It may be useful to refer to the physical chemistry sections on orbital theory and “electron in a box” diagrams before continuing with this section if they are not familiar concepts.

The analogy of the harmonics (standing waves) of a piece of string fastened at both ends to an electron in an energy level is useful in this instance.  It lets us visualise what we mean by a node (a part of the string that does not move), and look at the concept of phases.

In the example of the piece of string, the phases are defined by whether the string is above or below the nodal plane.  So too, if we imagine the orbital diagram of our electron in a one-dimensional box, the parts of the diagram that appear above the nodal plane are positive phase, and those below are negative.

 

We can use this analogy to make assertions about the phase of molecular orbitals.  For example, we already know that the atomic 2p orbital (having one nodal plane) will have lobes of differing phase.  This is normally represented by shading the lobes in different colours; the black is positive, and white is negative:

We can see therefore that the molecular orbitals of butadiene can be represented as shown above.  Where two adjacent atomic orbitals have different phases, between them is a node.  We can see at a glance therefore, that this agrees with the previous representation; no nodes for Ψ1, one node for Ψ2, two for Ψ3 etc.  We can also note that as the representation of Ψis all in phase, this will be the strongest bonding orbital (hence it is lowest in energy).  Conversely, Ψwill be completely antibonding.

Bonding interactions can only occur between lobes of equal phase; antibonding ones occur between those of different phase.

Looking at the symmetry of the orbitals involved lead Woodward and Hoffman to create a set of rules to explain the behaviour of pericyclic reactions.  To obtain an understanding of these rules, we should ideally consider the symmetry of all the orbitals involved in the reaction, however, for our purposes, it is enough to describe the frontier orbitalsHOMO (Highest Occupied Molecular Orbital), and the LUMO (Lowest Unoccupied Molecular Orbital).  The names are fairly self-explanatory, but suffice to say that a HOMO is a potential electron donor, and a LUMO is a potential electron acceptor.