  Coarse-grained (CG) models are often parametrized to reproduce
one-dimensional structural correlation functions of an atomically-detailed
model along the degrees of freedom governing each interaction potential. While
cross correlations between these degrees of freedom inform the optimal set of
interaction parameters, the correlations generated from the higher-resolution
simulations are often too complex to act as an accurate proxy for the CG
correlations. Instead, the most popular methods determine the interaction
parameters iteratively, while assuming that individual interactions are
uncorrelated. While these iterative methods have been validated for a wide
range of systems, they also have disadvantages when parametrizing models for
multi-component systems or when refining previously established models to
better reproduce particular structural features. In this work, we propose two
distinct approaches for the direct (i.e., non-iterative) parametrization of a
CG model by adjusting the high-resolution cross correlations of an atomistic
model in order to more accurately reflect correlations that will be generated
by the resulting CG model. The derived models more accurately describe the
low-order structural features of the underlying AA model, while necessarily
generating inherently distinct cross correlations compared with the
atomically-detailed reference model. We demonstrate the proposed methods for a
one-site-per-molecule representation of liquid water, where pairwise
interactions are incapable of reproducing the true tetrahedral solvation
structure. We then investigate the precise role that distinct cross-correlation
features play in determining the correct pair correlation functions, evaluating
the importance of the placement of correlation features as well as the balance
between features appearing in different solvation shells.
