Lowest excited configuration of harmonium

The harmonium model has long been regarded as an exactly solvable laboratory bench for quantum chemistry [W. Heisenberg, Z. Phys. 38, 411 (1926)]. For studying correlation energy, only the ground state of the system has received consideration heretofore. This is a spin singlet state. In this work we exhaustively study the lowest excited (spin triplet) harmonium state, with the main purpose of revisiting the relation between entanglement measures and correlation energy for this quite different species. The task is made easier by working with Wigner quasiprobabilities on phase space