An essential approach to the architecture of diatomic molecules: 1. Basic theory
Tolga Yarman · Optics and Spectroscopy · 2004
We consider the quantum-mechanical description of a diatomic molecule of electronic mass m 0e , internuclear distance R 0, and total electronic energy E 0e . We apply to it the Born-Oppenheimer approximation, together with the relation E 0e m 0e R 0 2 ∼ h 2 (which we established previously), written for the electronic description (with fixed nuclei). Our approach yields an essential relationship for T 0, the classical vibration period, at the total electronic energy E 0e ; i.e., $$T_0 = [4\pi ^2 /(\sqrt {n_1 n_2 } h)]\sqrt {g\mathcal{M}_0 m_e } R_0^2$$ . Here, ℳ0 is the reduced mass of the nuclei; m e is the mass of the electron; g is a dimensionless and relativistically invariant coefficient, roughly around unity (this quantity is associated with the particular electronic structure under consideration; thus, it remains practically the same for bonds bearing similar electronic configurations); and n 1 and n 2 are the principal quantum numbers of electrons making up the bond(s) of the diatomic molecule in hand; because of quantum defects, they are not integer numbers. The above relationship holds generally, although the quantum numbers n 1 and n 2 need to be refined. This task is undertaken in our next article, yielding a whole new systematization regarding all diatomic molecules.