8. Molecular Dynamics

7/26/2019 8. Molecular Dynamics

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MOLECULAR DYNAMICS

Molecular dynamics (MD) calculations are dynamic studies of many-body systems inwhich equations of motion of individual particles are solved explicitly. These

equations may be either classical or quantum mechanical. As a result, we obtain the

structural as well as dynamical properties of the system studied.

The approach taken by MD is to solve equations of motion numerically on a

computer. Thus in an MD simulation we compute trajectories of a collection of

particles (which may but need not have intrinsic degrees of freedom) in the phase

space (defined by positions and momenta or velocities of the particles).

Two types of molecular dynamics studies

1.

Studies of the thermodynamical equilibrium of a system of particles subject to

certain boundary conditions. This approach is usually adopted when studying

dependence of certain physical quantities on temperature, pressure, applied loads

etc. Examples are calculation of the dependence of the specific heat, diffusivity,

elastic moduli and other physical quantities on temperature. However, it also

includes investigation of the structure (e. g. crystal vs liquid) as a function of

temperature but not the dynamics of processes of structural transformations.

2.

Investigations of the dynamical development of a system computer

experiments. This may involve study of the development from a non-

equilibrium state to a final equilibrium state, investigation of the changes of thesystem induced by externally applied fields exerting forces on particles, studies

of the mechanisms of phase transformations induced by changes of temperature

or various external parameters. e. g. pressure.

STAGES OF MD SIMULATION

Construction of the relaxed block and boundary conditionsThe initial construction of the block of particles proceeds as described in the section

dealing with General Aspects of computer modeling. Non-periodic, periodic or semi-

periodic boundary conditions can be used.

Initialization

Assignment of initial conditions for the equations of motion.

This may be done in two different ways:


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(i) Coordinates and velocities of the particles are given at the beginning, chosen astime = 0.

(ii) Coordinates of the particles are given at the time = 0 and at a time !t.

The precise choice of initial conditions may not be crucial since, ultimately, the

system will loose memory of the initial state. However, this depends on whether

the initial state is a physically well defined state and we investigate the dynamic

development of the system from this state or whether it is only a startingconfiguration used to attain the thermodynamic equilibrium during the calculation. In

the former case we are performing a computer experiment and in the latter case we

study the thermodynamic equilibrium of a system of particles subject to certain

boundary conditions.

Equilibration

When studying the thermodynamic equilibrium the initial state is usually not an

equilibrium state in mechanical equilibrium as defined by equation (G5). Hence, the

system must be relaxed (develop in the phase space, i. e. when both particlecoordinates and velocities change) by integrating equations of motion for a number of

time steps until the system has settled to definite mean values of the kinetic and

potential energies. In this process the choice of !t, which plays the role of the timestep used in the numerical integration of equations of motion, is crucial. The mostappropriate choice of !t is such that it assures the stability of the solution and at thesame time the maximum possible speed of calculations. This can only be achieved

by trial and error and by gradually building up the experience.

In MD calculations the total vector of velocity V = v ii

! ,where v iis the velocity of

the particle I (a vector), must be permanently set equal to zero. This means that no

rigid body motion of the assembly takes place.

EQUATIONS OF MOTION AND THEIR INTEGRATION

All the molecular dynamics algorithms for the study of a classical (not quantum)

system integrate Newtons equations of motion

mid 2 r

i

!

dt2=Fi

!

(! =1,2,3; i =1,2,....,N) (MD1)

where the vector ri determines position of the particle i and Fi is the force acting on this

particle (see also equations G3 and G4). Here i numbers particles and "coordinates 1,


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2, 3 in the 3D case and 1, 2 in the 2D case). Since the energy of the system, equal to the

sum of the potential and kinetic energy, is the first integral of equations of motion, it

must remain constant during integration. If it does not it indicates that the calculation is

erroneous. However, the kinetic and potential energies vary during the integration

process and settle to certain mean values when the equilibrium has been attained. This

is shown schematically in Fig. 1.

