Article

Archive for Rational Mechanics and Analysis

, Volume 199, Issue 2, pp 407-433

The Electronic Structure of Smoothly Deformed Crystals: Wannier Functions and the Cauchy–Born Rule

  • Weinan EAffiliated withDepartment of Mathematics and Program in Applied and Computational Mathematics, Princeton University Email author 
  • , Jianfeng LuAffiliated withDepartment of Mathematics and Program in Applied and Computational Mathematics, Princeton UniversityCourant Institute of Mathematical Sciences, New York University

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Abstract

The electronic structure of a smoothly deformed crystal is analyzed for the case when the effective Hamiltonian is a given function of the nuclei by considering the regime when the scale of the deformation is much larger than the lattice parameter. Wannier functions are defined by projecting the Wannier functions for the undeformed crystal to the space spanned by the wave functions of the deformed crystal. The exponential decay of such Wannier functions is proved for the case when the undeformed crystal is an insulator. The celebrated Cauchy–Born rule for crystal lattices is extended to the present situation for electronic structure analysis.