introduction to scientiﬁc computing ii...michael bader – sccs: introduction to scientiﬁc...

Technische Universitat Munchen

Introduction to Scientific Computing II

Molecular Dynamics Simulation

Michael Bader – SCCSSummer Term 2012

Michael Bader – SCCS: Introduction to Scientific Computing II

Molecular Dynamics Simulation, Summer Term 2012 1

Molecular Dynamics and N-Body Problems – An IntroductionMicro and Nano SimulationsAstrophysicsParticle-oriented Numerical MethodsLaws of Motion

Molecular Dynamics – the Physical ModelQuantum vs. Classical MechanicsVan der Waals AttractionLennard Jones Potential

Molecular Dynamics – the Mathematical ModelSystem of ODEInitial and Boundary ConditionsComputational Domain

The Simulation Pipeline – What Did We Cover So Far?

phenomenon, process etc.

mathematical model?

modelling

numerical algorithm?

numerical treatment

simulation code?

implementation

results to interpret?

visualization

��

HHHHj embedding

statement tool

validation

The Seven Dwarfs of HPC – Dwarf # 4

“dwarfs” = key algorithmic kernels in many scientific computingapplications

P. Colella (LBNL), 2004:

1. dense linear algebra

2. sparse linear algebra

3. spectral methods

4. N-body methods5. structured grids

6. unstructured grids

7. Monte Carlo

Molecular Dynamics – Overview

• modelling aspects of molecular dynamics simulations:• why to leave the classical continuum mechanics point of view?• where appropriate?• which models, i.e. which equations?

• numerical aspects of molecular dynamics simulations?• how to discretize the resulting modelling equations?• efficient algorithms?

• implementation aspects of molecular dynamics simulations?• suitable data structures?• parallelisation?

Hierarchy of Models

Different points of view for simulating human beings:

issue level of resolution model basis (e.g.!)

global increase inpopulation

countries, regions population dynamics

local increase inpopulation

villages, individuals population dynamics

man circulations, organs system simulatorblood circulation pump/channels/valves network simulatorheart blood cells continuum mechanicscell macro molecules continuum mechanicsmacro molecules atoms molecular dynamicsatoms electrons or finer quantum mechanics

Scales – an Important Issue

• length scales in simulations:• from 10−9m (atoms)• to 1023m (galaxy clusters)

• time scales in simulations:• from 10−15s• to 1017s

• mass scales in simulations:• from 10−24g (atoms)• to 1043g (galaxies)

• obviously impossible to take all scales into acount in an explicit andsimultaneous way

• first molecular dynamics simulations reported in 1957

More General: Particle-Oriented Simulation Methods

General Approach:

• “N-body problem”→ compute motion paths of many individual particles

• requires modelling and computation of inter-particle forces• typ. leads to ODE for particle positions and velocities

Examples:

• Molecular dynamics• Astrophysics• Particle-oriented discretisation techniques

Applications for Micro and Nano Simulations

Lab-on-a-chip, used in brewing technology (Siemens)

Flow through a nanotube (where the assumptions of continuum mechanicsare no longer valid)

Protein simulation: actin, important component of muscles (overlay ofmacromolecular model with electron density obtained by X-ray

crystallography (brown) and simulation (blue))

Protein simulation: human haemoglobin (light blue and purple: alpha chains;red and green: beta chains; yellow, black, and dark blue: docked stabilizers

or potential docking positions for oxygen)

Material science: hexagonal crystal grid of Bornitrid

HPC Example – Gordon Bell Prize 2005

• Gordon-Bell-Prize 2005 (most important annual supercomputing award)• phenomenon studied: solidification processes in Tantalum and Uranium• method: 3D molecular dynamics, up to 524,000,000 atoms simulated• machine: IBM Blue Gene/L, 131,072 processors (world’s #1 in

November 2005)• performance: more than 101 TeraFlops (almost 30% of the peak

performance)

(Streitz et al., 2005)

HPC Example – Millennium-XXL Project

(Springel, Angulo, et al., 2010)

• N-body simulation with N = 3 · 1011 “particles”• study gravitational forces

(each “particles” corresp. to ∼ 109 suns)• simulates the generation of galaxy clusters

served to “validate” the cold dark matter model

Millennium-XXL Project (2)

