1. 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

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

3. 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

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

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

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

7. 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.

8. 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

9. 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

10. 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

11. 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.

12. 1. MD dynamics

13. 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)

14. 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

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

16. 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.

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

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

19. 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

20. 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

21. 3. Historical Perspective on MD

22. 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  coll.  2.3 years
N=32: collisions / h
N=500: collisions / h
Production run ~20000 steps

23. 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)

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

25. 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.

26. 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

27. 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

28. 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

29. 4. Verlet algorithm- velocity form

30. 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

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

32. 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!

33. 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

34. 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

35. 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):

36. 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

37. 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

38. 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)

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

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

41. 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

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

43. 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.

44. 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)

45. 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)

46. 4. MD scheme

47. 5. Analysis of MD Configurations Averages Fluctuations
Time Correlations

48. 5. Time averages and ensemble averages
macroscopic numbers of atoms or molecules (of the order of 1023, Avogadro's number is × 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:

49. 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:

50. 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

51. 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

52. 5. Pair correlation function

53. 5. Outputs

54. 5. Equilibration of energy

55. 5. Time variation of energies
kinetic energies
potential energies

56. 5. Time variation of pressure
Equilibration of pressure with time

57. 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

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

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

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

61. 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

62. 7. Ewald Sum Method

63. 7. Ewald Sum Method

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

65. 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

66. 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)

67. 7. Multigrid

68. 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

69. 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

70. 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

71. 7. FMM/RESPA Performance

72. 7. P3ME/RESPA Performance

73. 7. P3ME/RESPA Performance

74. 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

75. 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

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

77. 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)

78. 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 ~ Ångstroms
Micro-void sizes ~ 5-10 Ångstroms
Multi-scale problem in space (1cm to Ångstroms) and time (pico-seconds to seconds)

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

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

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

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

83. 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

84. 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

85. 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

86. 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

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

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

89. 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)

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

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

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