Perturbation theories for the calculation of molecular interaction energies. II. Application to H2+

The many perturbation expansions reviewed in Paper I [D. M. Chipman, J. D. Bowman, and J. O. Hirschfelder, J. Chem. Phys. 59, 0000 (1973)] are applied to the interaction of a ground state hydrogen atom with a proton to form the 1s σ_g and 2p σ_u electronic states of ${\mathrm{H}}_2^{+}$ . The calculations were made with high precision for the range of separations R = 0.2–15a_o using a large basis set of Slater‐type orbitals. For the polarization and symmetrized polarization methods the energies were obtained analytically, these results providing a check on the completeness of the basis set. Also, long range and short range asymptotic formulas are given for the first order wavefunctions and the perturbation energies. The HS and MSMA methods gave the best second order energies. The EL‐HAV second order energy is good at intermediate separations but becomes bad at large separations. The expectation value of the Hamiltonian using the wavefunction truncated after the first order (where the coefficient of the zero order function is energy optimized) gave very good results in the OPT, COR, CH, HAV, EL, HS, DEM‐I, AM, and SYM‐P methods in the order listed; the MSMA and P results were poor due to the use of unsymmetrized wavefunctions. The OPT treatment, which energy optimizes the coefficients of three separate first order functions, has an error of less than 0.04 kcal/mole for the 2p σ_u state, over the full range of separations, and for the 1s σ_g state has an error of only 1.4 kcal/mole at the equilibrium separation R = 2a_o, becoming much better than this at larger separations. Reasonably good interaction energies can be calculated from the polarization function ${\mathrm{\phi \hspace{0.17em}}}_p^{\text{(1)}}$ without using the auxiliary functions θ and ω. In the limit of large separations, we find that the EL‐HAV second order energy approaches 41/54 = 0.759 ··· times the exact Coulombic energy, rather than the previously estimated factor of one‐half. Also, at large R, the SYM‐P, HS, and DEM‐I exchange energy agrees with Herring's exact asymptotic expression to within 0.1%.