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A Monte Carlo Resampling Approach for the Calculation of Hybrid Classical and Quantum Free Energies
Hybrid
free energy methods allow estimation of free energy differences
at the quantum mechanics (QM) level with high efficiency by performing
sampling at the classical mechanics (MM) level. Various approaches
to allow the calculation of QM corrections to classical free energies
have been proposed. The single step free energy perturbation approach
starts with a classically generated ensemble, a subset of structures
of which are postprocessed to obtain QM energies for use with the
Zwanzig equation. This gives an estimate of the free energy difference
associated with the change from an MM to a QM Hamiltonian. Owing to
the poor numerical properties of the Zwanzig equation, however, recent
developments have produced alternative methods which aim to provide
access to the properties of the true QM ensemble. Here we propose
an approach based on the resampling of MM structural ensembles and
application of a Monte Carlo acceptance test which in principle, can
generate the exact QM ensemble or intermediate ensembles between the
MM and QM states. We carry out a detailed comparison against the Zwanzig
equation and recently proposed non-Boltzmann methods. As a test system
we use a set of small molecule hydration free energies for which hybrid
free energy calculations are performed at the semiempirical Density
Functional Tight Binding level. Equivalent ensembles at this level
of theory have also been generated allowing the reverse QM to MM perturbations
to be performed along with a detailed analysis of the results. Additionally,
a previously published nucleotide base pair data set simulated at
the QM level using <i>ab initio</i> molecular dynamics is
also considered. We provide a strong rationale for the use of the
Monte Carlo Resampling and non-Boltzmann approaches by showing that
configuration space overlaps can be estimated which provide useful
diagnostic information regarding the accuracy of these hybrid approaches