## Abstract

The solution to the Kohn-Sham equation in the density functional theory of the quantum many-body problem is studied in the context of the electronic structure of smoothly deformed macroscopic crystals. An analog of the classical Cauchy-Born rule for crystal lattices is established for the electronic structure of the deformed crystal under the following physical conditions: (1) the band structure of the undeformed crystal has a gap, i.e. the crystal is an insulator, (2) the charge density waves are stable, and (3) the macroscopic dielectric tensor is positive definite. The effective equation governing the piezoelectric effect of a material is rigorously derived. Along the way, we also establish a number of fundamental properties of the Kohn-Sham map.

Original language | English (US) |
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Pages (from-to) | 1-109 |

Number of pages | 109 |

Journal | Memoirs of the American Mathematical Society |

Volume | 221 |

Issue number | 1040 |

DOIs | |

State | Published - Jan 1 2013 |

## All Science Journal Classification (ASJC) codes

- General Mathematics
- Applied Mathematics