The Very Basics of Molecular Dynamics · 2016-05-08 · The World of Molecular Mechanics 8 Ali...

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The Very Basics of Molecular Dynamics

Ali Hassanali, ICTP.

Where and What is ICTP

1 Ali Hassanali

“Scientific thought is the common heritage of mankind”

Molecular Dynamics

Trieste, Italy

Going beyond the Zero-K World

2 Ali Hassanali Molecular Dynamics

What is the structure of the water molecule?

O-H bond length?

H-O-H angle?

What is its dipole moment?

Imagine how the water molecule moves at 150K

The importance of fluctuations:

Structural and dipole

distributions

How quickly/slowly do things diffuse?

What is Classical Molecular Dynamics?

F(t) = MR..(t) = −∇V ( R(t){ })

Numerical Integration

V ( R(t){ })

Trajectory: positions and velocities over time

Molecular Dynamics: Make molecular movies

How does water dance on the surface of silica?

What is the diffusion constant of water around silica?

Once you have a trajectory …

Essential Ingredients to cook an MD simulation

F(t) = MR..(t) = −∇V ( R(t){ })

Interaction potential between the particles

(Empirical potential or Ab Initio – More later …)

Initial coordinates or structure of the physical system

(Chemical Intuition or Experiments: X-Ray)

How to model matter on the computer?

neutron proton electron

Quantum Mechanics

The World of Molecular Mechanics

Use classical mechanics to model molecular interactions

An empirical function is used to approximate the Born Oppenheimer potential energy surface as a function of

nuclear positions

ETotal = Ebonded +Enon−bonded

How to treat bonded Interactions

Harmonic distance interaction between neighboring atoms:

( )202

)( ijijijharm rrkrv −=

2)( θθθ −=kvang

Harmonic angle potentials:

2, 3 and 4 particles bonded to each other

Dihedral angle potential:

vdihedral (ϕ ) = cos n ϕ −ϕ0( )⎡

⎣⎤⎦

Peptide Backbone

Non-bonded interactions between molecules

How does the potential energy change as a function of the different geometries shown?

The Potential Energy Surface of the Water Dimer

Empirical potential: distance and angular dependence

Behler and co-workers

How to treat non-bonded interactions?

Enon−bonded = Evan−der−Waals +Eelectrostatics

Lennard-Jones Potential

⎥⎥⎦

⎢⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛−⎟⎠

⎞⎜⎝

⎛=612

rvLJσσ

vLJ(r) σ

21/6 σ

Actually, repulsive part of potential better described by an exponential: Ae-br

Electrostatic Interactions

Define atomic-based charges: RESP, Lowdin …

qqvrr −

= ∑∑−

molecule secondon sites

moleculefirst on sites

Example: SPC/E water

qH = +.4238 qH = +.4238

qO = -.8476

Effect of polarization in molecular systems

Dipole moment of isolated water molecule

D8.1OH2=µ

1D(Debye) = .208194e Ao

Dipole moment of water molecule in water or ice ≈ 2.6-3.0 D.

What about polarization?

E electric field of other molecules +

possible external applied field

electron cloud shifts, causing induced dipole

Eαµ ⋅=ind α is the polarizability tensor.

If molecule close to isotropic, the tensor α can be replaced by the scalar α, the polarizability.

Eµ α=ind

Force-Field: Potentials used to get Forces

F(t) = MR..(t) = −∇V ( R(t){ })

Potential (force-field) to derive forces

vLJ (r) = 4εσr

⎝⎜

⎠⎟12

−σr

⎝⎜

⎠⎟6⎡

⎣⎢⎢

⎦⎥⎥

−∂vLJ (r)∂r

= 4ε 12 σr

⎝⎜

⎠⎟13

− 6 σr

⎝⎜

⎠⎟7⎡

⎣⎢⎢

⎦⎥⎥

Numerical Integration: Verlet Algorithm

xk+1 = xk + hx 'k = xk + hf (tk, xk )Remember Euler’s Method?

