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Module 2 : Molecular Structure

Lecture 7 : Homonuclear Diatomic Molecules

There are two main observations in this figure 7.10(b). One is that the energies of the MOs are distance dependent and the other is that as the distance (r) between the nuclei is increased, the difference in the energies of the bonding and antibonding MOs diminish and vanishes as . At very large r, there is no binding and the energies are simply the energy levels of two separated non-interacting atoms. In fig 7.10(a) you will notice that the energy of the antibonding MO, is greater than zero (the energy of the AO is taken as the reference value of zero) throughout. The reference value of zero is actually the energy of the atomic orbital (AO). The energy of the bonding orbital is lower than the energy of the AO for all but the very short distances. At very short distances, the energies of the MOs rise very steeply; far more steeply than 1/r, which is the formula for Coulomb repulsion between the electrons. The origin of this steep repulsion is not Coulombic, but the Pauli exclusion principle. When electrons are forced to be very close to one another, there arises a possibility that all the four quantum numbers of two electrons may be the same. Since this is forbidden by the exclusion principle, and furthermore since the electrons are indistinguishable as well (ie labeling the electrons as 1234 is no different from labeling them as 1432 since one can not associate any labels with these “tiny” indistinguishable particles), the energy rises sharply at short distances.

In Fig 7.10 (b), the potential energies of H₂, H_{2_⁺}, He_2⁺ and He₂ are shown. He₂ does not form a stable molecule at room temperature, because its bond order is zero. Only at very low temperatures, such clusters like He₂, Ar₂, or Ar_n, n > 2) are found. These are called van der Waals clusters because their binding energy is very small, less than 1 kcal/mol. The species He_2⁺, H_2⁺ do have bound states with respectable amounts of stabilities. The bond lengths are all near 1. Similar to the sketch in Fig 7.10(b), the potential energies of other homonuclear diatomics can be drawn.

While concluding this lecture, we need to make an important point. Molecular orbitals are approximate constructs. They describe each electron as moving “independently” of other electrons in an effective field. In reality the motion of all electrons is correlated, ie the concept of an effective field is unreal as the electrons are moving all the time and the fields can not be averaged and the instantaneous influence of electrons on one another contributes (about 15%) to the binding energies and bond lengths. A common way to take into account this feature in the MO framework is through the inclusion of “higher” orbitals for calculating binding energies. This means that orbitals 2s, 2p, 3s, 3p, 3d and so on also make significant contributions to the binding energy in H₂ . These aspects are of prime concern in the computations of binding energies of molecules and solids.