# Parameterization of Coarse-Grained Molecular Interactions through Potential of Mean Force Calculations and Cluster Expansion Techniques

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- (a)
- In structural, or correlation-based, methods, the main goal is to find effective CG potentials that reproduce the pair radial distribution function $g\left(r\right)$, and the distribution functions of bonded degrees of freedom (e.g., bonds, angles, dihedrals) for CG systems with intramolecular interaction potential [6,7,9,10,21,22]. The CG effective interactions in such methods are obtained using the direct Boltzmann inversion, or reversible work, method [10,26,27,28], or iterative techniques, such as the iterative Boltzmann inversion (IBI) [7,29] and the inverse Monte Carlo (IMC); or inverse Newton approach [22,30].
- (b)
- (c)
- The relative entropy (RE) [8,18,34] method employs the minimization of the relative entropy, or Kullback–Leibler divergence, between the microscopic Gibbs measure $\mu $ and ${\mu}^{\theta}$, representing approximations to the exact coarse space Gibbs measure. In this case, the microscopic probability distribution can be thought of as the observable. The minimization of the relative entropy is performed through Newton–Raphson approaches and/or stochastic optimization techniques [19,35].

## 2. Molecular Models

#### 2.1. Atomistic and “Exact” Coarse-Grained Description

#### 2.2. Coarse-Grained Approximations

- (a)
- The correlation-based (e.g., DBI, IBI and IMC) methods that use the pair radial distribution function $g\left(r\right)$, related to the two-body potential of mean force for the intermolecular interaction potential, as well as distribution functions of bonded degrees of freedom (e.g., bonds, angles, dihedrals) for CG systems with intramolecular interaction potential [6,7,9,10,21,22]. These methods will be further discussed below.
- (b)
- Force matching (FM) methods [5,16,31] in which the observable function is the average force acting on a CG particle. The CG potential is then determined from atomistic force information through a least-square minimization principle, to variationally project the force corresponding to the potential of mean force onto a force that is defined by the form of the approximate potential.
- (c)

- (a)
- One method is by fixing the distance ${r}_{1,2}:={r}_{1}-{r}_{2}$ between two molecules and performing molecular dynamics with such forces that maintain the fixed distance ${r}_{1,2}$. In this way, we sample atomistic potential energy (and forces) over the constrained phase space and obtain the conditional partition function as the integral in (8). Alternatively, by integration of the constrained force ($-{\int}_{{r}_{min}}^{{r}_{cut}}{\langle f\rangle}_{{r}_{1,2}=R}dR$), the two-body effective potential can be obtained. These are the ${W}^{\left(2\right),\mathrm{full},U}$ and ${W}^{\left(2\right),\mathrm{full},F}$ terms, respectively. Note that we have not used any kind of fitting or projection over a basis as in [64,65]; the data are in tabulated form.
- (b)
- Upon inverting $g\left(r\right)$ in (11) for two isolated molecules, the two-body effective potential can be directly obtained, since for such a system, $\gamma \left(r\right)=1$; i.e., the low $\rho $ regime. This method is only used for comparison with (a)as it uses the $g\left(r\right)$.

#### 2.3. Thermodynamic Consistency

## 3. Cluster Expansion

#### 3.1. Full Calculation of the PMF

## 4. Model and Simulations

#### 4.1. The Model

#### 4.2. Simulations

_{4}and 500 CH

_{3}–CH

_{3}molecules. We note here that the BBK integrator used for Langevin dynamics exhibits pressure fluctuations of the order of $\pm 40$ atm in the liquid phase, whereas temperature fluctuations have small variance, and the system is driven to the target temperature much faster than with conventional MD. Long production atomistic and CG simulations have been performed, from 10 ns up to 100 ns, depending on the system. The error bars (computed via the standard deviation) in all reported data are about 1–2% of the actual values. Details about the simulation parameters are shown in Table 1.

#### 4.2.1. Constrained Runs

_{4}molecules are oriented according to the highly repulsive forces and rotate around the axis connecting the two COMs. Due to this specific reason, we utilized stochastic (Langevin) dynamics in order to better explore the subspace of the phase space, as a random kick breaks this alignment. We determine the minimum amount of steps needed for the ensemble average to converge, in a semi-empirical manner upon inspection of the error-bars.

