A QUANTUM MECHANICAL INTERPRETATION OF THE FORCE CONSTANT INTERNUCLEAR DISTANCE RELATIONSHIPS FOR DIATOMIC MOLECULES

Abstract

“The change in energy arising from the displacement of the nuelei of diatomic molecule from their equilibrium positions can be obtained by perturbation theory. By considering the second-order chnage in energy involved in both a symmetrical and an anti-symmetrical displacement of the nuclei along the molecular axix, and expression for the force constant of the molecule can be obtained as follows:$$K/2Z_{A}Z_{B}=e^{2}/r_{0}^{3}-e^{2}\Sigma_{u}(D_{on}^{(i)}|\cos\theta_{ai}/r_{ai}^{2})(D_{oc}^{(1)}\cos\theta_{bi}/r_{bi}^{2}j/(E_{u}-E_{u}))$$ In this expression the contribution from the nuclear repulsion, $e^{2}/r^{3}_{0}$ appears as the largest term, $r_{0}$ being the equilibrium separation. The sum over all the exeited states of the molecule can be regarded as the electronic contribution it appears as a result of allowing the electrons to follow the nuclei during the vibration. Individual terms in this sum are determined by the interaction of the one-electron transition density function $D^{(i)}_{oc}$ between the ground state and the $n^{th}$ excited state, and the dipolar fields $\cos\theta_{ai}/r^{2}_{ai}$ and $\cos\theta_{bi}/r^{2}_{bi}$ centred on the two nuclei. This sum contains both positive and negative terms (unlike the usual second-order perturbation equation) and should converge quite rapidly. Plotting a graph of $K/2Z_{A}Z_{B}$ against $e^{2}/r^{2}_{0}$ for a number of diatomic molecules, shows that certain regularities exist in the magnitude of the electronic contribution. It is very nearly constant for molecules belonging to the same isoelectronic series, and varies regularly in a given period or group of the periodic table. An explanation of these regularities can to some extent be found by analysing the individual integrals involving atomic orbitals, which go to make up the electronic contribution.”