Dipole moment fluctuation formulas in computer simulations of polar systems

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Abstract
The fundamental equation of electrostatics, i.e. the integral equation for the polarization of a macroscopic dielectric in an arbitrary external field, is written in a form that allows explicit inclusion of the toroidal boundary conditions as well as the cutoff of dipolar interactions often used in the computer simulation of polar systems. The toroidal boundary conditions are accounted for in a natural way if the integral equation is formulated (and solved) in Fourier space. Rigorous expressions for the polarization induced by a homogeneous field as well as for the equilibrium dipole moment fluctuations, both as a function of the dielectric constant, are then easily derived for general systems in two and three dimensions. The equations obtained for spherical cutoff geometry with reaction field are identical to those valid for an infinite periodic system (Ewald sum plus reaction field). When applied to the case of a highly polar Stockmayer system (μ*2=3·0, θ*=0·822, T*=1·15) the formulas lead to a consistent value of the dielectric constant that is independent of the precise value of the reaction field—if the number of particles is sufficiently large. In the thermodynamic limit the dielectric constant obtained with the reaction field method seems to be much higher than one would expect from the Ewald sum results published for the Stockmayer system at the same thermodynamic state.