#### 4.2.2. Geometric Direct Computation of PMF

_{3}–CH

_{3}model, we have also calculated the two-body PMF (constraint partition function) directly, through “full sampling” of all possible configurations using a geometrical method proper for rigid bodies. In more detail, the geometric averaged constrained two-body effective potential ${W}^{\left(2\right),geom}\left(r\right)$ given in (24) is obtained by rotating the two (methane or ethane) molecules around their COMs, through their Eulerian angles and taking account of all of the possible (up to a degree of angle discretization) orientations. The main idea is to cover every possible (discretized) orientation and associate it with a corresponding weight. The Euler angles proved to be the easiest way to implement this; each possible orientation is calculated via a rotation matrix using three (Euler) angles in spherical coordinates.

_{4}and simple spherical coordinate sampling with $d\varphi =\pi /20,d\theta =\pi /45$ for CH

_{3}–CH

_{3}(as it is diagonally symmetric in the united atom description). Note, however, that in this case, the molecules are treated as rigid bodies; i.e., bond lengths and bond angles are kept fixed; essentially, it is assumed that intra-molecular degrees of freedom do not affect the intermolecular (non-bonded potential) ones. However, for this system, there is no considerable difference in the resulting non-bonded effective potential. The advantage of this method is that we avoid long (and more expensive) molecular simulations of the canonical ensemble, which might also get trapped in local minima and inadequately sample the phase space. We should also state that this method is very similar to the one used by McCoy and Curro in order to develop a CH

_{4}united-atom model from all-atom configurations [58].

## 5. Results

#### 5.1. Calculation of the Effective Two-Body CG Potential

- (a)
- A calculation of the PMF using the constraint force approach, ${W}^{\left(2\right),\mathrm{full},\phantom{\rule{4.pt}{0ex}}\mathrm{f}}$, as described in Section 4.2.1. Alternatively, through the same set of atomistic configurations, the two-body PMF, ${W}^{\left(2\right),\mathrm{full},\phantom{\rule{4.pt}{0ex}}\mathrm{u}}$, can be directly calculated through Equation (24).
- (b)
- A direct calculation of the PMF, ${W}^{\left(2\right),geom}$, using a geometrical approach as described in Section 4.2.2.
- (c)
- DBI method: The CG effective potential, ${W}^{\left(2\right),g\left(r\right)}$, is obtained by inverting the pair (radial) correlation function, $g\left(r\right)$, computed through a stochastic LD run with only two methane (or ethane) molecules freely moving in the simulation box. The pair correlation function, $g\left(r\right)$, of the two methane molecules is also shown in Figure 3a.

_{4}and CH

_{3}–CH

_{3}, respectively. As discussed in Section 3, cluster expansion is expected to be more accurate at high temperatures and/or lower densities. For this reason, we examine both systems at higher temperatures, than of the data shown in Figure 3; values of $-{k}_{B}T$ are shown with full lines. Both systems show the same behaviour. First, it is clear that the agreement between ${W}^{\left(2\right)}$ and the (more accurate) ${W}^{\left(2\right),\mathrm{full}}$ is very good only at long distances (not surprisingly, since the logarithm expansion holds for every $\beta $, as $\overline{V}\to 0$ in $C\left(\beta \right)$), whereas there are strong discrepancies in the regions where the potential is minimum, as well as in the high energy regions (short distances); Second, it is evident that adding terms up to the second order with respect to $\beta $, we obtain a better approximation of ${W}^{\left(2\right),\mathrm{full}}$. All of the above suggest that geometric averaging is the most accurate and computationally less expensive.

#### 5.1.1. Effect of Temperature-Density

_{4}molecules; see Figure 6. In principle, Equation (22) is a calculation of free energy; hence, it incorporates the temperature of the system, and thus, both approximations to the exact two-body PMF, ${W}^{\left(2\right)}$ and ${W}^{\left(2\right),\mathrm{full}}$, are not transferable. Indeed, we observe slight differences in the CG effective interactions (free energies) for the various temperatures, which become larger for the highest temperature.

