Radial distribution function simulation methods

A very important aspect of both these methods is the means to obtain radial distribution functions. Radial distribution functions are the best description of liquid structure at the molecular level. This is because they reflect the statistical nature of liquids. Radial distribution functions also provide the interface between these simulations and statistical mechanics. 

In structure matching methods, potentials between the CG sites are determined by fitting structural properties, typically radial distribution functions (RDF), obtained from MD employing the CG potential (CG-MD), to those of the original atomistic system. This is often achieved by either of two closely related methods, Inverse Monte Carlo [12-15] and Boltzmann Inversion [5, 16-22], Both of these methods refine the CG potentials iteratively such that the RDF obtained from the CG-MD approaches the corresponding RDF from an atomistic MD simulation. 

A different approach to mention here because it has some similarity to QM/MM is called RISM-SCF [5], It is based on a QM description of the solute, and makes use of some expressions of the integral equation of liquids (a physical approach that for reasons of space we cannot present here) to obtain in a simpler way the information encoded in the solvent distribution function used by MM and QM/MM methods. Both RISM-SCF and QM/MM use this information to define an effective Hamiltonian for the solute and both proceed step by step in improving the description of the solute electronic distribution and solvent distribution function, which in both methods are two coupled quantities. There is in this book a contribution by Sato dedicated to RISM-SCF to which the reader is referred. Sato also includes a mention of the 3D-RISM approach [6] which introduces important features in the physics of the model. In fact the simulation-based methods we have thus far mentioned use a spherically averaged radial distribution function, p(r) instead of a full position dependent function p(r) expression. For molecules of irregular shape and with groups of different polarity on the molecular periphery the examination of the averaged p(r) may lead to erroneous conclusions which have to be corrected in some way [7], The 3D version we have mentioned partly eliminates these artifacts. 

The use of radial distribution functions is one of the costs paid by simulations methods to the high computational cost of this approach. The ever increasing availability of computer power has allowed a sizable portion of these shortcomings to be eliminated. In a few years the description of the QM part of QM/MM applications has progressed from a rather crude semiempirical description to ab initio levels now sufficiently accurate to describe with reasonable accuracy solvent effects on molecular properties and reaction mechanisms. A greater availability of computer power has also permitted the introduction of some improvements in the formulation of the site-site potentials we briefly characterized above. 

The MFA [1] introduces the perturbation due to the solvent effect in an averaged way. Specifically, the quantity that is introduced into the solute molecular Hamiltonian is the averaged value of the potential generated by the solvent in the volume occupied by the solute. In the past, this approximation has mainly been used with very simplified descriptions of the solvent, such as those provided by the dielectric continuum [2] or Langevin dipole models [3], A more detailed description of the solvent has been used by Ten-no et al. [4], who describe the solvent through atom-atom radial distribution functions obtained via an extended version of the interaction site method. Less attention has been paid, however, to the use of the MFA in conjunction with simulation calculations of liquids, although its theoretical bases are well known [5]. In this respect, we would refer to the papers of Sese and co-workers [6], where the solvent radial distribution functions obtained from MD [7] calculations and its perturbation are introduced a posteriori into the molecular Hamiltonian. 

Molecular simulation methods provide an acceptable picture of the solvent structure around a solute. For small spherical solutes, the solvent structure can be represented by the radial distribution function (RDF), g(r), defined as... 

Lyubartsev has also developed a multiscale parameterisation method that has been used to systematically build a CG model of a DMPC bilayer. Lyubartsev uses an inverse Monte Carlo method to generate the CG parameters from an underlying atomistic simulation. The atomistic simulation trajectory is analysed to generate the radial distribution functions (RDFs) for the CG bead model. These RDFs can be converted into pairwise interaction potentials between the beads. The... 

Figure 6 The radial distribution function for a Lennard-Jones model of liquid argon at a temperature T = 300 K. A simulation cell of 35 A containing 864 atoms with periodic boundary conditions was used. The simulation was carried out by coupling each degree of freedom to an MTK thermostat, and the equation of motion was integrated using the methods discussed in Ref. 28.

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