L. Verlet, Phys. Rev. 159:98 (1967)

!+Δ+Δ−=Δ− )(21)()()( 2 ttttttt NNNN avrr

!+Δ+=Δ−+Δ+ )()(2)()( 2 ttttttt NNNN arrr

Forwards and backwards Taylor expansions

!+Δ+Δ−−=Δ+ )()()(2)( 2 ttttttt NNNN arrrVerlet algorithm (1st version)

!+Δ+Δ+=Δ+ )(21)()()( 2 ttttttt NNNN avrr

ADD 2 eq.’s

Obtaining the velocities

Ali Hassanali

L. Verlet, Phys. Rev. 159:98 (1967)

!+Δ+Δ+=Δ+ )(21)()()( 2 ttttttt NNNN avrr

!+Δ+Δ−=Δ− )(21)()()( 2 ttttttt NNNN avrr

!+Δ=Δ−−Δ+ )(2)()( tttttt NNN vrr

Forwards and backwards Taylor expansions

• Velocity: t

tttttNN

Δ−−Δ+=

2)()()( rrv

SUBTRACT 2 eq.’s

Molecular Dynamics 19

Numerical Errors in the Verlet Algorithm

Ali Hassanali

rN (t +Δt) = 2rN (t)− rN (t −Δt)+Δt2 aN (t)+O(Δt)Verlet algorithm (1st version)

vN (t) = rN (t +Δt)− rN (t −Δt)

2Δt+O(Δt)

Velocity verlet and Leap-frog verlet are variants that allow for better accuracy of the velocities.

Molecular Dynamics

Time-step in MD simulations (More Later)

Long Stability of Verlet Algorithm

Ali Hassanali

Compare Verlet algorithm to higher order ODE

Verlet, Δt = 1fs, 1.0 x 106 force evaluations.

tol=10-4, Δt = 2fs, 1.9 x 106 force evaluations

Gray, Noid, Sumpter J.Chem.Phys. 101:4062 (1994)

Periodic Boundary Conditions (in Liquids)

Ali Hassanali

Fluid can be strongly perturbed by wall out to ~5 molecular diameters.

Periodic Boundary Conditions (PBC)

Ali Hassanali

… …

. . . Molecular Dynamics 23

Minimum Image Convention

Ali Hassanali

Neglect all n except the one for which |ri - rj + nL| is smallest.

The minimum image between particles i and j in the drawing

are within the same cell. i i

Molecular Dynamics

Consider determining the interaction between two particles with PBCs

Minimum Image Convention

Ali Hassanali

Neglect all n except the one for which |ri - rj + nL| is smallest.

The minimum image between particles i and j in the drawing are across two different cells.

Calculating force between two particles

Ali Hassanali

( )jiv rr −Suppose there is an interaction between particles i and j .

Calculate the force on particle i from this interaction

Molecular Dynamics

( )jii

f rr −∂

∂−=, ( ) ji

v rrrr −∂

∂−ʹ−=

xxx rr

−=−

( ) .,ji

jijiix

−−ʹ−=( )jiv rr −If , then

Force discontinuity under minimum image

Ali Hassanali Molecular Dynamics

j i j i

( )jii

f rr −∂

∂−=,

−−ʹ−=

( )Lvx

xi xrr ˆ, +−∂

∂−=

( )LLxx

jiji xrrxrr

+−+−ʹ−=

Force on particle i from interaction with j is calculated here.

opposite sign

The minimum image switches when the distance between particles is ½ L along any direction.

Calculating force between two particles

Ali Hassanali

j i j i

Min.image typically used for r-n n ≥ 6, potentials.

Min.image only used for short range potentials for which v(L/2) ≈ 0 .

point where min.image neighbor switches

Molecular Dynamics

Truncation of potential → discontinuity of the force → numerical errors.

Treatment of short-range interactions

Example: 2.5σ is a traditional cut-off distance for Lennard-Jones interactions.

⎥⎥⎦

⎢⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛−⎟⎠

⎞⎜⎝

⎛=612

rvLJσσ

)(1 rvLJεσr

Shifted LJ Potential

Ali Hassanali

Massage the LJ Potential

)(1 rvLJε

⎪⎭

⎪⎬

⎪⎩

⎪⎨

<−⎥⎥⎦

⎢⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛−⎟⎠

⎞⎜⎝

cutcutLJcutLJ

rrrvrrrv,0

),(4)(

σσε

)(1, rv cutLJε

Force has a discontinuity at rcut .

Switching Function

Ali Hassanali

⎪⎪⎪⎪

<⎥⎥⎦

⎢⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛−⎟⎠

⎞⎜⎝

cutswitchLJ

σσε

)(1, rv switchLJε σ

cubic polynomial switching function, r1=2.2σ, rcut=2.5σ.

Verlet Neighbor Lists

Ali Hassanali

Calculation of N(N-1)/2 distances rij can be avoided at most simulation steps using neighbor lists.

All neighbors within rlist are listed every nlist steps. (rlist ≈ 10Å)

At each simulation step, only distances between listed neighbors are calculated.

Interactions calculated for rij < rcut .

Valid when particles diffuse less than (rlist - rcut) in nlist steps.

How about Long-Range Interactions?

Ali Hassanali

( )∑ ∑ +−=∞→

jijiNcell LvE

'21lim

omit i=j when n = 0.

