# Symmetry of Polyatomic Molecular Orbitals

#### Symmetry Theory

The importance of the symmetry of a molecule in determining the form of the molecular orbitals can be investigated by considering the molecular orbitals of the H_{3} molecule.

There are two possible geometries for H_{3}, linear and triangular.

#### Linear H_{3}

The basis set of atomic orbitals are the 1s orbitals in each H atom, named A, B and C, of which there are three, and so there are three molecular orbitals. One of these will be bonding, another antibonding, and the last with intermediate bonding character.

An in phase overlap of all three 1s orbitals gives the lowest energy bonding orbital. Here, there is a bonding interaction between A and B and between B and C.

The orbital which has the maximum out of phase overlap is the highest energy antibonding orbital. There is an antibonding interaction between A and B and between B and C.

The intermediate orbital has contributions only from the atomic orbitals on the A and C atoms, and no contribution from the central atom. The atomic orbitals on the outer atoms are too far apart to overlap, and this molecular orbital is essentially non-bonding.

All three molecular orbitals are spherically symmetric about the H-H-H axis, and so they are all σ orbitals. The order in energy of the molecular orbitals can be predicted by noting that the energy of an orbital increases with the number of nodes, or changes in sign of the phase, in the wavefunction. The orbitals above have 0, 1, and 2 nodes for the 1σ, 2σ and 3σ orbitals respectively.

#### Triangular H_{3}

Here, again, the basis orbitals are the 1s atomic orbitals on the three H atoms, A, B and C.

Now, there is a bonding orbital which results from the in phase overlap of all three atomic orbitals. In addition to the bonding interaction between A and B, and B and C, there is now an additional bonding interaction between A and C, and so the 1σ orbital moves to lower energy. This orbital is labeled a

_{1}.Hence both the energies and the forms of the molecular orbitals depend upon the geometry, or the symmetry, of the molecule, even when the basis orbitals are the same.Now, the non-bonding and antibonding orbitals form the linear case become degenerate in the triangular case. The 2σ orbital becomes antibonding between A and C, and so rises to higher energy, but the 3σ orbital, whilst remaining antibonding between A and B, and B and C, becomes bonding between A and C, and so moves to lower energy. These new molecular orbitals are degenerate, and are labeled e

_{g}as a pair.The energies of the orbitals in the two geometries can be related by a correlation diagram, or a Walsh diagram. This shows, for the H

_{3}molecule, the variation in the energies of the molecular orbitals with the H-H-H angle.

Walsh Diagram for H_{3}

#### Symmetry in Octahedral molecules

One of the most important geometries for molecules in inorganic chemistry is the octahedral complex.

The molecular orbitals in an octahedral geometry can be formed by considering the overlap of the orbitals on the central atom, and suitable combinations of the orbitals on the atoms at the vertices of the octahedron.

A discussion of the molecular orbitals in an octahedral d-metal-ligand complex shows the importance of symmetry adapted linear combinations (SALCs) of atomic orbitals. The six ligand orbitals, one from each of the six ligands, combine to form a new basis set of orbitals which have the same symmetry as the overall complex (a_{1g}+e_{g}+t_{1u}). Similarly, the nine d-metal valence orbitals are combined to produce a basis set of six orbitals of the same symmetry as the new ligand basis set (a_{1g}+e_{g}+t_{1u}), and of three orbitals which do not have the same symmetry as any of the new basis set of ligand orbitals (t_{2g}).

The bonding and antibonding interactions in the octahedral complex come from the overlap of the metal orbitals from the first set with the ligand orbitals, whilst the second set of metal orbitals, the t_{2g} set, remain non-bonding.

In these complexes, the conversion of the s, p, and d orbitals from the metal, and the σ(and sometimes π) orbitals from the ligands to the a_{1g}+e_{g}+t_{1u} and t_{2g} sets give us the SALCs. The SALCs are the a_{1g}, e_{g}, t_{1u}, and t_{2g} orbitals.