Time

Energy

Potential energy

Total energy

Kinetic energy

Fig. 1. Total, potential and kinetic energies as functions of time in MD simulations.

Furthermore, in the integration algorithms summarized below, the total number of

particles and the total volumeare also preserved. Consequently, in these algorithms

the system is treated from the statistical mechanics point of view as a microcanonical

ensemble. However, later on we shall discuss the situation when the volume of the

system may be changing.

Verlet algorithm

Procedure employing two successive positions of the particles

At the start we specify ri t = 0( ) and ri t = !t( ) . Using the central difference methodfor numerical evaluation of the second derivative, equation (MD1) can be written as

d2ri t( )

dt2 !

1

"t( )2 ri t +"t( )#2ri t( ) + ri t# "t( )$% &' =

1

mi

Fi t( ) (MD2)

and therefore

ri t + !t( ) = 2ri t( ) " ri t " !t( ) +!t( )

2

mi

Fi t( ) (MD3.1)

This is the basic recursion formula for the MD simulation, which then proceeds as

follows:


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At the time step J, when particles are at positions ri J!t( ) , we evaluate the force

acting on each particle, Fi J!t( ) . Positions of particles at the time step J+1,

ri (J +1)!t( ) , are then found using (MD3.1) with t = J!t. as

ri ( J + 1)!t( ) =2ri J!t( )" ri (J " 1)!t( ) +

!t( )2

m iF

i t( ) (MD3.2)

Velocities v i J!t( ) at the time step J are then

v i J!t( ) =ri (J +1)!t( )" ri (J "1)!t( )

2!t (MD4)

Procedure employing positions and velocities of the particles

At the start we specify initial positions ri t = 0( ) and initial velocities v i t = 0( ) .

Integration of equation (MD1) over a short period of time !t during which the force

Fi

t( ) and the velocity vi

t( ) can be considered to be constant gives:

rit + !t( ) = r

it( )+ v

it( )!t +

!t( )2

2mi

Fit( )

Following this equation the MD simulation can then proceeds as follows: At the time

step J, when particles are at positions ri J!t( ) and have velocities vi J!t( ) , weevaluate the force acting on each particle, Fi J!t( ) . Positions ri (J +1)!t( ) at the timestep J+1 are then evaluated as

ri(J+1)!t( ) =ri J!t( )+ vi J!t( )!t+

!t( )2

2mi

FiJ!t( ) (MD5)

and velocities at the time step J+1 as

v i (J + 1)!t( ) = v i J!t( ) + !t( )2m i

Fi (J + 1)!t( ) + Fi J!t( )( ) (MD6)

In (MD6) the average of forces in the Jth and (J+1)th iterations are used rather than

the force in the (J+1)th iteration only. (If positions in the (J+1)th iteration are already

determined then the force in the (J+1)th can be evaluated).


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The two procedures are, of course, equivalent. Equation (MD3.2) can be written as

2ri (J +1)!t( ) =2ri J!t( ) + ri (J +1)!t( ) " ri (J "1)!t( )+

!t( )2

mi

Fi t( )

and using equation (MD4) we obtain

2ri (J +1)!t( )=2ri J!t( ) +2!tv i J!t( ) +

!t( )

2

mi

Fi t( )

which, when divided by 2, is the same as equation (MD5).

