  Motivated by the discovery of superconductivity in beta-pyrochlore oxides, we
study property of rattling motion coupled with conduction electrons. We derive
the general expression of the Green's function of fully anharmonic lattice
vibration within the accuracy of the second order perturbation of electron-ion
interaction by introducing self-energy, vertex-correction, and normalization
factor for each transition. Using the expression, we discuss the characteristic
properties of the spectral function in the entire range from weakly anharmonic
potential to double-well case, and calculate NMR relaxation rate due to the two
phonon Raman process.
