© copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 Ch121a Atomic Level...

© copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 1

Ch121a Atomic Level Simulations of Materials and Molecules

William A. Goddard III, [email protected] and Mary Ferkel Professor of Chemistry,

Materials Science, and Applied Physics, California Institute of Technology

Lecture 5b and 61, April 14 and 16, 2014MD2: dynamics

Room BI 115Lecture: Monday, Wednesday Friday 2-3pm

TA’s Jason Crowley and Jialiu Wang


Homework and Research Project

First 5 weeks: The homework each week uses generally available computer software implementing the basic methods on applications aimed at exposing the students to understanding how to use atomistic simulations to solve problems.

Each calculation requires making decisions on the specific approaches and parameters relevant and how to analyze the results.

Midterm: each student submits proposal for a project using the methods of Ch121a to solve a research problem that can be completed in the final 5 weeks.

The homework for the last 5 weeks is to turn in a one page report on progress with the project

The final is a research report describing the calculations and conclusions


Previously we develop the force fields and we minimized the geometry now we do dynamics


Classical mechanics – Newton’s equations

Newton’s equations are

Mk (d2Rk/dt2) = Fk = - RkE

where Rk, Fk, and are 3D vectors

and where goes over every particle k

Here Fk = Sn≠k Fnk where Fnk is the force acting on k due to particle n

From Newton’s 3rd law Fnk = - Fkn

Solving Newton’s equations as a function time gives a trajectory with the positions and velocities of all atoms

Assuming the system is ergotic, we can calculate the properties using the appropriate thermodynamic average over the coordinates and momenta.


Consider 1D – the Verlet algorithmNewton’s Equation in 1D is M(d2x/dt2)t = Ft = - (dE/dx)t

We will solve this numerically for some time step d. First we consider how velocity is related to distancev(t+d/2) = [x(t+d) – x(t)]/ ;d that is the velocity at point t+d/2 is the difference in position at t+d and at t divided by the time increment d. Also v(t-d/2) = [x(t) – x(t-d)]/d Next consider how acceleration is related to velocitiesa(t) = [v(t+d/2) - v(t-d/2)]/d = that is the acceleration at point t is the difference in velocity at t+ /2d and at t- /2d divided by the time increment d. Now combine to get a(t)=F(t)/M= {[x(t+d) – x(t)]/ ]d - [x(t) – x(t-d)]/d)}/d ={{[x(t+d) – 2x(t) + x(t-d)]}/d2 Thus x(t+d) = 2x(t) - x(t-d) + d2 F(t)/MThis is called the Verlet (pronounced verlay) algorithm, the error is proportional to d4. At each time t we calculate F(t) and combine with the previous x(t-d) to predict the next x(t+d)


Initial condition for the Verlet algorithmThe Verlet algorithm, x(t+d) = 2x(t) - x(t-d) + d2 F(t)/Mstarts with x(t) and x(t-d) and uses the forces at t to predict x(t+d) and all subsequent positions.But typically the initial conditions are x(0) and v(0) and then we calculate F(0) so we need to do something special for the first point.Here we can write, v(d/2) = v(0) + ½ d a(0) and thenx(d) = x(0) + d v(d/2) = x(0) + v(0) + ½ d F(0)/MAs the special form for getting x(d) Then we can use the Verlet algorithm for all subsequent points.


Consider 1D – the Leap Frog Verlet algorithm

Since we need both velocities and coordinates to calculate the various properties, I prefer an alternative formulation of the numerical dynamics.Here we write v(t+d/2) = v(t-d/2) + d a(t) and x(t+d) = x(t) + v(t+d/2) This is called leapfrog because the velocities leap over positions and the positions leap over the velocitiesHere also there is a problem at the first point where we have v(0) not v(d/2).Here we write v(0) = {v(d/2) + v(-d/2)]/2 andv(d/2) - v(-d/2) = d a(0) So that 2v(d/2) = d a(0) + 2 v(0) orv(d/2) = v(0) + ½ d F(0)/M


Usually we start with a structure, maybe a minimized geometry, but where do we get velocities?

For an ideal gas the distribution of velocities follows the Maxwell-Boltzmann distribution,

For N particles at temperature T, this leads to an average kinetic

energy of KE= Sk=1,3N (Mk vk2/2)= (3N)(kBT/2) where kB is

Boltzmann’s constant.

In fact our systems will be far from ideal gases, but it is convenient to start with this distribution.

