Molecular Orbitals

This concept presumes a substantial degree of covalency (i.e. electron sharing) and, since we shall be considering homonuclear diatomic molecules of the first main period only, this presumption seems fair enough.

We cannot easily calculate the electron distribution in a molecule. It would require the solution of the Schrodinger equation for a specified geometric array of nuclei, and the determination of the geometric and energy distribution of the electron density of a multi-electron system. It is important to remember that the Schrodinger equation can only be exactly solved for a one-electron atom or ion!

Simplification of the Schrodinger Equation

In order to simplify the Schrodinger equation we make an approximation. We say that the orbitals found in a molecule (the Molecular Orbitals) can be approximated by a "Linear Combination of Atomic Orbitals" (LCAO) which themselves are assumed to be closely similar to hydrogen-like orbitals. This may sound like "some approximation"; but the test is always "does it predict the correct solutions" and perhaps surprisingly, it generally does.

Covalent molecules are ones where the bond results from electrons shared between the participating atoms. Imagine two hydrogen atoms with 1s orbitals (denoted by f1 and f2) approaching each other. At a certain distance apart, they merge (rather like the film images of "cell division" but in reverse) and the electron from each hydrogen has a probability of being found around either nucleus, and along the line joining them. This overlap forms a sigma (s) bond between the atoms, in which the maximum probability of finding the electron lies on the inter-nuclear line; the resulting orbital can be written as a linear combination (f1+f2) which, strictly speaking, needs to be normalised to (1/2)(f1+f2) to be a "one-electron wavefunction " - i.e. to have the same capacity as an atomic orbital to accommodate two electrons of opposite spins.

Now having started with two atomic orbitals (f1 and f2), and having produced one molecular orbital (1/2)(f1+f2) by linear combination, there must be a second (we cannot gain or lose from the total number of orbitals by linear combination). The only realistic option is a subtractive combination (1/2)(f1-f2): this is the sigma antibonding (s*) molecular orbital. The energies may be illustrated on an energy level diagram for di-hydrogen.

Dihydrogen

For H₂⁺, you lose one s-bonding electron, thereby reducing bond strength and increasing bond length. For H₂^-, you add one electron to the s* (antibonding) orbital which reduces the bond strength and increases the bond length again. Thus H₂ (neutral) is the optimum situation. For a postulated He₂, both s and s* orbitals would be doubly occupied, cancelling out (almost) all bond energy. Thus helium is monatomic He, not He₂.