^{1}and W. G. Noid

^{1,a)}

### Abstract

Coarse-grained (CG) models provide a computationally efficient means for investigating biological and soft-matter processes that evolve on long time scales and large length scales. The present work introduces an extended ensemble framework for calculating transferable CG potentials that accurately reproduce the structure of atomistic models for multiple systems. This framework identifies a generalized potential of mean force (PMF) as the appropriate CG potential for reproducing the structural correlations of an atomistic extended ensemble. A variational approach is developed for calculating transferable potentials that provide an optimal approximation to this PMF. Calculations for binary mixtures of alkanes and alcohols demonstrate that the extended ensemble potentials provide improved transferability relative to potentials calculated for a single system.

W.G.N. gratefully acknowledges many stimulating conversations with Dr. Vinod Krishna and also Professor Roger Loring and Professor Lasse Jensen for helpful criticism of drafts of the present manuscript. Numerical calculations presented in this work were performed using resources administered and maintained by the Penn State High Performance Computing Center.

I. INTRODUCTION

II. THEORY

A. Atomistic and coarse-grained extended ensembles

B. CG mappings for an extended ensemble

C. Consistency between atomistic and CG extended ensembles

D. Variational principle

E. Calculation of the CG force field

III. RESULTS

A. Binary methanol-neopentane mixtures

B. Transferable united atom potentials for alkane and alcohol mixtures

IV. DISCUSSION

### Key Topics

- Topology
- 147.0
- Classical ensemble theory
- 34.0
- Molecular dynamics
- 29.0
- Subspaces
- 9.0
- Thermodynamic properties
- 8.0

## Figures

Molecular RDFs calculated from atomically detailed simulations of methanol-neopentane mixtures for (a) pairs of methanol molecules, (b) pairs of methanol and neopentane molecules, and (c) pairs of neopentane molecules. In these figures, black, red, green, blue, and orange curves correspond to binary methanol-neopentane mixtures with 100%, 80%, 50%, 20%, and 0% methanol, respectively.

Molecular RDFs calculated from atomically detailed simulations of methanol-neopentane mixtures for (a) pairs of methanol molecules, (b) pairs of methanol and neopentane molecules, and (c) pairs of neopentane molecules. In these figures, black, red, green, blue, and orange curves correspond to binary methanol-neopentane mixtures with 100%, 80%, 50%, 20%, and 0% methanol, respectively.

Pair forces and differences in pair forces calculated for the M1N1 CG extended ensemble. (a)(1), (b)(1), and (c)(1) present the optimal transferable pair forces for interactions between pairs of M sites, pairs of M and N sites, and pairs of N sites, respectively. (a)(2), (b)(2), and (c)(2) present, for each pair of site types, the difference between the MS-CG pair force calculated separately for each topology and the optimal transferable pair force calculated for the entire extended ensemble. In these latter figures, black, red, green, blue, and orange curves correspond to binary methanol-neopentane mixtures with 100%, 80%, 50%, 20%, and 0% methanol, respectively.

Pair forces and differences in pair forces calculated for the M1N1 CG extended ensemble. (a)(1), (b)(1), and (c)(1) present the optimal transferable pair forces for interactions between pairs of M sites, pairs of M and N sites, and pairs of N sites, respectively. (a)(2), (b)(2), and (c)(2) present, for each pair of site types, the difference between the MS-CG pair force calculated separately for each topology and the optimal transferable pair force calculated for the entire extended ensemble. In these latter figures, black, red, green, blue, and orange curves correspond to binary methanol-neopentane mixtures with 100%, 80%, 50%, 20%, and 0% methanol, respectively.

Differences between molecular RDFs calculated from CG MD simulations employing the optimal transferable pair forces determined for each extended ensemble and those calculated from atomistic MD simulations. Each row presents the differences between the RDFs calculated from CG MD simulations using the transferable potentials determined for different extended ensembles and the RDFs from the corresponding row of Fig. 1 for the same pair of sites. Rows (a), (b), and (c) correspond to differences in molecular RDFs for pairs of methanol molecules, pairs of methanol and neopentane molecules, and pairs of neopentane molecules, respectively. Each column presents differences between RDFs calculated from CG MD simulations employing the transferable pair potentials for a particular extended ensemble and the RDFs calculated from atomistic simulations. Columns 1, 2, and 3 correspond to CG MD simulations for the M1N1, M2N1, and M3N1 extended ensembles. The M-M and M-N molecular RDFs were calculated using the center of mass for the one, two, and three sites used to represent methanol in the M1N1, M2N1, and M3N1 ensembles, respectively. In these panels, black, red, green, blue, and orange curves correspond to binary methanol-neopentane mixtures with 100%, 80%, 50%, 20%, and 0% methanol, respectively.

