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

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 −=

( )20

2)( θθθ −=kvang

Harmonic angle potentials:

θ

2, 3 and 4 particles bonded to each other


Dihedral angle potential:

vdihedral (ϕ ) = cos n ϕ −ϕ0( )⎡

⎣⎤⎦

φ

Peptide Backbone

Non-bonded interactions between molecules


i j

i j

i j

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

4)(rr

rvLJσσ

ε

r

vLJ(r) σ

ε

21/6 σ

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

Electrostatic Interactions


Define atomic-based charges: RESP, Lowdin …

ba

ba

base

qqvrr −

= ∑∑−

molecule secondon sites

moleculefirst on sites

Example: SPC/E water

model

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.

Why?

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

N

Δ

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)

4

vN (t) = rN (t +Δt)− rN (t −Δt)

2Δt+O(Δt)

2

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)

20

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.

i j

The minimum image between particles i and j in the drawing

are within the same cell. i i

i

i i i

i i

j j

j j j

j j j

Molecular Dynamics

Consider determining the interaction between two particles with PBCs

24

Minimum Image Convention

Ali Hassanali

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

i

j

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

i i

i i i

i i i

j j

j j j

j j j


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

i j

Molecular Dynamics

( )jii

ix vx

f rr −∂

∂−=, ( ) ji

iji x

v rrrr −∂

∂−ʹ−=

ji

jiji

i

xxx rr

rr−

−=−

∂

∂

( ) .,ji

jijiix

xxvf

rrrr

−

−−ʹ−=( )jiv rr −If , then

26

Force discontinuity under minimum image

Ali Hassanali Molecular Dynamics

L

j i j i

j i

j i j i

j i

( )jii

xi vx

f rr −∂

∂−=,

( )ji

jiji

xxv

rrrr

−

−−ʹ−=

( )Lvx

f jii

xi xrr ˆ, +−∂

∂−=

( )LLxx

Lvji

jiji xrrxrr

ˆˆ

+−

+−+−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.

27

Calculating force between two particles

Ali Hassanali

L

j i j i

j i

j i j i

j i

xj

fx,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.

28

Treatment of short-range interactions


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

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛−⎟⎠

⎞⎜⎝

⎛=612

4)(rr

rvLJσσ

ε

)(1 rvLJεσr

29

Shifted LJ Potential

Ali Hassanali

Massage the LJ Potential

)(1 rvLJε

σr

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

≥

<−⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛−⎟⎠

⎞⎜⎝

⎛=

cut

cutcutLJcutLJ

rr

rrrvrrrv,0

),(4)(

612

,

σσε

)(1, rv cutLJε

Force has a discontinuity at rcut .


Switching Function

Ali Hassanali

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

≥

<≤

<⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛−⎟⎠

⎞⎜⎝

⎛

=

cut

cutswitchLJ

rr

rrrrS

rrrr

rv

0

)(

4

)( 1

1

612

,

σσε

)(1, rv switchLJε σ

r

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.

rcut

rlist

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

( )∑ ∑ +−=∞→

cell

cell

N N

jijiNcell LvE

nnrr

,

'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

( )∑ ∑ +−=∞→

cell

cell

N N

jijiNcell LvE

nnrr

,

'21lim

( ) ( )∑ ∑ ∑⎥⎥⎦

⎤

⎢⎢⎣

⎡++−++−=

++ −+∈∈ ∈∈∞→

cell

cell

N N

ji

N

jijijiN

LvLvn

nrrnrrNa,Na Cl,Na

''21lim !

??=∞−∞+∞−∞=

rrdr

r

140

2∫∞

π

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

2

2 2/1

0

2/1 111

α

α

ππtrtr etdtetdt

r

long-range short-range

α will be chosen for numerical convenience.


Ewald Method

Ali Hassanali

∫∫∞

−−−− +=2

2

2

2 2/1

0

2/1 111

α

α

ππtrtr etdtetdt

r

long-range short-range rr)(erf α

rr

rr )(erfc)(erf-1 αα=

r rr)(erf α

−

r1

−

rr)(erfc α

−


Ewald Method


( )∑∑

+−

+−=

∞→

N

ji ji

jiji

N

Nreal L

LqqE

cell

cell ,

erfc'21lim

nrr

nrr

n

α

∑ ∫∑ −+−−−

→∞→=

N

ji

LLtN

jiNrecipji

cell

cell

etdtqqE, 0

2/1

0

2221

21limlim

αζ

ζ πnnrr

n

⎥⎥⎦

⎤

⎢⎢⎣

⎡⋅+⎟

⎠

⎞⎜⎝

⎛−⎟⎠

⎞⎜⎝

⎛= ∑∑∞

+−∞

rmmm

nr

n Li

LxxLe Lx π

ππ 2exp1 222/3

3

2

Jacobi sum formula.

