The continuum limit and QM-continuum approximation of quantum mechanical models of solids. (English) Zbl 1141.35046

E, Weinan; Lu, Jianfeng

Commun. Math. Sci. 5, No. 3, 679-696 (2007).

Summary: 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.

MSC:

35Q40	PDEs in connection with quantum mechanics
74Q05	Homogenization in equilibrium problems of solid mechanics
82D20	Statistical mechanics of solids
81V25	Other elementary particle theory in quantum theory

Keywords:

density functional theory; Kohn-Sham models; Tomas-Fermi-von Weizsäcker model; QM-continuum approximation; continuum limit