Transition metal dichalcogenides are at the center of intense scientific activity due to their promising applications, as well as the growing interest in basic research related to their electronic and dielectric properties. The layered structure of single-(ML) and two-layer (2ML) samples presents exciting features for light-matter interaction, electron transport, and electronic and optoelectronic applications. Lattice vibrations and electron-phonon interactions are essential for studying the above mentioned topics. Phonon spectra in ML and 2ML of MoX2 and WX2 (X = S, Se, and Te) families are studied using first principles calculations. A comprehensive analysis of the two-dimensional optical-phonon dispersion laws is performed, and the results illustrate the main differences between ML and 2ML for each considered semiconductor. Taking advantage of ab initio calculations, a generalization of the phenomenological Born-Huang dielectric model for long-wavelength vibrational modes around the Γ-point of the Brillouin zone (BZ) in 2ML structures is implemented. Explicit expressions are derived for the optical phonon dispersion of in-plane and out-of-plane normal modes. The set of characteristic parameters describing each long-wavelength optical branch is resolved from a direct comparison with the exact dispersion laws provided using the first principles calculations. The long-range electron-phonon Pekar-Fröhlich (PF) interaction and intra-valley electron scattering rates at the K-point of the BZ via E' (LO) and Eul longitudinal optical oscillations are examined for the ML and 2ML structures, respectively. The non-local macroscopic screening and the coupling between the in-plane electric field and longitudinal optical mechanical oscillation, profoundly affect the PF Hamiltonian and the carrier inverse relaxation time.
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