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

What is commonly called the Gutzwiller wave function, 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 D0 spin correlations in 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 reproduces the exact energy with remarkable accuracy.