^{1,a)}, Pu Liu

^{1}, Yanting Wang

^{1,b)}, Jhih-Wei Chu

^{1,c)}, Gary S. Ayton

^{1}, Sergei Izvekov

^{1}, Hans C. Andersen

^{2}and Gregory A. Voth

^{1,d)}

### Abstract

The multiscale coarse-graining (MS-CG) method [S. Izvekov and G. A. Voth, J. Phys. Chem. B109, 2469 (2005);J. Chem. Phys.123, 134105 (2005)] employs a variational principle to determine an interaction potential for a CG model from simulations of an atomically detailed model of the same system. The companion paper proved that, if no restrictions regarding the form of the CG interaction potential are introduced and if the equilibrium distribution of the atomistic model has been adequately sampled, then the MS-CG variational principle determines the exact many-body potential of mean force (PMF) governing the equilibrium distribution of CG sites generated by the atomistic model. In practice, though, CG force fields are not completely flexible, but only include particular types of interactions between CG sites, e.g., nonbonded forces between pairs of sites. If the CG force field depends linearly on the force field parameters, then the vector valued functions that relate the CG forces to these parameters determine a set of basis vectors that span a vector subspace of CG force fields. The companion paper introduced a distance metric for the vector space of CG force fields and proved that the MS-CG variational principle determines the CG force force field that is within that vector subspace and that is closest to the force field determined by the many-body PMF. The present paper applies the MS-CG variational principle for parametrizing molecular CG force fields and derives a linear least squares problem for the parameter set determining the optimal approximation to this many-body PMF. Linear systems of equations for these CG force field parameters are derived and analyzed in terms of equilibrium structural correlation functions. Numerical calculations for a one-site CG model of methanol and a molecular CG model of the ionic liquid are provided to illustrate the method.

This research was supported by a Collaborative Research in Chemistry grant from the National Science Foundation (CHE-0628257). W.G.N. acknowledges funding from the National Institutes of Health through a Ruth L. Kirschstein National Research Service Award postdoctoral fellowship (Grant No. 5 F32 GM076839-02). Allocations of computer time from the Lonestar supercomputer at the Texas Advanced Computing Center are gratefully acknowledged. W.G.N. gratefully acknowledges Dr. V. Krishna for many stimulating conversations and also Dr. B. Hopkins for a critical reading of the manuscript.

I. INTRODUCTION

II. COMPUTATION OF THE CG POTENTIAL

A. Basis functions to represent the coarse-grained potential

B. Linear equations for the MS-CG force field

III. RESULTS

A. MS-CG methanol model

B. MS-CG ionic liquidmodel

IV. DISCUSSION

V. CONCLUDING REMARKS

### Key Topics

- Ionic liquids
- 27.0
- Statistical mechanics models
- 23.0
- Phase space methods
- 18.0
- Variational principles
- 18.0
- Chemical bonds
- 12.0

## Figures

The RDFs computed from simulations of the atomistic (dashed curve) and MS-CG (solid curve) model of methanol are compared. The difference between the two curves results from the use of basis functions that describe the CG force field as a sum of pairwise additive terms. This small difference indicates that the many-body PMF for the one-site CG model of methanol is well represented by a sum of pair-additive interactions between sites.

The RDFs computed from simulations of the atomistic (dashed curve) and MS-CG (solid curve) model of methanol are compared. The difference between the two curves results from the use of basis functions that describe the CG force field as a sum of pairwise additive terms. This small difference indicates that the many-body PMF for the one-site CG model of methanol is well represented by a sum of pair-additive interactions between sites.

The distribution of the velocities of CG sites in the direction computed from MD simulations of the atomistic (dashed curve) and MS-CG (solid curve) model of methanol.

The distribution of the velocities of CG sites in the direction computed from MD simulations of the atomistic (dashed curve) and MS-CG (solid curve) model of methanol.

The inset presents the pair CG force calculated using the linear spline basis and a conjugate gradient algorithm involving the normal matrix to iteratively minimize the MS-CG residual. This force function provides the most accurate approximation to the many-body PMF obtained in the present calculations. See Table I. The main figure presents the difference between this calculated force and force calculated in various other ways. “Normal” refers to the use of the normal matrix and a conjugate gradient algorithm, whereas “block-averaged” refers to the use of nonsymmetric matrix, a biconjugate gradient algorithm, and the BA approximation. “Delta” refers to the use of the delta function basis, “linear” refers to the use of the linear spline basis, and “cubic” refers to the use of the cubic spline basis representation. All the calculations used approximately the same large number of configurations and so all are subject to approximately the same statistical error. The small differences among the curves for demonstrate that in this case the systematic error in the BA approximation is negligible for the block size used and that the calculated CG force function is insensitive to the basis set used and the algorithm used to minimize the residual.

The inset presents the pair CG force calculated using the linear spline basis and a conjugate gradient algorithm involving the normal matrix to iteratively minimize the MS-CG residual. This force function provides the most accurate approximation to the many-body PMF obtained in the present calculations. See Table I. The main figure presents the difference between this calculated force and force calculated in various other ways. “Normal” refers to the use of the normal matrix and a conjugate gradient algorithm, whereas “block-averaged” refers to the use of nonsymmetric matrix, a biconjugate gradient algorithm, and the BA approximation. “Delta” refers to the use of the delta function basis, “linear” refers to the use of the linear spline basis, and “cubic” refers to the use of the cubic spline basis representation. All the calculations used approximately the same large number of configurations and so all are subject to approximately the same statistical error. The small differences among the curves for demonstrate that in this case the systematic error in the BA approximation is negligible for the block size used and that the calculated CG force function is insensitive to the basis set used and the algorithm used to minimize the residual.

