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The Kohn-Sham Equation for Deformed Crystals
Weinan E, Princeton University, NJ, and Jianfeng Lu, Duke University, Durham, NC

Memoirs of the American Mathematical Society
2013; 97 pp; softcover
Volume: 221
ISBN-10: 0-8218-7560-4
ISBN-13: 978-0-8218-7560-5
List Price: US$69
Individual Members: US$41.40
Institutional Members: US$55.20
Order Code: MEMO/221/1040
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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, the authors also establish a number of fundamental properties of the Kohn-Sham map.

Table of Contents

  • Introduction
  • Perfect crystal
  • Stability condition
  • Homogeneously deformed crystal
  • Deformed crystal and the extended Cauchy-Born rule
  • The linearized Kohn-Sham operator
  • Proof of the results for the homogeneously deformed crystal
  • Exponential decay of the resolvent
  • Asymptotic analysis of the Kohn-Sham equation
  • Higher order approximate solution to the Kohn-Sham equation
  • Proofs of Lemmas 5.3 and 5.4
  • Appendix A. Proofs of Lemmas 9.3 and 9.9
  • Bibliography
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