Evaluation of second and third dielectric virial coefficients for non-polarisable molecular models

The dielectric constant, ε, of a dilute vapour can be estimated from the dielectric virial equation of state (VEOS), but the long-ranged nature of the electrostatic interactions complicates the evaluation of coefficients of this series. We propose a formulation of the second and third dielectric coefficients of a general non-polarisable molecular model that permits their reliable calculation using Mayer sampling Monte Carlo. We demonstrate for three models: dipolar hard spheres, dipolar Lennard–Jones, and TIP4P water. The coefficients are used to compute ε for each model as a function of density, which are compared to molecular-simulation data. The form of the VEOS relating ε to density depends on the dielectric constant ε′ of the embedding medium. Three choices are examined: vacuum (ε′ = 1), self-consistent (ε′ = ε) and tin foil (ε′ = ∞). The vacuum-boundary form is found to be unreliable, losing accuracy at low density and yielding divergent results for ε at moderate densities. In contrast, the series formulated using the tin-foil boundary produces accurate and stable values of ε for almost all conditions and models examined here, even when truncated at second order (which itself is shown to be a large improvement over the first-order Clausius–Mossotti–Debye formula).

稀薄蒸气的介电常数（dielectric constant）ε可通过介电维里状态方程（dielectric virial equation of state，VEOS）进行估算，但静电相互作用的长程特性使得该级数的系数求值过程变得复杂。我们提出了一种针对一般非极化分子模型的二阶与三阶介电系数公式，该公式可借助梅耶抽样蒙特卡洛（Mayer sampling Monte Carlo）方法实现可靠计算。我们针对三类模型展开了验证：偶极硬球（dipolar hard spheres）、偶极伦纳德-琼斯（dipolar Lennard–Jones）模型以及TIP4P水模型（TIP4P water）。利用所得到的系数，可计算每种模型的介电常数随密度的变化关系，并将其与分子模拟所得数据进行对比。将介电常数ε与密度关联的维里状态方程形式，取决于嵌入介质的介电常数ε′。本文考察了三种取值场景：真空环境（ε′=1）、自洽介质环境（ε′=ε）以及锡箔边界环境（ε′→∞）。研究表明，真空边界形式并不可靠：在低密度区间精度大幅下降，且在中等密度下会给出介电常数ε的发散结果。与之相对，采用锡箔边界构建的级数形式，即使仅截断至二阶（该二阶截断本身已被证实较一阶克劳修斯-莫索提-德拜公式（Clausius–Mossotti–Debye formula）有显著改进），也能在本文考察的几乎所有条件与模型下，给出准确且稳定的介电常数ε数值。