This is a comprehensive path about theoretical approaches to the vibrational properties in periodic and multilayered structures. The phonons modes taking a large part of our work are detailed beyond the limit offered by the area targeted by this sort of paper. In fact, in many cases of this type of materials, metallic, semiconductor or insulator, these modes are investigated. Although, the electronic properties are devoted, generally, to the semiconductor ones as well as the optical properties will be derived in the second paper of this volume. Another topic can be touched which concerns the coupling effect between phonons and electrons or photons and electrons giving rise to interesting results; but it is not addressed here. Throughout this paper, we bring forth the theoretical model currently used by us, and the results obtained in consequence. This model is called the response function based on the Green function and its properties are stressed as well as the results obtained.This is a comprehensive path about theoretical approaches to the vibrational properties in periodic and multilayered structures. The phonons modes taking a large part of our work are detailed beyond the limit offered by the area targeted by this sort of paper. In fact, in many cases of this type of materials, metallic, semiconductor or insulator, these modes are investigated. Although, the electronic properties are devoted, generally, to the semiconductor ones as well as the optical properties will be derived in the second paper of this volume. Another topic can be touched which concerns the coupling effect between phonons and electrons or photons and electrons giving rise to interesting results; but it is not addressed here. Throughout this paper, we bring forth the theoretical model currently used by us, and the results obtained in consequence. This model is called the response function based on the Green function and its properties are stressed as well as the results obtained
