Contents Online

# 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

DOI: http://dx.doi.org/10.4310/MAA.2016.v23.n3.a4

#### Authors

#### Abstract

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 $Z_c = 2 \alpha^{-1} \pi^{-1}$.

#### Keywords

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

#### 2010 Mathematics Subject Classification

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