Question #44508

In the Walsh and Coulson-Moffitt models of cyclopropane ($\ce{C3H6}$) at $298\text{ K}$, carbon atoms rehybridize to minimize Baeyer strain. NMR data ($J_{\ce{C-H}} \approx 160\text{ Hz}$) indicates that the two $\ce{C-H}$ bonds on each carbon are formed using exactly $sp^2$ hybrid orbitals. The $\ce{C-C}$ bonds are consequently formed by the 'bent' overlap of $sp^n$ orbitals. Due to the spatial delocalization of these bent bonds (increasing positional uncertainty $\Delta x$), the Heisenberg Uncertainty Principle dictates a lower confinement energy for the bonding electrons, which raises the HOMO energy. Identify the correct tuple for: (1) the hybridization index $n$ for the $\ce{C-C}$ bonding orbitals, (2) the cosine of the inter-orbital angle ($\cos \theta_{\text{inter-orbital}}$) between the two $\ce{C-C}$ bonds on the same carbon, and (3) the ratio of the lowest electronic excitation energy of cyclopropane to that of propane ($\frac{\Delta E_{\text{cyclo}}}{\Delta E_{\text{propane}}}$).