Coarse-grained ͑CG͒ models provide a computationally efficient method for rapidly investigating the long time-and length-scale processes that play a critical role in many important biological and soft matter processes. Recently, Izvekov and Voth introduced a new multiscale coarse-graining ͑MS-CG͒ method ͓J. Phys. Chem. B 109, 2469 ͑2005͒; J. Chem. Phys. 123, 134105 ͑2005͔͒ for determining the effective interactions between CG sites using information from simulations of atomically detailed models. The present work develops a formal statistical mechanical framework for the MS-CG method and demonstrates that the variational principle underlying the method may, in principle, be employed to determine the many-body potential of mean force ͑PMF͒ that governs the equilibrium distribution of positions of the CG sites for the MS-CG models. A CG model that employs such a PMF as a "potential energy function" will generate an equilibrium probability distribution of CG sites that is consistent with the atomically detailed model from which the PMF is derived. Consequently, the MS-CG method provides a formal multiscale bridge rigorously connecting the equilibrium ensembles generated with atomistic and CG models. The variational principle also suggests a class of practical algorithms for calculating approximations to this many-body PMF that are optimal. These algorithms use computer simulation data from the atomically detailed model. Finally, important generalizations of the MS-CG method are introduced for treating systems with rigid intramolecular constraints and for developing CG models whose equilibrium momentum distribution is consistent with that of an atomically detailed model.

The multiscale coarse-graining ͑MS-CG͒ method ͓S. Izvekov and G. A. Voth, J. Phys. Chem. B 109, 2469 ͑2005͒; J. Chem. Phys. 123, 134105 ͑2005͔͒ 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 ͑PMF͒ 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 EMIM + / NO 3 − ionic liquid are provided to illustrate the method.

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(2005)]. If no approximations are made, the MS-CG method yields a many-body multi-dimensional potential of mean force describing the interactions between CG sites. However, numerical applications of the MS-CG method typically employ a set of pair potentials to describe non-bonded interactions. The analogy between coarse-graining and the inverse problem of liquid state theory clarifies the general significance of three-particle correlations for the development of such CG pair potentials. It is demonstrated that the MS-CG methodology incorporates critical three-body correlation effects and that, for isotropic homogeneous systems evolving under a central pair potential, the MS-CG equations are a discretized representation of the well-known Yvon-Born-Green equation. Numerical calculations validate the theory and illustrate the role of these structural correlations in the MS-CG method.

We show that arbitrary phase space vector fields can be used to generate phase functions whose ensemble averages give the thermodynamic temperature. We describe conditions for the validity of these functions in periodic boundary systems and the molecular dynamics (MD) ensemble, and test them with a short-ranged potential MD simulation.