Correction for Nuclear Motion inH2+

Abstract

The forced separation of variables usually employed in the quantum-mechanical treatment of molecular problems introduces certain small errors into the wave function. If the exact Hamiltonian is used, energy values can be computed very accurately because first-order errors in a wave function give rise to second-order errors in the energy. The energy of $\mathrm{H}_{2}^{}{}_{}{}^{+}$ is ordinarily computed by using a separable, approximate Hamiltonian instead of an exact one. From a consideration of the terms which must be added to the approximate Hamiltonian to make it exact, Van Vleck has derived the correction terms needed to reduce the first-order error in the computed energy to a second-order error. The correction term is a function of $R$, the internuclear distance, and its calculation requires a knowledge of the wave function. In this paper the ground state wave function of $\mathrm{H}_{2}^{}{}_{}{}^{+}$ is accurately determined over values of $R$ from 1.20 to 2.75 atomic units and a table of the wave function coefficients is given, along with the corresponding energy values. Then, correction terms are calculated for a set of values of $R$. Including the proper correction term, the total negative energy of $\mathrm{H}_{2}^{}{}_{}{}^{+}$ for the equilibrium internuclear distance is found to be $1.20472\ifmmode\pm\else\textpm\fi{}0.00001{E}_{H}=132,132\ifmmode\pm\else\textpm\fi{}10$ ${\mathrm{cm}}^{\ensuremath{-}1}$. This result is compared with the consequences of certain experimental data.