Introduction

Polarizability

When an electric field E is applied to a liquid, it induces a dipole in the constituent molecules. The fluid becomes polarized, with polarization density \(P = \epsilon_0 \chi E\). The polarizability \(\chi\) has three contributions:  electronic (electron clouds distort), configurational (molecular shapes change), and rotational (permanent dipoles orient).  For polar molecules with large permanent dipoles, the rotational contribution to polarizability dominates.  Only the rotational contribution is represented by MD simulations of nonpolarizable molecules.

The dielectric constant \(\epsilon\) of a material is determined by its polarizability.  A slab between charged plates polarizes in response to the external field; the polarization gives rise to a layer of surface charge that partially cancels the external field.  Analysis of this parallel-plate geometry gives

\[ \epsilon/\epsilon_0 = 1 + \chi \quad \quad (1) \]

Polarizability from simulations

In this tutorial, we aim to simulate the dielectric constant of three polar fluids: dichloromethane, acetone, and acetonitrile. 

methylene chloride

acetone

acetonitrile

Using molecular dynamics simulations, polarizability can be measured in two ways — linear response, and fluctuations.

In the linear response method, a constant electric field E is applied to a simulated fluid.  With periodic boundary conditions, a constant polarization density P results, with no accumulation of charges on the boundary and hence no reduction of the applied field.  For small enough E, the induced P is proportional to E — the response is linear.

In the fluctuation method, a fluid is simulated with no applied field.  Thermal fluctuations result in a time-varying polarization M (equal to the integral of P over the system).  By the fluctuation-response theorem, the response of M to E is related to the fluctuations of M:

\[\frac{\partial \langle M_i \rangle}{\partial E_j} = \beta \langle M_i M_j \rangle \quad \quad (2) \]

Here \( \langle \ldots \rangle\) indicate equilibrium average, and \(\beta = 1/(k_B T)\).  In essence, the more readily the polarization fluctuates, the more easily an applied field can bias those fluctuations in a particular direction.  For an isotropic fluid, this leads to

\[\chi = \frac{\beta}{3 \epsilon_0 V} \langle M^2 \rangle \quad \quad (3)  \]

where V is the system volume and \(\epsilon_0\) the vacuum permittivity.

Partial charge adjustment

Classical MD simulations represent the electron clouds of molecules in terms of partial charges on each atom.  Partial charges for molecules in vacuum can be computed using quantum chemistry packages.  However, realistic partial charges for molecules in a liquid are renormalized, by effects of electronic polarization and electrostatic screening.  

Recent work¹ suggests that simulations using nonpolarizable potentials and vacuum-derived partial charges significantly underestimate the polarizability.  Ref. 1 suggests scaling up the vacuum partial charges by a factor of f = 1/0.7 = 1.43, based on the adjustment made for the successful SPC/E model of water.  We explore how \(\epsilon\) for our three fluids depends on this scaling, by performing simulations with  f = 1, 1.2, and the suggested factor of 1.43.