When potential is long range (Coulomb,…), the full sum must be calculated.

Full sum is often slowly (and conditionally) convergent.

Ewald method: Original slowly convergent sum → two rapidly convergent sums.

Coulomb Sums are Conditionally Convergent

Ali Hassanali

( )∑ ∑ +−=∞→

jijiNcell LvE

'21lim

( ) ( )∑ ∑ ∑⎥⎥⎦

⎢⎢⎣

⎡++−++−=

++ −+∈∈ ∈∈∞→

jijijiN

nrrnrrNa,Na Cl,Na

''21lim !

??=∞−∞+∞−∞=

2∫∞

Coulomb sum with only like

or unlike charges ~

does not converge.

The Ewald Summation

Ali Hassanali

deLeeuw, Perram, Smith, Proc.RoySoc.Lond.A373:27 (1980)

Introduce integral representation for r - 1 .

∫∞

−−=0

2/1 211 tretdtr π

Break this integral into two parts.

∫∫∞

−−−− +=2

2/1 111

ππtrtr etdtetdt

long-range short-range

α will be chosen for numerical convenience.

Ewald Method

Ali Hassanali

∫∫∞

−−−− +=2

2/1 111

ππtrtr etdtetdt

long-range short-range rr)(erf α

rr )(erfc)(erf-1 αα=

r rr)(erf α

rr)(erfc α

Ewald Method

( )∑∑

∞→

Nreal L

cell ,

erfc'21lim

∑ ∫∑ −+−−−

→∞→=

jiNrecipji

etdtqqE, 0

21limlim

ζ πnnrr

⎥⎥⎦

⎢⎢⎣

⎡⋅+⎟

⎞⎜⎝

⎛−⎟⎠

⎞⎜⎝

⎛= ∑∑∞

+−∞

LxxLe Lx π

ππ 2exp1 222/3

Jacobi sum formula.

is a reciprocal lattice vector for a cubic cell.

small x: left side is slowly convergent, right side rapidly convergent.

Ewald Method: Hidden Assumption

( ).01∑=

Simulations can be run with

( )[ ]

/2expexp21

αππ

παπ

jijirecip

−⎟⎠

⎞⎜⎝

⎛−

−⋅⎥⎥⎦

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎞⎜⎝

∑ ∑

rrmmmm

Tacit assumption of a uniform neutralizing charge density with total charge - Q.

Effect is to add an overall constant to Erecip .

Connecting MD to Thermodynamic Ensembles

39 Ali Hassanali

isolated system (N,V,E) closed system in contact with heat bath (N,V,T) – (Rossi)

closed system in contact with heat bath and volume reservoir (N,P,T)

open system in contact with heat bath (μ,V,T)

Microcanonical ensemble Canonical ensemble

Isothermal-isobaric ensemble Grand-Canonical Ensemble

Molecular Dynamics

Fluctuations of System Energy in NVE and NVT

40 Ali Hassanali

isolated system

system maintained at a temperature T

time energy exchange between system

and surroundings

system energy E

EU =What are the properties of these energy fluctuations?

Molecular Dynamics

Fluctuations in Thermodynamic Properties

Ideal atomic gas example

β23NE =

2⎟⎟⎠

⎞⎜⎜⎝

∂−=

βββNN

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

∂−=

βσ22

Relative fluctuations of energy/temperature in MD simulation

Constant Volume (NVT) vs Constant Pressure (NPT)

rigid container

Constant Volume Constant Pressure

Fluctuations in Volume at Constant Pressure

Consequences of

NVV 1∝

1 VVPV

−=⎟⎟⎠

⎞⎜⎜⎝

∂−β

BTNT VTkP

σκ =⎟⎟⎠

⎞⎜⎜⎝

∂−=

Some general theoretical considerations with classical MD

MD: Time vs Ensemble Averages

43 Ali Hassanali

The Ergodic Hypothesis

ATime=T −>∞lim 1

TdtA(p(t), x(t))

A(pN ,rN )ensemble

dpN drN∫ A(pN ,rN )δ H (pN ,rN )−E( )