Thus we pick a distribution of random numbers {a1,…,a3N} that has a Gaussian distribution and write vi =ai sqrt[(2kT/M)

This will lead to KE ~ (3N/2)(kBT), but usually we want to start the initial KE to be exactly the target bath TB.

Thus we scale all velocities by l such that KE = (3N/2)(kBTB)


Equilibration

Although our simulation system may be far from an ideal gas, most dense systems will equilibrate rapidly, say 50 picoseconds, so our choice of Maxwell-Boltzmann velocities will not bias our result

Often we may start the MD with a minimized structure. If so some programs allow you to start with double Tbath.

The reason is that for a harmonic system in equilibrium the Virial theorem states that

<PE>=<KE>.

Thus if we start with <PE>=0 and the KE corresponding to some Tinitial, the final temperature, after equilibration will be T ~½ Tinitial.


The next 4 slides are background material The Virial Theorem - 1

For a system of N particles, the Virial (a scalar) is defined as

G = Sk (pk ∙ rk) where p and r are vectors and we take the dot or inner product. Then

dG/dt = Sk (pk ∙ drk/dt) + Sk (dpk/dt ∙ rk)

= Sk Mk(drk/dt ∙ drk/dt) + Sk (Fk ∙ rk)

= 2 KE + Sk (Fk ∙ rk)

Where we used Newton’s equation: Fk = dpk/dt

Here Fk = Sn’ Fnk is the sum over all forces from

the other atoms acting on k, and the prime on the sum indicates that n≠k


The Virial Theorem - 2

Thus Sk (Fk ∙ rk) = Sk Sn’ (Fn,k ∙ rk)

= Sk Sn<k (Fn,k ∙ rk) + Sk Sn>k (Fn,k ∙ rk)

= Sk Sn<k (Fn,k ∙ rk) + Sk Sk>n (Fk,n ∙ rn)

= Sk Sn<k (Fn,k ∙ rk) + Sk Sk>n (-Fn,k ∙ rk)

= Sk Sn<k [Fn,k ∙ (rk – rn)]

Where line 3 relabeled k,n in 2nd term and line 4 used Newton’s 3rd law Fk,n = - Fn,k .

But Fn,k = - rk PE = - [d(PE)dr] [(rk – rn)/rnk]

Sk (Fk ∙ rk) = - Sk Sn<k [d(PE)dr] [(rk – rn)]2/rnk

= - Sk Sn<k [d(PE)dr] rnk



Using Sk (Fk ∙ rk) = - Sk Sn<k [d(PE)dr] rnk we get

dG/dt = 2 KE - Sk Sn<k [d(PE)dr] rnk

Now consider a system in which the PE between any 2 particles has the power law form,

PE(rnk) = a (rnk)p for example: p=2 for a harmonic system and p=-1 for a Coulomb system.

Then dG/dt = 2KE – p PEtot which applies to every time t.

Averaging over time we get

<dG/dt> = 2<KE> – p <PEtot>



<dG/dt> = 2<KE> – p <PEtot>

Integrating over time ʃ0t (dG/dt) dt = G( ) - t G(0)

For a stable bound system G(∞)=G(0)

Hence <dG/dt> = 0, leading to the Virial Theorem

2<KE> = p <PEtot>

Thus for a harmonic system

<KE> = <PEtot> = ½ Etotal

And for a Coulomb system

<KE> = -1/2 <PEtot>

So that Etotal = ½ <PEtot> = - <KE>


microcanonical Dynamics (NVE)

Solving Newton’s equations for N particles in some fixed volume, V, with no external forces, leads to conservation of energy, E.

This is referred to as microcanonical Dynamics (since the energy is fixed) and denoted as (NVE).

Since no external forces are acting on this system, there is no heat bath to define the temperature.


canonical Dynamics (NVT) – isokinetic energyWe will generally be interested in systems in contact with a heat bath at Temperature TB. In this case the states of the system should have (q,p) such that P(p,q) = exp[-H(p,q)/kBTB]/Z is the probability of the system having coordinates p,q. This is called a canonical distributions of states.However, though we started with

KE= Sk=1,3N (Mk vk2/2)= (3N/2)(kBTB) there is no guarantee that the

KE will correspond to this TB at a later time.A simple fix to this is velocity scaling, where all velocities are scaled by a factor, l, such that the KE remains fixed (isokinetic energy dynamics). Thus on each iteration we calculate the temperature corresponding to the new velocities,

Tnew = (2kB/3N) Sk=1,3N (Mk vk2/2), then we multiply each v by l, so

that (2kB/3N) Sk=1,3N (Mk (lvk)2/2) = TB where l =sqrt(TB/Tnew)Thus if Tnew is too high, we slow down the velocities.


