Faddeev-Born-Oppenheimer equations for molecular three-body systems: Adiabatic calculation of 9Be

The low-lying states of ⁹Be are calculated in the α-particle cluster model. The calculation is performed using a rotationally invariant molecular formulation of the three-body problem based on the Faddeev equations, which are solved for the α + n + α system in the adiabatic limit with the α-α interaction turned off. The resulting two-centre wave function is used to formulate an ansatz for the solution of the full hamiltonian of the system where all three particles interact. Unlike the traditional molecular approach, the ansatz we propose allows for the coupling between the movement of the light particle and the rotational motion of the heavy particles. This leads to a set of coupled ordinary differential equations for the three-body wave function that has good total angular momentum and parity. Although only one Born-Oppenheimer molecular energy curve is considered, all adiabatic corrections due to Coriolis coupling effects, mass polarization and derivatives of the two-centre wave function with respect to the separation distance between the α-particles are taken into account. Comparison with exact Faddeev results is presented for the ground-state energies of ⁹Be in a model problem where the α-α interaction is turned off. The validity of the molecular approach for small mass ratios between the heavy particle and the light particle is studied in a very general framework.