Predictor-corrector algorithm

Using Taylor expansion to the third order we can write the predictedvalues of the

positions of the particles, ri

p, velocities, v

i

p, and accelerations, a

i

p in the J+1 time

step as

rip(J +1)!t( ) = ri J!t( ) + !tv i J!t( ) +

1

2!t

2a i J!t( ) +

1

6!t

3b i J!t( )

v ip(J + 1)!t( ) = v i J!t( ) + !ta i J!t( )+

1

2!t

2b i J!t( )

a ip (J + 1)!t( ) = a i J!t( ) + !tb i J!t( )

(MD7.1)

where ri , vi and a i are the positions, velocities and accelerations in the step J; b i isthe time derivative of the acceleration in the step J. In the predictor step we take

b ip

(J + 1)!t( ) = b i J!t( ) (MD7.2)

Equation of motion is now introduced by evaluating the accelerations in accordance

with the forces:

a i J!t( ) =1

mi

Fi J!t( ) (MD8)

Hence, the 'correct' acceleration at the step J+1 can be evaluated using (MD8) as

ai(J + 1)!t( ) = Fi (J + 1)!t( ) / m i and the error in the predictor step is

!a i (J + 1)!t( ) = a i (J + 1)!t( ) " a ip

(J + 1)!t( ) (MD9)


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The corrector stepis then introduced that determines the values of ri,v i,a i and b i inthe step (J+1) as follows:

ri (J + 1)!t( ) =rip(J + 1)!t( ) +c0!a i (J + 1)!t( )

v i (J +1)!t( ) = v ip (J + 1)!t( ) +c1!a i (J + 1)!t( )

a i (J +1)!t( ) = a ip (J + 1)!t( ) +c2!a i (J + 1)!t( )

b i (J + 1)!t( ) = b ip (J +1)!t( ) +c3!a i (J + 1)!t( )

(MD10)

The coefficients c0 ,c1,c 2 and c3 are 'suitably' chosen such as to assure stability and

fast convergence. This is again achieved through experience when using this

algorithm.

What is the best algorithm?This cannot be decided generally since it depends on the problem. The following are

the general rules when choosing an algorithm.

1) Calculation must be fast and require as little memory as possible.

2) The time step !t should be as long as possible but the solution must remainstable. This means: Solution follows the classical trajectory along which

the energy and momentum are conserved- the procedure is thus stable.

3) Must be time reversible.

AVERAGE VALUES OF PHYSICAL QUANTITIES

When comparing MD calculations with experiments or other calculations the physical

quantities measured always correspond to values averaged over a long enough period

of time. Computation of average values of physical quantities is done by continuing

the MD relaxation along the trajectory of the system in the phase space, i. e. space of

positions and velocities of particles, for a sufficiently long time. The time average of

a quantity A is then generally defined as

A = lim(! " #)1

!A(t)

0

!

$ dt (MD11.1)

and in molecular dynamics calculations this corresponds to


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A =1

M A(J

J=1

M

! "t) (MD11.2)

where M is the total number of MD steps (after equilibration!)and !t the time step;

this means ! =M"t. This formula is exact only when M!" but in practice large

values of M are used. However, how many MD steps are needed to obtain theaverage value with a sufficient precision cannot be easily decided and most

commonly, it is again done by testing for several different lengths of time M!t .

Connection with Statistical Mechanics - Ergodic Theorem

In order to make a link with statistical mechanics we consider that the accessible

space for the system studied is the 3N dimensional space defined by the 3N

dimensional vector X = (r1,r2,....,r

N) , where r

1,r2,....,r

Nare the position vectors of the

particles. This vector describes possible spatial configuration of the system studied.Similarly, accessible velocities of all the particles are described by the 3N

dimensional vector !X = (!r1,!r2,....,!r

N) . The energy of the system (kinetic plus

potential) is H(X,!X) and depends, in general, on both positions and velocities of

particles. Positions and velocities !X define a 6N dimensional space, called the

phase spaceof the system studied, and we denote vectors in this space Q ! (X, !X) .

If ) is the distribution function1 in the phase space, i. e. the probability that the

particles are in the phase space at a point defined by the vector Q! (X, !X) , then the

average value of a physical quantity A, evaluated within the statistical physics

approach, is

A =Z!1

A(Q)F(Q)dQ

Phase space

" (MD12.1)

where

Z = F(Q)dQPhasespace

! (MD12.2)

is the so called partition function; the integration extends over the whole phase

space and dQ = dXd!X .