Why scale the velocity rather than say, KE (ie v2)

We will take the center of mass of our system to be fixed, so that the sum of the momenta must be zero

P = Sk mkvk = 0 where P and v (and 0) are vectors.

Thus scaling the velocities we have

P= Sk mk (lvk) = lP = 0

If instead we used some other transformation on the velocities this would not be true.


Advanced Classical Mechanics Lagrangian Formulation

The Lagrangian equation of motion become

∂L (q, q)/∂q = (d/dt) [∂L(q, q)/∂q]

Where the momentum is p = [∂L(q, q)/∂q]

and dp/dt = [∂L(q, q)/∂q]

Letting KE=1/2 Mq2 this leads to

p = Mq and

dp/dt = -PE = F or F = m a, Newton’s equation

The Frenchman Lagrange (~1795) developed a formalism to describe complex motions with noncartesian generalized coordinates, q and velocities q=(dq/dt) (usually this is q with a dot over it)

L (q, q) = KE(q) – PE (q)


Advanced Classical Mechanics Hamiltonia Formulation

The Hamiltonian equation of motion become

∂H(p, q)/∂p = q

∂H(p, q)/∂q = - dp/dt

Letting KE=p2/2M this leads to

q = p/M

dp/dt = -PE = F or F = m a, Newton’s equation

The irishman Hamilton (~1825) developed an alternate formalism to describe complex motions with noncartesian generalized coordinates, q and momenta p. Here the Hamiltonian H is defined in terms of the Lagrangian L as

H(p, q) = pq– L (q, q)


Statistical Ensembles - Review

There are 4 main classes of systems for which we may want to calculate properties depending on whether there is a heat bath or a pressure bath.

Microcanonical Ensemble (NVE)

1

1

[ ( , ; ) ] ( , ; )( , , )

! ( , , )

[ ( , ; ) ]where ( , , )

!

N N N N N N

VNVE

N N N N

V

dr dp r p V E F r p VF N V E

N N V E

dr dp r p V EN V E

N

Canonical Ensemble (NVT)

1

1

( , ; )exp[ ] ( , ; )

( , , )! ( , , )

( , ; )exp[ ]

where ( , , ) !

N NN N N N

V

NVT

N NN N

V

r p Vdr dp F r p V

kTF N V TN Q N V T

r p Vdr dp

kTQ N V TN

Isothermal-isobaric Ensemble (NPT)

1

0

1

0

[ ( , ; )]exp[ ] ( , ; )

( , , )! ( , , )

[ ( , ; )]exp[ ]

where ( , , ) !

N NN N N N

V

NPT

N NN N

V

PV r p VdV dr dp F r p V

kTF N P T

N N P T

PV r p VdV dr dp

kTN P T

N

Isoenthalpic-isobaric Ensemble (NPH)

1

0

1

0

[ ( , ; ) ] ( , ; )

( , , )! ( , , )

[ ( , ; ) ]

where ( , , ) !

N N N N N N

V

NPH

N N N N

V

dV dr dp r p V PV H F r p V

F N P HN N P H

dV dr dp r p V PV H

N P HN

Solve Newton’s Equations

Normal experimental conditions

Usual way to solve MD Equations


Calculation of thermodynamics properties

Depending on the nature of the heat and pressure baths we may want to evaluate the ensemble of states accessible by a system appropriate for NVE, NVT, NPH, or NPT ensembles

Here NVE corresponds to solving Newton’s equations and NPT describes the normal conditions for experiments

We can do this two ways using our force fields

Monte Carlo sampling considers a sequence of geometries in such a way that a Boltzmann ensemble, say for NVT is generated. We will discuss this later.

