Walsh Cyclopropane Molecular Orbitals

Rigorous Derivations

Rigorous derivation of the A₁' cyclopropane MO from the A₁ MOs of the methylene fragments.

There is one identical σ-type orbital on each of the three carbon atoms. Let the orbital on C₁ be φ_1s, that on C₂ be φ_2s, and that on C₃ be φ_3s.

We now apply the A₁' projection operator P^A1' to one of the three orbitals, let us say φ_1s. In order to do this, we apply each symmetry operation in turn, multiplying the result by the character of the A₁' representation.

PA1'φ1s	≈	φ1s + φ2s + φ3s + φ1s + φ2s + φ3s + φ1s + φ2s + φ3s + φ2s + φ3s + φ1s
	=	4(φ1s + φ2s + φ3s)
	≈	φ1s + φ2s + φ3s
The MO is therefore
Normalized:	ψA1'  =  1  (φ1s + φ2s + φ3s) √3

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Rigorous derivation of the σ-type E' cyclopropane MOs from the A₁ MOs of the methylene fragments.

There is one identical σ-type orbital on each of the three carbon atoms. Let the orbital on C₁ be φ_1s, that on C₂ be φ_2s, and that on C₃ be φ_3s.

We now apply the E' projection operator P^E' to one of the three orbitals, let us say φ_1s. In order to do this, we apply each symmetry operation in turn, multiplying the result by the character of the E' representation.

PE'φ1s	≈	2φ1s - φ2s - φ3s + 2φ1s - φ2s - φ3s
	=	2(2φ1s - φ2s - φ3s)
	≈	2φ1s - φ2s - φ3s
The MO is therefore
Normalized:	ψE',a  =  1  (2φ1s - φ2s - φ3s) √6

But we need another orbital. In order to do this, we need to pick a symmetry operation belonging to the D_3h group which will convert ψ_E',a into something other than ±1 times itself. For example, C₃:

The new function is not orthogonal to ψ_E',a, and so it must be a linear combination of ψ_E',a with some other function, ψ_E',b. To find ψ_E',b we multiply the new function by some appropriate factor, then subtract ψ_E',a from it:

	2(2φ2s - φ3s - φ1s) + (2φ1s - φ2s - φ3s)
=	4φ2s - 2φ3s - 2φ1s + 2φ1s - φ2s - φ3s
=	3φ2s - 3φ3s ≈ φ2s - φ3s
The MO is therefore
Normalized:	ψE',b  =  1  (φ2s - φ3s) √2

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Rigorous derivation of the A₂' cyclopropane MO from the B₂ MOs of the methylene fragments.

There is one identical π-type orbital on each of the three carbon atoms. Let the orbital on C₁ be φ_1p, that on C₂ be φ_2p, and that on C₃ be φ_3p.

We now apply the A₂' projection operator P^A2' to one of the three orbitals, let us say φ_1p. In order to do this, we apply each symmetry operation in turn, multiplying the result by the character of the A₂' representation.

PA2'φ1p	≈	φ1p + φ2p + φ3p - (-φ1p - φ2p - φ3p) + φ1p + φ2p + φ3p - (-φ1p - φ2p - φ3p)
	=	4(φ1p + φ2p + φ3p)
	≈	φ1p + φ2p + φ3p
The MO is therefore
Normalized:	ψA2'  =  1  (φ1p + φ2p + φ3p) √3

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Rigorous derivation of the π-type E' cyclopropane MOs from the B₂ MOs of the methylene fragments.

There is one identical π-type orbital on each of the three carbon atoms. Let the orbital on C₁ be φ_1p, that on C₂ be φ_2p, and that on C₃ be φ_3p.

We now apply the E' projection operator P^E' to one of the three orbitals, let us say φ_1p. In order to do this, we apply each symmetry operation in turn, multiplying the result by the character of the E' representation.

PE'φ1p	≈	2φ1p - φ2p - φ3p + 2φ1p - φ2p - φ3p
	=	2(2φ1p - φ2p - φ3p)
	≈	2φ1p - φ2p - φ3p
The MO is therefore
Normalized:	ψE',c  =  1  (2φ1p - φ2p - φ3p) √6

But we need another orbital. In order to do this, we need to pick a symmetry operation belonging to the D_3h group which will convert ψ_E',c into something other than ±1 times itself. For example, C₃:

The new function is not orthogonal to ψ_E',c, and so it must be a linear combination of ψ_E',c with some other function, ψ_E',d. To find ψ_E',d we multiply the new function by some appropriate factor, then subtract ψ_E',c from it:

	2(2φ2p - φ3p - φ1p) + (2φ1p - φ2p - φ3p)
=	4φ2p - 2φ3p - 2φ1p + 2φ1p - φ2p - φ3p
=	3φ2p - 3φ3p ≈ φ2p - φ3p
The MO is therefore
Normalized:	ψE',d  =  1  (φ2p - φ3p) √2

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Copyright © 1997 by Daniel J. Berger. This work may be copied without limit if its use is to be for non-profit educational purposes. Such copies may be by any method, present or future. The author requests only that this statement accompany all such copies. All rights to publication for profit are retained by the author.