  We revisit the two-site Hubbard-Holstein model by using extended phonon
coherent states. The nontrivial singlet bipolaron is studied exactly in the
whole coupling regime. The ground-state (GS) energy and the double occupancy
probability are calculated. The linear entropy is exploited successfully to
quantify bipartite entanglement between electrons and their environment
phonons, displaying a maximum entanglement of the singlet-bipolaron in strong
coupling regime. A dramatic drop in the crossover regime is observed in the GS
fidelity and its susceptibility. The bipolaron properties is also characterized
classically by correlation functions. It is found that the crossover from a
two-site to single-site bipolaron is more abrupt and shifts to a larger
electron-phonon coupling strength as electron-electron Coulomb repulsion
increases.
