Toward an Understanding of Potential-Energy Functions for Diatomic Molecules

The problem of understanding potential‐energy functions for diatomic molecules is discussed. First, starting from the virial theorem, the Born‐Oppenheimer electronic energy $\mathrm{W\hspace{0.17em}(\hspace{0.17em}R)}$ is shown to satisfy the differential equation $$R^2{\mathrm{(d}}^2{\mathrm{W\hspace{0.17em}/\hspace{0.17em}dR}}^2{\text{)\hspace{0.17em}+\hspace{0.17em}4R(dW\hspace{0.17em}/\hspace{0.17em}dR)\hspace{0.17em}+\hspace{0.17em}2W\hspace{0.17em}=\hspace{0.17em}−\hspace{0.17em}(1\hspace{0.17em}/\hspace{0.17em}R) (d\hspace{0.17em}/\hspace{0.17em}dR)[R}}^2\mathrm{T(R)]\hspace{0.17em}\equiv \hspace{0.17em}Q(R)}$$ , where $R$ is the internuclear distance and $\mathrm{T(R)}$ is the electronic kinetic energy, as a function of distance. Then the assumption $\mathrm{Q(R)\hspace{0.17em}=\hspace{0.17em}}\mathrm{const}$ for $R$ near $R_e$ , equivalent to the assumption $$\mathrm{T(R)\hspace{0.17em}=\hspace{0.17em}}\mathrm{const}\mathrm{\hspace{0.17em}+\hspace{0.17em}}\mathrm{const}{\mathrm{\hspace{0.17em}/\hspace{0.17em}R}}^2\mathrm{for}R\mathrm{near}{R}_e$$ , is considered. This assumption is shown to lead to simple formulas relating quadratic ${\mathrm{(k}}_e)$ , cubic ${\mathrm{(l}}_e)$ , quartic ${\mathrm{(m}}_e)$ , and higher potential constants, e.g., $l_e{\text{\hspace{0.17em}=\hspace{0.17em}−\hspace{0.17em}6k}}_e$ , $m_e{\text{\hspace{0.17em}=\hspace{0.17em}36k}}_e$ , ${\mathrm{(k}}_e{m}_e{\mathrm{\hspace{0.17em}/\hspace{0.17em}l}}_e^2{)}^{\text{1\hspace{0.17em}/\hspace{0.17em}2}}\text{\hspace{0.17em}=\hspace{0.17em}1}$ . Ratios of potential constants are calculated for many diatomic species, including several electronically excited states. They are found to agree well with experiment. The quadratic force constants $k_e$ are estimated for a number of species, using a modified overlap‐population concept. An approximate formula is developed which gives the force constant in terms of overlap populations, the equilibrium distance, the dissociation energy, and atomic‐orbital kinetic energies. Calculated and experimental values agree within about 15%. Finally, a well‐known approximate rule that $R_e^2{\omega }_e\mathrm{\hspace{0.17em}=\hspace{0.17em}}\mathrm{const}$ , relating internuclear distance and vibrational frequency through a series of different electronic states of one molecule or different molecules, is elucidated using a free‐electron or uncertainty‐principle interpretation of the R‐dependent part of $\mathrm{T(R)}$ .