Comparison of four methods to compute the dielectric permittivity of liquids from molecular dynamics simulations

Four methods to compute the dielectric permittivity ε of a liquid from molecular simulations are compared in the context of the simple point charge (SPC) water model. In the first method (unrestrained method), ε is evaluated from the fluctuations of the box dipole moment M, monitored during a single equilibrium simulation. In the three other methods, ε is evaluated from the probability distribution p(M) of the dipole moment norm. This distribution is itself evaluated in three different ways: (i) from multiple simulations involving a M-dependent biasing potential (umbrella-sampling method), (ii) from multiple simulations involving a constrained dipole moment norm (M-constraint method), or (iii) from fitting of incomplete p(M) estimates to a Maxwell distribution (fitting method). The four methods are shown to converge to an identical estimate of ε=61±1 for SPC water (256 molecules, reaction-field electrostatics). The convergence properties, advantages, and drawbacks of the different methods are analyzed in detail.