Molecular dynamics simulations in the grand canonical ensemble: Formulation of a bias potential for umbrella sampling

Abstract

An extended Hamiltonian technique for performing grand canonical ensemble molecular dynamics simulations has been reformulated to include umbrella sampling, thus improving the efficiency of particle creation and annihilation processes. This was accomplished through incorporation of a bias potential in the Hamiltonian that modifies the free energy contour between integer particle number states. The extended Hamiltonian includes a continuous particle number variable that is the sum of the integer particle number and a coupling parameter, the latter being used in a scaling function for a fractional molecule’s interactions with the rest of the system. Equations of motion for the coupling parameter, derived from the Hamiltonian, integrate to yield density fluctuations at constant chemical potential. This new method may be adapted to a wide range of potential scaling functions. The technique was applied to calculations of extended simple point charge water density versus chemical potential, using both linear and nonlinear scaling. For each scaling function, a bias potential was constructed using a thermodynamic integration technique. Grand canonical ensemble simulations then yielded results in agreement with independent calculations.