Overview of Molecular Dynamics Simulation Theory Dr. Mohamed Shajahan Gulam Razul

Introduction In the 1950’s two landmark pioneering publications: The first by Metropolis et al. N. Metropolis, A.W. Rosenbluth, M. N. Rosenbluth, A.H. Teller and E. Teller, J. Chem. Phys., 21, 1087 (1953).   The second by Alder and Wainwright B. J. Alder and T. E. Wainwright, J. Chem. Phys., 27, 1208 (1957). Indelibly laid the foundation for the Monte Carlo method and molecular dynamics, respectively. The molecular simulation world was forever changed Computer simulations have provided a bridge between the abstract theoretical construct of statistical mechanics to the macroscopic level of experimental measurements of thermodynamics

So we have coordinates in Theory Consider a system of N atoms Positions Momenta So we have coordinates in 6N Dimensional space or phase space

Theory so a point in phase space can be given by and an instantaneous property, Aint Aint() therefore an experimentally observable “macroscopic”property Aobs ,over a long time, tobs

Theory The Ergodic Hypothesis states that the time average in the previous slide can be given by an ensemble average. The ensemble average for a closed system can be regarded as a collection of points, , where in phase space

Theory Where Aobs via the ensemble average is given by Partition function Where Aobs via the ensemble average is given by Now lets replace rens by a weight function, wens()

Theory Therefore in the the canonical ensemble (constant N,V,T) rens is proportional to with the partition function expressed in quasi-classical form given by

Theory Many other connections are also made in different ensembles (the basis of equilibrium statistical mechanics) Connection of statistical mechanics and thermodynamics The connection to a thermodynamic function is made through the Helmholtz free energy

Theory Many other connections are also made in different ensembles (the basis of equilibrium statistical mechanics) Connection of statistical mechanics and thermodynamics The connection to a thermodynamic function is made through the Helmholtz free energy

Theory No let’s turn our attention to the link between statistical mechanics and molecular dynamics Liouville’s theorem: the rate of change of phase space density is given by An alternative formulation can be derived from a phase point traveling in space, the rate of change of phase space density, , in this case is now 0,

Theory Force exerted on a particle i that includes all particles in the system calculated by pairwise interactions

Theory and 6N first order differential equations as opposed to Newton’s 3N second order equations The Newtonian equation can be written in Hamiltonian form, and

Computation There are two principal techniques utilized to solve the equations of motion: the predictor-corrector and the Taylor series methods. All these methods allow one to obtain numerical solutions to differential equations. Which means time had to be discretized, thus the origin of the MD timestep

Computation As an example, a general predictor-corrector algorithm is outlined below: at a time t, use the current positions and current velocities and their time derivatives to predict new positions and velocities at a new time (t + t); at the new positions, forces are evaluated; utilizing these forces, correct the positions, velocities and their derivatives; d) finally, variables of interest are calculated and time averages are performed; the process now iterates by returning to step a).

Computation As an example, a general predictor-corrector algorithm is outlined below: at a time t, use the current positions and current velocities and their time derivatives to predict new positions and velocities at a new time (t + t); at the new positions, forces are evaluated; utilizing these forces, correct the positions, velocities and their derivatives; d) finally, variables of interest are calculated and time averages are performed; the process now iterates by returning to step a).

Computation As an example, a general predictor-corrector algorithm is outlined below: at a time t, use the current positions and current velocities and their time derivatives to predict new positions and velocities at a new time (t + t); at the new positions, forces are evaluated; utilizing these forces, correct the positions, velocities and their derivatives; d) finally, variables of interest are calculated and time averages are performed; the process now iterates by returning to step a).

Computation As an example, a general predictor-corrector algorithm is outlined below: at a time t, use the current positions and current velocities and their time derivatives to predict new positions and velocities at a new time (t + t); at the new positions, forces are evaluated; utilizing these forces, correct the positions, velocities and their derivatives; d) finally, variables of interest are calculated and time averages are performed; the process now iterates by returning to step a).

Simple Verlet algorithm Computation Simple Verlet algorithm

Computation PBC

Computation PBC

Computation PBC

PBC helps model bulk systems, restricting the PBC to 2D or none at all Computation PBC PBC helps model bulk systems, restricting the PBC to 2D or none at all is also possible!

Computation Therefore the simulation “box” shape is important PBC Therefore the simulation “box” shape is important other shapes are supported as well example-dodecahedron

The minimum image convention Computation PBC The minimum image convention

Computation PBC

Computation PBC Usually half the box Length for short range interactions PBC

Computation Short range Long range

Computation Simple Short range Complicated Long range Ewald method Particle mesh ewald

Computation Simple Short range Complicated Long range Ewald method Particle mesh ewald

Simulation

Simulation How?

Simulation How?

Simulation How?

Simulation

Key Properties Temperature Pressure