I have actually never before heard/read around something together a \$sp^5\$ hybridization. Today, Henry Rzepa"s blog article made me conscious of the existance of such a bonding system. That made me search a tiny bit and also I uncovered an entry in a german barisalcity.org forum, where this concern was likewise asked ... They answered it v a mathematics construction\$^ast\$:

Cyclopropane has the following angles:

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

The orbitals in the direction of the protons space \$sp^2\$ since of the \$120^circ\$ angles. The orbitals towards the carbons originate in the following relation:

\$\$1 + a cos~alpha = 0\$\$ ... Whereby \$alpha\$ is the shortcut angle and also \$a\$ is the p-amount in sp\$^a\$ for the orbitals, which make up the angle.

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This way for the orbitals, which expectations the 102 degree 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 solitary sp\$^a\$ orbital, the s-amount is: \$frac11+a\$, due to the fact that \$1+a\$ equals the amount of all amounts 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 protons, the s-amount is \$frac11+2 = frac13\$ In the orbitals that room oriented towards the carbons, the s-amount is \$frac11+5 = frac16\$ addition of every s-amounts at a solitary carbon through all 4 bond orbitals yield: \$frac13+frac13+frac16+frac16=1\$, which is correct, due to the fact that there is only one solitary s-orbital in ~ every carbon atom. for p: In the orbitals that are oriented towards the protons, the p-amount is \$frac21+2 = frac23\$ In the orbitals that space oriented towards the carbons, the p-amount is \$frac51+5=frac56\$ enhancement of every p-amounts in ~ a single carbon with four bond orbitals yield: \$frac23+frac23+frac56+frac56\$, i m sorry is correct, since there room 3 p-orbitals in ~ every carbon atom.

This means, the the bent bonds with \$21^circ\$ native the \$ceC-C\$-bond space spanned by sp\$^5\$ orbitals.

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

A fast calculation (\$omega\$B97X-D/def2-TZVPP) and a subsequent analysis of the isosurface the the Laplacian that the electron density, proved at the very least the intended "nonlinear", slightly bent bond in between the carbon atoms.

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Whoever might want to view the electron localization function (ELF), which likewise shows the bent bonds fairly good:

\$^ast\$ While i tried to interpret it to mine best, some errors could have been introduced by this . . . please exactly me, whereby I"m wrong.