  We investigate the origin of the inter-band coupling in MgB2 by focusing on
its unusual phononic features, namely, the strong anharmonicity of the phonons
and the presence of both linear and quadratic electron-phonon interactions of
the Su-Schrieffer-Heeger (SSH) type. The bare electronic Hamiltonian has two
bands with intra- and inter-band hopping, which lead to two decoupled
hybridized bands. The phonon Hamiltonian including the anharmonic terms is
diagonalized approximately by a squeezing transformation, which causes the
softening of the phonon frequency. The linear SSH coupling amplitude is
reduced, consistently with the estimates from first-principle calculations.
Additionally, the quadratic coupling generates an effective phonon-induced
interaction between the hybridized bands, which is non-vanishing even in the
limit of vanishing inter-bare-band hopping amplitude.