The ergodic theorem of the statistical mechanics links the averages over time with

averages over statistical ensembles. It states that for !" #

1 The distribution function is defined by the physics of the problem studied. An example of the

distribution function is the Boltzmann distribution exp(!H / k BT), where T is the temperature

and kB

the Boltzmann constant.


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A = A (MD13)

This means that the average over time is the same as the average over the phase space

with appropriate distribution function. When evaluating average quantities it is

always assumed that #=M!t

is 'long enough' so that (MD13) is valid. Averagequantities of interest in MD simulations are the average energy (preserved in the

microcanonical case), average kinetic and potential energies, average stresses and

pressure and any other physical quantity one may be investigating. The instantaneous

values of some of these quantities have been defined in the Chapter General aspects

of atomistic computer modeling.

Temperature

The kinetic energy of the system determines its temperature whenv

ii!

= 0.

Following the equipartition theorem, the average kinetic energy of a three

dimensional system of particles with no internal degrees of freedom is

1

2m i v i

2

i=1

N

! =3

2NkBT , (MD14)

where N is the total number of particles in the system, mi the mass of the particle i

andv

i

2

the average of the square of the velocity of this particle. In general, if there

are !degrees of freedom per particle Kinetic energy =!

2NkBT . (See derivation in

the Appendix). The situation when !> 3 may arise, for example, when considering

molecules with internal rotational and vibrational degrees of freedom. For a two-

dimensional system of particles with no internal degrees of freedom != 2.

Use of MD for structural studies not involving temperature:Molecular statics calculations using MD

The energy of the system studied, that is composed of the kinetic and potential

energy, is for the microcanonical case fixed by the starting conditions that are not

necessarily physically meaningful. Since it does not change during the relaxation

process this value of the energy is imposed on the system. If the goal is to find a

minimum potential energystructure as in molecular statics calculations, we need to


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deal with the situation corresponding to 0K when all the velocities are zero. This canbe achieved using MD by decreasing the velocities and thus the kinetic energy

gradually to zero in the following way.

We first carry out MD simulation for a number of steps until the thermodynamic

equilibrium has been attained. At this point the temperature is finite but arbitrary.

We then set the velocities to zero and continue the MD simulations. The velocitiesagain become finite. After some equilibration we again set the velocities to zero and

continue the MD simulations. This procedure is repeated until the desired precision

of zero velocities is attained. At this point the potential energy is minimized.

MOLECULAR DYNAMICS AT CONSTANT

TEMPERATURE

Quantities conserved: Temperature, T, total number of particles, N, and total

volume, V. In statistical mechanics this corresponds to the canonical ensemble

(N,V,T).

In order to achieve a constant temperature the system studied has to be conceptually

coupled to a heat bath that introduces energy variations needed to keep the

temperature fixed. The energy of the system alone is thus not conserved.

In order to keep a fixed temperature, T, in an MD calculation we must keep the

kinetic energy of the system fixed. For the case of particles with three degrees of

freedom we must achieve that

1

2Nm

i < v

i

2

i=1

N

! >=3

2kBT (MD15)

Scaling of velocities: isokinetic MD

Starting with an unrelaxed configuration, the potential energy first decreases and if

the energy is preserved the kinetic energy increases and, consequently, also the

temperature. This increase must be compensated by the removal of the kinetic

energy, leading to a decrease of the total energy.

The simplest procedure that preserves the kinetic energy involves scaling of

velocities and proceeds as follows.

