  In QM/MM indirect free energy simulation, QM/MM corrections can be obtained
from integration of partial derivatives of alchemical Hamiltonians or from
perturbation-based estimators including free energy perturbation (FEP) and
acceptance ratio methods. With FEP or exponential averaging, researchers tend
to only sample MM states and calculate single point energy to get the free
energy estimates. In this case the sample size hysteresis arises and the
convergence is determined by bias elimination rather than variance
minimization. Various criteria are proposed to evaluate the convergence issue
and numerical studies are reported. It has been found that criteria including
variance of distribution, effective sample size, information entropies and so
on can be used and they are variance-of-distribution-dependent. However, no
theoretical interpretation is presented. In this paper we present theoretical
interpretations to dig the underlying statistical nature behind the problem.
The convergence criteria are proven to be related with variance of distribution
in Gaussian approximated Exponential averaging. Further, we prove that these
estimators are nonlinearly dependent on the variance of the free energy
estimate. As these estimators are often orders of magnitude overestimated, the
variance of the FEP estimate is orders of magnitude underestimated. Hence,
computing this statistical uncertainty is meaningless. In numerical calculation
from timeseries data the effective sample size is bounded by 1 and N and thus
the variance of the free energy estimate is proven to be bounded by 0 and 1
(kBT)2 for EXP and 0 and 2 (kBT)2 for BAR, which indicates an inevitable
underestimation. Specifically, the upper bounds for these estimators are
sample-size dependent. The effective sample size is proven to be a function of
the overlap scalar, from which the range of the overlap scalar can also be
derived.