#### 5.2. Bulk CG CH_{4} Runs Using a Pair Potential

#### 5.2.1. Effect of Temperature-Density

## 6. Effective Three-Body Potential

#### 6.1. Calculation of the Effective Three-Body Potential

_{4}at T = 80 K for different COM distances (): (a) ${r}_{12}=3.9$, ${r}_{13}=3.9$; (b) ${r}_{12}=4.0$${r}_{13}=4.0$; (c) ${r}_{12}=4.3$, ${r}_{13}=4.0$; (d) ${r}_{12}=3.8$, ${r}_{13}=5.64$. At smaller distances, the potential of the triplet deviates from the sum of the three pairwise potentials, and this is where improvement in accuracy can be obtained. As shown in Figure 10, improvement is needed for close distances around the (three-dimensional) well. We used a three-dimensional cubic polynomial to fit the potential data (conjugate gradient method), which means that 20 constants should be determined. A lower order polynomial cannot capture the curvature of the forces upon differentiation. The benefit of this fitting methodology (over partial derivatives for instance) is the analytical solution of the forces with respect to any of ${r}_{12},{r}_{13},{r}_{23}$ in contrast to tabulated data that induce some small error.

#### 6.2. CG Runs with the Effective Three-Body Potential

## 7. Discussion and Conclusions

- (a)
- The hierarchy of the cluster expansion formalism allowed us to systematically define the CG effective interaction as a sum of pair, triplets, etc., interactions. Then, CG effective potentials can be computed as they arise from the cluster expansion. Note, that for this estimation, no information from long simulations of n-body (bulk) systems is required.
- (b)
- The two-body coarse-grained potentials can be efficiently computed via the cluster expansion giving comparable results with the existing methods, such as the conditional reversible work. In addition, we present a more efficient direct geometric computation of the constrained partition function, which also alleviates sampling noise issues. No basis function is needed in any of these methods.
- (c)
- The obtained pair CG potentials were used to model methane and ethane systems in various regimes. The derived $g\left(r\right)$ data were compared against the all-atom ones. Clear differences between methane and ethane systems were observed; for the (almost spherical) methane, pair CG potentials seem to be a very good approximation, whereas much larger differences between CG and atomistic distribution functions were observed for ethane.
- (d)
- We further investigated different temperature and density regimes and in particular cases where the two-body approximations are not good enough compared to the atomistic simulations. In the latter case, we considered the next term in the cluster expansion, namely the three-body effective potentials, and we found that they give a small improvement over the pair ones in the liquid state.

