Calculating the pressure in simulations using periodic boundary conditions

Abstract

Because it is not immediately clear how to write down a proper Hamiltonian for a system in periodic boundary conditions, particularly with Coulombic interactions, we consider a large, finite array of copies of a basic simulation cell containingN particles with some interaction between them. We also putN independent copy particles in each of the copy cells of the array and write down a constrained Lagrangian for the whole system. Constraints on the velocities of the particles of the whole array together with an appropriate initial condition implement the periodic structure in the cells of the array of copies. We derive a Hamiltonian for the whole system with constraints and then derive the equations of motion and a virial expression for the pressure tensor in terms of the forces on the system. In the limit as the array of cell copies becomes large, the equations of motion become the standard ones used in periodic-boundaryconditions simulations. The method also provides an unequivocal algorithm for the pressure in this limit in terms of a virial expression. Particular attention is paid to the case of Coulombic interactions.