  Using a microscopic dipole lattice model including electronic polarization
(EP) of ions and local field effects (LFEs) self-consistently, two sets of
three equations are deduced for long wavelength in-plane and out-of-plane
optical lattice vibrations in monolayer (ML) transition metal dichalcogenides
(TMDs). Expressions for the lattice vibrational energy density are obtained for
the two-dimensional (2D) crystals. The linear coefficients of the lattice
equations can be determined by first-principles calculation, making these
equations very useful for studying the lattice dynamical properties. The two
pairs of equations describing the polar vibrations have the same forms as those
for ML hexagonal BN and also resemble Huang's equations for bulk [Proc. Roy.
Soc. A 208, 352 (1951)]. Each pair of the equations are solved simultaneously
with the equation of electrostatics, and explicit expressions are obtained for
the in-plane longitudinal and transverse optical (LO and TO) modes and
out-of-plane optical (ZO) modes. The LO phonon dispersion relation is identical
to the analytical expression of Sohier et al. [Nano Lett. 17, 3758 (2017)]. All
the optical phonon branches except for LO are nondispersive at the long
wavelengths. A 2D longitudinal lattice dielectric function $\epsilon(k,\omega)$
is deduced, allowing one to rederive the LO phonon dispersion simply from
$\epsilon(k,\omega)=0$. A 2D Lyddane--Sachs--Teller relation and a
frequency--susceptibility relation are obtained for in-plane and out-of-plane
vibrations, respectively, through which the phonon frequencies are related to
the 2D dielectric functions or susceptibilities. The explicit expressions are
applied to calculate various dynamical properties when knowing three
first-principles calculated parameters, and the ionic EP and LFEs are studied
thoroughly.
