Knowledge Bank

The Hartree-Fock equations for selected open-shell states of $N2$ and $O2$ were solved in the LCAO MO approximation in both restricted and unrestricted forms. The unrestricted Hartree-Fock method for systems with unpaired electrons results in different orbitals for different spin. The restricted forms of the secular equations, involving the additional approximation of identical orbitals for paired electrons, are simpler to solve and lead to a state function of the proper spin character. It is shown that the results of the two methods differ only slightly. The basis set chosen to describe the electronic states of $N2$ and $O2$ was the set of primitive symmetry orbitals constructed from the orthogonalized Is, 2s and 2p Slater AO's on each atom. The open-shell ionized and excited states chosen for this study were those which could be represented reasonably well by a single Slater determinant. These were the states $X\mathrm{<mrow\; data-mjx-texclass="ORD">1</mrow>}\Sigma g+,A\mathrm{<mrow\; data-mjx-texclass="ORD">3</mrow>}\Sigma g+,B\mathrm{<mrow\; data-mjx-texclass="ORD">3</mrow>}IIg,C\mathrm{<mrow\; data-mjx-texclass="ORD">3</mrow>}IIu$ of $N2+;X\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}\Sigma g+,B\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}\Sigma u+$ of $N2+;X\mathrm{<mrow\; data-mjx-texclass="ORD">3</mrow>}\Sigma g-,a\mathrm{<mrow\; data-mjx-texclass="ORD">1</mrow>}\Delta g$ of $O2$; and $X\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}IIg$ of $O2+$. Calculations of all states of each molecule were performed at the equilibrium internuclear distance of the neutral ground state, 1.094 {\AA} for $N2$ and 1.2074 {\AA} for $O2$. The ionization and excitation energies found by direct calculation are compared with the observed values and those predicted by Koopmans' theorem. Agreement is within 1-2 ev. in each case.