Close Relation between Localized-Electron Magnetism and the Paramagnetic Wave Function of Completely Itinerant Electrons

What is commonly called the Gutzwiller wave function ψ, an approximate ground state of the single-band Hubbard Hamiltonian, is considered here for N-site N-electron rings, when the number D of doubly occupied sites is small ($N=6, 10, 14, 18$). For D→0 spin correlations in ψ surprisingly close to the exact values at zero bandwidth are found (i.e., those of the antiferromagnetic Heisenberg-model ground state). But the energy is grossly in error. A simple modification of ψ reproduces the exact energy with remarkable accuracy.