Simulation Figures:• N-body simulation with N = 3 · 1011 particles• 10 TB RAM required only to store positions and velocities (single

precision)• entire memory requirements: 29 TB• JuRoPa Supercomputer (Jlich)• computation on 1536 nodes

(each 2x QuadCore, i.e., 12 288 cores)• hybrid parallelisation: MPI plus OpenMP/Posix threads• execution time: 9.3 days; ca. 300 CPU years

Example – Smoothed Particle Hydrodynamics

• approximate functions using kernel functions W :

f (x) ≈∫V

f (r ′)W (|r − r ′|, h) dV ′

• for h→ 0: W → δ (Dirac function)• approximation of derivatives:

∇f (x) ≈∫V

f (r ′)∇W (|r − r ′|, h) dV ′

Example – Smoothed Particle Hydrodynamics (2)

• approximate integrals at particle positions:

f (ri ) ≈N∑

ρ(rj )f (rj )W (|ri − rj |, h)

• similar for derivatives:

∇f (ri ) ≈N∑

ρ(rj )f (rj )∇W (|ri − rj |, h)

• leads to N-body problem (based on Navier-Stokes equations, e.q.)

Laws of Motion

• force on a molecule: ~Fi =∑

j 6=i~Fij

• leads to acceleration (Newton’s 2nd Law):

~ri =~Fi

∑j 6=i~Fij

mi= −

∑j 6=i

∂U(~ri ,~rj )∂|rij |

• system of dN ODE (2nd order)(N: number of molecules, d : dimension),

• reformulated into a system of 2dN 1st-order ODEs:

~pi := mi~ri (2a)

~pi = ~Fi (2b)

Example: Hooke’s Law

• ”‘harmonic potential”’: Uharm (rij ) = 12 k (rij − r0)2

• potential energy of a spring of length r0 when extended or compressedto length rij

• resulting force:

1D : ~Fij = −gradU (rij ) = −∂U∂rij

= −k (rij − r0)

allg. : ~Fij = −k (rij − r0)

Example: Gravity

• attractive force due to the mass of two bodies (planets, etc.)• gravity potential: Ugrav (rij ) = −g mi mj

1D : ~Fij = −gradU (rij ) = −gmimj

Example: Coulomb Potential

• attractive or repulsive force between charged particles• Coulomb potential: Ugrav (rij ) = 1

4πε0

qi qjrij

1D : ~Fij = −gradU (rij ) =1

4πε0

2. Molecular Dynamics – the Physical ModelQuantum Mechanics – a “Tour de Force”

• particle dynamics described by the Schrodinger equation• its solution (state or wave function ψ) only provides probability

distributions for the particles’ (i.e. nuclei and electrons) position andmomentum

• Heisenberg’s uncertainty principle: position and momentum can not bemeasured with arbitrary accuracy simultaneously

• there are discrete values/units (for the energy of bonded electrons, e.g.)• in general, no analytical solution available

Quantum Mechanics – a “Tour de Force” (2)

• high dimensional problems: dimensionality corresponds to number ofnuclei and electrons

Ψ = Ψ(R1, . . . ,RN , r1, . . . , rK , t)

ψ - wave functionR - position of nucleusr - position of electront - time

• hence, numerical solution is possible for rather small systems only• therefore, various (simplifying and approximating) approaches such as

density functional method or Hartree-Fock approach (ab-initio MolecularDynamics, see next slide)

Classical Molecular Dynamics

• Quantum mechanicsapproximation−−−−−−−→ classical Molecular Dynamics

• classical Molecular Dynamics is based on Newton’s equations of motion• molecules are modelled as particles; simplest case: point masses• there are interactions between molecules• multibody potential functions describe the potential energy of the system;

the velocities of the molecules (kinetic energy) are a composition of• the Brownian motion (high velocities, no macroscopic movement),• flow velocity (for fluids)

• ab-initio Molecular Dynamics uses quantum mechanical calculations todetermine the potential hypersurface, apart from semi-empirical potentialfunctions (cf. Car Parrinello Molecular Dynamics (CPMD) methods)

• total energy is constant↔ energy conservation

Fundamental Interactions

• Classification of the fundamentalinteractions:

• strong nuclear force• electromagnetic force• weak nuclear force• gravity

• interaction→ potential energy• the total potential of N particles is the sum of multibody potentials:

• U :=∑

0<i<N U1(ri ) +∑

∑i<j<N U2(ri , rj )

∑i<j<N

∑j<k<N U3(ri , rj , rk ) + . . .