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications, 2nd ed.; Academic Press: New York, NY, USA, 2002. [Google Scholar]
- Allen, M.P.; Tildesley, D.J. Computer Simulation of Liquids; Oxford University Press: Oxford, UK, 1987. [Google Scholar]
- Harmandaris, V.A.; Mavrantzas, V.G.; Theodorou, D.; Kröger, M.; Ramírez, J.; Öttinger, H.C.; Vlassopoulos, D. Dynamic crossover from Rouse to entangled polymer melt regime: Signals from long, detailed atomistic molecular dynamics simulations, supported by rheological experiments. Macromolecules
**2003**, 36, 1376–1387. [Google Scholar] [CrossRef] - Kotelyanskii, M.; Theodorou, D.N. Simulation Methods for Polymers; Taylor & Francis: Abingdon, UK, 2004. [Google Scholar]
- Izvekov, S.; Voth, G.A. Multiscale coarse-graining of liquid-state systems. J. Chem. Phys.
**2005**, 123, 134105. [Google Scholar] [CrossRef] [PubMed] - Tschöp, W.; Kremer, K.; Hahn, O.; Batoulis, J.; Bürger, T. Simulation of polymer melts. I. Coarse-graining procedure for polycarbonates. Acta Polym.
**1998**, 49, 61–74. [Google Scholar] [CrossRef] - Müller-Plathe, F. Coarse-Graining in Polymer Simulation: From the Atomistic to the Mesoscopic Scale and Back. ChemPhysChem
**2002**, 3, 754–769. [Google Scholar] [CrossRef] - Shell, M.S. The relative entropy is fundamental to multiscale and inverse thermodynamic problems. J. Chem. Phys.
**2008**, 129, 144108. [Google Scholar] [CrossRef] [PubMed] - Briels, W.J.; Akkermans, R.L.C. Coarse-grained interactions in polymer melts: A variational approach. J. Chem. Phys.
**2001**, 115, 6210. [Google Scholar] - Harmandaris, V.A.; Adhikari, N.P.; van der Vegt, N.F.A.; Kremer, K. Hierarchical Modeling of Polystyrene: From Atomistic to Coarse-Grained Simulations. Macromolecules
**2006**, 39, 6708. [Google Scholar] [CrossRef] - Harmandaris, V.A.; Kremer, K. Dynamics of Polystyrene Melts through Hierarchical Multiscale Simulations. Macromolecules
**2009**, 42, 791. [Google Scholar] [CrossRef] - Harmandaris, V.A.; Kremer, K. Predicting polymer dynamics at multiple length and time scales. Soft Matter
**2009**, 5, 3920. [Google Scholar] [CrossRef] - Johnston, K.; Harmandaris, V. Hierarchical simulations of hybrid polymer–solid materials. Soft Matter
**2013**, 9, 6696–6710. [Google Scholar] [CrossRef] [Green Version] - Noid, W.G.; Chu, J.W.; Ayton, G.S.; Krishna, V.; Izvekov, S.; Voth, G.A.; Das, A.; Andersen, H.C. The multiscale coarse-graining method. I. A rigorous bridge between atomistic and coarse-grained models. J. Chem. Phys.
**2008**, 128, 4114. [Google Scholar] [CrossRef] [PubMed] - Lu, L.; Izvekov, S.; Das, A.; Andersen, H.C.; Voth, G.A. Efficient, Regularized, and Scalable Algorithms for Multiscale Coarse-Graining. J. Chem. Theor. Comput.
**2010**, 6, 954–965. [Google Scholar] [CrossRef] [PubMed] - Rudzinski, J.F.; Noid, W.G. Coarse-graining entropy, forces, and structures. J. Chem. Phys.
**2011**, 135, 214101. [Google Scholar] [CrossRef] [PubMed] - Noid, W.G. Perspective: Coarse-grained models for biomolecular systems. J. Chem. Phys.
**2013**, 139, 090901. [Google Scholar] [CrossRef] [PubMed] - Chaimovich, A.; Shell, M.S. Anomalous waterlike behaviour in spherically-symmetric water models optimized with the relative entropy. Phys. Chem. Chem. Phys.
**2009**, 11, 1901–1915. [Google Scholar] [CrossRef] [PubMed] - Bilionis, I.; Zabaras, N. A stochastic optimization approach to coarse-graining using a relative-entropy framework. J. Chem. Phys.
**2013**, 138, 044313. [Google Scholar] [CrossRef] [PubMed] - Coifman, R.R.; Kevrekidis, I.G.; Lafon, S.; Maggioni, M.; Nadler, B. Diffusion Maps, Reduction Coordinates, and Low Dimensional Representation of Stochastic Systems. Multiscale Model. Simul.
**2008**, 7, 842–864. [Google Scholar] [CrossRef] - Soper, A.K. Empirical potential Monte Carlo simulation of fluid structure. Chem. Phys.
**1996**, 202, 295–306. [Google Scholar] [CrossRef] - Lyubartsev, A.P.; Laaksonen, A. On the Reduction of Molecular Degrees of Freedom in Computer Simulations. In Novel Methods in Soft Matter Simulations; Karttunen, M., Lukkarinen, A., Vattulainen, I., Eds.; Springer: Berlin, Germany, 2004; Volume 640, pp. 219–244. [Google Scholar]
- Harmandaris, V.A. Quantitative study of equilibrium and non-equilibrium polymer dynamics through systematic hierarchical coarse-graining simulations. Korea Aust. Rheol. J.