Molecular dynamics sampling follows a trajectory with the idea that a long enough trajectory will eventually sample close enough to every relevant part of phase space (ergotic hypothesis)


Now we have a minimized structure, let’s do Molecular Dynamics

This is just solving Newton’s Equation with time

Mk (∂2Rk)/∂t2) = Fk = - -(∂E/∂Rk) for k=1,..,3N

Here we start with 3N coordinates {Rk}t0 and 3N velocities {Vk}t0 at time, t0, then we calculate the forces {Fk}t0 and we use this to predict the 3N {Rk}t1 and velocities {Vk}t1 at some later time step t1 = t0 + dt


Ergodic Hypothesis

Time/Trajectory average MD

Ensemble average

MC

ThermodynamicsKinetics


Canonical Dyanamics (NVT) - Langevin Dynamics

Consider a system coupling to a heat bath with fixed reference temperature, TB.. The Langevin formalism writes Mk dvk/dt = Fk – gk Mk vk + Rk(t)

0( ) ( ) 2 ( )i j i i ijR t R t m kT

The damping constants gk determine the strength of the coupling to the heat bath and where Ri is a Gaussian stochastic

variable with zero mean and with intensity

The Langevin equation corresponds physically to frequent collisions with light particles that form an ideal gas at temperature TB.Through the Langevin equation the system couples both globally to a heat bath and also subjected locally to random noise.


canonical Dynamics (NVT)-Berendsen Thermostat

The problem with isokinetic energy dynamics is that the KE is constant, whereas a system of finite size N, in contact with a heat bath at temperature TB, should exhibit fluctuations in temperature about TB, with a distribution scaling as 1/sqrt(N)A simple fix to this problem is the Berendsen Thermostat. Here the velocities are scaled at each step so that the rate of change of the T is proportional to the difference between the instantaneous T and TB, dT(t)/dt = [TB – T(t)]/ twhere the damping constant t determines the relaxation timeThis leads to a change in T between successive steps spaced by d is DT = ( / )d t [TB-T(t)], which leads to l2=1+ ( / )d t [TB/T(t-d/2) – 1], since we use leap frog for the velocities. (Note that = t d leads to isokinetic energy dynamics)The Berendsen Equation of motion isMk dvk/dt = Fk – (Mk/t)(TB/T – 1)vk


Canonical dynamics (NVT): Andersen Method

( ) tP t e where n is the collision frequency (default 10 fs-1 in CMDF).It is reasonable to require the stochastic collision frequency to be the actual collision frequency for a particle.

Andersen NVT does produce a Canonical distribution.Indeed Andersen NVT generates a Markov chain in phase space, so it is irreducible and aperiodoic.Moreover, Andersen NVT does not generate continuous (real) dynamics due to stochastic collision, unless the collision rate is chosen so that the time scale for the energy fluctuation in the simulation equal to correct values for the real system

Hans Christian Andersen (the one at Stanford, not the one that wrote Fairy tales) suggested using Stochastic collisions with the heat bath at the probability


Size of Heat bath damping constant t

The damping of the instantaneous temperature of our system due to interaction with the heat bath should depend on the nature of the interactions (heat capacity,thermal conductivity, ..) but we need a general guideline.

One criteria is that the coupling with the heat bath be slow compared to the fastest vibrations. For most systems of interest this might be CH vibrations (3000 cm-1) or OH vibrations (3500 cm-1). We will take (1/l)=3333 cm-1

Since ln=c the speed of light, the Period of this vibration is

T = 1/n = ( /l c) = 1/[c(1/l)] = 1/[(3*1010 cm/sec)(3333)] = =1/[1014] = 10-14 sec = 10 fs (femtoseconds)

Thus we want t >> 10 fs = 0.01 ps. A good value in practice is t = 100 fs = 0.1 ps. Better would be 1 ps, but this means we must wait ~ 20 t = 20 ps to get equilibrated properties.


Canonical Dynamics (NVT) – Nose-HooverThe methods described above do not lead properly to a canonical ensemble of states, which for a Hamiltonian H(p,q) has a Boltzmann distribution of states. That is the probability for p,q scales like exp[- H(p,q)/kBTB]. Thus these methods do not guarantee that the system will evolve over time to describe a canonical distribution of states, which might invalidate the calculation of thermodynamic and other properties.

S. Nose formulated a more rigorous MD in which the trajectory does lead to a Boltzmann distribution of states. S. Nosé, J. Chem. Phys. 81, 511 (1984); S. Nosé, Mol. Phys. 52, 255 (1984)

Nosé introduced a fictitious bath coordinate s, with a KE scaling like Q(ds/dt)2 where Q is a mass and PE energy scaling like [gkBTB ln s] where g=Ndof + 1.

