^{1}and Hans C. Andersen

^{1,a)}

### Abstract

The multiscale coarse-graining (MS-CG) method, proposed by Izvekov and Voth [J. Phys. Chem. B109, 2469 (2005); Izvekov and VothJ. Chem. Phys.123, 134105 (2005)], is a method for determining the effective potential energy function for a coarse-grained model of a fluid using data obtained from molecular dynamics (MD) simulation of the corresponding atomically detailed model. The method has been given a rigorous statistical mechanical basis [Noid *et al.*J. Chem. Phys.128, 244114 (2008); Noid *et al.*,J. Chem. Phys.128, 244115 (2008)]. The coarse-grained (CG) potentials obtained using the MS-CG method are an approximate variational solution for the exact many-body potential of mean force for the coarse-grained sites. In this paper we apply this method to study the many-body potential of mean force among solutes in a simple model of a solution of Lennard-Jones particles. We use a new set of basis functions for the variational calculation that is useful when the coarse-grained potential is approximately equal to an arbitrarily complicated pairwise additive, central interaction among the sites of the coarse-grained model. For this model, pairwise additivity of the many-body potential of mean force is a very good approximation when the solute concentration is low, and it becomes less accurate for high concentrations, indicating the importance of many-body contributions to the coarse-grained potential. The best possible pairwise additive CG potential of the solute particles is found to be quite long ranged for all concentrations except those for which the mole fraction of solute is very close to unity. We discuss strategies for construction of short-ranged potentials for efficient but accurate CG MD simulation. We also discuss how the choice of basis functions for the variational calculation can be used to provide smoothing of the calculated CG potential function to overcome statistical sampling error in the atomistic simulation data used for the generation of the potential.

This research was supported by a Collaborative Research in Chemistry grant from the National Science Foundation under Grant No. CHE-0628257. It arose out of ongoing collaboration and interaction with Professor Gregory A. Voth of the University of Utah and his research group. We acknowledge many stimulating discussions with them on the matters discussed here.

I. INTRODUCTION

II. BASIS FUNCTIONS

A. Representation of the CG force field

B. Choice of new basis functions

C. Comparison with other basis sets

III. MODEL SYSTEM

IV. DETAILS OF CALCULATIONS

V. RESULTS AND DISCUSSION

A. The range of the CG pair potential

B. Test of pairwise additivity of the CG potential

VI. STRATEGIES FOR CONSTRUCTING SHORTER RANGED CG POTENTIALS

VII. CONCLUSION

### Key Topics

- Statistical mechanics models
- 21.0
- Polynomials
- 10.0
- Molecular dynamics
- 8.0
- Particle distribution functions
- 5.0
- Solvents
- 5.0

## Figures

Basis functions defined using a set of grid points with nonuniform spacings. Vertical lines show the positions of the grid points, the first grid point being at . The top panel shows functions for even , and the bottom panel shows functions for odd . Each basis function is nonzero in two adjacent intervals and zero elsewhere. Solid curves are the basis functions (for ) associated with first interior grid point (0.9 for the present case) and dashed curves are examples of basis functions associated with other interior grid points.

Basis functions defined using a set of grid points with nonuniform spacings. Vertical lines show the positions of the grid points, the first grid point being at . The top panel shows functions for even , and the bottom panel shows functions for odd . Each basis function is nonzero in two adjacent intervals and zero elsewhere. Solid curves are the basis functions (for ) associated with first interior grid point (0.9 for the present case) and dashed curves are examples of basis functions associated with other interior grid points.

The CG force function for large distances. The solid line is the average of the results of five variational calculations using statistically independent atomistic data. The error bars are estimated from the variance of the five results in the usual way. Note the expanded vertical scale.

The CG force function for large distances. The solid line is the average of the results of five variational calculations using statistically independent atomistic data. The error bars are estimated from the variance of the five results in the usual way. Note the expanded vertical scale.

Dependence of the calculated CG force and potential functions on the parameter for solute mole fraction of 0.25. Results for values of 5.2, 8.0, and 10.5 were obtained from the variational calculation using data from atomistic simulations of systems of 1000, 4000, and 8000 particles, respectively. The main figures show the results for and the insets give the difference between the functions for and those for a smaller .