**2014**, 26, 15–28. [Google Scholar] [CrossRef] - Espanol, P.; Zuniga, I. Obtaining fully dynamic coarse-grained models from MD. Phys. Chem. Chem. Phys.
**2011**, 13, 10538–10545. [Google Scholar] [CrossRef] [PubMed] - Padding, J.T.; Briels, W.J. Uncrossability constraints in mesoscopic polymer melt simulations: Non-Rouse behaviour of C
_{120}H_{242}. J. Chem. Phys.**2001**, 115, 2846–2859. [Google Scholar] [CrossRef] - Deichmann, G.; Marcon, V.; van der Vegt, N.F.A. Bottom-up derivation of conservative and dissipative interactions for coarse-grained molecular liquids with the conditional reversible work method. J. Chem. Phys.
**2014**, 141, 224109. [Google Scholar] [CrossRef] [PubMed] - Fritz, D.; Harmandaris, V.; Kremer, K.; van der Vegt, N. Coarse-grained polymer melts based on isolated atomistic chains: Simulation of polystyrene of different tacticities. Macromolecules
**2009**, 42, 7579–7588. [Google Scholar] [CrossRef] - Brini, E.; Algaer, E.A.; Ganguly, P.; Li, C.; Rodriguez-Ropero, F.; van der Vegt, N.F.A. Systematic Coarse-Graining Methods for Soft Matter Simulations—A Review. Soft Matter
**2013**, 9, 2108–2119. [Google Scholar] [CrossRef] - Reith, D.; Pütz, M.; Müller-Plathe, F. Deriving effective mesoscale potentials from atomistic simulations. J. Comput. Chem.
**2003**, 24, 1624–1636. [Google Scholar] [CrossRef] [PubMed] - Lyubartsev, A.P.; Laaksonen, A. Calculation of effective interaction potentials from radial distribution functions: A reverse Monte Carlo approach. Phys. Rev. E
**1995**, 52, 3730–3737. [Google Scholar] [CrossRef] - Izvekov, S.; Voth, G.A. Effective force field for liquid hydrogen fluoride from ab initio molecular dynamics simulation using the force-matching method. J. Phys. Chem. B
**2005**, 109, 6573–6586. [Google Scholar] [CrossRef] [PubMed] - Noid, W.G.; Liu, P.; Wang, Y.; Chu, J.; Ayton, G.S.; Izvekov, S.; Andersen, H.C.; Voth, G.A. The multiscale coarse-graining method. II. Numerical implementation for coarse-grained molecular models. J. Chem. Phys.
**2008**, 128, 244115. [Google Scholar] [CrossRef] [PubMed] - Lu, L.; Dama, J.F.; Voth, G.A. Fitting coarse-grained distribution functions through an iterative force-matching method. J. Chem. Phys.
**2013**, 139, 121906. [Google Scholar] [CrossRef] [PubMed] - Katsoulakis, M.A.; Plechac, P. Information-theoretic tools for parametrized coarse-graining of non-equilibrium extended systems. J. Chem. Phys.
**2013**, 139, 4852–4863. [Google Scholar] - Chaimovich, A.; Shell, M.S. Coarse-graining errors and numerical optimization using a relative entropy framework. J. Chem. Phys.
**2011**, 134, 094112. [Google Scholar] [CrossRef] [PubMed] - Cho, H.M.; Chu, J.W. Inversion of radial distribution functions to pair forces by solving the Yvon–Born–Green equation iteratively. J. Chem. Phys.
**2009**, 131, 134107. [Google Scholar] [CrossRef] [PubMed] - Noid, W.G.; Chu, J.; Ayton, G.S.; Voth, G.A. Multiscale coarse-graining and structural correlations: Connections to liquid-state theory. J. Phys. Chem. B
**2007**, 111, 4116–4127. [Google Scholar] [CrossRef] [PubMed] - Mullinax, J.W.; Noid, W.G. Generalized Yvon–Born–Green theory for molecular systems. Phys. Rev. Lett.
**2009**, 103, 198104. [Google Scholar] [CrossRef] [PubMed] - Mullinax, J.W.; Noid, W.G. Generalized Yvon–Born–Green theory for determining coarse-grained interaction potentials. J. Phys. Chem. C
**2010**, 114, 5661–5674. [Google Scholar] [CrossRef] - McCarty, J.; Clark, A.J.; Copperman, J.; Guenza, M.G. An analytical coarse-graining method which preserves the free energy, structural correlations, and thermodynamic state of polymer melts from the atomistic to the mesoscale. J. Chem. Phys.
**2014**, 140, 204913. [Google Scholar] [CrossRef] [PubMed] - Li, Z.; Bian, X.; Li, X.; Karniadakis, G.E. Incorporation of memory effects in coarse-grained modeling via the Mori-Zwanzig formalism. J. Chem. Phys.
**2015**, 143, 243128. [Google Scholar] [CrossRef] [PubMed] - McCarty, J.; Clark, A.J.; Lyubimov, I.Y.; Guenza, M.G. Thermodynamic Consistency between Analytic Integral Equation Theory and Coarse-Grained Molecular Dynamics Simulations of Homopolymer Melts. Macromolecules
**2012**, 45, 8482–8493. [Google Scholar] [CrossRef] - Clark, A.J.; McCarty, J.; Guenza, M.G. Effective potentials for representing polymers in melts as chains of interacting soft particles. J. Chem. Phys.
**2013**, 139, 124906. [Google Scholar] [CrossRef] [PubMed] - Stell, G. The Percus-Yevick equation for the radial distribution function of a fluid. Physica
**1963**, 29, 517–534. [Google Scholar] [CrossRef] - Kuna, T.; Tsagkarogiannis, D. Convergence of density expansions of correlation functions and the Ornstein-Zernike equation. arXiv, 2016; arXiv:1611.01716. [Google Scholar]
- Bolhuis, P.G.; Louis, A.A.; Hansen, J.P. Many-body interactions and correlations in coarse-grained descriptions of polymer solutions. Phys. Rev. E
