Sum Rule Relating Optical Properties to the Charge Distribution

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Abstract
A sum rule is constructed which relates the third moment of the imaginary part of the dielectric function at zero wave vector to an integral of the product of the Laplacian of the crystal potential and the fluctuation of the electron density. This sum rule, though rigorous, will be of real use only when the real solid can be replaced by a solid of pseudoatoms having only valence electrons. A method is found for obtaining an "experimental" third moment from other known moments. In this approximation some of the recent empiricism about bonding in crystals can be given a more rigorous basis. A quantitative application of the theorem to the prediction of the dielectric constant of GaAs is sketched.