F.2.1 The Phonon GREEN's Function

(F.38) |

where denotes the Cartesian components. This expansion reduces the density-density correlation function (F.34) to

where the phonon GREEN's function in real space is

Owing to the lattice periodicity of the ionic charge densities, the spatial FOURIER transformation of (F.39) takes the form

The FOURIER expansion of the lattice displacement can be written as

where are the mass of the atoms and is the number of atoms in the unit cell. By means of (F.42), the FOURIER transformation of (F.40) is given by

By diagonalizing the dynamical matrix [283], one obtains the eigen-modes and eigen-frequencies of the lattice vibrations. These eigen-vectors can be used to expand the FOURIER components of the displacement in terms of phonon operators

where these operators have the time dependence

This eigen-vector expansion allows one to factorize (F.43) for each phonon mode according to

into a weight factor

(F.47) |

and the phonon GREEN's function (Appendix D.2)

where , , and are the annihilation and creation operators for Bosons. This factorization allows one to evaluate the coupling for any combination of phonon branch indices.