Nose showed that microcanonical dynamics over g dof leads to a probability of exp[- H(p,q)/kBTB] over the normal 3N dof.


Canonical Dynamics (NVT) – Nose-Hoover

Bill Hoover W. G. Hoover, Phys. Rev. A 31, 1695 (1985)) modified the Nose formalism to simplify applications, leading to

d2Rk/dt2 = Fk/Mk – g (dRk/dt)

(dg/dt) = (kB Ndof/Q)T(t){(g/Ndof) [1 - TB/T(t))]}

where g derives from the bath coordinate.

The 1st term shows that g serves as a friction term, slowing down the particles when g>0 and accelerating them when g<0.

The 2nd term shows that TB/T(t) < 1 (too hot) leads to (dg/dt) > 0 so that g starts changing toward positive friction that will eventually start to cool the system. However the instantaneous g might be negative so that the system still heats up (but at a slower rate) for a while.

Similarly TB/T(t) > 1 (too cold) leads to (dg/dt) > 0 so that g starts changing toward negative friction.


Canonical Dynamics (NVT) – discussionThis evolution of states from Nose-Hoover has a damping variable g that follows a trajectory in which it may be positive for a while even though the system is too cold or negative for a while even though the system is to hot. This is what allows the ensemble of states over the trajectory to describe a canonical distribution.

But this necessarily takes must longer to converge than Berendsen which always has positive friction, moving T toward TB at every step

I discovered the Nose and Hoover papers in 1987 and immediately programmed it into Biograf/Polygraf which evolved into the Cerius and Discover packages from Accelrys because I considered it a correct and elegant way to do dynamics.

However my view now is that for most of the time we just want to get rapidly a distribution of states appropriate for a given T, and that the distribution from Berendsen is accurate enough.


The next 3 slides are background material details about Nose-Hoover - I

S. Nose introduced (1984) an additional degree of freedom s to a N particle system:

i iv s r

2 2 2( ) ( 1) ln2 2i

i eqi

m QL s r r s f kT s

.( )d L L

dt AA

2( )i ii

dm s r

dt r

2

1 2i i

i i

sr r

m s r s

2 ( 1) eqii

i

f kTQs m s r

s

The Lagrangian of the extended system of particles and s is postulated to be:

Equation of motion:

For particles:

For s variable:

or


details about Nose-Hoover - II

2 2

1( 1)

i ii

eq

m s rf kT

s s

2i i i

i

Lp m s r

r

s

Lp Q s

s

2 2

1 2( ) ( 1) ln

2 2i s

eqi i

p pH r f kT s

m s Q

The averaged kinetic energy coincides with the external TB:

Momenta of particle:

Momenta of s:

The Hamiltonian of the extended system (conserved quantity, useful for error checking):


details about Nose-Hoover - IIIAn interpretation of the variable s: a scaling factor for the time

step Dt ' tt

s

The real time step Dt’ is now unequal because s is a variable.Q has the unit of “mass”, which indicates the strength of coupling. Large Q means strong coupling.

Nose NVT yields rigorous Canonical distribution both in coordinate and momentum space.

Hoover found Nose equations can be further simplified by introducing a thermodynamics friction coefficient x: ' ' /ss p Q Nose-Hoover formulation is most rigorous and widely used thermostat.S. Nose, Molecular Physics, 100, 191 (2002) reprint; Original paper was published in 1983.


250 260 270 280 290 300 310 320 3300

20

40

60

80

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Co

un

t

Temperature (K)

Nose-Hoover Andersen Berendsen

NVT (T=300 K) 216 Water

All 3 thermostats yield an average

temperature matching the target TB. All 3 allow similar

instantaneous fluctuations in T around target TB (300 K). Nose

should be most accurate

250 260 270 280 290 300 310 320 3300

20

40

60

80

100

120

140

Co

un

t

Temperature (K)



250 260 270 280 290 300 310 320 3300

20

40

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80

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120

140

Co

un

t

Temperature (K)



250 260 270 280 290 300 310 320 3300

20

40

60

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140

Count

Temperature (K)



216 waters at 300K

Instantaneous Temperature Fluctuations


Stopped April 14, 2014


What about volume and pressure?

For MD on a finite molecule, we consider that the molecule is in some box, but if the CM of the molecule is zero the CM stays fixed so that we can ignore the box.

