### Abstract

We consider the continuum limit for models of solids that arise in density functional theory and the QM-continuum approximation of such models. Two different versions of QM-continuum approximation are proposed, depending on the level at which the Cauchy-Born rule is used, one at the level of electron density and one at the level of energy. Consistency at the interface between the smooth and the non-smooth regions is analyzed. We show that if the Cauchy-Born rule is used at the level of electron density, then the resulting QM-continuum model is free of the so-called "ghost force" at the interface. We also present dynamic models that bridge naturally the Car-Parrinello method and the QM-continuum approximation.

Original language | English (US) |
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Pages (from-to) | 679-696 |

Number of pages | 18 |

Journal | Communications in Mathematical Sciences |

Volume | 5 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2007 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Keywords

- Continuum limit
- Density functional theory
- QM-continuum approximation