Faddeev-Born-Oppenheimer equations for molecular three-body systems: Application to ${\mathrm{H}}_2$${\mathrm{}}^{+}$

A nonvariational parameter-free molecularlike approach is developed for the three-body problem based on the Faddeev equations. Considering a system of two identical heavy particles (atomic nuclei) and a light one (electron), we study the adiabatic limit of the corresponding Faddeev equation in the absence of interaction between the heavy particles and using general heavy-light potentials that are represented in a separable form through the Hilbert-Schmidt method. The resulting rotationally invariant Faddeev two-center eigenfunctions are used to formulate an ansatz for the solution of the full Hamiltonian where all three particles interact. A set of coupled differential Born-Oppenheimer-like equations is obtained for the movement of the heavy particles. Numerical calculations are shown for the 1sσg, 2sσg, 3dσg, 2pσu, and 2pπu electronic states in ${\mathrm{H}}_2$${\mathrm{}}^{+}$. The resulting molecular energy curves appear to converge to the exact ones when up to fifteen terms are used in the Hilbert-Schmidt expansion of the Coulomb potential. The noncrossing rule for 2sσg and 3dσg curves is verified in our work.