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Screening in the Finite-Temperature Reduced Hartree–Fock Model

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Abstract

We prove the existence of solutions of the reduced Hartree–Fock equations at finite temperature for a periodic crystal with a small defect, and show the total screening of the defect charge by the electrons. We also show the convergence of the damped self-consistent field iteration using Kerker preconditioning to remove charge sloshing. As a crucial step of the proof, we define and study the properties of the dielectric operator.

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Notes

  1. This reasoning also holds true for more complex methods. The Jacobian of the system has a condition number proportional to \(L^{2}\), and therefore we expect simple methods to require a number of iterations proportional to \(L^{2}\), and Krylov-type methods such as Anderson acceleration to require a number of iterations proportional to L [23, 26].

  2. Note that if other occupation functions are used, the contour may need to be modified. For instance, Gaussian smearing [5] decays exponentially only if \(b < 1\). Our technique is less general than that of [20] based on the Helffer–Sjöstrand formula, which does not require any analyticity in \(f_{\varepsilon _{F}}\).

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Acknowledgements

Stimulating discussions with Eric Cancès, Thierry Deutsch and Domenico Monaco are gratefully acknowledged.

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Correspondence to Antoine Levitt.

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Communicated by A. Levitt

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Levitt, A. Screening in the Finite-Temperature Reduced Hartree–Fock Model. Arch Rational Mech Anal 238, 901–927 (2020). https://doi.org/10.1007/s00205-020-01560-0

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