# The Kohn-Sham Equation for Deformed Crystals

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*Weinan E; Jianfeng Lu*

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

# Table of Contents

## The Kohn-Sham Equation for Deformed Crystals

- Chapter 1. Introduction 18 free
- Chapter 2. Perfect crystal 714 free
- Chapter 3. Stability condition 1118
- Chapter 4. Homogeneously deformed crystal 1522
- Chapter 5. Deformed crystal and the extended Cauchy-Born rule 1724
- Chapter 6. The linearized Kohn-Sham operator 2330
- Chapter 7. Proof of the results for the homogeneously deformed crystal 4350
- Chapter 8. Exponential decay of the resolvent 4754
- Chapter 9. Asymptotic analysis of the Kohn-Sham equation 5158
- Chapter 10. Higher order approximate solution to the Kohn-Sham equation 7582
- Chapter 11. Proofs of Lemmas 5.3 and 5.4 8996
- Appendix A. Proofs of Lemmas 9.3 and 9.9 93100
- Bibliography 97104