Molecular orbital theory applied to the bonding in H₂

An approximate description of the MOs in H₂ can be obtained by considering them as linearcombinations of atomic orbitals (LCAOs). Each of the H atoms has one 1s atomic orbital; let the two associated wavefunctions be ψ ₁ and  ψ ₂. The sign of the wavefunction associated with the 1s atomic orbital may be either ‏ + or -.

Just as transverse waves interfere in a constructive (inphase) or destructive (out-of-phase) manner, so too do orbitals. Mathematically, we represent the possible combinations of the two 1s atomic orbitals by equations 1.1 and 1.2, where N and N^* are the normalization factors. Whereas _MO is an in-phase (bonding) interaction ψ ^* MO  is an out-ofphase (antibonding) interaction.

The values of N and N^* are determined using equations 1.3 and 1.4 where S is the overlap integral. This is a measure of the extent to which the regions of space described by the two wavefunctions ψ 1 and ψ 2 coincide. Although we mentioned earlier that orbital interaction is efficient if the region of overlap between the two atomic orbitals is significant, the numerical value of S is still much less than unity and is often neglected giving the approximate results shown in equations 1.3 and 1.4.

The interaction between the H 1s atomic orbitals on forming H₂ may be represented by the energy level diagram in Figure 1. The bonding MO, ψ _MO, is stabilized with respect to the 1s atomic orbitals, while the antibonding MO, ψ ^*_MO, is destabilized. Each H atom contributes one electron and, by the aufbau principle, the two electrons occupy the lower of the two MOs in the H₂ molecule and are spin-paired (Figure 1).

Fig. 1.1 An orbital interaction diagram for the formation of H₂ from two hydrogen atoms. By the aufbau principle, the two electrons occupy the lowest energy (bonding) molecular orbital.

It is important to remember that in MO theory we construct the orbital interaction diagram first and then put in the electrons according to the aufbau principle.