text: For ordinary alkenes, [2+2] cycloadditions only observed under photochemical activation.
  The rationale for the non-observation of thermal [2+2] cycloadditions begins with the analysis of the four possible stereochemical consequences for the [2+2] cycloaddition: [2 + 2], [2 + 2], [2 + 2], [2 + 2]. The geometrically most plausible [2 + 2] mode is forbidden under thermal conditions, while the [2 + 2], [2 + 2] approaches are allowed from the point of view of symmetry but are rare due to an unfavorable strain and steric profile.  
  Considering the [2 + 2] cycloaddition.  This mechanism leads to a retention of stereochemistry in the product, as illustrated to the right.  Two symmetry elements are present in the starting materials, transition state, and product: σ and σ.  σ is the mirror plane between the components perpendicular to the p-orbitals; σ splits the molecules in half perpendicular to the σ-bonds. These are both local-symmetry elements in the case that the components are not identical.
  To determine symmetry and asymmetry with respect to σ and σ, the starting material molecular orbitals must be considered in tandem.  The figure to the right shows the molecular orbital correlation diagram for the [2 + 2] cycloaddition.  The two π and π molecular orbitals of the starting materials are characterized by their symmetry with respect to first σ and then σ.  Similarly, the σ and σ molecular orbitals of the product are characterized by their symmetry.  In the correlation diagram, molecular orbitals transformations over the course of the reaction must conserve the symmetry of the molecular orbitals.  Thus π correlates with σ, π correlates with σ, π correlates with σ, and finally π correlates with σ.  Due to conservation of orbital symmetry, the bonding orbital π is forced to correlate with the antibonding orbital σ.  Thus a high barrier is predicted.
  This is made precise in the state correlation diagram below.  The ground state in the starting materials is the electronic state where π and π are both doubly populated – i.e. the state (SS)(AS).  As such, this state attempts to correlate with the electronic state in the product where both σ and σ are doubly populated – i.e. the state (SS)(AS).  However, this state is neither the ground state (SS)(SA) of cyclobutane, nor the first excited state ES-1=(SS)(SA)(AS), where an electron is promoted from the HOMO to the LUMO.(AS).
  Similarly, the ground state of the product cyclobutane, as can be seen in the molecular orbital diagram above, is the electronic state where both σ and σ are doubly populated – i.e. the state (SS)(SA).  This attempts to correlate with the state where π and π are both doubly populated – i.e. a second excited state ES-2=(SS)(SA).
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  Finally, the first excited state of the starting materials is the electronic configuration where π is doubly populated, and π and π are both singly occupied – i.e. the state (SS)(AS)(SA).  The first excited state of the product is also the state (SS)(SA)(AS) as σ is doubly populated, and σ and σ are both singly occupied.  Thus these two excited states correlate.