Dependence of the calculated CG force and potential functions on the parameter for solute mole fraction of 0.25. Results for values of 5.2, 8.0, and 10.5 were obtained from the variational calculation using data from atomistic simulations of systems of 1000, 4000, and 8000 particles, respectively. The main figures show the results for and the insets give the difference between the functions for and those for a smaller .

Test of the pairwise additivity approximation of the CG potential for a system with solute mole fraction of 0.25. The top panel shows the atomistic and CG functions, and the bottom panel shows the difference between the two.

Test of the pairwise additivity approximation of the CG potential for a system with solute mole fraction of 0.25. The top panel shows the atomistic and CG functions, and the bottom panel shows the difference between the two.

Test of the pairwise additivity approximation for various solute mole fractions for the model system. The top panel shows the CG functions for various mole fractions of solute as well as the atomistic . The bottom panel shows the difference between each CG and the atomistic , Numbers shown in the legends of both the figures give the mole fractions of solute ( particles) in the atomistic systems.

Test of the pairwise additivity approximation for various solute mole fractions for the model system. The top panel shows the CG functions for various mole fractions of solute as well as the atomistic . The bottom panel shows the difference between each CG and the atomistic , Numbers shown in the legends of both the figures give the mole fractions of solute ( particles) in the atomistic systems.

Comparison of CG potentials obtained from force matching calculations of systems with different compositions. Numbers shown in the legends give the mole fractions of solute particles in the atomistic systems. The thinner solid line in the inset is the truncated and shifted Lennard-Jones potential, which is the correct CG potential in the limit of solute mole fraction of 1.

Comparison of CG potentials obtained from force matching calculations of systems with different compositions. Numbers shown in the legends give the mole fractions of solute particles in the atomistic systems. The thinner solid line in the inset is the truncated and shifted Lennard-Jones potential, which is the correct CG potential in the limit of solute mole fraction of 1.

Test of the first strategy for the construction of short-ranged CG potentials (for details, see the text). Comparison of radial distribution functions from atomistic and CG simulations with short-ranged potentials constructed using the first strategy for a system with solute mole fraction of 0.25.

Test of the first strategy for the construction of short-ranged CG potentials (for details, see the text). Comparison of radial distribution functions from atomistic and CG simulations with short-ranged potentials constructed using the first strategy for a system with solute mole fraction of 0.25.

Test of the second strategy for the construction of short-ranged CG potentials (for details, see the text). Comparison of radial distribution functions from atomistic and CG simulations with short-ranged potentials constructed using the second strategy for a system with solute mole fraction of 0.25. is the position of the last grid point or the distance beyond which the basis function part of the total CG force [i.e., , see Eq. (1)] is zero.

Test of the second strategy for the construction of short-ranged CG potentials (for details, see the text). Comparison of radial distribution functions from atomistic and CG simulations with short-ranged potentials constructed using the second strategy for a system with solute mole fraction of 0.25. is the position of the last grid point or the distance beyond which the basis function part of the total CG force [i.e., , see Eq. (1)] is zero.

Effects of grid spacing on the quality of fitting in the force matching calculation. The same atomistic data set is used with three different choices of grid spacings. The top panel shows the overall fit. The middle and bottom panels show the effects on the large and short distance parts of the CG force. Neither of the uniform grid spacings led to an adequate representation of the force function both in the first peaks, where the function varies rapidly, and at large distances where there is more statistical error in the force information. (In the middle panel, with its expanded scale, the dashed curve is almost completely coincident with the solid curve. In the bottom panel, the dotted curve is almost completely coincident with the solid curve.)

Effects of grid spacing on the quality of fitting in the force matching calculation. The same atomistic data set is used with three different choices of grid spacings. The top panel shows the overall fit. The middle and bottom panels show the effects on the large and short distance parts of the CG force. Neither of the uniform grid spacings led to an adequate representation of the force function both in the first peaks, where the function varies rapidly, and at large distances where there is more statistical error in the force information. (In the middle panel, with its expanded scale, the dashed curve is almost completely coincident with the solid curve. In the bottom panel, the dotted curve is almost completely coincident with the solid curve.)

Article metrics loading...

Full text loading...

Commenting has been disabled for this content