Reaction field method

The reaction field method is used in molecular simulations to simulate the effect of long range dipole-dipole interactions for simulations with periodic boundary conditions. Around each molecule there is a 'cavity' or sphere within which the Couloumb interactions are treated explicitly. Outside of this cavity the medium is assumed to have a uniform dielectric constant. The molecule induces polarization in this media which in terms create a reaction field, sometimes called the Onsager reaction field. Although Onsager's name is often attached to the technique, because he considered such a geometry in his theory of the dielectric constant, the method was first introduced by Barker and Watts in 1973.

The effective pairwise potential becomes:

$U_{AB}=q_{A}q_{B}\left[{\frac {1}{r_{AB}}}+{\frac {(\varepsilon _{RF}-1)r_{AB}^{2}}{(2\varepsilon _{RF}+1)r_{c}^{3}}}\right]$

where $r_{c}$ is the cut-off radius.

When a molecule enters or leaves the sphere defined by the cut-off radius, there is a discontinuous jump in energy. When all of these jumps in energy are summed, they do not exactly cancel, leading to poor energy conservation, a deficiency found whenever a spherical cut-off is used. The situation can be improved by tapering the potential energy function to zero near the cut-off radius. Beyond a certain radius $r_{t}$ the potential is multiplied by a tapering function $f(r)$. A simple choice is linear tapering with $r_{t}=.95r_{c}$, although better results may be found with more sophisticated tapering functions.

Another potential difficulty of the reaction field method is that the dielectric constant must be be known a priori. However, it turns out that in most cases, dynamical properties are fairly insensitive to the choice of $\varepsilon _{RF}$. It can be put in by hand, or calculated approximately using any of a number of well-known relation between the dipole fluctuations inside the simulation box and the macroscopic dielectric constant.

Comparison to Ewald summation

The reaction field method is an alternative to the popular technique of Ewald summation. Monte Carlo simulations have been performed for dipolar anisotropic models (hard spherocylinders and the Gay-Berne model for liquid crystals) both indicating that the results for the reaction field and the Ewald summation are consistent. However, the reaction field presents a considerable reduction in the computer time required. The reaction field should be applied carefully, and becomes complicated or impossible to implement when studying liquid-vapour coexistence or phase transitions.

Further reading

• Martin Neumann and Othmar Steinhauser "The influence of boundary conditions used in machine simulations on the structure of polar systems", Molecular Physics 39 pp. 437-454 (1980)
• Martin Neumann, Othmar Steinhauser and G. Stuart Pawley "Consistent calculation of the static and frequency-dependent dielectric constant in computer simulations", Molecular Physics 52 pp. 97-113 (1984)
• Andrij Baumketner "Removing systematic errors in interionic potentials of mean force computed in molecular simulations using reaction-field-based electrostatics", Journal of Chemical Physics 130 104106 (2009)
• Reaction Field method