Quantum-Mechanical Study of Some Three-Center Two-Electron Systems. II. A Natural-Orbital, Electron-Population, and Energy Analysis

Abstract

Configuration‐interaction wavefunctions for a series of pseudomolecular ions of the form ${\mathrm{ZHZ}}^{\text{+2Z−1}}$ , where $\text{Z\hspace{0.17em}=\hspace{0.17em}1.0 (0.4) 2.2}$ , were reformulated in terms of natural orbitals. Consequently, changes in the electron density could be investigated, as a function of $Z$ and the bond angle $\mathrm{ZĤZ}$ , by means of the population analysis of Mulliken. Contour diagrams were also determined for the charge density evaluated in the plane of the molecule. The total energy for each system was analyzed in terms of the kinetic energy, nuclear attraction energy, and the electron and nuclear repulsion energies. The results are presented graphically as a function of the bond angle $\theta$ for each value of $Z$ . For $\text{Z\hspace{0.17em}=\hspace{0.17em}1.0}$ , i.e., H₃⁺, the occupation numbers for the natural orbitals revealed the united atom character of this system, and the contour diagrams indicated that, in the equilibrium configuration, the “bonds” were directed from each nucleus towards the centroid of the positive charges. As $\theta$ was increased beyond 60°, it was found that the “central” proton within H₃⁺ began to dominate the system. For $\text{Z\hspace{0.17em}=\hspace{0.17em}2.2}$ , the ions approach most closely to their separated systems of minimum energy as $\theta$ tends to 180°. The contour diagrams and the electron‐population analysis also show that the ${\mathrm{ZHZ}}^{\text{+2Z−1}}$ ions change from a three‐center to, basically, a two‐center system and a strongly perturbing proton as the values for $Z$ and $\theta$ increase and decrease, respectively. The $\theta$ dependence of the kinetic energy is perhaps the most interesting of the energy curves. By associating a decrease in kinetic energy with an increase in spatial freedom for the electrons, it was possible to understand the $\theta$ variations of all the energy components for each value of $Z$ . The present examination has allowed us to observe, in detail, changes which occur in the electron density and energy components for a series of three‐center two‐electron systems when $\theta$ and $Z$ are allowed to vary.