For the molecule methylenecyclopropene (structure given below), the roots obtained from the Hückel secular determinant can be approximated as x = -2.0, -0.30, +1.0, +1.5, where x = \(\frac{\alpha-E}{\beta}\) with E being the energy of a π orbital.

The delocalization energy of methylenecyclopropene is:

(Given the energy of the ground state π orbital of ethylene is E = α + β)

Concept:

The Huckel secular determinant is used to solve for the molecular orbital energies (eigenvalues) and coefficients (eigenvectors) for a molecule. It is a determinant formed from a matrix representing the Hamiltonian operator for the molecule in the π electron system. The Hamiltonian operator describes the total energy of the electrons in the molecular system.

Given:

• \(x = {𝛼-E \over β}\)
• x = -2, -0.3, +1, +1.5
• \(E = {𝛼-βx}\)

Explanation:

Putting the value of x in above energy equation gives four energy levels which are arranged as follows-

𝛼+2β, 𝛼+0.3β, 𝛼-β, 𝛼-1.5β

In methylenecyclopropene molecule total 4πe^- are present

⇒ Calculated π electrons energy

= 2 x (𝛼+0.3β) + 2x(𝛼+2β)

= 4𝛼 + 4.6β

⇒ Expected π electron energy

= number of electrons (𝛼+β)

= 4(𝛼+β)

= 4𝛼 + 4β

⇒ Delocalization energy

= calculated energy - expected energy

= (4𝛼+4.6β)-(4𝛼+4β)

= 0.6β

Conclusion:

The delocalization energy of methylenecyclopropene is 0.6β.