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The birefringence in calcite is huge (possibly the largest?). The question is, why is it so special? why is such kind of materials so rare?

In other words, is it explained by first principles?

By google, I just found a paper predicting the birefringence in some material. See http://pubs.acs.org/doi/abs/10.1021/jp506744s

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The size of the birefringence of calcite is not surprising. It has a crystal structure in which all the planar carbonate ions have the same orientation. The polarization of planar ions is much larger for electrical fields in the plane (light coming in perpendicular to the plane) than for electrical fields perpendicular to the plane.

A simple calculation would start out by doing this for a single CO$_3^{2-}$ ion (or for an ion with a 1+ charge), and then apply Clausius-Mosotti.

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