Theoretical Approach to Potential Energy Functions for Linear AB2 and ABC and Bent AB2 Triatomic Molecules

Abstract

Let the coordinate system for a linear AB₂ triatomic molecule have its origin on the A nucleus with R₁ the distance to one B nucleus, R₂ the distance to the other, and θ the apex angle. Then a working formula for the Born‐Oppenheimer potential energy near equilibrium, W(R₁, R₂, θ), is $${\mathrm{W(R}}_1{\mathrm{,\; R}}_2{\mathrm{,\; \theta )\; =W}}_D{\mathrm{(R}}_1{\mathrm{)+W}}_D{\mathrm{(R}}_2\mathrm{)+A/|}{\mathbf{R}}_3{|}^N{\mathrm{-B/(R}}_1{\mathrm{+R}}_2{)}^N$$ , where W _D(R₁) and W _D(R₂) are potential functions for the ground state diatomic molecule AB, R₃ is the vector sum ${\mathbf{R}}_1+{\mathbf{R}}_2$ , A and B are constants and N is an integer. The potential energy for linear ABC tri‐atomic molecules is given by Eq. (i) with R₂ [or R₁] scaled: $R_2{\mathrm{\to \; \eta \; R}}_2$ . Equation (i) is tested for CO₂, CS₂, OCS, HCN, and N₂O by predicting all force constants up to fourth order except for the harmonic bending constant which is used in the parameterization. For bent AB₂ molecules the working formula for the potential energy is $${\mathrm{W(R}}_1{\mathrm{,\; R}}_2{\mathrm{,\; \theta )\; =\; W}}_D{\mathrm{(R}}_1{\mathrm{)+W}}_D{\mathrm{(R}}_2\mathrm{)+A/|}{\mathbf{R}}_3{|}^N{\mathrm{-B/(R}}_1{\mathrm{+R}}_2{)}^N\mathrm{+C/|}{\mathbf{R}}_3{|}^M{\mathrm{-A}}_{\theta }\hspace{0.17em}\cos {\mathrm{(\theta \; -\; \theta }}_e\mathrm{),\; (M\ne \; N)}$$ , where θ_e is the equilibrium angle. Equation (ii) is tested for H₂O, SO₂, and O₃ by predicting all force constants up to fourth order except for the harmonic bending and stretch‐bend interaction constants which are used in the parameterization. A theory is presented for the presence of the W _D(R) components in Eqs. (i) and (ii). It is suggested that the inverse $|{\mathbf{R}}_3|$ terms partially represent averaged multipole interaction energies while the inverse ${\mathrm{(R}}_1{\mathrm{+R}}_2)$ terms and angular term in Eq. (ii) partially represent valence orbital effects. The formulas for the parameters in Eqs. (i) and (ii) are given. Suggestions for writing down potential functions for larger molecules are included.