Formally Exact and Practically Useful Analytic Solution of Harmonium

We provide a novel exact analytic solution of harmonium with arbitrary Coulomb interaction strength, for ground as well as all the excited states, using our recently developed method for solving Schrödinger equations. By comparing three formally exact analytic representations of the wave function including the one that utilizes biconfluent Heun function, we find that the best and practically useful representation for the ground state is given by an exact factorized form involving a noninteger power pre-exponential factor, an exponentially decaying term and a modulator function. For excited states, additional factors are needed to account for the nodal information. We show that our method is far more efficient than basis-expansion-based methods in representing the wave function. With the exact wave functions, we have also analyzed the evolution trends of the electron density and natural occupation numbers with increasing interaction strength, which gives insight into the interesting physics in the strong correlation limit.