Methods and Applications of Analysis

Volume 23 (2016)

Number 3

Minimizers for open-shell, spin-polarised Kohn–Sham equations for non-relativistic and quasi-relativistic molecular systems

Pages: 269 – 292



C. Argaez (School of Science and Engineering, Reykjavik University, Reykjavik, Iceland)

M. Melgaard (Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Brighton, United Kingdom)


We study the open-shell, spin-polarized Kohn–Sham models for non-relativistic and quasi-relativistic $N$-electron Coulomb systems, that is, systems where the kinetic energy of the electrons is given by either the non-relativistic operator $-\Delta_{x_n}$ or the quasi-relativistic operator $\sqrt{- \alpha^{-2} \Delta_{x_n} + \alpha^{-4}} - \alpha^{-2}$. For standard and extended Kohn–Sham models in the local density approximation, we prove existence of a ground state (or minimizer) provided that the total charge $Z_{\mathrm{tot}}$ of $K$ nuclei is greater than $N-1$. For the quasi-relativistic setting we also need that $Z_{\mathrm{tot}}$ is smaller than a critical charge $Z_c = 2 \alpha^{-1} \pi^{-1}$.


open-shell, spin-polarised Kohn–Sham equations, ground state, variational methods, concentration-compactness

2010 Mathematics Subject Classification

Primary 35J60. Secondary 47J10, 58Z05, 81V55.

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Published 9 November 2016