The wavefunction when antiparallel spin electrons coincide and its relation to the ground-state electron density in the Hookean atom

For a specific force constant, the spatial form of the ground-state wavefunction Ψ(r1,r2) is known analytically for the Hookean atom in which two electrons repel coulombically. Here, it is first shown that the wavefunction Ψ(r,r) for the antiparallel spin electrons at the same position r is sufficient to construct the ground-state density in this model example. An off-diagonal generalization is then effected, which relates the one-particle density matrix to the electron pair correlation function, again for this same model.