  In the molecular dynamics calculations for the free energy of ions and ionic
molecules, we often encounter wet charged molecular systems where electrical
neutrality condition is broken. This causes a problem in the evaluation of
electrostatic interaction under periodic boundary condition. A standard remedy
for the problem is to consider a hypothetical homogeneous background charge
density to neutralize the total system. Here, we present a new expression for
the evaluation of electrostatic interactions for the system including the
background charge by fast multipole method (FMM). Further, an efficient scheme
to evaluate solute-solvent interaction energy by FMM has been developed to
reduce the computation of far-field part. We have calculated hydration free
energy of ions, Mg$^{2+}$, Na$^{+}$, and Cl$^{-}$ dissolved in neutral solvent
using the new expression. The calculated free energy showed a good agreement
with the result using well-established particle mesh Ewald method,
demonstrating the validity of the present expression in the framework of FMM.
An advantage of the present scheme is in an efficient free energy calculation
of a large-scale charged systems (particularly over million particles) based on
highly parallel computations.
