Solving non-Born–Oppenheimer Schrödinger equation for hydrogen molecular ion and its isotopomers using the free complement method

The Schrödinger equations for the hydrogen molecular ion $(H2+)$ and its isotopomers (⁠$D2+$⁠, $T2+$⁠, $HD+$⁠, $HT+$⁠, and $DT+$⁠) were solved very accurately using the free iterative complement interaction method, which is referred to in short as the free complement (FC) method, in the non-Born–Oppenheimer (non-BO) level, i.e., in the nonrelativistic limit. Appropriate complement functions for both electron and nuclei were generated automatically by the FC procedure with the use of the non-BO Hamiltonian, which contains both electron and nuclear operators on an equal footing. Quite accurate results were obtained not only for the ground state but also for the vibronic excited states. For example, we obtained the ground-state energy of $H2+$ as $−0.597139063123405074834134096025974142a.u.$⁠, which is variationally the best in literature. The difference in the nuclear spin states of $S1$ (para) and $P3$ (ortho) of $H2+$ and some physical expectation values for several of the isotopomers shown above were also examined. The present study is the first application of the FC method to molecular systems with the non-BO Hamiltonian.