I have never before prior to heard/read around something as a \$sp^5\$ hybridization. Today, Henry Rzepa"s blog short article made me mindful of the existance of such a bonding mechanism. That made me search a small little and I found an entry in a germale slrfc.org forum, wright here this question was likewise asked ... they answered it through a mathematical construction\$^ast\$:

Cyclopropane has the complying with angles:

\$angle ceHCH=118^circ~ extresp.~gg 120^circ\$ \$angle ceCCC~ extvia bent bonds:~60 + 2 cdot 21 = 102^circ\$

The orbitals towards the prolots are \$sp^2\$ bereason of the \$120^circ\$ angles. The orbitals towards the carbons originate in the following relation:

\$\$1 + a cos~alpha = 0\$\$ ... wbelow \$alpha\$ is the bond angle and \$a\$ is the p-amount in sp\$^a\$ for the orbitals, which consist of the angle.

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This suggests for the orbitals, which expectancy the 102 level angle: \$\$1 + a cos 102^circ = 0\$\$ \$\$1 + a cdot (-0.20) = 0\$\$ \$\$a = frac-1-0.20 = 5\$\$ \$\$Rightarrow extsp^5 ext-orbitals\$\$

Test:

In a single sp\$^a\$ orbital, the s-amount is: \$frac11+a\$, because \$1+a\$ equates to the sum of all quantities of s and p In a solitary sp\$^a\$ orbital, the p-amount is: \$fraca1+a\$

For s:

In the orbitals that are oriented in the direction of the proloads, the s-amount is \$frac11+2 = frac13\$ In the orbitals that are oriented towards the carbons, the s-amount is \$frac11+5 = frac16\$ Addition of all s-quantities at a single carbon with all four bond orbitals yield: \$frac13+frac13+frac16+frac16=1\$, which is correct, bereason tbelow is only one single s-orbital at eincredibly carbon atom. For p: In the orbitals that are oriented in the direction of the prolots, the p-amount is \$frac21+2 = frac23\$ In the orbitals that are oriented towards the carbons, the p-amount is \$frac51+5=frac56\$ Addition of all p-amounts at a solitary carbon via four bond orbitals yield: \$frac23+frac23+frac56+frac56\$, which is correct, because tright here are 3 p-orbitals at eincredibly carbon atom.

This implies, that the bent bonds with \$21^circ\$ from the \$ceC-C\$-bond are spanned by sp\$^5\$ orbitals.

So math-magically this appears to make sense, however is there one more explanation that might base more on chemical intuition or "real" chemical concepts?

A quick calculation (\$omega\$B97X-D/def2-TZVPP) and also a succeeding evaluation of the isosurface of the Laplacian of the electron density, proved at leastern the intended "nonlinear", slightly curved bond in between the carbon atoms.

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Whoever before might want to watch the electron localization feature (ELF), which additionally reflects the bent bonds quite good:

\$^ast\$ While I tried to translate it to my best, some errors could have been introduced by this . . . please correct me, where I"m wrong.