Fundamentals of Classical Molecular...

2549-1

ADGLASS Winter School on Advanced Molecular Dynamics Simulations

M P Allen

18 - 21 February 2013

University of Warwick Department of Physics

United Kingdom

Fundamentals of Classical Molecular Dynamics

Fundamentals ofClassical Molecular Dynamics

M P Allen

Department of PhysicsUniversity of Warwick

Trieste ICTP 18 Feb 2013

1 Fundamentals of MD Trieste ICTP 18 Feb 2013

A General Bibliography

D. Frenkel and B. Smit, Understanding MolecularSimulation: from Algorithms to Applications, 2nd edition(Academic Press, 2002).

D. C. Rapaport, The Art of Molecular DynamicsSimulation, 2nd ed (Cambridge University Press, 2004).

M. E. Tuckerman, Statistical Mechanics: Theory andMolecular Simulation, (Oxford University Press, 2010).

M. P. Allen and D. J. Tildesley, Computer Simulation ofLiquids (Clarendon Press, 1989).

B. J. Leimkuhler and S. Reich, Simulating HamiltonianDynamics (Cambridge University Press, 2004).


Molecular Interactions

Outline

1 Molecular Interactions

2 Molecular Dynamics Algorithms

3 Periodic Boundaries: Short and Long-Ranged Forces

4 Thermostats




Molecular dynamics: numerical, step-by-step, solutionof the classical equations of motion.

Newton’s or Hamilton’s Equations

m!r ! = ƒ ! or

!

r ! = p!/m!

p! = ƒ !where ƒ ! = !

"

"r !U = !!!U

System of coupled ordinary differential equations.

Need to be able to calculate the forces ƒ !usually derived from a potential energy U(r)

r = r1, r2, . . . rN = {r !} are atomic coordinates



Non-bonded Interactions

The non-bonded potential energy Unb is traditionallysplit into 1-body, 2-body, 3-body . . . terms:

Non-bonded Potential

Unb(r) ="

!

#(r !) +"

!,j>!

$(r !, r j) +"

!jk

%(r !, r j, rk) + . . . .

The external field or container wallsusually dropped for simulations of bulk systems

The interatomic pair potential

Usually neglect higher order interactions.

There is an extensive literature on the experimentaldetermination of these potentials.



Lennard-Jones Potential

Sometimes sufficient to use the simplest modelsthat faithfully represent the essential physics.

The Lennard-Jones potential is the most commonlyused form, developed for studies of inert gases.

Lennard-Jones

0.5 1 1.5 2-2

-1

0

1

2

3

4

$(r)/!

r/"

LJ

#12#6

Lennard-Jones

$LJ(r) = 4!

#

$

"

r

%12

!

$

"

r

%6&

= #LJ12(r) + #

LJ6 (r)

" = diameter

! = well depth



Electrostatics

Electrostatic charges interact via long-ranged potentials

Coulomb Potential

#qq(r) =q1q2

4$%0r

q1, q2 are the charges

%0 is the permittivity of free space.

The correct handling of long-range forces provokesmuch discussion in the literature.

Interactions involving dipole moments andhigher-order multipoles expressed in similar way.



Atoms to Molecules

For molecular systems, we simply build themolecules out of Lennard-Jones site-site potentials,or similar.

Typically, a single-molecule quantum-chemicalcalculation may be used to estimate the electrondensity throughout the molecule.

This may then be modelled by a distribution ofpartial charges,

or more accurately by a distribution of electrostaticmultipoles.



Bonding Potentials

Lengths and Angles

3

1

2

4

r23

&1234

'234

For molecules we must alsoconsider the intramolecularbonding interactions.Consider this geometry ofan alkyl chain (just showingthe carbons).

interatomic distance r23

bend angle '234

torsion angle &1234



Bonding Potentials

A very simple example:

Intramolecular Bonding Potentials

Uint =12

"

bonds

kr(j

'

r(j ! req(2

+ 12

"

bendangles

k'(jk

'

'(jk ! 'eq(2

+ 12

"

torsionangles

"

m

k&,m(jk)

)

1+ cos(m&(jk) ! *m)*

Packaged force fields specify kr(j, k'

(jk, k

&,m(jk) , req, 'eq, *m

or similar parameters.



Bond Stretching Potentials

Harmonic Potential

12kr(j

'

r(j ! req(2

The “bonds” involve separations r(j = |r ( ! r j|between atoms in a molecular framework.

We assume a harmonic form with specifiedequilibrium separation - not the only possibility.

