Sigma - pi separability

There is nothing magical about the separation of σ and π electrons. It is a simple consequence of symmetry. If we have a planar compound, then reflection in the plane that contains all the atoms is a symmetry operation in the point group that the molecule belongs to. For example, ethene (ethylene) has D_2h symmetry and reflection in the plane of the 6 atoms is one of the 3 reflection operations of the D_2h point group.

All molecular orbitals must transform like one of the irreducible representations of the group. This jargon simply indicates that for these planar compounds the molecular orbitals must be symmetric or antisymmetric with respect to reflection in the plane. Molecular orbitals which are symmetric, i.e the same above and below the plane, are called σ orbitals. Those that are antisymmetric, i.e. they change sign going through the plane, are called π orbitals. σ molecular orbitals must be formed from symmetric atomic orbitals. This means that only s orbitals and the p_x and p_y orbitals (we specify the xy axis as being in the plane and the z axis being perpendicular to the plane) can contribute to the σ orbitals. Similarly only antisymmetric atomic orbitals can contribute to the π molecular orbitals.

This means that only the 2p_z orbital contributes to the π molecular orbitals. Note that in these empirical methods we are only using atomic orbitals that are occupied in the free atom. This is called a minimal basis set. We are not using d orbitals.

The simple conclusion from this is that the π molecular orbitals are built from just one atomic orbital (the 2p_z orbital) on each heavy atom (carbon for hydrocarbons). The hydrogen atomic orbitals only contribute to the σ orbitals.

In a proper treatment, the form of the π orbitals is determined by the nature of the σ orbitals and vice versa because of the repulsion between the π and σ electrons. However, in an empirical method, we simply fudge in the repulsion from the σ electrons into the terms we include in the calculation of the π orbitals.

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