Introduction into Molecular Dynamics Ralf Schneider, Abha Rai, Amit Rai Sharma Outline: 1. Basics 2. Potentials 3. History 4. Numerics 5. Analysis of MD runs (6. Physics extensions= 7. Numerical extensions 8. Summary

1. Molecular Dynamics Solve Newton’s equation for a molecular system: Or equivalently solve the classical Hamiltonian equation:

1. Molecular Dynamics method deterministic method: state of the system at any future time can be predicted from its current state MD cycle for one step: 1) force acting on each atom is assumed to be constant during the time interval 2) forces on the atoms are computed and combined with the current positions and velocities to generate new positions and velocities a short time ahead

1. Molecular Dynamics method K. Nordlund U. Helsinki

1. Molecular Dynamics method K. Nordlund, U. Helsinki

1. Molecular Dynamics method K. Nordlund U. Helsinki

1. Molecular Dynamics method RuBisCO protein simulations Paul Crozier, Sandia Important for converting CO2 to organic forms of carbon and in the photosynthetic process. Even though the pocket is closed, a CO2 molecule escapes, which was a surprise.

1. Molecular Dynamics method H. Wang U. California, Santa Cruz ATP Synthase: within the mitochondria of a cell a rotary engine uses the potential difference across the bilipid layer to power a chemical transformation of ADP into ATP

1. Motivation: Why atomistic MD simulations? MD simulations provide a molecular level picture of structure and dynamics (biological systems!)  property/structure relationships Experiments often do not provide the molecular level information available from simulations Simulators and experimentalists can have a synergistic relationship, leading to new insights into materials properties

1. Motivation: Why atomistic MD simulations? MD simulations allow prediction of properties for Novel materials which have not been synthesized Existing materials whose properties are difficult to measure or poorly understood Model validation

1. Timescales Bond vibrations - 1 fs Collective vibrations - 1 ps Conformational transitions - ps or longer Enzyme catalysis - microsecond/millisecond Ligand Binding - micro/millisecond Protein Folding - millisecond/second Molecular dynamics: Integration timestep - 1 femtosecond Set by fastest varying force. Accessible timescale about 10 nanoseconds.

1. MD dynamics

2. Molecular Dynamics: Potential We need to know The motion of the atoms in a molecule, x(t) and therefore, the potential energy, V(x)

2. Molecular Dynamics: Potentials How do we describe the potential energy V(x) for a molecule? Potential Energy includes terms for Bond stretching Angle Bending Torsional rotation Improper dihedrals

2. Molecular Dynamics: Potentials Potential energy includes terms for (contd.) Electrostatic Interactions van der Waals

2. Molecular Interaction Types – Non-bonded Energy Terms Lennard-Jones Energy. Coloumb Energy. Coloumb law (rough detail) All pairs of atoms LJ eve for the noncharge interactions.

2. Molecular Interaction Types – Bonded Energy Terms Bond energy: Bond Angle Energy:

2. Molecular Interaction Types – Bonded Energy Terms Improper Dihedral Angle Energy: Dihedral Angle Energy: Emphasis the potential as a function of the coordinates.

2. Scaling Scaling by model parameters size s energy e mass m taken from Dr. D. A. Kofke’s lectures on Molecular Simulation, SUNY Buffalo http://www.eng.buffalo.edu/~kofke/ce530/index.html

2. L-J: dimensionless form Dimensions and units - scaling Lennard-Jones potential in dimensionless form Dimensionless properties must also be parameter independent convenient to report properties in this form, e.g. P*(r*) select model values to get actual values of properties Equivalent to selecting unit value for parameters taken from Dr. D. A. Kofke’s lectures on Molecular Simulation, SUNY Buffalo http://www.eng.buffalo.edu/~kofke/ce530/index.html

3. Historical Perspective on MD

3. First molecular dynamics simulation (1957/59) Hard disks and spheres (calculation of collision times) IBM-704: solid phase liquid phase liquid-vapour-phase N=32  6.5x105 coll.  4 days N=500  107 coll.  2.3 years N=32: 7000 collisions / h N=500: 500 collisions / h Production run ~20000 steps

3. First MD simulation using continuous potentials (1964) CDC-3600 VACF RDF MSD 864 particles Time / Step ~ 45s Production run ~20000 steps  10 days ! (standard PC [Pentium 1.2 GHz]: ½ hour)

3. MD – development (aus T. Schlick, „Molecular Modelling and Simulation“, 2002)

4. Verlet algorithm Let then Starting from and all subsequent positions are determined For the kinetic energy we need the velocities Note: the velcoities are one step behind. Therefore: Specify positions and Compute the forces at timestep n: 3. Compute the positions at timestep (n+1) as in (1.1): 4. Compute velocities at timestep n as in (1.2); then increment n and goto 2.

