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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)

DOI: https://doi.org/10.1090/S0065-9266-2012-00659-9

Published electronically: May 21, 2012

MSC: Primary 74B20; Secondary 35Q40

### Table of Contents

**Chapters**

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

### Abstract

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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