Molecular Dynamics

What we can get from an MD simulation

Ensemble Average: Experimental measurement

ATime ≈ A(pN ,rN )ensemble

Classical Statistical Mechanics

44 Ali Hassanali

Classical statistical mechanics for 1 particles in 1-dimension

⎥⎥⎦

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−∝ ∑

exp),(1

iNN Vm

P rppr β

1=β N

N rrrrr …,,, 321↔

NN ppppp …,,, 321↔

Classical stat. mech. for N particles in 3-dimensions

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−∝ )(

2exp),(

xVmppxP β

Molecular Dynamics

45 Ali Hassanali

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−∝ )(

2exp),(

xVmppxP β

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−= )(

2exp1),(

CpxP β

Normalization: 1),( =∫∫∞

∞−

pxPdpdx

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−= ∫∫

∞−

xVmpdpdxC β

Molecular Dynamics

46 Ali Hassanali

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−= )(

2exp1),(

CpxP β

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−= ∫∫

∞−

xVmpdpdxC β

[ ] ⎥⎦

⎤⎢⎣

⎡−×−= ∫∫

∞−

∞− mpdpxVdx2

exp)(exp2

px CC ×=

[ ]px Cmp

CxV ⎥

⎤⎢⎣

⎡−

exp)(exp

)()( pPxP ×=

Probability factors

Molecular Dynamics

Properties of classical statistical mechanics

47 Ali Hassanali

Classical statistical mechanics

⎥⎥⎦

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−∝ ∑

exp),(1

iNN Vm

P rppr β

)()(),( NNNN PPP prpr =

[ ])(exp2

exp),(1

iNN Vm

P rppr ββ −⎥⎦

⎤⎢⎣

⎡−∝ ∑

Probability distribution factors.

In classical statistical mechanics, momentum and configurational fluctuations are strictly uncorrelated (statistically independent).

Molecular Dynamics

Properties of classical statistical mechanics

∫ ∑

∑∞

∞− =

⎥⎦

⎤⎢⎣

⎡−

⎥⎦

⎤⎢⎣

⎡−

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎞⎜⎜⎝

⎛= ∑

2exp2 p

ββπ

For simplicity, all mi = m here.

[ ])(exp1)( NN VZ

P rr β−= [ ]∫ −=V

NN VdZ )(exp rr β

Z is the configuration integral (partition function)

Most interesting features of equilibrium classical statistical mechanics arise from the configurational distribution .

Classical MD simulations of Light and Heavy Water

Classical statistical mechanics

)()(),( NNNN PPP prpr =

[ ])(exp2

exp),(1

iNN Vm

P rppr ββ −⎥⎦

⎤⎢⎣

⎡−∝ ∑

Compare simulations of H2O and D2O.

In classical stat. mech., is there a difference in their boiling or freezing point?

Is there a difference in their configurational distribution functions?

The Lagrangian and the Action

)(21),( 2

NN VmL rrrr −=∑=

),()]([t

NNN LdttA rrr !

∂L∂ri

−ddt∂L∂!ri

⎝⎜

⎠⎟ ⎟⎟

⎞⎜⎜⎝

∂−

∂−= ∑

1 21)( jj

mdtdV rr

Principle of least action

Important information extracted from MD simulations

The Pair-Correlation Function: RDF

ρ (2)(r, ʹr ) Probability that any particle is as r and any other particle is at r′.

)()()(ˆ)(ˆ),( )1()1(large )2( rrrrrr rr ʹ=ʹ⎯⎯⎯ →⎯ʹ ʹ− ρρρρρ

When large, densities at r and r′ are uncorrelated. rr ʹ−

)()(),(),( )1()1(

)2()2(

rrrrrr

ρρρg

1),( large )2( ⎯⎯⎯ →⎯ʹ ʹ−rrrrg

Define:

The RDF for LJ Liquid, Hard-Spheres, Water

)()2( rr ʹ−ρ

rr ʹ−

Hard Spheres/LJ Liquid Water

Number of Neighbors

54 Ali Hassanali

rgrdrRN0

2 )(4)( πρ = number of neighbors out to distance R.

1st solvation shell

Molecular Dynamics

What else can we calculate from the radial distribution function?

Thermodynamic Properties from RDF

E = 32NkT + N

24πρ drr2u(r)g(r)

= ρ −ρ2

6kTdr4πr3u '(r)g(r)

Total Energy

Pressure

Free energies from RDF

)( 21 rr −g

21 rr −

0 )( 21 rr −w

Under what conditions would the w(r) → v(r), the pair potential?

),(),(ln1 2121 rrrr wg ≡−β

Some Very Important References

B. J. Alder and T. E. Wainwright (1957). "Phase Transition for a Hard Sphere System". J. Chem. Phys. 27 (5): 1208.

A. Rahman (1964). "Correlations in the Motion of Atoms in Liquid Argon". Physical Review 136: A405-A411.

A. Rahman and Frank Stillinger (1964). “Molecular Dynamics Study of Liquid Water". Journal of Chemical Physics 55(7): 3336

The Very Basics of Molecular Dynamics · 2016-05-08 · The World of Molecular Mechanics 8 Ali...

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