1. Molecular Dynamics Simulations An Introduction TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA A A A A Pingwen Zhang

2. Molecular Dynamics Definitions, Motivations Force fields Algorithms and computations Analysis of Data

3. Molecular dynamics (MD) is a computer simulation technique: the time evolution of interacting atoms is followed by integrating their equations of motion. We follow the laws of classical mechanics, and most notably Newton's law: Molecular dynamics - Introduction

4.  F = MA exp(-  E/kT) domain quantum chemistry molecular dynamics Monte Carlo mesoscale continuum Length Scale Time Scale 10 -10 M10 -8 M10 -6 M10 -4 M 10 -12 S 10 -8 S 10 -6 S Scale in Simulations

6. Modeling the motion of a complex molecule by solving the wave functions of the various subatomic particles would be accurate… But it would also be very hard to program and take more computing power than anyone has! Why Not Quantum Mechanics?

7. Given an initial set of positions and velocities, the subsequent time evolution is in principle completely determined. Atoms and molecules will ‘move’ in the computer, bumping into each other, vibrating about a mean position (if constrained), or wandering around (if the system is fluid), oscillating in waves in concert with their neighbours, perhaps evaporating away from the system if there is a free surface, and so on, in a way similar to what real atoms and molecules would do. Molecular dynamics - Introduction

8. The computer experiment. In a computer experiment, a model is still provided by theorists, but the calculations are carried out by the machine by following a recipe (the algorithm, implemented in a suitable programming language). In this way, complexity can be introduced (with caution!) and more realistic systems can be investigated, opening a road towards a better understanding of real experiments. Molecular dynamics -Motivation

9. The computer calculates a trajectory of the system 6N-dimensional phase space (3N positions and 3N momenta ). A trajectory obtained by molecular dynamics provides a set of conformations of the molecule, They are accessible without any great expenditure of energy (e.g. breaking bonds) MD also used as an efficient tool for optimisation of structures (simulated annealing).

10. Molecular dynamics - Motivation MD allows to study the dynamics of large macromolecules Dynamical events control processes which affect functional properties of the bio- molecule (e.g. protein folding). Drug design is used in the pharmaceutical industry to test properties of a molecule at the computer without the need to synthesize it.

11. Molecular dynamics – Time Limitations Typical MD simulations are performed on systems containing millions of atoms Simulation times: picoseconds to nanoseconds. A simulation is reliable when the simulation time is much longer than the relaxation time of the quantities we are interested in.

12. Historical Perspective on MD

13. Procedure of MD

14. Initialization Position –X-ray, NMR, simulation or analytical calculation Velocity –The initial velocities are assigned taking them from a Maxwell distribution at a certain temperature T Another possibility is to take the initial positions and velocities to be the final positions and velocities of a previous MD run

15. Choosing the Time Step  t No MD follows the true trajectories for very many time steps – errors always accumulate. The time over which the MD trajectory = true trajectory is called correlation time. No MD truly conserves energy since there are always errors. The goal is to have a constant average E with fluctuations as small as possible Time step  t should be as large as possible to still get accurate trajectories (on the time scale needed) and conserve of energy In general,  t should be ≈0.01 x the fastest behavior of your system (E.g., atoms oscillate about once every 10 -12 s in a solid  MD time steps are ≈ 10 -14 s in simulations of solids

16. Potentials ab initio potential –Quantum calculation, DFT, OFDFT, Tight-binding, etc. Empirical potential –Comes from quantum –Consistent with continuum –Fit the database: elastic moduli, surface energy, etc. Connection with continuum –Constitutive relation

17. The Lennard-Jones Potential The truncated and shifted Lennard-Jones potential The truncated Lennard-Jones potential The Lennard-Jones potential

18. FENE potential FENE stands for: Finitely extensible nonlinear elastic

19. EAM potential Embedded Atom Method works for metallic solids Two contributions nuclear-nuclear interaction embedding an atom to the electron cloud

20. Potential for Covalent Carbon The Stillinger-Weber potential The Tersoff Potential Not only accounts for the contribution of bond lengths, but also for the bond angles

21. Non-Bonded Atoms There are two potential functions we need to be concerned about between non-bonded atoms: van der Waals Potential Electrostatic Potential

22. The van der Waals Potential Atoms with no net electrostatic charge will still tend to attract each other at short distances Atoms tend to repel when they get too close The Constants A and C depend on the atom types, and are derived from experimental data

23. The Electrostatic Potential: Coulomb’s Law Opposite Charges Attract Like Charges Repel The force of the attraction is inversely proportional to the square of the distance

24. Bonded Atoms There are three types of interaction between bonded atoms: Stretching along the bond Bending between bonds Rotating around bonds

25. Bond Length Potentials Both the spring constant and the ideal bond length are dependent on the atoms involved.

26. Bond Angle Potentials The spring constant and the ideal angle are also dependent on the chemical type of the atoms.

27. Torsional Potentials Described by a dihedral angle and coefficient of symmetry (n=1,2,3), around the middle bond.

28. Effects of solvents Implicit models –“Generalized Born” solvent Model coarse-graining the effects of solvent by approximately solving the Poisson equation Explicit models –Explicitly adding the water molecules which are regarded as rigid bodies

29. Integrator: Verlet Algorithm Start with {r(t), v(t)}, integrate it to {r(t+  t), v(t+  t)}: {r(t), v(t)} {r(t+  t), v(t+  t)} The new position at t+  t: Similarly, the old position at t-  t: (1) (2) Add (1) and (2): Thus the velocity at t is: (3) (4)

30. Verlet Scheme Is time reversible Does conserve volume in phase space (Is symplectic) Doe not suffer from energy drift

31. Velocity Verlet scheme Velocity calculated explicitly Possible to control the temperature Stable in long time simulation Most commonly used algorithm

32. Central Simulation box rcrc Periodic Boundary Conditions Minimum Image

33. Ewald Sum: split into two teems, real space sum and k-space sum Periodic Boundary Conditions Ewald Method

34. Saving CPU time Cell listVerlet list

35. Ensembles NVE – micro-canonical ensemble NVT – canonical ensemble NPT – grand-canonical ensemble Temperature control –Berendsen thermostat (velocity rescaling) –Andersen thermostat (velocity resampling) –Nose-Hoover chain Pressure control –Berendsen volume rescaling –Andersen piston

36. MD as Optimization tool Simulated Annealing Most popular global optimization algorithm Start at high T, decrease T in small steps (cooling schedule) Easy to understand & implement Drawback: might be easily trapped in local minima Cooling Schedules:

37. Molecular dynamics – Analyses The simplest way of analyzing the system during (or after) its dynamic motion is looking at it. One can assign a radius to the atoms, represent the atoms as balls having that radius, and have a computer program construct a ‘photograph’ of the system. We may also color the atoms according to its properties (charge, displacement, ‘temperature’…)

38. Molecular dynamics – Analyses We also can measure instantaneous and time averages of various physically important quantities To measure time averages: If the instantaneous values of some property A at time t is then its average is where N T is the number of steps in the trajectory

39. Softwares AMBER CHARMM VASP (DFT) XMD CPMD(DFT) (Car-Parrinello MD)

40. References M. P. Allen, D. J. Tildesley (1989) Computer simulation of liquids. Oxford University Press. Frenkel Daan; Smit, Berend [2001]. Understanding Molecular Simulation : from algorithms to applications. Academic Press. D. C. Rapaport (1996) The Art of Molecular Dynamics Simulation. Tamar Schlick (2002) Molecular Modeling and Simulation. Springer.