fundamentals of classical molecular...
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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).
2 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Interactions
Outline
1 Molecular Interactions
2 Molecular Dynamics Algorithms
3 Periodic Boundaries: Short and Long-Ranged Forces
4 Thermostats
3 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Interactions
Molecular Interactions
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
4 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Interactions
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.
5 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Interactions
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
6 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Interactions
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.
7 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Interactions
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.
8 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Interactions
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
9 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Interactions
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.
10 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Interactions
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.
11 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Interactions
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.
12 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Dynamics Algorithms
Outline
1 Molecular Interactions
2 Molecular Dynamics Algorithms
3 Periodic Boundaries: Short and Long-Ranged Forces
4 Thermostats
13 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Dynamics Algorithms
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.
14 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Dynamics Algorithms
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.
15 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Dynamics Algorithms
The Verlet Algorithm
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.
16 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Dynamics Algorithms
The Verlet Algorithm
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.
17 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Dynamics Algorithms
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
18 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Dynamics Algorithms
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.
19 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Dynamics Algorithms
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.
20 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Dynamics Algorithms
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(
21 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Dynamics Algorithms
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.
22 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Dynamics Algorithms
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
23 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Dynamics Algorithms
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.
24 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Dynamics Algorithms
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.
25 Fundamentals of MD Trieste ICTP 18 Feb 2013
Molecular Dynamics Algorithms
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.
26 Fundamentals of MD Trieste ICTP 18 Feb 2013
Periodic Boundaries: Short and Long-Ranged Forces
Outline
1 Molecular Interactions
2 Molecular Dynamics Algorithms
3 Periodic Boundaries: Short and Long-Ranged Forces
4 Thermostats
27 Fundamentals of MD Trieste ICTP 18 Feb 2013
Periodic Boundaries: Short and Long-Ranged Forces
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.
28 Fundamentals of MD Trieste ICTP 18 Feb 2013
Periodic Boundaries: Short and Long-Ranged Forces
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.
29 Fundamentals of MD Trieste ICTP 18 Feb 2013
Periodic Boundaries: Short and Long-Ranged Forces
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.
30 Fundamentals of MD Trieste ICTP 18 Feb 2013
Periodic Boundaries: Short and Long-Ranged Forces
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
31 Fundamentals of MD Trieste ICTP 18 Feb 2013
Periodic Boundaries: Short and Long-Ranged Forces
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
32 Fundamentals of MD Trieste ICTP 18 Feb 2013
Periodic Boundaries: Short and Long-Ranged Forces
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
33 Fundamentals of MD Trieste ICTP 18 Feb 2013
Thermostats
Outline
1 Molecular Interactions
2 Molecular Dynamics Algorithms
3 Periodic Boundaries: Short and Long-Ranged Forces
4 Thermostats
34 Fundamentals of MD Trieste ICTP 18 Feb 2013
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).
35 Fundamentals of MD Trieste ICTP 18 Feb 2013
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.
36 Fundamentals of MD Trieste ICTP 18 Feb 2013
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,/).
37 Fundamentals of MD Trieste ICTP 18 Feb 2013
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.
38 Fundamentals of MD Trieste ICTP 18 Feb 2013
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 ..
39 Fundamentals of MD Trieste ICTP 18 Feb 2013
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).
40 Fundamentals of MD Trieste ICTP 18 Feb 2013
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.
41 Fundamentals of MD Trieste ICTP 18 Feb 2013
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
42 Fundamentals of MD Trieste ICTP 18 Feb 2013
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