Differences between molecular RDFs calculated from CG MD simulations employing the optimal transferable pair forces determined for each extended ensemble and those calculated from atomistic MD simulations. Each row presents the differences between the RDFs calculated from CG MD simulations using the transferable potentials determined for different extended ensembles and the RDFs from the corresponding row of Fig. 1 for the same pair of sites. Rows (a), (b), and (c) correspond to differences in molecular RDFs for pairs of methanol molecules, pairs of methanol and neopentane molecules, and pairs of neopentane molecules, respectively. Each column presents differences between RDFs calculated from CG MD simulations employing the transferable pair potentials for a particular extended ensemble and the RDFs calculated from atomistic simulations. Columns 1, 2, and 3 correspond to CG MD simulations for the M1N1, M2N1, and M3N1 extended ensembles. The M-M and M-N molecular RDFs were calculated using the center of mass for the one, two, and three sites used to represent methanol in the M1N1, M2N1, and M3N1 ensembles, respectively. In these panels, black, red, green, blue, and orange curves correspond to binary methanol-neopentane mixtures with 100%, 80%, 50%, 20%, and 0% methanol, respectively.

Differences between the RDFs calculated from CG MD simulations employing either extended ensemble potentials or MS-CG potentials and RDFs calculated from atomistic MD simulations. Rows a, b, and c present differences in the M-M, M-N, and N-N RDFs, respectively. Columns 1, 2, and 3 correspond to MD simulations of 80%, 50%, and 20% methanol mixtures, respectively. Panels (a) and (c) correspond to MD simulations of pure methanol and pure neopentane, respectively. In each panel, the red, green, and blue lines correspond to RDFs from CG MD simulations employing the MS-CG pair potentials calculated for the 80%, 50%, and 20% methanol mixtures, respectively. In each panel the solid black line corresponds to RDFs from CG MD simulations employing the transferable potentials calculated via the extended ensemble approach for the M1N1 ensemble.

Differences between the RDFs calculated from CG MD simulations employing either extended ensemble potentials or MS-CG potentials and RDFs calculated from atomistic MD simulations. Rows a, b, and c present differences in the M-M, M-N, and N-N RDFs, respectively. Columns 1, 2, and 3 correspond to MD simulations of 80%, 50%, and 20% methanol mixtures, respectively. Panels (a) and (c) correspond to MD simulations of pure methanol and pure neopentane, respectively. In each panel, the red, green, and blue lines correspond to RDFs from CG MD simulations employing the MS-CG pair potentials calculated for the 80%, 50%, and 20% methanol mixtures, respectively. In each panel the solid black line corresponds to RDFs from CG MD simulations employing the transferable potentials calculated via the extended ensemble approach for the M1N1 ensemble.

Representative examples of transferable pair forces calculated for the M2N1 and M3N1 extended ensembles. (a)–(c) present the transferable Me-Me, Me-OH, and OH–OH pair forces, respectively, calculated for the M2N1 extended ensemble. (d)–(f) present the transferable Me-Me, Me-O, and O–O pair forces, respectively, calculated for the M3N1 extended ensemble.

Representative examples of transferable pair forces calculated for the M2N1 and M3N1 extended ensembles. (a)–(c) present the transferable Me-Me, Me-OH, and OH–OH pair forces, respectively, calculated for the M2N1 extended ensemble. (d)–(f) present the transferable Me-Me, Me-O, and O–O pair forces, respectively, calculated for the M3N1 extended ensemble.

Optimal transferable pair forces calculated for the CG extended canonical ensemble of alcohols and alkanes described by the first five rows of Table III.

Optimal transferable pair forces calculated for the CG extended canonical ensemble of alcohols and alkanes described by the first five rows of Table III.