**2001**, 64, 021801. [Google Scholar] [CrossRef] [PubMed] - Mayer, J.E.; Mayer, M.G. Statistical Mechanics; John Wiley & Sons: Hoboken, NJ, USA, 1940. [Google Scholar]
- Morita, T.; Hiroike, K. The statistical mechanics of condensing systems. III. Prog. Theor. Phys.
**1961**, 25, 537. [Google Scholar] [CrossRef] - Frisch, H.; Lebowitz, J. Equilibrium Theory of Classical Fluids; W.A. Benjamin: New York, NY, USA, 1964. [Google Scholar]
- Hansen, J.P.; McDonald, I.R. Theory of Simple Lipquids; Academic Press: London, UK, 1986. [Google Scholar]
- Pulvirenti, E.; Tsagkarogiannis, D. Cluster expansion in the canonical ensemble. Commun. Math. Phys.
**2012**, 316, 289–306. [Google Scholar] [CrossRef] - Louis, A.A. Beware of density dependent pair potentials. J. Phys. Condens. Matter
**2002**, 14, 9187–9206. [Google Scholar] [CrossRef] - Katsoulakis, M.; Plecháč, P.; Rey-Bellet, L.; Tsagkarogiannis, D. Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems. ESAIM Math. Model. Numer. Anal.
**2007**, 41, 627–660. [Google Scholar] [CrossRef] - Katsoulakis, M.; Plecháč, P.; Rey-Bellet, L.; Tsagkarogiannis, D. Coarse-graining schemes for stochastic lattice systems with short and long range interactions. Math. Comput.
**2014**, 83, 1757–1793. [Google Scholar] [CrossRef] - Katsoulakis, M.; Plecháč, P.; Rey-Bellet, L.; Tsagkarogiannis, D. Mathematical strategies and error quantification in coarse-graining of extended systems. J. Non Newton. Fluid Mech.
**2008**, 152, 101–112. [Google Scholar] [CrossRef] - Trashorras, J.; Tsagkarogiannis, D. Reconstruction schemes for coarse-grained stochastic lattice systems. SIAM J. Numer. Anal.
**2010**, 48, 1647–1677. [Google Scholar] [CrossRef] - Kremer, K.; Müller-Plathe, F. Multiscale problems in polymer science: Simulation approaches. MRS Bull.
**2001**, 26, 205–210. [Google Scholar] [CrossRef] - McCoy, J.D.; Curro, J.G. Mapping of Explicit Atom onto United Atom Potentials. Macromolecules
**1998**, 31, 9352–9368. [Google Scholar] [CrossRef] - Tsourtis, A.; Harmandaris, V.; Tsagkarogiannis, D. Effective coarse-grained interactions: The role of three-body terms through cluster expansions. under preparation.
- Kirkwood, J.G. Statistical Mechanics of Fluid Mixtures. J. Chem. Phys.
**1935**, 3, 300–313. [Google Scholar] [CrossRef] - McQuarrie, D.A. Statistical Mechanics; University Science Books: Sausalito, CA, USA, 2000. [Google Scholar]
- Kalligiannaki, E.; Chazirakis, A.; Tsourtis, A.; Katsoulakis, M.; Plecháč, P.; Harmandaris, V. Parametrizing coarse grained models for molecular systems at equilibrium. Eur. Phys. J.
**2016**, 225, 1347–1372. [Google Scholar] [CrossRef] - Henderson, R. A uniqueness theorem for fluid pair correlation functions. Phys. Lett. A
**1974**, 49, 197–198. [Google Scholar] [CrossRef] - Larini, L.; Lu, L.; Voth, G.A. The multiscale coarse-graining method. VI. Implementation of three-body coarse-grained potentials. J. Chem. Phys.
**2010**, 132, 164107. [Google Scholar] [CrossRef] [PubMed] - Das, A.; Andersen, H.C. The multiscale coarse-graining method. IX. A general method for construction of three body coarse-grained force fields. J. Chem. Phys.
**2012**, 136, 194114. [Google Scholar] [CrossRef] [PubMed] - Lelièvre, T.; Rousset, M.; Stoltz, G. Free Energy Computations: A Mathematical Perspective; Imperial College Press: London, UK, 2010. [Google Scholar]
- Tsourtis, A.; Pantazis, Y.; Katsoulakis, M.A.; Harmandaris, V. Parametric sensitivity analysis for stochastic molecular systems using information theoretic metrics. J. Chem. Phys.
**2015**, 143, 014116. [Google Scholar] [CrossRef] [PubMed] - Kuna, T.; Lebowitz, J.; Speer, E. Realizability of point processes. J. Stat. Phys.
**2007**, 129, 417–439. [Google Scholar] [CrossRef]