Deriving forces from this is straightforward.

Vibration frequencies relatively highE.g. for C—H bonds, period " 10fsin a step-by-step solution of the equations ofmotion, need timestep "t # 5fs.



Bond Angle Bending Potentials

Quadratic Approximation or Trigonometric Form

12k'(jk

'

'(jk ! 'eq(2

or 12k'(jk

+

1! cos2'

'(jk ! 'eq(

,

The “bend angles” '(jk are between successivebond vectors such as r ( ! r j and r j ! rk.

Therefore, they involve three atom coordinates:

Bend Angle Definition

cos'(jk = r (j · r jk ='

r (j · r (j(!1/2'

r jk · r jk(!1/2'

r (j · r jk(

Derived forces affect all three atoms.Calculated using the chain rule.Angle-bend timescales, e.g. in H2O, are " 20fs.


Molecular Dynamics Algorithms

Outline




4 Thermostats



The MD Algorithm

Calculating forces is expensive, typically a pairwise sumover atoms, so we need to perform this as infrequentlyas possible.

Wish to make the timestep as large as possible.Hence, simulation algorithms tend to be low order(i.e. do not use high derivatives of r);this allows the time step to be increased as much aspossible without jeopardizing energy conservation.

Cannot accurately follow true trajectory for verylong times trun.

Classical trajectories are ‘ergodic’ and ‘mixing’;trajectories diverge from each other exponentially;however long-term energy conservation is possible.



The Verlet Algorithm

There are various, essentially equivalent, versions ofthe Verlet algorithm, including this one:

Velocity Verlet Equations

p(t + 12"t) = p(t) + 1

2"t ƒ (t)

r(t + "t) = r(t) + "tp(t + 12"t)/m

p(t + "t) = p(t + 12"t) + 1

2"t ƒ (t + "t)

r = {r (} (all coordinates), p = {p(} (all momenta) andƒ = {ƒ (} (all forces).After the middle step, a force evaluation is carried out,to give ƒ (t+"t) for the last step. This scheme advancesthe coordinates and momenta over a timestep "t.




A piece of pseudo-code illustrates how this works:

Velocity Verlet Algorithm

do step = 1, nstepp = p + 0.5*dt*fr = r + dt*p/mf = force(r)p = p + 0.5*dt*f

enddo

The force routine carries out the time-consumingcalculation of all the forces, and potential energy U.

The kinetic energy K can be calculated after thesecond momentum update.

At this point the total energy is U+ K.




Important features of the Verlet algorithm are:1 it is exactly time reversible;2 it is symplectic (to be discussed later);3 it is low order in time, permitting long timesteps;4 it requires just one force evaluation per step;5 it is easy to program.



Propagators and Molecular Dynamics

Formally, for A = r, p, or any A(r,p):

A = iLA (Liouville operator);

A(t + "t) = eiL"tA(t) (propagator).

Useful approximations arise from splitting iL in two:

Split Propagator

iL = iLp + iLr

iLp = ƒ ·+

+peiLp"tp = p+ ƒ"t kick

iLr = m!1p ·+

+reiLr"tr = r +m!1p"t drift



Propagators and the Verlet Algorithm

The following approximation is asymptotically exact inthe limit "t$ 0.

Symmetric Splitting

eiL"t = e(iLp+iLr)"t # eiLp"t/2 eiLr"t eiLp"t/2

For nonzero "t this is an approximation to eiL"t

because in general iLp and iLr do not commute,

but it is still exactly time reversible and symplectic.Symplectic (roughly) implies conserving phasespace volume dr(t + "t)dp(t + "t) = dr(t)dp(t).

It is then easy to see that the three successive stepsembodied in the above equation, with the above choiceof operators, generate the velocity Verlet algorithm.



Propagators and the Verlet Algorithm

The trajectories generated by the above schemeare approximate, and will not conserve the trueenergy H.

Nonetheless, they do exactly conserve a“pseudo-hamiltonian” or “shadow hamiltonian” H‡

H and H‡ differ from each other by a small amount,H‡ = H+O("t2).

This means that the system will remain on ahypersurface in phase space which is “close” to thetrue constant-energy hypersurface.

Such a stability property is extremely useful in MD,since we wish to sample constant-energy states.



