This letter attempts to clarify the meaning of three closely related mean-field approximations: random phase approximation (RPA), local molecular field (LMF) approximation, and symmetry-preserving mean-field (SPMF) approximation, and their use of reliability and validity in the field of theory and simulation of liquids when the long-ranged component of the intermolecular interaction plays an important role in determining density fluctuations and correlations. The RPA in the framework of classical density functional theory (DFT) neglects the higher order correlations in the bulk and directly applies the long-ranged part of the potential to correct the pair direct correlation function of the short-ranged system while the LMF approach introduces a nonuniform mimic system under a reconstructed static external potential that accounts for the average effect arising from the long-ranged component of the interaction. Furthermore, the SPMF approximation takes the viewpoint of LMF but instead instantaneously averages the long-ranged component of the potential over the degrees of freedom in the direction with preserved symmetry. The formal connections and the particular differences of the viewpoint among the three approximations are explained and their performances in producing structural properties of liquids are stringently tested using an exactly solvable model. We demonstrate that the RPA treatment often yields uncontrolled poor results for pair distribution functions of the bulk system. On the other hand, the LMF theory produces quite reasonably structural correlations when the pair distribution in the bulk is converted to the singlet particle distribution in the nonuniform system. It turns out that the SPMF approach outperforms the other two at all densities and under extreme conditions where the long-ranged component significantly contributes to the structural correlations.

Artifacts arise when the long-ranged electrostatic interaction is inappropriately treated in molecular simulations of electrolytes. When the usual Ewald3D sum method with the tinfoil boundary condition (e3dtf) is used for simulations of an interfacial liquid under an external electric field, a straightforward analysis of the liquid structure often suggests unphysical dielectric properties as a consequence of the inaccurate treatment of the electrostatics. In order to understand the underlying mechanism that leads to this apparent violation of thermodynamics, we now derive a new equation in the weak-field limit that, in a mean field view, accounts for the average effect arising from the difference between e3dtf and the sophisticated Ewald2D sum method (e2d). Numerical simulations of a water system in slab geometry confirm the validity of the weak-field limit equation for a series of parameter setup associated with e3dtf. Moreover, a similar procedure applied to a spherically confined water system suggests that corrections to the seemingly inappropriate treatment of the electrostatics in fact vanish. This cancellation of the boundary effect due to symmetry immediately sheds light on the long-lasting problem of the validity of the ad hoc application of e3dtf for bulk systems. In total, we argue that artifacts arising from e3dtf are often predictable and analytical corrections to the straightforward analysis might be applied to reveal consistent thermodynamic properties in liquid simulations.

We present a rigorous Ewald summation formula to evaluate the electrostatic interactions in two-dimensionally periodic planar interfaces of three-dimensional systems. By rewriting the Fourier part of the summation formula of the original Ewald2D expression with an explicit order N2 complexity to a closed form Fourier integral, we find that both the previously developed electrostatic layer correction term and the boundary correction term naturally arise from the expression of a rigorous trapezoidal summation of the Fourier integral part. We derive the exact corrections to the trapezoidal summation in a form of contour integrals offering precise error bounds with given parameter sets of mesh size and system length. Numerical calculations of Madelung constants in model ionic crystals of slab geometry have been performed to support our analytical results.

We present a unified derivation of the Ewald sum for electrostatics in a three-dimensional infinite system that is periodic in one, two, or three dimensions. The derivation leads to the Ewald3D sum being expressed as a sum of a real space contribution and a reciprocal space contribution, as in previous work. However, the k → 0 term in the reciprocal space contribution is analyzed further and found to give an additional contribution that is not part of previous reciprocal space contributions. The transparent derivation provides a unified view of the existing conducting infinite boundary term, the vacuum spherical infinite boundary term and the vacuum planar infinite boundary term for the Ewald3D sum. The derivation further explains that the infinite boundary term is conditional for the Ewald3D sum because it depends on the asymptotic behavior that the system approaches the infinite in 3D but it becomes a definite term for the Ewald2D or Ewald1D sum irrespective of the asymptotic behavior in the reduced dimensions. Moreover, the unified derivation yields two formulas for the Ewald sum in one-dimensional periodicity, and we rigorously prove that the two formulas are equivalent. These formulas might be useful for simulations of organic crystals with wirelike shapes or liquids confined in uniform cylinders. More importantly, the Ewald3D, Ewald2D, and Ewald1D sums are further written as sums of well-defined pairwise potentials overcoming the difficulty in splitting the total Coulomb potential energy into contributions from each individual group of charges. The pairwise interactions with their clear physical meaning of the explicit presence of the periodic images thus can be used to consistently perform analysis based on the trajectories from computer simulations of bulk or interfaces.