• there are ( Nn ) = N!

n!(N−n)! ∈ O(Nn) n-body potentials Un, particularyN one-body and 1

2 N(N − 1) two-body potentials

• force ~F = −gradU

Van der Waals Attraction

• intermolecular, electrostatic interactions• electron motion in the atomic hull may result in a temporary asymmetric

charge distribution in the atom (i.e. more electrons (or negative charge,resp.) on one side of the atom than on the opposite one)

• charge displacement⇒ temporary dipole• a temporary dipole

• attracts another temporary dipole• induces an opposite dipole moment for a non-dipole atom and

attracts it• dipole moments are very small and the resulting electric attraction forces

(van der Waals or London dispersion forces) are weak and act in a shortrange only

• atoms have to be very close to attract each other, for a long distance thetwo dipole partial charges cancel each other

• high temperature (kinetic energy) breaks van der Waals bonds

Well-Known Potentials

• some potentials from mechanics:• harmonic potential (Hooke’s law): Uharm (rij ) = 1

2 k (rij − r0)2;potential energy of a spring with length r0, stretched/clinched to alength rij

• gravitational potential: Ugrav (rij ) = −g mi mjrij

;potential energy caused by a mass attraction of two bodies (planets,e.g.)

• the resulting force is ~Fij = −gradU (rij ) = − ∂U∂rij

integration of the force over the displacement results in the energy or a potentialdifference

• Newton’s 3rd law (actio=reactio):~Fij = −~Fji

Intermolecular Two-Body Potentials

−1.5

−0.5

0 0.5 1 1.5 2 2.5 3

distance r

hard sphere potentials

hard sphereSquare−well

Sutherland

−1.5

−0.5

0 0.5 1 1.5 2 2.5 3po

distance r

soft sphere potentials

soft sphereLennard−Jonesvan der Waals

Intermolecular Two-Body Potentials

• hard sphere potential: UHS (rij ) =

{∞ ∀ rij ≤ d0 ∀ rij > d

Force: Dirac Funktion• soft sphere potential: USS (rij ) = ε

(σrij

• Square-well potential: USW (rij ) =

∞ ∀ rij ≤ d1

−ε ∀ d1 < rij < d2

0 ∀ rij ≥ d2

• Sutherland potential: USu (rij ) =

∞ ∀ rij ≤ d−εr6ij∀ rij > d

• Lennard Jones potential

• van der Waals potential UW (rij ) = −4εσ6(

• Coulomb potential: UC (rij ) = 14πε0

qi qjrij

ε = energy parameterσ = length parameter (corresponds to atom diameter, cmp. van der Waalsradius)

Lennard Jones Potential

• Lennard Jones potential: ULJ (rij ) = αε((

)n−(σrij

)m)with n > m and α = 1

) 1n−m

• continuous and differentiable (C∞), since rij > 0

Lennard Jones Potential (2)

LJ 12-6 potential

ULJ (rij ) = 4ε((

)12−(σrij

• m = 6: van der Waals attraction (van der Waals potential)

• n = 12: Pauli repulsion (softsphere potential): heuristic• application: simulation of inert gases (e.g. Argon)

• force between 2 molecules:

Fij = − ∂U(rij )∂rij

= 24εrij

(2(σrij

)12−(σrij

• very fast fade away⇒ short range (m = 6 > 3 = d dimension)

LJ Atom-Interaction Parameters

atom ε σ

[1.38066 · 10−23J]a [10−1nm]b

H 8.6 2.81He 10.2 2.28C 51.2 3.35N 37.3 3.31O 61.6 2.95F 52.8 2.83

Ne 47.0 2.72S 183.0 3.52Cl 173.5 3.35Ar 119.8 3.41Br 257.2 3.54Kr 164.0 3.83

aBoltzmann-constant: kB := 1.38066 · 10−23 JK

b10−1nm = 1A (Angstom)

ε = energy parameterσ = length parameter (cmp.van der Waals radius)→ parameter fitting to realworld experiments

