Molecular Dynamics A brief overview

2 Notes - Websites "A Molecular Dynamics Primer", F. Ercolessi "Scientific computing and visualization", A. Nakano University of Southern California "The Art of Molecular Dynamics Simulation", D. C. Rapaport, CUP, 1997 “Computer simulation of liquids”, M. P. Allen, D. J. Tildesley, OUP, 1990

3 What is molecular dynamics? Solving the classical equations of motion For a system of N (N>>3) particles Which interact through a “given” potential And then apply some “tricks” … Deterministic technique  Monte Carlo

4 What is molecular dynamics? However, errors in trajectories always accumulate: MD is a statistical mechanics method  thermodynamic properties To obtain set of configurations according to statistical ensemble -Microcanonical ensemble (NVE) -Canonical ensemble (NVT) -Isobaric-Isothermal Ensemble (NPT) -Grand canonical ensemble (also number of particles can change, VT) Ergodic hypothesis Also used for the optimization of structures (simulated annealing)

5 Applications of molecular dynamics Properties of liquids Plasma physics Defects in solids Fracture Surface properties Friction Molecular clusters Biomolecules Dynamics of galaxies Formation of stellar clusters

6 Overview I Model -System Hamiltonian -Interaction potentials: bonded and non-bonded interactions -Finite system – infinite system IntegratorSymplecticity? Statistical ensemble Collecting results

7 Overview II Collecting data

8 First MD simulation (1957/1959) weeks

9 First MD simulation using continuous potentials (1960) IBM 704

10 First MD simulation using Lennard-Jones potential (1964)

11 Limitations Use of classical forces  quantum effects -MD only valid if (at 300 K, t > 6 ps) Realism of forces Time and size limitations -Thousands to millions of atoms -Time step t should be as large as possible while conserving total energy -In general, t ≈0.01 x fastest behavior of your system Atoms oscillate about once every s in a solid  t ≈ s -Total time: picoseconds to hundreds of nanoseconds -Simulation only reliable if simulation time is much longer than relaxation time of quantities of interest -Idem for correlation length

12 Initialisation Positions -Random positions -Regular pattern, e.g. fcc lattice -Previous run Velocities -Random velocity or from Maxwell distribution -Previous run -Linked with temperature -No drift condition -Rescale velocities to realize desired temperature

13 Interaction potentials Origins are quantum mechanical Easiest model: Lennard-Jones potential Truncated (but with energy conservation)

14 Interaction potentials Distinguish between long range  short range interactions Distinguish between intermolecular  intramolecular forces

15 Interaction potentials Stretch energy Bending energy

16 Interaction potentials Interactions between charge inhomogeneities Approaches -Point charges -Point multipoles Screened Coulomb interaction

17 Interaction potentials Multi-body interactions e.g. Tersoff and Brenner potentials

18 Infinite systems Periodic boundary conditions Minimum image criterion for short range potentials -At most one among all pairs formed by a particle i in the box and the set of all periodic images of another particle j will interact Central Simulation box rcrc Number of interacting pairs increases enormously

19 Infinite systems Ewald method for Coulomb interaction

20 Integrators How should a good integration scheme look like? -High accuracy (reproduces true trajectories well) -Good stability (conservation of energy) -Time reversible -Robust (allow for large time steps) -Conservation of phase space density (Liouville’s theorem)(symplecticity) Simple Euler method is not time reversible and not symplectic.

21 Integrators Verlet algorithm -Positions -Velocities -Properties Time reversible Symplectic Does not suffer from energy drift But no info on velocity untill the next step is made +

22 Integrators Comparison between Euler and Verlet Test system consists of 7 Lennard-Jones atoms (Ar) Time step is 10 fs Time step is 1 fs

23 Integrators Velocity Verlet -Properties Velocity calculated explicitly Possible to control the temperature Stable Most commonly used algorithm

24 Neighbour lists Complexity of force calculations ~O(N 2 ) But there are only interactions, often n << N

25 Neighbour lists Verlet lists -Idea: introduce a list, where particles are included which are located within interaction sphere -Also introduce a “reservoir”, where particles outside R c are stored, so that unknown particles cannot become neighbors in next steps -Do an update of the list every n steps Either statically with fixed n Or dynamically with an update criterion -There exists an optimal R skin

26 Neighbour lists Linked lists method ~ O(N) Interacts with atoms in 26 neighbour cells

27 Measuring Kinetic energy + Potential energy = Total energy Temperature per degree of freedom The caloric curve E(T) Mean square displacement !Periodic boundary conditions Diffusion coefficient Correlation functions

28 MD (and MC) as optimization tool Simulated annealing -Start at high T, decrease T in small steps (cooling schedule) -Easy to understand & implement -Drawback: might be easily trapped in local minima Cooling schemes

29 Parallel strategies Atom decomposition -Atoms are distributed among processors -All coordinates are exchanged before computing forces -OK for long range interactions -Easy to implement Force decomposition -Each processor calculates the interactions for certain atom pairs Spatial decomposition -Subdivides space and assigns each processor a particular subregion -Atoms are allowed to move from one processor to its neighbours -More complex to implement (similar to linked lists)

30 Analyzing the sequential code md.c and lmd.c "Scientific computing and visualization", A. Nakano University of Southern California Code description, details at -“Basic molecular dynamics algorithms” -“Linked-list cell MD algorithm”

31 Guidelines for final report Test energy conservation as function of t Compare speed of md.c and lmd.c Add possibility to save configuration, and to start from a previous run. Allow temperature rescaling and equilibration. Study the behavior of the caloric curve E(T) by means of constant energy runs at a fixed density, starting from a crystalline arrangement (= , T max =1.5). Insert calculations of the total linear and angular momenta. Check their conservation. Insert calculation of the mean square displacement. Remove periodic boundary conditions and study a free cluster.