Example

Consider a simple harmonic oscillator, of naturalfrequency ,, representing perhaps an interatomic bondin a diatomic molecule. The equations of motion andconserved hamiltonian are

Harmonic Oscillator Equations

r = p/m , p = !m,2r , H(r, p) = p2/2m+ 12m,2r2

For this system, velocity Verlet exactly conserves:

The Shadow Hamiltonian

H‡(r, p) = p2/2m+ 12m,2r2'

1! (,"t/2)2(



Example

Harmonic Oscillator

-1 0 1

-1

0

1

p

r

1 1

2

2

3

3

44

5

5 6

6In a “phase portrait”,the simulated systemremains on theconstant-H‡ ellipse(dashed line) whichdiffers only slightly(for small ,"t) fromthe true constant-Hellipse (full line), e.g.here for ,"t = $/3.



Multiple Timesteps

One approach to handling the fast bond vibrations is touse a shorter timestep for them.

Use Liouville operator formalism to generatetime-reversible Verlet-like multiple-timestepalgorithm.

Suppose there are “slow” F, and “fast” ƒ , forces.

Momentum satisfies p = F+ ƒ .

Break up Liouville operator iL = iLp + i)p + iLr:

Multiple Timestep Liouville Operator

iLp = F ·+

+p, i)p = ƒ ·

+

+p, iLr = m!1p ·

+

+r



Multiple Timesteps

The propagator approximately factorizes

Long Timestep Splitting

eiL"t # eiLp"t/2 e(i)p+iLr)"t eiLp"t/2

where "t represents a long time step. The middle partis then split again, using the conventional separation asusual, iterating over small time steps -t = "t/n:

Short Timestep Splitting

e(i)p+iLr)"t #+

ei)p-t/2 eiLr-t ei)p-t/2,n

Fast-varying forces computed at short intervals.

Slow forces computed once per long timestep.



Multiple Timesteps

Multiple Timestep Algorithm

do STEP = 1, NSTEPp = p + (DT/2)*Fdo step = 1, n

p = p + (dt/2)*fr = r + dt*p/mf = force(r)p = p + (dt/2)*f

enddoF = FORCE(r)p = p + (DT/2)*F

enddo

Some pseudo-codeillustrates how simplethis is.The simulation runconsists of NSTEP longsteps, of length DT, eachconsisting of n sub-stepsof length dt, whereDT = n*dt.



Multiple Timesteps

Non-bonded interactions may be calculated moreefficiently this way too. A typical approach is to split theinteratomic force law into a succession of componentscovering different ranges:

the short-range forces change rapidly with time andrequire a short time step,

the long-range forces vary more slowly, so we use alonger time step and less frequent evaluation.

Multiple-time-step algorithms are still under activestudy, and there is some concern that resonances mayoccur between the natural frequencies of the systemand the various timesteps used in schemes of this kind.The area remains one of active research.


Periodic Boundaries: Short and Long-Ranged Forces

Outline




4 Thermostats



Periodic Boundary Conditions

Cubic System

488

10x10x10=10008x8x8= 512

Consider N = 103 atomsarranged in a cube.

Nearly half the atomsare on the outer faces,

will have a large effecton the measuredproperties.

Even forN = 1003 = 106 atoms,6% of atoms are onsurface.



Periodic Boundary Conditions

Periodic Boundaries Surround the cube withreplicas

For short-range potentials,adopt the minimum imageconvention: each atominteracts with the nearestatom or image in theperiodic array.

If an atom leaves thebasic simulation box,attention can be switchedto the incoming image.



Neighbour Lists

For short-range potentials #(r) = 0, r > rcut, speed upsearch for those interactions which are in range.

Cell Structure Divide L% L% L simulationbox into n% n% n sub-cells

Side of the cell

) = L/n & rcut

In searching for atoms withinrange, examine the atom’sown cell, and nearestneighbour cells.



Long-range forces

In periodic systems, the evaluation of the Coulombinteractions is non-trivial.

U =N"

(=1

"

j>(

q(qj

4$!0r(j

( and j vary over all ions in all cells: no cutoff

Formally, this sum is only conditionally convergentthe result depends on the ordering of the terms

There are some subtleties associated with thedielectric medium assumed to be “outside” theinfinitely periodic system



Ewald Sum

Ewald Sum

position

char

ge

Point chargesShieldingCompensation

A trick allows us to evaluatethe sum. Add to the systemof point charges a set ofpositive and negativeGaussian distributions.

