# Spin-polarized Thomas-Fermi theory

## Abstract

Of concern is a rigorous Thomas-Fermi theory of electron densities for spin-polarized quantum-mechanical systems. The number N↑, N↓ of spin-up and spin-down electrons are specified in advance, and one seeks to minimize the energy functional E(P↑ ,p↓) = c1∫ R3(p↑ (X)5/3 +P↓ (x)5/3)dx + C2∫R3∫R3 [p(x)p(y)/|x - y|]dx dy + ∫R3 V(x)p(x)dx, where c 1, C2 are given positive constants, p↑, and p↓ are non-negative functions, p = p↑ + p↓ is the total electron density, ∫R3P↑ (x)dx = N↑, ∫R3P↓ (x)dx = N↓, and V is a given potential. These results are analogous to the classical rigorous (spin-unpolarized) Thomas-Fermi theory developed by Lieb and Simon [Phys. Rev. Lett. 33, 681 (1973)] and by Bénilan and Brezis ("The Thomas-Fermi problem," in preparation). © 1988 American Institute of Physics.

## Publication Title

Journal of Mathematical Physics

## Recommended Citation

Goldstein, J., & Rieder, G.
(1988). Spin-polarized Thomas-Fermi theory.* Journal of Mathematical Physics**, 29* (3), 709-716.
https://doi.org/10.1063/1.528011