Simple Bond-Charge Model for Potential-Energy Curves of Homonuclear Diatomic Molecules

Abstract

For a homonuclear diatomic molecule near its equilibrium internuclear distance $R_e$ , in some bound electronic state, a potential‐energy function $\mathrm{W(R)}$ of the form ${\mathrm{W\hspace{0.17em}=\hspace{0.17em}W}}_0{\mathrm{\hspace{0.17em}+\hspace{0.17em}W}}_1{\mathrm{\hspace{0.17em}/\hspace{0.17em}R\hspace{0.17em}+\hspace{0.17em}W}}_2{\mathrm{\hspace{0.17em}/\hspace{0.17em}R}}^2$ has previously been shown to be a good approximation to the true potential. From this equation and the molecular virial theorem, there follow expressions for the total electronic potential energy $\mathrm{V(R)}$ and the total electronic kinetic energy ${\text{T(R), V\hspace{0.17em}=\hspace{0.17em}2W}}_0{\mathrm{\hspace{0.17em}+\hspace{0.17em}W}}_1{\mathrm{\hspace{0.17em}/\hspace{0.17em}R,\; T\hspace{0.17em}=\hspace{0.17em}-W}}_0{\mathrm{\hspace{0.17em}+\hspace{0.17em}W}}_2{\mathrm{\hspace{0.17em}/\hspace{0.17em}R}}^2$ . The R‐dependent, Coulombic part of $V$ is modeled by locating a positive charge $\mathrm{Ze}$ at each nucleus and a negative charge $\mathrm{-qe}$ at the bond center, with $\text{q\hspace{0.17em}=\hspace{0.17em}2Z}$ . The Rdependent, free‐electron‐like part of $T$ is modeled by assuming that the charge $q$ moves freely in a one‐dimensional box of length $\mathrm{\nu R}$ . Thus $W_1{\mathrm{\hspace{0.17em}/\hspace{0.17em}R\hspace{0.17em}=\hspace{0.17em}e}}^2{\mathrm{(Z}}^2{\text{\hspace{0.17em}−\hspace{0.17em}4Zq)\hspace{0.17em}/\hspace{0.17em}R, W}}_2{\mathrm{\hspace{0.17em}/\hspace{0.17em}R}}^2{\mathrm{\hspace{0.17em}=\hspace{0.17em}h}}^2{\text{q\hspace{0.17em}/\hspace{0.17em}8mν}}^2{R}^2$ , and ${\mathrm{W\hspace{0.17em}=\hspace{0.17em}W}}_0{\mathrm{\hspace{0.17em}+\hspace{0.17em}(e}}^2{\mathrm{\hspace{0.17em}/\hspace{0.17em}R)\; (Z}}^2{\text{\hspace{0.17em}−\hspace{0.17em}4Zq)\hspace{0.17em}+\hspace{0.17em}(1\hspace{0.17em}/\hspace{0.17em}R}}^2{\mathrm{)[(h}}^2{\text{\hspace{0.17em}/\hspace{0.17em}8m) (q\hspace{0.17em}/\hspace{0.17em}ν}}^2\mathrm{)]}$ . For 17 molecules in 63 different electronic states, parameters $q$ and $\nu$ are given that reproduce exactly the experimental equilibrium distance $R_e$ and harmonic force constant $k_e$ . The $\nu$ values obtained vary little from state to state in a given molecule, or through a given row of the periodic table. The average $\nu$ values are $\text{ν\hspace{0.17em}=\hspace{0.17em}1.0, 0.80, 0.75, 0.65}$ for first‐, second‐, third‐, and fourth‐row homonuclear diatomics, respectively. A relation between $R_e$ and $q$ is derived, $R_e{\text{(Å)\hspace{0.17em}=\hspace{0.17em}2.98\hspace{0.17em}/\hspace{0.17em}qν}}^2$ , and this, together with the observed trends in the $q$ values, shows that $q$ is a reasonable measure of the charge accumulated in the bond region of these molecules. It is suggested that the formula ${\text{q\hspace{0.17em}=\hspace{0.17em}(4R}}_e^3{k}_e{\text{\hspace{0.17em}/\hspace{0.17em}7e}}^2{)}^{\text{1\hspace{0.17em}/\hspace{0.17em}2}}$ may be a useful definition of the bond order for a given state of a homonuclear diatomic molecule. For fixed $\nu$ , this simple point‐charge model, and certain generalizations of it, predict $R_e$ to be proportional to $\text{(1\hspace{0.17em}/\hspace{0.17em}q)}$ , and the quantity $R_e^5{k}_e$ to be constant. The one‐dimensional‐box interpretation is given a justification based on separate virial theorems for the parallel and perpendicular components of the kinetic energy.