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Using two pairs of lattice equations deduced from a microscopic dipolelattice model taking into account electronic polarization (EP) of ions andlocal field effects (LFEs) self-consistently, in-plane and out-of-plane opticalvibrations in two-dimensional (2D) hexagonal BN are studied theoretically. Thethree mutually independent coefficients of either pair of lattice equations aredetermined by a set of three generally known quantities such as the 2Delectronic and static susceptibilities and phonon frequency, making the latticeequations very useful for calculating the lattice dynamical properties.Explicit expressions are obtained for lattice vibrational energy density, andphonon dispersion, group velocity and density of states. The transparent phonondispersion relations describe the previous numerical calculations very well,and the longitudinal optical (LO) phonon dispersion relation is identical tothe analytical expression of Sohier {\it et al}. [Nano Lett. {\bf 17}, 3758(2017)]. The out-of-plane phonon frequency is finite owing to ionic EP. A 2Dlattice dielectric function $\epsilon(k,\omega)$ is derived--due solely to theLO vibrations--which also allows the LO phonon dispersion to be rederivedsimply from $\epsilon(k,\omega)=0$, similar to the bulk case. A 2DLyddane--Sachs--Teller relation and a frequency--susceptibility relation areobtained for the in-plane and out-of-plane vibrations, respectively, connectingthe phonon frequencies to the 2D dielectric functions or susceptibilities.Using three first-principles calculated parameters, the lattice dynamicalproperties are studied comprehensively, particular attention being paid to theEP and LFEs. The ionic EP and LFEs should be included simultaneously, butotherwise neglecting either or both causes large discrepancies to thecalculated dynamical properties.

Long optical lattice vibrations and dielectric function of monolayer hexagonal boron nitride