To describe the bulk states of a gas, liquid, or solid, we usually want to consider an infinite volume with no surfaces, but we want the number of independent molecules to be limited, (say 1000 independent molecules)

In this case we imagine a box (the unit cell), which for a gas or liquid could be cubic, repeated in the x, y, and z directions through all space.

If the cell has fixed sizes then there will be a pressure (or stress) on the cell that fluctuates with time (NVE, NVT)

We can also adjust the cell sizes from step to step to keep the pressure constant (NPH or NPT)


Periodic Boundary Conditions for solids

Consider a periodic system with unit cell vectors, {a, b, c} not necessarily orthogonal but noncoplanarWe define the atom positions within a unit cell in terms of scaled coordinates, where the position along the a, b, c axes is (0,1). We define the transformation tensor h={a,b,c} that converts from scaled coordinates S to cartesian coordinate r, r=hS, In the MD, we expect that the atomic coordinates will adjust rapidly compared to the cell coordinates. Thus for each time step the forces on the atoms lead to changes in the particle velocities and positions, which are expressed in terms of scaled coordinates, then the stresses on the cell parameters are used to predict new values for the cell parameters {a, b, c} and their time derivatives.Here the magnitude of rk is ri

2= (si|G|si) where G=hTh is the metric tensor for the nonorthogonal coordinate system and T indicates a transpose so that G is a 3 by 3 metric tensor


NPH Dynamics: Berendsen Method

Berendsen NPT is similar to Berendsen NVT, but the scaling factor scales atom coordinates and cell length by a factor l each time step

1/30[1 ( )]

P

tP P

Note that Berendsen NPH does NOT allow the change of cell shape. This is appropriate for a liquid, where the unit cell is usually a cube.

The cell length and atom coordinates will gradually adjust so that the average pressure is P0. The time scale for reaching equilibrium is controlled by tP, the coupling strength. tP is related to the sound speed and heat capacity.larger tP weaker coupling slower relaxationI recommend tP = 20 tT = 400 d (time step) Thus d = 1 fs (common default) tT = 0.02 ps tP = 0.4 ps.


The next 3 slides are background materialNPH: Andersen Method I

1/3/ , 1, 2,..., .i ir V i N

2/3 1/3 2

1 1

1 1L , , ,

2 2

N NN N

ii iji i j

Q Q mQ u Q M Q Q

2/32i i

i

LmQ

2L M QQ

Use scaled coordinates for the particles. For a cubic cell V1/3 = L the cell length,ri =(0,1)

Andersen defined (1980) a Lagrangian by introducing a new variable Q, related to the cell volume

Particle momentum:

Volume momentum:

aQ ~ PVCell KEatom KE u ~ atom PE

Replaces p = mv

Similar to atom momentum but for the cell


NPH: Andersen Method II

21

2/3 1 1/3 1 2

1 1

( , , , ) ( , , , )

(2 ) ( ) (2 )

NN N N N

i ii

N N

i i iji i j

H Q Q L Q Q

mQ u Q M Q

22/3

i i

i

d H

dt mQ

1/3

1/32

( ) 1

( )

Nij iji

j ii ij

u Qd HQ

dt

2HdQ

dt M

1 2/3 1 1/3 1/32

1

(3 ) 2(2 ) ( ) 3N

i i ij iji i j

HdQ mQ Q u Q Q

dt Q

Equations of motion:

Hamiltonian of the extended system:

Internal stress scalar

External stress scalar

Replaces v = p/m

Replaces dp/dt = F = -E

High M slow Cell changes

aQ ~ PVCell KEatom KE atom PE


NPH: Andersen Method III

The variable M can be interpreted as the “mass” of a piston whose motion expands or compresses the studied system.

M controlls the time scale of volume fluctuations. We expect M ~ L/c where c is the speed of sound in the sample and L the cell length

Andersen NPH dynamics yields isoenthalpic-isobaric ensemble.

In 1980 Andersen suggested “It might be possible to do this (constant temperature MD) by inventing one or more additional degree of freedom, as we did in the constant pressure case. We have not been able to do this in a practical way.”

~ 3 years later, Nose figured out how to implement Andersen’s idea!

