Molecular Orbital Theory

Molecular orbital theory does not consider the electrons in a bond to be localised between two nuclei. Rather it considers them to occupy a molecular orbital. This orbital, the region of space in which the electron is most likely to be found, covers the whole molecule, and the atoms that make up the molecule are bound together by the electrons that move around them. Thus the electrons may be found anywhere in the molecule, though they are most likely to be found between the nuclei that VB theory fixes them between.

It is possible to solve the Schrodinger equation exactly for only one system, the H2+ ion (and even then the Born-Oppenheimer approximation must be made). Thus an approximate procedure that may be readily be applied to even quite complex molecules, has been developed. This procedure is known as the Linear Combination of Atomic Orbitals (LCAO), and will be illustrated using the H2+ ion before extensions to other systems will be discussed.

A “linear combination of atomic orbitals” merely indicates that the molecular orbitals are formed from the sum or difference (which is the linear combination) of the wavefunctions for atomic orbitals. i.e. If an electron can be found in an atomic orbital on atom A and also in an atomic orbital on atom B, then the overall wavefunction for the electron is a superposition of the two atomic orbitals:

where A and B represent the atomic orbitals on the appropriate atom, and N is a normalisation factor.

Note there are two possible wavefunctions to describe the state of the electron. This equation applies to the electron in H2+ when  A and B are hydrogen 1s orbitals. The above equation is a linear combination of two atomic orbitals. The approximate molecular orbitals formed (the + and – combinations) both have cylindrical symmetry about the internuclear axis, so are termed σ orbitals.

According to the Born interpretation, the probability density of the electron in H2+ is proportional to the square modulus of its wavefunction. Thus for the wavefunction Ψ+ (N(A+B)), the corresponding probability density is given by:

which, when plotted, looks like the following:

Once again, note the buildup of electron density in the internuclear region.

When we look in detail at the form of Ψ+2, we can see that there are three separate contributions to the probability density:
1. A2, the probability density of the electron in the atomic orbital on A.
2. B2, the probability density of the electron in the atomic orbital on B.
3. 2AB, an extra contribution that arises from the superposition of the two wavefunctions A and B.

This last contribution is termed the overlap density, and represents the increased probability of finding the electron in the internuclear region, which arises from constructive interference between the two wavefunctions. This accumulation of electron density between the nuclei means that the likelihood is that the electron will be found in a region where it can interact with both nuclei and bind them together.

This σ orbital is a bonding orbital; one which, if occupied, binds the two atoms together. Electrons in such orbitals have a lower energy relative to the separated atoms. For the H2+ ion, where the two orbitals are the 1s orbitals of the two hydrogen atoms, this the lowest energy bonding orbital in the molecule. It is thus labeled as the 1σ orbital. (The σ orbital that is next highest in energy would be the 2σ orbital, and so on.) The electronic configuration of the H2+ ion is thus written as 1σ1.

The linear combination Ψ has a higher energy than Ψ+, but it is also a σ orbital. We can thus label it as the 2σ orbital. The probability density for this linear combination is given by:

It turns out that this orbital has a nodal plane between the nuclei (a plane in which there is zero probability that the electron will be found). The probability density diagram for this linear combination looks like this:

Note that the probability density in the internuclear region has been reduced in this case. This can also be seen from the -2AB term in the expression for the probability density.

This linear combination is thus termed an antibonding orbital, one which, if occupied, destabilises the molecule. Electrons in such orbitals have a greater energy relative to the situation in which the atoms are separated. Note that antibonding orbitals are often labeled separately from the bonding orbitals, with a * to distinguish them. Thus the orbital above could also be labeled 1σ* , as the lowest energy antibonding orbital.

This is the convention we shall adopt.

The final point to be made is that when 2 atomic orbitals overlap, they give rise to one bonding and one antibonding orbital. The antibonding orbital orbital is more antibonding than the bonding orbital is bonding. i.e. if both orbitals are fully occupied, the net effect is antibonding (and the molecule will likely exist only briefly, if at all, before separating into atoms).