Communications in Mathematical Sciences

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

Weinan E and Jianfeng Lu

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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.


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