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The Kohn-Sham equation for deformed crystals

About this Title

Weinan E, Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544 and Jianfeng Lu, Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, New York 10012

Publication: Memoirs of the American Mathematical Society
Publication Year: 2013; Volume 221, Number 1040
ISBNs: 978-0-8218-7560-5 (print); 978-0-8218-9466-8 (online)
Published electronically: May 21, 2012
MSC: Primary 74B20; Secondary 35Q40

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Perfect crystal
  • Chapter 3. Stability condition
  • Chapter 4. Homogeneously deformed crystal
  • Chapter 5. Deformed crystal and the extended Cauchy-Born rule
  • Chapter 6. The linearized Kohn-Sham operator
  • Chapter 7. Proof of the results for the homogeneously deformed crystal
  • Chapter 8. Exponential decay of the resolvent
  • Chapter 9. Asymptotic analysis of the Kohn-Sham equation
  • Chapter 10. Higher order approximate solution to the Kohn-Sham equation
  • Chapter 11. Proofs of Lemmas 5.3 and 5.4
  • Appendix A. Proofs of Lemmas 9.3 and 9.9


The solution to the Kohn-Sham equation in the density functional theory of the quantum many-body problem is studied in the context of the electronic structure of smoothly deformed macroscopic crystals. An analog of the classical Cauchy-Born rule for crystal lattices is established for the electronic structure of the deformed crystal under the following physical conditions: (1) the band structure of the undeformed crystal has a gap, i.e. the crystal is an insulator, (2) the charge density waves are stable, and (3) the macroscopic dielectric tensor is positive definite. The effective equation governing the piezoelectric effect of a material is rigorously derived. Along the way, we also establish a number of fundamental properties of the Kohn-Sham map.

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