4. A widely-used algorithm: Leap-frog Verlet Using accelerations of the current time step, compute the velocities at half-time step: v(t+Dt/2) = v(t – Dt/2) + a(t)Dt v t-Dt/2 t t+Dt/2 t+Dt t+3Dt/2 t+2Dt

4. A widely-used algorithm: Leap-frog Verlet Using accelerations of the current time step, compute the velocities at half-time step: v(t+Dt/2) = v(t – Dt/2) + a(t)Dt Then determine positions at the next time step: X(t+Dt) = X(t) + v(t + Dt/2)Dt v X t-Dt/2 t t+Dt/2 t+Dt t+3Dt/2 t+2Dt

4. A widely-used algorithm: Leap-frog Verlet Using accelerations of the current time step, compute the velocities at half-time step: v(t+Dt/2) = v(t – Dt/2) + a(t)Dt Then determine positions at the next time step: X(t+Dt) = X(t) + v(t + Dt/2)Dt v v X t-Dt/2 t t+Dt/2 t+Dt t+3Dt/2 t+2Dt

4. Verlet algorithm- velocity form

4. Advantages Positions and velocities are now in step => kinetic and potential energies are in step Numerical stability is enhanced Eq (1.2) gives velocity as difference of 2 r’s of the same order of magnitude => round-off errors important for long runs With reasonable h, Verlet’s algorithm conserves energy

4. Beeman algorithm Better velocities, better energy conservation More expensive to calculate

4. Predictor-corrector algorithms 1. Predictor. From the positions and their time derivatives up to a certain order q, all known at time t, one ``predicts'' the same quantities at time by means of a Taylor expansion. Among these quantities are, of course, accelerations . 2. Force evaluation. The force is computed taking the gradient of the potential at the predicted positions. The resulting acceleration will be in general different from the ``predicted acceleration''. The difference between the two constitutes an ``error signal''. 3. Corrector. This error signal is used to ``correct'' positions and their derivatives. All the corrections are proportional to the error signal, the coefficient of proportionality being a ``magic number'' determined to maximize the stability of the algorithm. Fifth-order Gear (requires more calculational effort and memory than Verlet, but needs only one calculation of the force per time step, wheras Verlet needs two!

4. Evaluate integration methods Fast, minimal memory, easy to program Calculation of force is time consuming Conservation of energy and momentum Time-reversible Long time step can be used

4. Choosing the time step Too small: covering small conformation space Too large: instability Suggested time steps Translation, 10 fs Flexible molecules and rigid bonds, 2fs Flexible molecules and bonds, 1fs

4. How do you run a MD simulation? Get the initial configuration Assign initial velocities At thermal equilibrium, the expected value of the kinetic energy of the system at temperature T is: This can be obtained by assigning the velocity components vi from a random Gaussian distribution with mean 0 and standard deviation (kBT/mi):

4. Periodic Boundary Conditions infinite system with small number of particles remove surface effects shaded box represents the system we are simulating, while the surrounding boxes are exact copies in every detail whenever an atom leaves the simulation cell, it is replaced by another with exactly the same velocity, entering from the opposite cell face (number of atoms in the cell is conserved) rcut is the cutoff radius when calculating the force between two atoms

4. Minimum Image Bulk system modeled via periodic boundary condition not feasible to include interactions with all images must truncate potential at half the box length (at most) to have all separations treated consistently Contributions from distant separations may be important These two are same distance from central atom, yet: Black atom interacts blue atom does not Only interactions considered

4. Potential cut-offs Bonded interactions: local, therefore O(N), where N is the number of atoms in the molecule considered) Non-bonded interactions: involve all pairs of Atoms, therefore O(N2) Reducing the computing time: use of cut-off in UNB The cutoff distance may be no greater than ½ L (L= box length)

4. Potential truncation common approach: cut-off the at a fixed value Rcut problem: discontinuity in energy and force possibility of large errors

4. Speed-up Tamar Schlick, “Molecular Modeling and Simulation”, Springer

4. Cutoff schemes for faster energy computation wij : weights (0< wij <1). Can be used to exclude bonded terms, or to scale some interactions (usually 1-4) S(r) : cut-off function. Three types: 1) Truncation: b

4. Cutoff schemes for faster energy computation 2. Switching a b with 3. Shifting or b

4. Neighbor lists Verlet requires O(N) operations Force needs O(N2) operations at each step Most of these are outside range and hence zero Time reduced by counting only those within range listed in a table (needs to be updated) - Verlet (1967) suggested keeping a list of the near neighbors for a particular molecule, which is updated periodically - Between updates of the list, the program does not check through all the molecules, just those on the list, to calculate distances and minimum images, check distances against cutoff, etc.