The molecular structure of the ion pair is represented with five CG sites. CG sites , , , and describe the cation, while site describes the anion. The coordinates of each site are defined by the center of mass coordinates for the atoms involved in the site.

The molecular structure of the ion pair is represented with five CG sites. CG sites , , , and describe the cation, while site describes the anion. The coordinates of each site are defined by the center of mass coordinates for the atoms involved in the site.

Bonded distribution functions calculated from MD simulations of the atomistic (dashed curve) and MS-CG (solid curve) model of the ion pair are presented for (a) the distribution of site bond displacements, (b) the distribution of site valence angles, and (c) the distribution of site dihedral angles.

Bonded distribution functions calculated from MD simulations of the atomistic (dashed curve) and MS-CG (solid curve) model of the ion pair are presented for (a) the distribution of site bond displacements, (b) the distribution of site valence angles, and (c) the distribution of site dihedral angles.

Nonbonded RDFs calculated from atomistic (dashed curve) and CG (soli d curve) MD simulations of the ion pair are presented for the distributions of (a) site pairs, (b) site pairs, (c) site pairs, and (d) site pairs.

Nonbonded RDFs calculated from atomistic (dashed curve) and CG (soli d curve) MD simulations of the ion pair are presented for the distributions of (a) site pairs, (b) site pairs, (c) site pairs, and (d) site pairs.

Short-ranged nonbonded interactions in the MS-CG force field calculated for the ionic liquid system are presented for (a) site pairs, (b) site pairs, (c) site pairs, and (d) site pairs.

Short-ranged nonbonded interactions in the MS-CG force field calculated for the ionic liquid system are presented for (a) site pairs, (b) site pairs, (c) site pairs, and (d) site pairs.

The analytic functional forms (solid curves) approximating the calculated MS-CG bonded interactions (dashed curves) in CG MD simulations of the ion pair are presented for the (a) the bond interaction, (b) the valence angle interaction, and (c) the dihedral angle interaction.

The analytic functional forms (solid curves) approximating the calculated MS-CG bonded interactions (dashed curves) in CG MD simulations of the ion pair are presented for the (a) the bond interaction, (b) the valence angle interaction, and (c) the dihedral angle interaction.

## Tables

Parameters describing calculations using various methods for minimizing the MS-CG residual function to obtain the methanol CG pair force. The total number of parameters, , the magnitude of the MS-CG residual, , and the condition number (both before and after preconditioning) obtained from iteratively minimizing the residual using either the normal matrix (normal) or the nonsymmetric matrix with the block-averaging (BA) approximation are presented for each basis set. The BA result for the cubic spline has been presented for both 980 (b) and 1000 (c) configurations. Poor sampling of the core region in the 99th block introduced statistical error into the BA calculation for the cubic spline basis set resulting in an abnormally high magnitude residual. Every other variational calculation described in the table employed the same 1000 configurations sampled from atomistic MD simulations of the methanol as discussed in the text. In each calculation employing the BA approximation, these 1000 configurations were partitioned into 100 disjoint sets of 10 configurations each, the MS-CG variational calculation was performed with the configurations in the block, and the resulting 100 calculated force curves were averaged.

Parameters describing calculations using various methods for minimizing the MS-CG residual function to obtain the methanol CG pair force. The total number of parameters, , the magnitude of the MS-CG residual, , and the condition number (both before and after preconditioning) obtained from iteratively minimizing the residual using either the normal matrix (normal) or the nonsymmetric matrix with the block-averaging (BA) approximation are presented for each basis set. The BA result for the cubic spline has been presented for both 980 (b) and 1000 (c) configurations. Poor sampling of the core region in the 99th block introduced statistical error into the BA calculation for the cubic spline basis set resulting in an abnormally high magnitude residual. Every other variational calculation described in the table employed the same 1000 configurations sampled from atomistic MD simulations of the methanol as discussed in the text. In each calculation employing the BA approximation, these 1000 configurations were partitioned into 100 disjoint sets of 10 configurations each, the MS-CG variational calculation was performed with the configurations in the block, and the resulting 100 calculated force curves were averaged.

Parameters describing the CG mapping for the ionic liquid system. The coordinates for each site were defined by the center of mass coordinates for the set of atoms involved in the definition of the site. The mass and charge for each CG site were determined by summing the total masses and charges for the atoms involved in each site.

Parameters describing the CG mapping for the ionic liquid system. The coordinates for each site were defined by the center of mass coordinates for the set of atoms involved in the definition of the site. The mass and charge for each CG site were determined by summing the total masses and charges for the atoms involved in each site.

Parameters describing the bonded MS-CG force field for the ionic liquid system. The bonded MS-CG interactions were determined without assuming any functional forms but were then approximately represented by the analytic functions described above in CG MD simulations with the DḺPOLY program (Ref. 57). The analytic functional form for each interaction and the parameters used in fitting the MS-CG interactions are presented.

Parameters describing the bonded MS-CG force field for the ionic liquid system. The bonded MS-CG interactions were determined without assuming any functional forms but were then approximately represented by the analytic functions described above in CG MD simulations with the DḺPOLY program (Ref. 57). The analytic functional form for each interaction and the parameters used in fitting the MS-CG interactions are presented.

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