Channel-coupling theory of molecular structure. Global basis-set expansions for ${\mathrm{H}}_2^{+}$, ${\mathrm{H}}_2$, and He${\mathrm{H}}^{+}$

Earlier work applying equations of the channel-coupling array theory of many-body scattering to ${\mathrm{H}}_2^{+}$ and ${\mathrm{H}}_2$ has been extended. Primary consideration has been given to the use of the Galerkin-Petrov method, by which the non-Hermitian matrix equations of the theory have been solved approximately with the use of globally defined bases as expansion sets. For ${\mathrm{H}}_2^{+}$, both hydrogenic and Hylleraas-Shull-Löwdin functions were used, but neither set led to minima in the ground-state (gerade) potential-energy curves which lie higher than the exact value nor indicated a convergence to the correct minimum value. Only hydrogenic functions were used for ${\mathrm{H}}_2$ in a two-channel truncation approximation. Slow convergence (from above) to the exact minimum in the singlet ground-state potential-energy curve was indicated. The inner minimum in the ${\mathrm{H}}_2$ $E {}_{}{}^1{}_{}{}^{}\Sigma _g^{+}{}_{}{}^{}$ curve originally calculated by Davidson was reproduced to within 0.002 a.u. by an extremely simple covalent approximate calculation; there was no indication of Davidson's outer, ionic minimum, but this is not surprising since an ionic component was not built into our approximation. The He${\mathrm{H}}^{+}$ ground state was also determined. The equilibrium separation and energy minimum for the ground state was relatively accurate given the simple (Eckart) wave function used to approximate the He ground state. In addition to these results, the nonphysical ${\mathrm{H}}_2^{+}$ ungerade and ${\mathrm{H}}_2$ triplet potential-energy curves found in the earlier one-state approximate calculations also occurred in the present computations. The failure to achieve accurate convergence and the persistence of nonphysical potential-energy curves remain serious problems that require further investigation. A resolution of both these problems for ${\mathrm{H}}_2^{+}$ has been achieved and is described in the following paper.