A charge granularity correction to electrostatic self-energies of planar charge distributions¹

¹Dedicated to Professor Andreas Savin on the occasion of his 65th birthday.

ABSTRACT

The cohesive (or Madelung) energy E_coh of N unit charges constitutes the charge granularity correction to the electrostatic self-energy of a continuous density $\rho \left(\overrightarrow{r}\right)$ that ensues at the N → ∞ limit. A rigorous upper bound to E_coh is derived for charges in a planar arrangement. It is given by a functional dependent on the density itself and an auxiliary quantity that is a solution of an eikonal equation involving $\rho \left(\overrightarrow{r}\right)$. For densities constant along one direction or possessing radial symmetry, it reduces to an expression of the local-density type.

GRAPHICAL ABSTRACT

Keywords:

• Madelung energy
• Coulombic systems
• local density approximation
• eikonal equation
• upper bound

Acknowledgements

The authors acknowledge valuable comments from Prof. Paola Gori-Giorgi.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ Dedicated to Professor Andreas Savin on the occasion of his 65th birthday.

Additional information

Funding

This work was supported by NCN (Poland) [grant number DEC-2012/07/B/ST4/00553].