Density-functional calculations of the phonon spectrum and electron-phonon coupling in $\mathrm{MgB}{}_2$ are presented. The $E_{2g}$ phonons, which involve in-plane B displacements, couple strongly to the $p_{x,y}$ electronic bands. The isotropic electron-phonon coupling constant is calculated to be about 0.8. Allowing for different order parameters in different bands, the superconducting $\lambda$ in the clean limit is calculated to be significantly larger. The $E_{2g}$ phonons are strongly anharmonic, and the nonlinear contribution to the coupling between the $E_{2g}$ modes and the $p_{x,y}$ bands is significant.

• Received 27 March 2001

DOI:https://doi.org/10.1103/PhysRevLett.87.087005