Lm

is a reciprocal lattice vector for a cubic cell.

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

37

Ewald Method: Hidden Assumption


∑=

=N

iiq

1

0

( ).01∑=

≠=N

ii Qq

Simulations can be run with

( )[ ]

23

22

13

2

, 0

2

3

232

/2expexp21

αππ

παπ

π

LQq

L

LiLL

qqL

E

N

iii

ji

N

jijirecip

−⎟⎠

⎞⎜⎝

⎛−

−⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠

⎞⎜⎝

⎛=

∑

∑ ∑

=

∞

≠

−

r

rrmmmm

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

Effect is to add an overall constant to Erecip .

+ Z -

+ Z -

38

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

time

U = E

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 =

VNE

E

,

2⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂−=

βσ

2,

23

23

βββNN

VN

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂−=

NN

N

N

EE 1

32

23

23

2

∝==

β

βσ22

3β

σN

E =

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∝

σ

( )2,

1 VVPV

NP

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂−β

2

,

11V

BTNT VTkP

VV

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))

0

T

∫

A(pN ,rN )ensemble

=1Ω

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

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−∝ ∑

=

)(2

exp),(1

2N

N

i i

iNN Vm

P rppr β

TkB

1=β N

N rrrrr …,,, 321↔

NN ppppp …,,, 321↔

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

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−∝ )(

2exp),(

2

xVmppxP β

Molecular Dynamics


45 Ali Hassanali

TkB

1=β

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−∝ )(

2exp),(

2

xVmppxP β

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−= )(

2exp1),(

2

xVmp

CpxP β

Normalization: 1),( =∫∫∞

∞−

∞

∞−

pxPdpdx

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−= ∫∫

∞

∞−

∞

∞−

)(2

exp2

xVmpdpdxC β

Molecular Dynamics


46 Ali Hassanali

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−= )(

2exp1),(

2

xVmp

CpxP β

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−= ∫∫

∞

∞−

∞

∞−

)(2

exp2

xVmpdpdxC β

[ ] ⎥⎦

⎤⎢⎣

⎡−×−= ∫∫

∞

∞−

∞

∞− mpdpxVdx2

exp)(exp2

ββ

px CC ×=

[ ]px Cmp

CxV ⎥

⎦

⎤⎢⎣

⎡−

×−

=2

exp)(exp

2

ββ

)()( pPxP ×=

Probability factors

Molecular Dynamics

Properties of classical statistical mechanics

47 Ali Hassanali

Classical statistical mechanics

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−∝ ∑

=

)(2

exp),(1

2N

N

i i

iNN Vm

P rppr β

)()(),( NNNN PPP prpr =

[ ])(exp2

exp),(1

2N

N

i i

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


∫ ∑

∑∞

∞− =

=

⎥⎦

⎤⎢⎣

⎡−

⎥⎦

⎤⎢⎣

⎡−

=N

i i

iN

N

i i

i

N

md

mP

1

2

1

2

2exp

2exp

)(pp

p

pβ

β

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛= ∑

=

− N

i

iN

mm

1

22/3

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

2N

N

i i

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

1

Nii

N

i

NN VmL rrrr −=∑=

!!

rN(t)

t

∫=2

1

),()]([t

t

NNN LdttA rrr !

∂L∂ri

−ddt∂L∂!ri

⎛

⎝⎜

⎞

⎠⎟ ⎟⎟

⎠

⎞⎜⎜⎝

⎛

∂

∂−

∂

∂−= ∑

=

2

1 21)( jj

N

ji

N

i

mdtdV rr

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

ʹ

ʹ=ʹ

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

∫=R

rgrdrRN0

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

)(rg

1

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)

0

∞

∫

PkT

= ρ −ρ2

6kTdr4πr3u '(r)g(r)

0

∞

∫

Total Energy

Pressure

Free energies from RDF


)( 21 rr −g

21 rr −

1

0 )( 21 rr −w

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

),(),(ln1 2121 rrrr wg ≡−β

1 2

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

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

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