Calculation of Coulomb-interaction parameters for ${\mathrm{La}}_2$${\mathrm{CuO}}_4$ using a constrained-density-functional approach

The constrained-density-functional approach is used to calculate the energy surface as a function of local charge fluctuations in ${\mathrm{La}}_2$${\mathrm{CuO}}_4$. This energy surface is then mapped onto a self-consistent mean-field solution of the Hubbard model which allows extraction of the Coulomb interaction parameters when combined with one-electron parameters derived from band-structure results. Variations in the local Cu d charges and in-plane O p charge are considered for the prototypical high-$T_c$ parent oxide ${\mathrm{La}}_2$${\mathrm{CuO}}_4$. To isolate the charge fluctuations, the calculations are done in a supercell of size up to 2×2 in the basal plane. The local density-functional calculations are done using the linear muffin-tin orbital approach with the atomic sphere approximation. In the Hubbard Hamiltonian, the Cu d($x^2$-$y^2$) and O p(x,y) orbitals are included in the pdσ configuration. The one-electron parameters consist of bare on-site energies (${\varepsilon }_p$,${\varepsilon }_d$) and first-neighbor hopping from Cu to O ($t_{\mathrm{pd}}$) and from O to O ($t_{\mathrm{pp}}$) while the Coulomb-interaction parameters include on-site ($U_d$,$U_p$) and intersite ($U_{\mathrm{pd}}$,$U_{\mathrm{pp}}$) terms. Results of the present calculation indicate that ${\mathrm{La}}_2$${\mathrm{CuO}}_4$ is intermediate between the extreme spin or charge fluctuation regimes. This places strong constraints on the available parameter space for theories of high-$T_c$ superconductivity based on the extended Hubbard model.