Real Space and Reciprocal Space Terms

Ur-space ="

(=1

"

j>(

q(qj

4$!0r(jerfc.r(j

Uk-space =1

2V!0

"

k

e!k2/4.2

k2

-

-

-

-

-

"

j

qje!ik·r j

-

-

-

-

-

2



Ewald Sum

The Coulomb energy is thus transformed into

a real-space sum, involving the muchshorter-ranged screened Coulomb interaction

a reciprocal-space sum, i.e. a sum overwave-vectors k, involving the interaction betweenGaussian charge clouds

some correction terms handling the self-interactionbetween the added Gaussians

Smoothed Particle-Mesh Ewald

Mapping onto a regular grid by interpolation.

The primary mathematical operation may now beperformed by Fast Fourier Transform

Greatly speeds up biological simulations


Thermostats

Outline




4 Thermostats


Thermostats

Constant-Temperature Dynamics

How to simulate systems at given T by MD?

Andersen Thermostat

Periodically reselect atomic velocities at randomfrom the Maxwell-Boltzmann distribution.

Like occasional random coupling with thermal bath.

The resampling may be done to individual atoms,or to the entire system.

Simple to implement and reliable.

Proven to sample the canonical ensembleif MD is accurate!

HC Andersen, J. Chem. Phys., 72, 2384 (1980).


Thermostats

Constant-Temperature Dynamics

An alternative, deterministic, approach:

Nosé-Hoover Equations

r = ! = p/m p = ƒ ! /!

/ = M!1."

#2 ! gkBT/m/ "

#2 '"

(.

#2(.

/: friction coefficient, allowed to vary in time;

M is a thermal inertia parameter, determining arelaxation rate for thermal fluctuations;

g # 3N is the number of degrees of freedom.

If the system is too hot (cold), then / will tend toincrease (decrease) tending to cool (heat) the system.


Thermostats

Hoover Derivation

WG Hoover, Molecular dynamics, Springer Berlin (1987).

Assumed Equations of Motion

r = p/m , p = ƒ (r)! /! , / = G(r,p)

G(r,p) is the object of the derivation.

Ansatz: G(r,p) depends only on r, p, not /.

Generalized Liouville Equation

" +

+r·'

0r(

+" +

+p·'

0p(

++

+/

'

0/(

= 0

Follows from continuity d1/dt = 0, and stationarity+1/+t = 0 of phase space distribution function 1(r,p,/).


Thermostats

Canonical Distribution

Try: 0(r,p,/) # exp0

!H(r,p)/kBT1

exp0

! 12M/2/kBT1

Required Form of G

G(r,p) = M!1"

(

2

p(

m(

· ! ( ! kBT+

+p(· ! (

3

= M!1"

(

.

#2(! 3kBT/m/

M arbitary constant (thermal inertia)

Term in square brackets vanishes if averaged overcanonical momentum distribution.


Thermostats

The Configurational Temperature

JO Hirschfelder, J. Chem. Phys., 33, 1462 (1960).

LD Landau and EM Lifshitz, Statistical Physics (1958).

Based on hypervirial theorem in canonical ensemble.

Configurational and Kinetic Temperature

kBTc =

4

'

+H/+r(.(25

6

+2H/+r2(.7

, ( = 1 . . . N, . = 2, y, z

kBTk =6

p2(./m7

Both sides may be averaged over ( and ..


Thermostats

The Configurational Temperature

The configurational temperature is useful:

to define T in the microcanonical ensemble

HH Rugh, Phys. Rev. Lett., 78, 772 (1997).

as a test for simulation nonequilibrium

BD Butler, et al, J. Chem. Phys., 109, 6519 (1998).

in experiments on colloidal suspensions

YL Han, DG Grier, Phys. Rev. Lett., 92, 148301(2004);

As the basis of a thermostat in MD.

C Braga and KP Travis, J. Chem. Phys., 123, 134101(2005).


Thermostats

Configurational Nosé-Hoover Thermostat

C Braga and KP Travis, J. Chem. Phys., 123, 134101(2005).

Equations of Motion

r = p/m+ 3ƒ , p = ƒ

3 = M!1

8

9

:

"

j

-

-

-

-

-

+H

+r j

-

-

-

-

-

2

! kBT+

+r j·+H

+r j

;

<

=

3 is a dynamical “mobility” coefficient.

The driving force is related to Tc.


Thermostats

Summary

We have discussed the fundamentals of classical MD:

specifying the molecular model;

a good algorithm to advance the system in time;

some techniques to improve efficiency;

modifications for different physical conditions.

We have not discussed:

how to analyse the resultsstructural and dynamical properties

how to efficiently use different hardwareparallel computers or GPUs

the relation between MD and Monte CarloHybrid Monte Carlo, Brownian/Langevin Dynamics


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