Cycloaddition reactions and FMO theory

There are certain reactions which take place through a cyclic transition state called Pericyclic reactions
These reactions generally are not influenced by a solvent or a catalyst.
They are Electrocyclic reactions, Sigmatropic rearrangements, Cheleotropic reactions and Cycloadditions.
The feasibility of these reactions can be ascertained by three methods.
1. By applying te Woodward-Hoffmann symmetry correlation of the molecular orbitals involved in the process
2. By using the PMO method ( Huckel-Mobius considerations)
3. By the FMO (Frontier Molecular Orbitals method).
In this article the FMO method is applied for [2+2] cyclo addition reactions and the possibility of the reaction under thermal and photo chemical conditions is determined.
The numbers denote the number of electrons involved in each of the molecules.

A nodal plane is a plane of zero electron density.

The following reaction between two ethene molecules leading to cyclobutane, is a [2+2]- cycloaddition

In the above π- Molecular Orbital (π-MO) picture of ethene:
π₁ is the Bonding and the Highest Occupied Molecular Orbital( BMO and HOMO).
π₁* is the Anti-Bonding and the Lowest Unoccupied Molecular Orbital.(ABMO and LUMO)
The two π electrons in ethene will occupy the lowest energy level which is π₁ and π₁* is unoccupied.
Under photochemical conditions an electron undergoes excitation to the next level so π₁* will then be the HOMO.
The lowest energy MO π₁ has zero nodal planes and the highest energy MO π₁* has one nodal plane.

The FMO theory.

This theory considers only the Frontier MOs, the HOMO of one ethene molecule and the LUMO of another.
Only if the interacting lobes are in phase the reaction is thermally feasible.

In the MO diagram the lobes on both sides are not in phase, hence this process is thermally not feasible.
On photo-irradiation one of the electrons from π₁ under goes excitation to π₁*. Thus under photochemical conditions the HOMO is π₁*.
The FMO picture is: