heptacity ($\eta^y$) Determination

The cyclopentadienyl ligand ($C_p$, $C_5H_5$) commonly binds to metals using all five carbon atoms, which is represented as $\eta^5$. This mode provides significant stabilization to the complex.

• Cp ligand heptacity: $\eta^5$

NO Ligation Mode Analysis

The nitrosyl (NO) ligand's coordination mode (linear or bent) influences its electronic contribution and the overall stability of the complex. The stability of the target complexes, $[(\eta^x\text{-allyl})Ru(CO)_2(NO)]$ and $[(\eta^y\text{-Cp})Ru(CO)_2(NO)]$, dictates the preferred NO binding.

• In $[(\eta^3\text{-allyl})Ru(CO)_2(NO)]$: The allyl ligand ($\eta^3$, 3e donor) and two CO ligands (2e + 2e donors) along with Ru (assumed Group 8, 8e). For stability, NO often acts as a linear, 2e donor (formally NO$^+$) in such systems, contributing to achieving favourable electron counts. Thus, NO is linear.
• In $[(\eta^5\text{-Cp})Ru(CO)_2(NO)]$: The Cp ligand ($\eta^5$, 5e donor) and two CO ligands (2e + 2e donors) along with Ru (8e). To satisfy the stable 18-electron rule (8 (Ru) + 5 (Cp) + 2 (CO) + 2 (CO) + 1 (NO$^-$) = 18e), the NO ligand typically binds in a bent fashion, acting as a 1e donor (formally NO$^-$). Thus, NO is bent.

Final Conclusion

Combining the determined heptacities and NO ligation modes:

• For $[(\eta^x\text{-allyl})Ru(CO)_2(NO)]$: ($\eta^3$, NO-linear)
• For $[(\eta^y\text{-Cp})Ru(CO)_2(NO)]$: ($\eta^5$, NO-bent)

Therefore, the correct combination is ($\eta^3$, NO-linear) and ($\eta^5$, NO-bent).