4. Simulating at constant T: the Berendsen scheme Bath supplies or removes heat from the system as appropriate Exponentially scale the velocities at each time step by the factor : where  determines how strong the bath influences the system system Heat bath T: “kinetic” temperature Berendsen et al. Molecular dynamics with coupling to an external bath. J. Chem. Phys. 81:3684 (1984)

4. Simulating at constant P: Berendsen scheme Couple the system to a pressure bath Exponentially scale the volume of the simulation box at each time step by a factor : where  : isothermal compressibility P : coupling constant system pressure bath where u : volume xi: position of particle i Fi : force on particle i Berendsen et al. Molecular dynamics with coupling to an external bath. J. Chem. Phys. 81:3684 (1984)

4. MD scheme

5. Analysis of MD Configurations Averages Fluctuations Time Correlations

5. Time averages and ensemble averages macroscopic numbers of atoms or molecules (of the order of 1023, Avogadro's number is 6.02214199 × 1023 ): impossible to handle for MD statistical mechanics (Boltzmann, Gibbs): a single system evolving in time is replaced by a large number of replications of the same system that are considered simultaneously time average is replaced by an ensemble average:

5. Ergodic hypothesis Classical statistical mechanics integrates over all of phase space {r,p}. The ergodic hypothesis assumes that for sufficiently long time the phase trajectory of a closed system passes arbitrarily close to every point in phase space. Thus the two averages are equal:

5. Statistical Mechanics Extracting properties from simulations static properties such as structure, energy, and pressure are obtained from pair (radial) distribution functions DME-water and DMP-water solutions

5. Pair correlation function g(r)dr is the probability of finding a particle in volume d3r around r given one at r =0 For isotropic system g(r) depends on r only g(r) -> 0 as r -> 0 i.e. atoms are inpenetrable g(r) tends to 1 as r goes to infinity 15

5. Pair correlation function

5. Outputs

5. Equilibration of energy

5. Time variation of energies kinetic energies potential energies

5. Time variation of pressure Equilibration of pressure with time

5. Statistical Mechanics Extracting properties from simulations dynamic and transport properties are obtained from time correlation functions ¥ ó · · ô T = dt ô < A (t) A (0)> õÿ = lim <[A(t) - A(0)]2> /2t t ® ¥ Polybutadiene, 353 K Mw = 1600

5. Outputs - Hydrogen in perfect crystal graphite 150K 900K

5. Outputs two diffusion channels no diffusion across graphene layers (150K – 900K) Lévy flights?

Non-Arrhenius temperature dependence 5. Outputs Non-Arrhenius temperature dependence

7. Bottlenecks in Molecular Dynamics Long-range electrostatic interactions O(N2): fast electrostatics algorithms (each method still needs fine-tuning for each system!) Ewald summation O(N 3/2 ) Ewald, 1921 Fast Multipole Method O(N) Greengard, 1987 Particle Mesh Ewald O(N log N) Darden, 1993 Multi-grid summation (dense mat-vec as a sum of sparse mat-vec) Brandt et al., 1990 Skeel et al., 2002 Izaguirre et al., 2003 Intrinsic different timescales, very small time step needed: multiple-time step methods

7. Ewald Sum Method

7. Ewald Sum Method

7. Ewald Sum Method additional corrections: arises from a gaussian acting on its own site (self-energy correction) or from a surface in vacuum

7. Particle Mesh Ewald Similar to Ewald method except that it uses FFT P3ME method has a very similar spirit with PME Assigning charges onto grids Use Fast Fourier Transform to speed up the k-space evaluation Differentiation to determine forces on the grids Interpolating the forces on the grid back to particles Calculating the real-space potential as normal Ewald

7. Fast Multipole Method Represent charge distributions in a hierarchically structured multipole expansion Translate distant multipoles into local electric field Particles interact with local fields to count for the interactions from distant charges Short-range interactions are evaluated pairwise directly CPU: O(N)

7. Multigrid

7. Multiple time step dynamics Reversible reference system propagation algorithm (r-RESPA) Forces within a system classified into a number of groups according to how rapidly the force changes Each group has its own time step, while maintaining accuracy and numerical stability