Over the past decade, there has been much controversy regarding the microscopic mechanism by which the π-electron-rich carbon nanomaterials such as graphene and carbon nanotubes can be dispersed in ionic liquids. Through a combination of a quantum mechanical calculation on the level of density functional theory, an extensive molecular dynamics study on the time scale of microseconds, and a kinetic analysis at the experimental time scale, we have demonstrated that collective van der Waals forces between ionic liquids and graphene are able to describe both the short-ranged cation−π interaction and the long-ranged dispersion interaction and this microscopic interaction drives two graphene plates trapped in their metastable state while two graphene plates easily self-assemble into graphite in water.

How structural features observed computationally are connected to excitation-wavelength dependent kinetics observed experimentally remains an unanswered question in the field of ionic liquids. Using molecular dynamics simulation methods and approximated models that simplify the electrostatic interactions in ionic liquids, we discovered that on the timescale shorter than the relaxation time of the photochemical process, the energetic heterogeneity in terms of distribution of excitation energies is the consequence of the structure heterogeneity formed by local arrangement of ions around the solute probe.

This letter attempts to clarify the meaning of three closely related mean-field approximations: random phase approximation (RPA), local molecular field (LMF) approximation, and symmetry-preserving mean-field (SPMF) approximation, and their use of reliability and validity in the field of theory and simulation of liquids when the long-ranged component of the intermolecular interaction plays an important role in determining density fluctuations and correlations. The RPA in the framework of classical density functional theory (DFT) neglects the higher order correlations in the bulk and directly applies the long-ranged part of the potential to correct the pair direct correlation function of the short-ranged system while the LMF approach introduces a nonuniform mimic system under a reconstructed static external potential that accounts for the average effect arising from the long-ranged component of the interaction. Furthermore, the SPMF approximation takes the viewpoint of LMF but instead instantaneously averages the long-ranged component of the potential over the degrees of freedom in the direction with preserved symmetry. The formal connections and the particular differences of the viewpoint among the three approximations are explained and their performances in producing structural properties of liquids are stringently tested using an exactly solvable model. We demonstrate that the RPA treatment often yields uncontrolled poor results for pair distribution functions of the bulk system. On the other hand, the LMF theory produces quite reasonably structural correlations when the pair distribution in the bulk is converted to the singlet particle distribution in the nonuniform system. It turns out that the SPMF approach outperforms the other two at all densities and under extreme conditions where the long-ranged component significantly contributes to the structural correlations.

How structural features observed computationally are connected to excitation-wavelength dependent kinetics observed experimentally remains an unanswered question in the field of ionic liquids. Using molecular dynamics simulation methods and approximated models that simplify the electrostatic interactions in ionic liquids, we discovered that on the timescale shorter than the relaxation time of the photochemical process, the energetic heterogeneity in terms of distribution of excitation energies is the consequence of the structure heterogeneity formed by local arrangement of ions around the solute probe.

Artifacts arise when the long-ranged electrostatic interaction is inappropriately treated in molecular simulations of electrolytes. When the usual Ewald3D sum method with the tinfoil boundary condition (e3dtf) is used for simulations of an interfacial liquid under an external electric field, a straightforward analysis of the liquid structure often suggests unphysical dielectric properties as a consequence of the inaccurate treatment of the electrostatics. In order to understand the underlying mechanism that leads to this apparent violation of thermodynamics, we now derive a new equation in the weak-field limit that, in a mean field view, accounts for the average effect arising from the difference between e3dtf and the sophisticated Ewald2D sum method (e2d). Numerical simulations of a water system in slab geometry confirm the validity of the weak-field limit equation for a series of parameter setup associated with e3dtf. Moreover, a similar procedure applied to a spherically confined water system suggests that corrections to the seemingly inappropriate treatment of the electrostatics in fact vanish. This cancellation of the boundary effect due to symmetry immediately sheds light on the long-lasting problem of the validity of the ad hoc application of e3dtf for bulk systems. In total, we argue that artifacts arising from e3dtf are often predictable and analytical corrections to the straightforward analysis might be applied to reveal consistent thermodynamic properties in liquid simulations.