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The Kohn-Sham Equation for Deformed Crystals
 
Weinan E Princeton University, Princeton, NJ
Jianfeng Lu Duke University, Durham, NC
The Kohn-Sham Equation for Deformed Crystals
eBook ISBN:  978-0-8218-9466-8
Product Code:  MEMO/221/1040.E
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $41.40
The Kohn-Sham Equation for Deformed Crystals
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The Kohn-Sham Equation for Deformed Crystals
Weinan E Princeton University, Princeton, NJ
Jianfeng Lu Duke University, Durham, NC
eBook ISBN:  978-0-8218-9466-8
Product Code:  MEMO/221/1040.E
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $41.40
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2212013; 97 pp
    MSC: Primary 74; Secondary 35;

    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
     
     
    • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2212013; 97 pp
MSC: Primary 74; Secondary 35;

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.

  • 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.