## Communications in Mathematical Sciences

- Commun. Math. Sci.
- Volume 5, Issue 3 (2007), 679-696.

### The continuum limit and QM-continuum approximation of quantum mechanical models of solids

Weinan E and Jianfeng Lu

#### 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.

#### Article information

**Source**

Commun. Math. Sci. Volume 5, Issue 3 (2007), 679-696.

**Dates**

First available in Project Euclid: 29 August 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.cms/1188405674

**Mathematical Reviews number (MathSciNet)**

MR2352337

**Zentralblatt MATH identifier**

1141.35046

**Subjects**

Primary: 35Q40: PDEs in connection with quantum mechanics 74Q05: Homogenization in equilibrium problems 34E05: Asymptotic expansions 74B20: Nonlinear elasticity

**Keywords**

continuum limit QM-continuum approximation density functional theory

#### Citation

E, Weinan; Lu, Jianfeng. The continuum limit and QM-continuum approximation of quantum mechanical models of solids. Commun. Math. Sci. 5 (2007), no. 3, 679--696. http://projecteuclid.org/euclid.cms/1188405674.