7. Multiple Time Step Algorithm Reversible Reference System Propagator Algorithm (r-RESPA) Tuckerman, Berne, Martyna, 1992 The Liouville Operator: {r(t+Dt), v(t+Dt)} {r(t), v(t)} The Liouville Propagator and the state of system is given by: Trotter expansion of the Liouville Propagator: Reference System Propagator: dt Correction Propagator: Dt= n dt

7. RESPA for Biosystems 5-stage r-RESPA decomposition for biological systems Zhou & Berne, 1997 The Liouville Operator decomposition for biosystems: Reference: Correction: time step 0.50 fs 1.0 fs 2.0 fs 4.0 fs 8.0 fs

7. FMM/RESPA Performance

7. P3ME/RESPA Performance

7. P3ME/RESPA Performance

8. MD as a tool MD is a powerful tool with different levels of sophistication in terms of physics and numerics EVERYTHING DEPENDS ON THE POTENTIAL! Time-step limitations require combinations with other methods, e.g. Kinetic Monte Carlo

Multi-scales sputtered and backscattered species and fluxes Max-Planck-Institut für Plasmaphysik, EURATOM Association Multi-scales sputtered and backscattered species and fluxes Plasma-wall interaction Molecular dynamics Binary collision approximation Kinetic Monte Carlo Kinetic model Fluid model impinging particle and energy fluxes

Max-Planck-Institut für Plasmaphysik, EURATOM Association From atoms to W7-X

Max-Planck-Institut für Plasmaphysik, EURATOM Association Diffusion in graphite Carbon deposition in divertor regions of JET and ASDEX UPGRADE Major topics: tritium codeposition chemical erosion JET Paul Coad (JET) ASDEX UPGRADE Achim von Keudell (IPP, Garching) V. Rohde (IPP, Garching)

Real structure of the material needs to be included Diffusion in graphite Max-Planck-Institut für Plasmaphysik, EURATOM Association Real structure of the material needs to be included Internal Structure of Graphite Granule sizes ~ microns Void sizes ~ 0.1 microns Crystallite sizes ~ 50-100 Ångstroms Micro-void sizes ~ 5-10 Ångstroms Multi-scale problem in space (1cm to Ångstroms) and time (pico-seconds to seconds)

5. Outputs - Hydrogen in perfect crystal graphite 150K 900K

5. Outputs two diffusion channels no diffusion across graphene layers (150K – 900K) Lévy flights?

Non-Arrhenius temperature dependence 5. Outputs Non-Arrhenius temperature dependence

8. Multi-scale approach Macroscales Mesoscales Microscales KMC and Monte Carlo Diffusion (MCD) Mesoscales Kinetic Monte Carlo (KMC) Microscales Molecular Dynamics (MD)

Max-Planck-Institut für Plasmaphysik, EURATOM Association Kinetic Monte Carlo 0 = jump attempt frequency (s-1) Em = migration energy (eV) T = trapped species temperature (K) Assumptions: Poisson process (assigns real time to the jumps) jumps are not correlated

Diffusion coefficients without knowledge of structure are meaningless Max-Planck-Institut für Plasmaphysik, EURATOM Association Multi-scale results standard graphites highly saturated graphite Large variation in observed diffusion coefficients Diffusion in voids dominates Strong dependence on void sizes and not void fraction Saturated H: 0~105s-1 and step sizes ~1Å (QM?) Diffusion coefficients without knowledge of structure are meaningless

Max-Planck-Institut für Plasmaphysik, EURATOM Association Effect of voids A: 10 % voids B: 20 % voids C: 20 % voids Larger voids Longer jumps Higher diffusion

Max-Planck-Institut für Plasmaphysik, EURATOM Association Downgrading TRIM, TRIDYN: much faster than MD (simplified physics: binary collision approximation) very good match of physical sputtering dynamical changes of surface composition

Max-Planck-Institut für Plasmaphysik, EURATOM Association 2D-TRIDYN

Max-Planck-Institut für Plasmaphysik, EURATOM Association 2D-TRIDYN

Max-Planck-Institut für Plasmaphysik, EURATOM Association Extension of TRIDYN Bombardment of tungsten with carbon (6 keV) steepening of surface structures (Ivan Biyzukov, IPP Garching)

Max-Planck-Institut für Plasmaphysik, EURATOM Association Extension of TRIDYN

Max-Planck-Institut für Plasmaphysik, EURATOM Association Extension of TRIDYN

Max-Planck-Institut für Plasmaphysik, EURATOM Association Extension of TRIDYN