**Figure 1.**Visualization of the partition in (19) for non-intersecting sets ${V}_{1}=\{2,3,4,5\}$, ${V}_{2}=\{6,7\}$, ${V}_{3}=\{9,10,11\}$ in each of which we display by solid lines the connected graphs ${g}_{i}\in {\mathcal{C}}_{{V}_{i}}$, $i=1,2,3$.

**Figure 2.**Snapshot of model systems in atomistic and coarse-grained description. (

**a**,

**b**) Two and three methanes used for the estimation of the coarse-graining (CG) effective potential from isolated molecules; (

**c**) bulk methane liquid.

**Figure 3.**Representation of the two-body PMF, for two isolated molecules, as a function of distance r, through different approximations: geometric averaging, (constrained) force matching and inversion of $g\left(r\right)$. (

**a**) CH

_{4}at T = 100 K; (

**b**) CH

_{3}–CH

_{3}at T = 150 K. For the methane, the corresponding $g\left(r\right)$ curve is also shown.

**Figure 4.**Relation of the PMF through cluster expansions and energy averaging at high temperatures, i.e., ${W}^{\left(2\right)}({r}_{1},{r}_{2})$ and ${W}^{\left(2\right),\mathrm{full}}({r}_{1},{r}_{2})$, through expansion over $\beta $ for (

**a**) CH

_{4}at T = 300 K; (

**b**) CH

_{3}–CH

_{3}at T = 650 K. As expected from the analytic form and the relation between the two formulas, ${W}^{\left(2\right)}$ and ${W}^{\left(2\right),\mathrm{full}}$ tend to converge to the same effective potential.

