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Properties of the phonon-induced pairing interaction in YBa$_2$Cu$_3$O$_7$ within the local density approximation

The properties of the phonon-induced interaction between electrons are studied using the local density approximation (LDA). Restricting the electron momenta to the Fermi surface we find generally that this interaction has a pronounced peak for large momentum transfers and that the interband contributions between bonding and antibonding band are of the same magnitude as the intraband ones. Results are given for various symmetry averages of this interaction over the Fermi surface. In particular, we find that the dimensionless coupling constant in the d-wave channel $λ^d$, relevant for superconductivity, is only 0.022, i.e., even about ten times smaller than the small value of the s-wave channel. Similarly, the LDA contribution to the resistivity is about a factor 10 times smaller than the observed resistivity suggesting that phonons are not the important low-energy excitations in high-T$_c$ oxides.