Bill Smith Computational Science and Engineering STFC Daresbury Laboratory Warrington WA4 4AD

1. MSc in High Performance ComputingComputational Chemistry ModuleIntroduction to Molecular Dynamics Bill Smith Computational Science and Engineering STFC Daresbury Laboratory Warrington WA4 4AD

2. What is Molecular Dynamics? • MD is the solution of the classical equations of motion for atoms and molecules to obtain the time evolution of the system. • Applied to many-particle systems - a general analytical solution not possible. Must resort to numerical methods and computers • Classical mechanics only - fully fledged many-particle time-dependent quantum method not yet available • Maxwell-Boltzmann averaging process for thermodynamic properties (time averaging).

3. rcut Example: Simulation of Argon Pair Potential: Lagrangian:

4. Lennard -Jones Potential V(r) s r e rcut

5. Equations of Motion Lagrange Newton Lennard- Jones

6. Periodic Boundary Conditions

7. Minimum Image Convention Use rij’ not rij L xij = xij - L* Nint(xij/L) rcut j’ j i Nint(a)=nearest integer to a rcut < L/2

8. r’ (t+Dt) r (t+Dt) v (t)Dt Net displacement r (t) f(t)Dt2/m [r (t+Dt), v(t+Dt), f(t+Dt)] [r (t), v(t), f(t)] Integration Algorithms: Essential Idea Time step Dt chosento balance efficiency and accuracy of energy conservation

9. Integration Algorithms (i) Verlet algorithm

10. Integration Algorithms (ii) Leapfrog Verlet Algorithm

11. Integration Algorithms Velocity Verlet Algorithm As Applied

12. Verlet Algorithm: Derivation

13. Initialise Forces Motion Properties Summarise Key Stages in MD Simulation • Set up initial system • Calculate atomic forces • Calculate atomic motion • Calculate physical properties • Repeat ! • Produce final summary

14. MD – Further Comments • Constraints and Shake • If certain motions are considered unimportant, constrained MD can be more efficient e.g. SHAKE algorithm - bond length constraints • Rigid bodies can be used e.g. Eulers methods and quaternion algorithms • Statistical Mechanics • The prime purpose of MD is to sample the phase space of the statistical mechanics ensemble. • Most physical properties are obtained as averages of some sort. • Structural properties obtained from spatial correlation functions e.g. radial distribution function. • Time dependent properties (transport coefficients) obtained via temporal correlation functions e.g. velocity autocorrelation function.

15. Thermodynamic Properties Kinetic Energy: Temperature: System Properties: Static (1)

16. Configuration Energy: Pressure: Specific Heat System Properties: Static (2)

17. Structural Properties Pair correlation (Radial Distribution Function): Structure factor: Note: S(k) available from x-ray diffraction System Properties: Static (3)

18. Radial Distribution Function R R

19. g(r) 1.0 separation (r) Typical RDF

20. All above calculable by molecular dynamics or Monte Carlo simulation. But NOT Free Energy: where is the Partition Function. But can calculate a free energy difference! Free Energies?

21. System Properties: Dynamic (1) • The bulk of these are in the form of Correlation Functions :

22. Mean squared displacement (Einstein relation) Velocity Autocorrelation (Green-Kubo relation) System Properties: Dynamic (2)

23. Liquid <|ri(t)-ri(0)|2> (A2) Solid time (ps) Typical MSDs

24. <vi(t).vi(0)> 0.0 t (ps) Typical VAF 1.0

25. Recommended Textbooks • The Art of Molecular Dynamics Simulation, D.C. Rapaport, Camb. Univ. Press (2004) • Understanding Molecular Simulation, D. Frenkel and B. Smit, Academic Press (2002). • Computer Simulation of Liquids, M.P. Allen and D.J. Tildesley, Oxford (1989). • Theory of Simple Liquids, J.-P. Hansen and I.R. McDonald, Academic Press (1986). • Classical Mechanics, H. Goldstein, Addison Wesley (1980)

26. The DL_POLY Package A General Purpose Molecular Dynamics Simulation Package

27. DL_POLY Background • General purpose parallel MD code to meet needs of CCP5 (academic collaboration) • Authors W. Smith, T.R. Forester & I. Todorov • Over 3000 licences taken out since 1995 • Available free of charge (under licence) to University researchers.

28. DL_POLY Versions • DL_POLY_2 • Replicated Data, up to 30,000 atoms • Full force field and molecular description • DL_POLY_3 • Domain Decomposition, up to 10,000,000 atoms • Full force field but no rigid body description. • I/O files cross-compatible (mostly) • DL_POLY_4 • New code under development • Dynamic load balancing

29. Rigid molecules Point ions and atoms Flexibly linked rigid molecules Polarisable ions (core+ shell) Rigid bond linked rigid molecules Flexible molecules Rigid bonds Supported Molecular Entities

30. M4 P4 M0 P0 M5 P5 M1 P1 M6 P6 M2 P2 M7 P7 M3 P3 DL_POLY is for Distributed Parallel Machines

31. Atomic systems Ionic systems Polarisable ionics Molecular liquids Molecular ionics Metals Biopolymers and macromolecules Membranes Aqueous solutions Synthetic polymers Polymer electrolytes DL_POLY: Target Simulations

32. DL_POLY Force Field • Intermolecular forces • All common van der Waals potentials • Finnis_Sinclair and EAM metal (many-body) potential (Cu3Au) • Tersoff potential (2&3-body, local density sensitive, SiC) • 3-body angle forces (SiO2) • 4-body inversion forces (BO3) • Intramolecular forces • bonds, angle, dihedrals, improper dihedrals, inversions • tethers, frozen particles • Coulombic forces • Ewald* & SPME (3D), HK Ewald* (2D), Adiabatic shell model, Neutral groups*, Bare Coulombic, Shifted Coulombic, Reaction field • Externally applied field • Electric, magnetic and gravitational fields, continuous and oscillating shear fields, containing sphere field, repulsive wall field * Not in DL_POLY_3

33. Algorithms Verlet leapfrog Velocity Verlet RD-SHAKE Euler-Quaternion* No_Squish* QSHAKE* [Plus combinations] *Not in DL_POLY_3 Ensembles NVE Berendsen NVT Hoover NVT Evans NVT Berendsen NPT Hoover NPT Berendsen NT Hoover NT PMF Algorithms and Ensembles

34. The DL_POLY Java GUI

35. The DL_POLY Website http://www.ccp5.ac.uk/DL_POLY/

36. The End