Electron-phonon coupling andd-wave superconductivity in the cuprates

We derive an effective single-band Hubbard-type Hamiltonian for ${\mathrm{CuO}}_2$ planes in the cuprate high-${\mathit{T}}_{\mathit{c}}$ superconductors. The Hamiltonian includes both electron-electron repulsion and electron-phonon coupling to oxygen vibrational modes. The effective Hamiltonian is derived by mapping from the multiband constrained-density-functional-theory Hamiltonian to a one-band model. A Hartree-Fock mapping leads to t=0.66 eV, t’=-0.14 eV, and U=4.0 eV. Very similar parameters are obtained by exact diagonalization of finite clusters. The electron-phonon coupling to oxygen breathing modes gives λ=0.57 for s-wave and λ=0.35 for ${\mathit{d}}_{\mathit{x}}^2$-${\mathit{y}}^2$ pairing. d-wave superconductivity is predicted to occur at 30 K for doped ${\mathrm{La}}_2$ ${\mathrm{CuO}}_4$, while the strong Coulomb repulsion suppresses the s-wave ${\mathit{T}}_{\mathit{c}}$ to 10 K.