Mean Field Approaches In Molecular Simulation

Modern molecular simulations often explicitly handle complicated interactions between lots of molecules.The classical force field mainly considers intramolecular and intermolecular interactions.Among them,the short-ranged Van der Waals interactions can be computed straightforwardly by setting a cutoff radius due to its fast decaying property.While developing accurate and fast method for long-ranged Coulomb interactions is a long-lasting issue that puzzling the practitioners.The most popular methods for treating electrostatic interactions contain truncation type method,Ewald type method and mean field type method.They all have advantages and drawbacks,and there are existing some relation among them.Truncation method is efficient and easy to implement,but there will generate artifacts when it is applied to handle many systems.Ewald type method is the most ‘accurate' method and is also most time-consuming.Since Coulomb lattice sum is conditional converging,there should exist a conditional term in the Ewald formula.But the popular ad hoc standard formula omits this term by covering the infinite lattice with a tinfoil.In the view of exact math and physical model,this term shall not be ignored when computing nonuniform planar or interface system,otherwise the dielectric properties need to be corrected.Moreover,for charged system,a correction term that indicates a uniform neutralizing charged background plasma should be added to the classical Ewald summation.The mean field methods in this thesis mainly include LMF(local molecular field)developed by John Weeks' s group in university of Maryland in America and SPMF(symmetry-preserving mean-field)method developed by our group,both of which cost less time compared with the other two types of methods.LMF can be applied to predict the structural and thermodynamic properties of the system.However,SPMF method introduces less approximation and can be used to describe the dynamic properties of the system.Theoretically speaking,with them a unified statistical mechanical framework can be constructed to study the long-ranged component of intermolecular interaction.We discuss the theorem of the two mean field methods and their applications in molecular simulations,including the following work:Firstly,in both theory and simulations of liquids,the long-ranged segment of intermolecular interaction acts as an important role to determine density fluctuation and correlation of molecules in the system.It is worth studying the reliability and validity of three closely related mean field approximation: random phase approximation(RPA),local molecular field(LMF)approximation and symmetry-preserving mean-field(SPMF)approximation.Starting from classical density functional theory(DFT),RPA ignores high-order correlation term and directly applies the long-ranged part of the potential to correct the pair direct correlation function of the short-ranged system.LMF introduces a nonuniform mimic system with a reconstructed static external potential,which consists of external potential and the average effect of long-ranged component of interactions.SPMF method originates from the core of LMF,but instantaneously average the impacts of the long-ranged component of interaction along the dimensions that the system owns symmetry.We test these three approximations for the structural properties of liquids with an exact solution model.The results indicate that RPA approximation generates uncontrolled and poor result for pair distribution function.On the other hand,LMF theory generates quite reasonable correlation of structure even though the pair correlation function of the real system is simply approximated by the singlet density distribution of the mimic system.And SPMF method surpasses the other two methods at all the densities and under extreme conditions where the long-ranged component significantly contributes to the structural correlations.Secondly,the energy computed by Ewald sum is the potential energy of the whole system,it cannot tell the potential energy of a pair of charges or between a group and the other group,while under many circumstances it is necessary to investigate these specific electrostatic potential energies.We rewrite the Ewald sum formula to a pairwise type,thus the total electrostatic potential energy can be interpreted as the sum of all the pairs of charges.Besides,from the analysis of components of the energy of a two dimensional periodic system of which the cell is composed by a pair of ions,we found that the result of e2 d was consistent with e3 dp method with a planar infinite boundary term.But the result of the standard e3 dtf method consisting of only real space term and reciprocal space term deviates a lot.Therefore,the planar infinite boundary term cannot be omitted when dealing with 2d periodic system.The analysis of e2 d,e3dp and e3 dtf methods and introducing formula of pairwise type provide a basis for the application of mean field methods.Thirdly,when the general Ewald3 D sum with tinfoil boundary condition is used to simulate the interface under an external field,a straightforward analysis of the structure of liquid will obtain unphysical dielectric property as a consequence of the inaccurate treatment of the electrostatics.To understand the mechanism of obviously violating thermodynamics,we obtain a new formula from mean field viewpoint at the limit of weak field,and explain the average effect of the difference between e3 dtf and sophisticated Ewlad2 D sum(e2d).The numerical simulation of a planar water system confirm the validity of the new formula.We also apply the similar method to the spherical confined water system,in which the correction of improper treatment for electrostatics vanishes.The result of eliminating “excess” effects by symmetry verify that the e3 dtf is proper to bulk system.And we also apply this method to ionic system,the results suggest e3 dtf method is improper to two dimensional periodic ionic system also.We give formula to explain the results.Anyway,the artifacts from e3 dtf are predictable and can be corrected.In total,we compare three mean field methods: RPA approximation,LMF theory and SPMF theory,which give different results for 1D exact solution system.We discuss their relations,differences and the superiority of SPMF and LMF methods.Then we mainly concentrate on the popular methods for calculating electrostatics in molecular simulations and the problems in.Meanwhile we exemplify the pairwise formula through a simple 2D-PBC model system of which the cell consists of a pair of ions.At last,we explain the electrostatic boundary condition with mean field method,and give proper illustration and correction for the unphysical dielectric properties generated by e3 dtf method.