Molecular orbitals resulting from combining atomic orbitals can be constructed in phase (ψ+) and out-of-phase (ψ-). In phase addition occurs when the wavefunctions of the two atomic orbitals combined are of the same sign. For example, in the simplest case of 1s AO which is positive, adding two such AO will lead in a constructive interference. A bonding MO is thus formed. Similarly subtracting the AOs leads to a destructive interference forming an antibonding MO.
In the region between the two nuclei in in phase addition the value of the MO wavefunctions is greater than that of the AO wavefunctions. Thus it is more like likely to find an electron in the inter-nuclear region.
The in-phase MO is lower in energy than the isolated AOs because:
1. When the MO is occupied, there is increased electron density between the nuclei. The attraction from both nuclei leads to a lowering of the potential energy of the system.
2. An electron in in-phase MO is less constrained compared when in an AO (i.e. it is more delocalized) leading to a decrease in its kinetic energy.
The in-phase combination MO is lower in energy than the original AO and occupancy of this orbital gives rise to bonding, thus this orbital is called the bonding molecular orbital.
For the out of phase addition, the anti-bonding combination has a node – where the electron density is zero. This is shown by the crossing point on the x axis (horizontal line).
The out-of-phase MO is higher in energy than the isolated AOs because:
1. When the MO is occupied, there is much more electron density outside the inter-nuclear region than in between the nuclei pulling them apart from each other. The potential energy in this MO is therefore higher than in the separate species.
2. The out-of-phase MO contains a node. This means that the kinetic energy of the electron in this orbital is greater than when it is in either the AO or in-phase MO.
The out-of-phase combination MO leads to an increase of the energy with respect of the original AO, thus this orbital is known as the anti-bonding molecular orbital.
This can be summarized graphically using an energy level diagram:
