Explicit Forms

So what are the combinations of hydrogen orbitals which form the A₁ and B₁ molecular orbitals? We can work this out by a process very similar to the one we used to generate the reducible representation in the first place.
Let us consider the two s-orbitals on the hydrogen atoms, and label them as s₁ and s₂. We then look at what happens to them under each transformation.

i.e. under the identity operation, s₁ remains in the same place. Under C₂ rotation, it exchanges places with s₂, etc.

To generate the combination of orbitals which has A₁ symmetry, we multiply by the A₁ irreducible representation:

So for the sigma-bond of A₁ symmetry, s(A₁),

The result of this must be normalised. To do this we take the square rot of the sum of the squares of the coefficients, and divide the explicit form by this, i.e.

The B₁ combination can be found by a similar process:

Hence,

and after normalisation,

We can now draw the orbitals which are used for the complete molecular orbital.
The A₁ bonding molecular orbital:

Note that both the hydrogen atomic orbitals have the same sign.
The A₁ antibonding molecular orbital:

Note that the hydrogen orbitals and the oxygen hybrid orbital have opposite signs.

The B₁ bonding molecular orbital:

Note that the hydrogen orbitals have opposite signs, and that there is a good bonding overlap with the oxygen p_x orbital, which is of B₁ symmetry.
The B₁ antibonding molecular orbital:

This is an antibonding interaction with the oxygen p_x orbital.

Such a molecular orbital is called a symmetry-adapted linear combination or SALC.