Molecular DynamicsA brief overview

2

Notes - Websites

• "A Molecular Dynamics Primer", F. Ercolessi

http://www.fisica.uniud.it/~ercolessi/md/

• http://cacs.usc.edu/education/cs596.html"Scientific computing and visualization", A. NakanoUniversity 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

1,i i iF m a i N

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)

EH

TkH B/exp

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

• Integrator Symplecticity?

• Statistical ensemble

• Collecting results

7

Overview II

Collecting data

8

First MD simulation (1957/1959)

weeks

9

First MD simulation usingcontinuous potentials (1960)

IBM 704

10

First MD simulation usingLennard-Jones potential (1964)

12 6

4u rr r

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 10-12 s in a solid t ≈10-14 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

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Interaction potentials

• Origins are quantum mechanical

• Easiest model: Lennard-Jones potential

• Truncated (but with energy conservation)

12 6

4u rr r

( ) ( ) ( )

'

0c

c c cr

c

duu r u r r r r r

dru rr r

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Interaction potentials

• Distinguish between long range short range interactions

• Distinguish betweenintermolecular intramolecular forces

15

Interaction potentials

• Stretch energy

• Bending energy

16

Interaction potentials

• Interactions between charge inhomogeneitiesApproaches- Point charges- Point multipoles

• Screened Coulomb interaction

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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

, 2x y cL L rCentral Simulation box

rc

Number of interacting pairs increases enormously

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Infinite systems

• Ewald method for Coulomb interaction

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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.

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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

)()()()(2)( 42 tOtatttrtrttr

)()(2

1)()()( 32 tOtatttvtrttr

)()(2

1)()()( 32 tOtatttvtrttr

)())()((2

1)()( 2tOttrttr

ttrtv

+

22

Integrators

• Comparison between Euler and VerletTest system consists of 7 Lennard-Jones atoms (Ar)

Time stepis 10 fs

Time stepis 1 fs

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Integrators

• Velocity Verlet

- Properties• Velocity calculated explicitly• Possible to control the temperature• Stable• Most commonly used algorithm

21/ 2 ( )

/ 2 1/ 2 ( )

1/

/ 2 1/ 2 ( )

i i i

i i

Ni

i i

r t t r t v t t a t t

v t t v t a t t

a t t m V r t t

v t t v t t a t t t

ttatvtT

Tttv

ttatvttv

iii

iii

)()2/1()()(

)2/(

)()2/1()()2/(

0

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Neighbour lists

• Complexity of force calculations ~O(N2)

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 Rc are stored, so that unknown particles cannot become neighbors in next steps

- Do an update of the listevery n steps

• Either statically with fixed n• Or dynamically with an update

criterion

- There exists an optimal Rskin

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Neighbour lists

• Linked lists method ~ O(N)

Interacts with atoms in 26 neighbour cells

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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

1

1 TN

tT

A A tN

i ji j i

V t u r t r t

21

2 i ii

K t m v t

/ 2Bk T

2)0()( rtrMSD

2)0()(

6

1lim rtr

tD t

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

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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)

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Analyzing the sequential code

• md.c and lmd.c"Scientific computing and visualization", A. Nakano

University of Southern California

• Code description, details at http://cacs.usc.edu/education/cs596.html

- “Basic molecular dynamics algorithms”- “Linked-list cell MD algorithm”

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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 (=0.6-0.8, Tmax=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.