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We present classical and quantum mechanical descriptions of lattice dynamics,from the atomic to the continuum scale, using atomic scale symmetry modes andtheir constraint equations. This approach is demonstrated for a one-dimensionalchain and a two-dimensional square lattice with a monatomic basis. For theclassical description, we find that rigid modes, in addition to thedistortional modes found before, are necessary to describe the kinetic energy.The long wavelength limit of the kinetic energy terms expressed in terms ofatomic scale modes is shown to be consistent with the continuum theory, and theleading order corrections are obtained. For the quantum mechanical description,we find conjugate momenta for the atomic scale symmetry modes. In direct space,graphical rules for their commutation relations are obtained. Commutationrelations in the reciprocal space are also calculated. As an example, phononmodes are analyzed in terms of symmetry modes. We emphasize that the approachbased on atomic scale symmetry modes could be useful, for example, for thedescription of multiscale lattice dynamics and the dynamics near structuralphase transition.

Symmetry-mode-based classical and quantum mechanical formalism of lattice dynamics