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.