From the dipole of a crystallite to the polarization of a crystal.

The quantum-mechanical expression for the polarization of a crystalline solid does not bear any resemblance to the (trivial) expression for the dipole of a bounded crystallite; in fact, it has been proved via a conceptually different path. Here, I show how to alternatively define the dipole of a bounded sample in a somewhat unconventional way; from such a formula, the crystalline polarization formula—as routinely implemented in electronic-structure codes—follows almost seamlessly. [ABSTRACT FROM AUTHOR]