Hans C. Andersen, J. Chem. Phys. 72, 2384 (1980)


NPH: Parrinello-Rahman Method I

1 1( ) ( )

2 2i i i ij exti j i

L m sG s r W Tr h h p

1 1

1

( )( ) , , =1,2,....,i i ij i j ij

s m r s s G G s i j N

1( )exth W p

Parrinello and Rahman (1980) defined a Lagrangian by involving the time dependence of the cell matrix h:

Leading to the equations of motion:

PVCell KEatom KE atom PEφ = (PE)

Replaces v = p/m

Rate of change in cell is proportional to difference in external stress and the internal stress

Proportional to v, thus is friction

Pext could be Sext


NPH: Parrinello-Rahman Method II

1 1

1

( )( ) , , =1,2,....,i i ij i j ij

s m r s s G G s i j N

1( )exth W p

( )( )( )i i i ij i j i ji i j i

mv v r r r r r

��

RP Equations of motion:

where

the internal stress tensor.

/ ijh /d rdr

Cell KE

Dynamics of cell changes

Newton’s Eqn Friction from cell changes


NPH: Parrinello-Rahman Method III

Hamiltonian of the extended system:. .

2 '1/ 2 ( ) 1/ 2ii ij

i i j i

H mv r WTr h h p

The Parrinello-Rahman method allows the variation of both size and shape of the periodic cell because it uses the full cell matrix (extension of Andersen method).

An appropriate choice for the value of W (Andersen) is such that the relaxation time is of the same order of magnitude as the time L/c, where L is the MD cell size and c is sound velocity. Actually this is much longer than defaults (tP = 0.4 to 2 ps)

Static averages are insensitive to the choice of W as in classical statistical mechanics (the equilibrium properties of a system are independent of the masses of its constituent parts).

PVCell KEatom KE atom PE φ = (PE)

dhT/dt

More general: pstress and

Vh dependent

term


NPT: combine thermostat and barostat

Barostat-Thermostat choicesBerendsen-BerendsenAndersen-AndersenAndersen-(Nose-Hoover)(Parinello-Rahman)-(Nose-Hoover)Other hybrid methods

Thermostats and barostats in CMDF are fully decoupled functionality modules and user can choose different thermostats and barostats to create a customized hybrid NPT method.


Martyna-Tuckerman-Klein Methods

The original Nose-Hoover method generates a correct Canonical distribution for molecular systems using a single coupling degree of freedom. This is appropriate if there is only one conserved quantity or if there are no external forces and the center of mass remains fixed. (This is normal case.)

• Martyna, Tuckerman and Klein [G. J. Martyna, M. E. Tuckerman, D. J. Tobias and M. L. Klein "Explicit reversible integrators for extended systems dynamics", Molecular Physics 87 pp. 1117-1157 (1996) extended the Nose-Hoover thermostat to use Nose-Hoover chains, where multiple thermostats couple each other linearly.

They also developed a reversible multiple time step integrator which can solve NPH dynamics explicitly without iteration (1996).

http://dx.doi.org/10.1080/00268979600100761

http://dx.doi.org/10.1080/00268979600100761

http://dx.doi.org/10.1080/00268979600100761

http://dx.doi.org/10.1080/00268979600100761

http://dx.doi.org/10.1080/00268979600100761

http://dx.doi.org/10.1080/00268979600100761

http://dx.doi.org/10.1080/00268979600100761

http://dx.doi.org/10.1080/00268979600100761

http://dx.doi.org/10.1080/00268979600100761


NPT dynamics at different temperatures (P= 0 GPa, T= 25K to 300K)

25K

300K


Compression becomes harder when pressure increases, as indicated by the smaller volume reduction.

<V> (A3)

588.7938

525.7325

489.6471

469.1264

452.1472439.3684432.4970

NPT dynamics under different pressures (300K, 0 to 12 GPa) for PETN


New ReaxFF can predict compressed structures well even without further fitting (same parameter as that in 100 K, 0 GPa fitting).

Exp. (300 K)

ReaxFF+Disp

How well can we predict volume as a function of pressure?


Conclusion

Most rigorous thermostat: Nose-Hoover (chain), but in practice often use Berendsen

Most rigorous barostat: Parrinello-Rahman, but in practice often use Berendsen

Stress calculations usually need higher accuracy than for normal fixed volume MD. Therefore, we often use choose NVT instead of NPT, but we use various sized boxes and calculate the free energy for various boxes to determine the optimum box size for a given external pressure and temperature (even though this may require more calculations).

© copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 Ch121a Atomic Level...

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Transcript of © copyright 2013-William A. Goddard III, all rights reservedCh120a-Goddard-L06 Ch121a Atomic Level...