Derivation of the Shell Model of Lattice Dynamics and its Relation to the Theory of the Dielectric Constant

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Abstract
A quantum-mechanical derivation of the dipolar models of lattice dynamics is presented, with particular reference to the shell model. The method is based on a systematic self-consistent Born-Oppenheimer perturbation expansion in powers of the lattice displacements, and utilizes a formalism previously developed by the author, whereby the perturbation in electronic charge density is obtained self-consistently in terms of a "bound" part moving rigidly with the cores and a "deformation part" representing the distortion effects. A certain approximation made in the relevant matrix elements is shown to lead to a self-consistent solution for the deformation part in terms of polarization waves, provided the dipole polarization vectors satisfy an equation which is exactly of the form of the shell equations of the shell model. The lattice equations of motion are then obtained and are exactly the same as the shell-model equations. The validity of the central assumption is discussed in terms of the band structure, and it is shown why the model should be reasonably good for insulators with large band gaps. It is shown how local-field corrections to the point-dipole approximation arise naturally out of the solution of the self-consistency conditions and may be incorporated into the theory. Explicit expressions are derived for the various bonding coefficients in terms of the general band structure of the solid. The close connection between the shell model and the pseudopotential theory for the lattice dynamics of metals is established, both being limiting cases of guessing at the nature of the solution for the self-consistent electron response to the lattice perturbation. Finally, the close relationship between the quantum-mechanical derivation of the shell model and the theory of the dielectric constant is pointed out. Rigorous relations are established between the dielectric constant obtained from the shell model and that calculated from the band structure in the random-phase approximation.

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