**Figure 5.**(

**a**) PMF through cluster expansions, using (22) and (25) for different temperatures for the CH

_{4}model; (

**b**) PMF through cluster expansions and energy averaging, i.e., ${W}^{\left(2\right)}({r}_{1},{r}_{2})$ and ${W}^{\left(2\right),\mathrm{full}}({r}_{1},{r}_{2})$ through expansion over $\beta $ for CH

_{4}at T = 150 K. The expansion is not valid at this temperature.

**Figure 6.**Potential of mean force at different temperatures (geometric averaging). Two CH

_{4}molecules at T = 80 K, 100 K, 300 K.

**Figure 7.**RDF from atomistic and CG using pair potential, ${W}^{\left(2\right)}$, for (

**a**) CH

_{4}at T = 80 K and (

**b**) CH

_{3}–CH

_{3}at T = 150 K. Spherical CG approximation to the non-symmetric ethane molecule induces discrepancy and implies that there is more room for improvement.

**Figure 8.**RDF of methane from atomistic data and CG models using pair potential at different temperatures: (

**a**) T = 300 K; (

**b**) T = 900 K. In both cases, the density is ${\rho}_{1}=0.399\frac{\mathrm{g}}{{\mathrm{cm}}^{3}}$.

**Figure 9.**RDF of methane from atomistic and CG using pair potential at different densities ${\rho}_{1}>{\rho}_{2}$. (

**a**) $T=300K$; (

**b**) T = 900 K. For this model, the pair approximation is sufficient, and in low $\rho $, high T conditions, ${W}^{\left(2\right)}$ converges to the reference $g\left(r\right)$.

**Figure 10.**Effective potential comparison between the ${W}^{\left(3\right),\mathrm{full}}$ three-body and $\sum {W}^{\left(2\right),\mathrm{full}}$ simulations (geometric averaging) for CH

_{4}at T = 80 K for different fixed centre of mass (COM) distances. (

**a**) ${r}_{12}=\phantom{\rule{3.33333pt}{0ex}}3.9,{r}_{13}=3.9$; (

**b**) ${r}_{12}=4.0,{r}_{13}=4.0$; (

**c**) ${r}_{12}=4.3,{r}_{13}=4.0$; (

**d**) ${r}_{12}=3.8,{r}_{13}=5.64$.

**Figure 11.**RDF from atomistic and CG using pair, ${W}^{\left(2\right),\mathrm{full}}$, and three-body, ${W}^{\left(3\right),\mathrm{full}}$, potential for CH

_{4}(T = 80 K). The three-dimensional cubic polynomial was used for the fitting.

System | N (Molecules) | T/K | Simulation Time/ns |
---|---|---|---|

CH_{4}, CH_{3}–CH_{3} | 2, 3 | 100–900 | 10–20 |

CH_{4} | 512 | 80, 100, 120, 300, 900 | 100 |

CH_{3}–CH_{3} | 500 | 150, 300, 650 | 100 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tsourtis, A.; Harmandaris, V.; Tsagkarogiannis, D.
Parameterization of Coarse-Grained Molecular Interactions through Potential of Mean Force Calculations and Cluster Expansion Techniques. *Entropy* **2017**, *19*, 395.
https://doi.org/10.3390/e19080395

**AMA Style**

Tsourtis A, Harmandaris V, Tsagkarogiannis D.
Parameterization of Coarse-Grained Molecular Interactions through Potential of Mean Force Calculations and Cluster Expansion Techniques. *Entropy*. 2017; 19(8):395.
https://doi.org/10.3390/e19080395

**Chicago/Turabian Style**

Tsourtis, Anastasios, Vagelis Harmandaris, and Dimitrios Tsagkarogiannis.
2017. "Parameterization of Coarse-Grained Molecular Interactions through Potential of Mean Force Calculations and Cluster Expansion Techniques" *Entropy* 19, no. 8: 395.
https://doi.org/10.3390/e19080395