 
Summary: Variational formulation for the electrostatic potential in dielectric continua
Phil Attard
School of Chemistry F11, University of Sydney, NSW 2006, Australia
Received 28 February 2003; accepted 16 April 2003
The thermodynamic potential for fixed charges in dielectric continua is given for arbitrary values of
the polarization density. Minimization of the functional gives the equilibrium polarization density
equivalently image charges and the electrostatic potential throughout the media. The functional,
which in general involves volume integrals, for the case of piecewise uniform dielectrica is reduced
to surface integrals of the constrained surface charge density at the dielectric discontinuities. A linear
integral equation for the induced surface charge, which can be solved by iteration, is given. The
conjugate thermodynamic force for the constrained surface charge is also given. The latter
formulation is suitable for CarParrinello or Lagrangian molecular dynamics in complex
geometries. © 2003 American Institute of Physics. DOI: 10.1063/1.1580805
I. INTRODUCTION
In classical continuum electrostatics, attention nowadays
has shifted from the original emphasis on analytic results in
simple geometries to the numerical solution of practical and
applicable problems. The advent of computer simulation
techniques, which give results in molecular detail, has in
creased the need for numerical procedures for dielectric con