Dimensionsless Formulation

using reference values such as σ, ε, reduced forms of the equations can bederived and implemented→ transformation of the problem• position, distance

~r∗ :=1σ~r (3a)

• time

t∗ :=1σ

mt (3b)

• velocity

~v∗ :=∆tσ~v (3c)

Dimensionsless Formulation (2)

• potential (atom-interaction parameters are eliminated!): U∗ := Uε

U∗LJ (rij ) :=ULJ (rij )

((r∗ij

2)−6−(

r∗ij2)−3

U∗kin :=Ukin

2∆t∗2 (4b)

• force~F∗ij :=

~Fijσ

ε= 24

r∗ij2)−6−(

r∗ij2)−3

)~r∗ijr∗ij

2 (4c)

Skipped: Multi-Centered Molecules

CA1 CA2

FA1B2FA2B1

• molecules can be composed with multipleLJ-centers→ rigid bodies without internal degrees offreedom

• additionally: orientation (quarternions), angularvelocity

• additionally: moment of inertia (principal axestransformation)

• calculation of the interactions between eachcenter of one molecule to each center of theother

• resulting force (sum) acts at the center of gravity,additional calculation of torque

Skipped: Multi-Centered Molecules (2)

• MBS (Multi Body System) point of view: instead of movingmulti-centered molecules, there is a holonomically constrained motion ofatoms (for a constraint to be holonomic it can be expressible as a function f (r, v, t) = 0)

• advantage: better approximation of unsymmetric molecules• there is not necessarily one LJ center for each atom

Skipped: Mixtures of Fluids

• simulation of various components (molecule types)• modified Lorentz-Berthelot rules for interaction of molecules of different

σAB :=σA + σB

εAB := ξ√εaεB (5b)

with ξ ≈ 1e.g. N2 + O2 → ξ = 1.007, O2 + CO2 → ξ = 0.979 . . .

e ,sA A

e ,sAB AB

3. Molecular Dynamics – the Mathematical ModelSystem of ODE

• resulting force acting on a molecule: ~Fi =∑

j 6=i~Fij

• acceleration of a molecule (Newton’s 2nd law):

~ir =~Fi

∑j 6=i~Fij

mi= −

∑j 6=i

∂U(~ri ,~rj )∂|rij |

• system of dN coupled ordinary differential equations of 2nd ordertransferable (as compared to Hamilton formalism) to 2dN coupledordinary differential equations of 1st order (N: number of molecules, d :dimension), e.g. independent variables q := r and p with

~pi := mi~ri (7a)

~pi = ~Fi (7b)

Boundary Conditions

Initial Value Problem:position of the molecules and velocities have to begiven;initial configuration e.g.:• molecules in crystal lattice (body-/face-centered

cell)• initial velocity

• random direction• absolute value dependent of the temperature

(normal distribution or uniform), e.g.32 NkBT = 1

∑Ni=1 mv2

i with vi := v0

⇒ v0 :=√

3kBTm resp. v∗0 :=

√3T ∗∆t∗

Time discretisation: t := t0 + i ·∆t→ time integration procedure

NVT-Ensemble, Thermostat

statistical (thermodynamics) ensemble: set of possible states a system mightbe in• for the simulation of a (canonical) NVT-ensemble, the following values

have to be kept constant:• N: number of molecules• V : volume• T : temperature

• a thermostat regulates and controls the temperature (the kinetic energy),which is fluctuating in a simulation

• the kinetic energy is specified by the velocity of the molecules:Ekin = 1

∑i mi~v2

• the temperatur is defined by T = 23NkB

(N: number of molecules, kB : Boltzmann-constant)

• simple method: the isokinetic (velocity) scaling:

vcorr := βvact mit β =√

TrefTact

• further methods e.g. Berendsen-, Nose-Hoover-thermostat

Domain

• Periodic Boundary Conditions (PBC):• modelling an infinite space, built from identical cells⇒ domain with torus topology

Domain

• Minimum Image Convention (MIC):• with PBC, each molecule and the associated interactions exist

several times• with MIC, only interactions between the closest representants of a

molecule are taken into consideration

introduction to scientiﬁc computing ii...michael bader – sccs: introduction to scientiﬁc...

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