Optimization of single-particle basis for exactly soluble models of correlated electrons

Abstract

We determine the explicit form of the single-particle wave functions ${{w}_{i}(\mathbf{r})}$ appearing in the microscopic parameters of models in the second-quantization representation. Namely, the general form of the renormalized wave equation is derived from the Lagrange-Euler principle by treating the system ground-state energy of an exact correlated state as a functional of ${{w}_{i}(\mathbf{r})}$ and their derivatives. The method is applied to three model situations with one orbital per atom. For the first example\char22{}the Hubbard chain\char22{}the optimized basis is obtained only after the electronic correlation has been included in the rigorous Lieb-Wu solution for the ground-state energy. The renormalized Wannier wave functions are obtained variationally starting from the atomic basis for the s-type wave functions. The principal characteristics such as the ground-state energy and the model parameters are calculated as a function of interatomic distance. Second, the atomic systems such as the ${\mathrm{H}}_{2}$ molecule or He atom can be treated in the same manner and the optimized orbitals are obtained to illustrate the method further. Finally, we illustrate the method by solving exactly correlated quantum dots of $Nl~8$ atoms with the subsequent optimization of the orbitals. Our method may be regarded as the next step in analyzing exactly soluble many-body models that provides properties as a function of the lattice parameter and defines at the same time the renormalized wave function for a single particle.