Comparison of representative intermolecular site-site RDFs calculated from atomistic and CG MD simulations of the propanol-butane mixture described by the sixth line of Table III. In each case, the RDFs calculated using configurations sampled from atomistic MD simulations are represented by solid lines, while RDFs calculated from CG MD simulations employing the optimal transferable potentials are represented by dashed lines. (a)–(c) present RDFs for CM-H, CP-H, and CT-CM site pairs in distinct propanol molecules. (d) and (e) present RDFs for CM-CM and O-CT site pairs that are in distinct propanol and butane molecules. (f) presents RDFs for CT-CT site pairs that are in distinct butane molecules.

Comparison of representative intermolecular site-site RDFs calculated from atomistic and CG MD simulations of the propanol-butane mixture described by the sixth line of Table III. In each case, the RDFs calculated using configurations sampled from atomistic MD simulations are represented by solid lines, while RDFs calculated from CG MD simulations employing the optimal transferable potentials are represented by dashed lines. (a)–(c) present RDFs for CM-H, CP-H, and CT-CM site pairs in distinct propanol molecules. (d) and (e) present RDFs for CM-CM and O-CT site pairs that are in distinct propanol and butane molecules. (f) presents RDFs for CT-CT site pairs that are in distinct butane molecules.

Comparison of representative intramolecular distributions calculated from atomistic and CG MD simulations of the propanol-butane mixture described by the sixth line of Table III. In each case, the distributions calculated from atomistic MD simulations are represented by solid lines, while distributions calculated from CG MD simulations employing the optimal transferable potentials are represented by dashed lines. (a)(1), (b)(1), and (c)(1) present the O–H bond stretch, CM-CP–O bond angle, and CM-CP-O–H dihedral angle distributions for propanol molecules. (a)(2), (b)(2), and (c)(2) present the CM-CM bond stretch, CT-CM-CM bond angle, and CT-CM-CM-CT dihedral angle distributions for butane molecules.

Comparison of representative intramolecular distributions calculated from atomistic and CG MD simulations of the propanol-butane mixture described by the sixth line of Table III. In each case, the distributions calculated from atomistic MD simulations are represented by solid lines, while distributions calculated from CG MD simulations employing the optimal transferable potentials are represented by dashed lines. (a)(1), (b)(1), and (c)(1) present the O–H bond stretch, CM-CP–O bond angle, and CM-CP-O–H dihedral angle distributions for propanol molecules. (a)(2), (b)(2), and (c)(2) present the CM-CM bond stretch, CT-CM-CM bond angle, and CT-CM-CM-CT dihedral angle distributions for butane molecules.

## Tables

Description of methanol-neopentane mixtures in the atomistic extended canonical ensemble. The first column identifies the mixture in the text, the second and third columns specify the number of methanol and neopentane molecules, respectively, in each mixture, and the last column specifies the length of each side in the cubic simulation cell.

Description of methanol-neopentane mixtures in the atomistic extended canonical ensemble. The first column identifies the mixture in the text, the second and third columns specify the number of methanol and neopentane molecules, respectively, in each mixture, and the last column specifies the length of each side in the cubic simulation cell.

Topology mappings for methanol-neopentane extended canonical ensembles. The first column identifies the particular topology mapping. Each mapping is determined by the CG representations of methanol and neopentane molecules, which are summarized by the second and third columns, respectively.

Topology mappings for methanol-neopentane extended canonical ensembles. The first column identifies the particular topology mapping. Each mapping is determined by the CG representations of methanol and neopentane molecules, which are summarized by the second and third columns, respectively.

Description of alkane-alcohol mixtures. The first and third columns identify the species included in each binary mixture. The second and fourth columns specify the number of molecules of each species. The last column specifies the length of each side in the cubic simulation cell. The first five rows describe the topologies included in the extended canonical ensemble employed to calculate the transferable CG potentials. The remaining six rows describe additional topologies used to test the potentials.

Description of alkane-alcohol mixtures. The first and third columns identify the species included in each binary mixture. The second and fourth columns specify the number of molecules of each species. The last column specifies the length of each side in the cubic simulation cell. The first five rows describe the topologies included in the extended canonical ensemble employed to calculate the transferable CG potentials. The remaining six rows describe additional topologies used to test the potentials.

Summary of topology mappings for the alkane-alcohol extended canonical ensemble. Each row describes the CG representation for each molecule.

Summary of topology mappings for the alkane-alcohol extended canonical ensemble. Each row describes the CG representation for each molecule.

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