MD calculation is carried out at constant temperature (microcanonical ensemble) for

a sufficient number of steps, until a thermodynamic equilibrium has been attained


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(within the precision required). We then evaluate the average kinetic energy

1

2m

i < v

i

2>

i=1

N

! and scale velocities (in the three-dimensional case) by the factor

!=

3NkBT

mi< v

i

2>

i=1

N

"

#

$

%%

%%%

&

'

((

(((

1/2

(MD16)

where T is the desired temperature. This scaling restores the chosen temperature T.The MD calculation is then restarted with velocities the magnitudes of which are

vi

new= ! "v

i

old and carried out again for a sufficient number of steps. We then repeat

the above procedure iteratively until the desired temperature is attained with required

precision, i. e. when ! " 1.

Warning: The adjustment of velocities must always be done after sufficientnumber of iterations, as close as possible to the equilibrium. If the adjustment

were done too frequently equations of motion would not be really solved.

This is the simplest approach that can be employed. More sophisticated schemes, so

called thermostats, have been developed for keeping the temperature constant

during MD simulations. Examples are Andersen thermostat and Nos-Hoover

thermostat in which the exchange of heat with a bath is explicitly included (see, for

example, Frenkel and Smit: Understanding Molecular Simulation, Academic Press,

1996). In thermostats the constant temperature of the studied system is attained via a

suitable choice of boundary conditions.

MD CALCULATIONS AT CONSTANT PRESSURE

Quantities conserved: Temperature, T, total number of particles, N, and pressure p.

This is a canonical isothermal-isobaric ensemble(N, p, T).

To keep a constant temperature, T, the kinetic energy has to be adjusted duringcalculations in the same way as described above, or using one of the thermostats.

However, in order to preserve the total pressure the volume of the system, V, must be

allowed to vary. To allow for the volume variation during MD calculations the

simplest approach is the following:

First we carry out calculation without any volume adjustment until the

thermodynamic equilibrium has been attained. If the boundaries are free the volume


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probably changed although not necessarily such that the hydrostatic pressure is zero.

If periodic boundary conditions are applied the volume, of course, remains the same.

When in equilibrium we evaluate the total hydrostatic pressure of the system:

p= pi

i=1

N

! , where pi is given by equation (G34) or (G36) when employing a pair

potential. If the pressure is positive we increase the volume and if negative we

decrease the volume. The variation of the volume can be done by appropriately

scaling the coordinatesof all the particles. We then carry out MD calculations at the

new volume and repeat the process of volume adjustment after reaching a new

equilibrium. This procedure is repeated iteratively until the hydrostatic pressure is

the imposed pressure (commonly zero) within a required precision.

However, a more sophisticated method in which the volume change is an integral part

of the MD calculations has been proposed by Andersen (H. C. Andersen, J. Chem.

Phys. 72, 2384, 1980). In this method, applicable for both periodic boundary

conditions and the block with free surfaces, we regard the volume V as a newdynamical variable and the block of atoms is allowed to expand and/or shrinkautomatically during the calculation. Since V is now a dynamical variable we have to

introduce an additional equation of motion for the volume V. For this purpose we

formally associate an 'effective mass', M, with the volume V and define, again

formally, its kinetic energy as

1

2M

dV

dt

!"

# $%

&2

(MD17)

The potential energy of the block associated with the volume V is

(p e! p)V (MD18)

where pe is the imposed external pressure (usually zero) and p= p ii=1

N

! is the current

pressure evaluated as summation of pressures at positions of individual particles,

given by equation (G34). Note that the dimension of the 'effective mass' M is not the

usual dimension of the mass and thus it is only a formally introduced quantity and thefinal result must not depend on M.

As the volume changes the distances between the particles change. In particular, the

volume V and positions rican be linked such that any length will scale with L = V

1/3

(in three dimensions, in two dimensions, when the volume is really area, L = V1/2

).

Hence, we introduce new scaled, dimensionless, position vectors


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i = r

i/ L (MD19)

Andersen proposed the following Lagrangian in order to generate equations of

motion foriand V

=

1

2m

i L

d

i

dt

"

#$%

&'

2

i=1

N

( )Ep (L1,L 2 ,...,LN ,V)+1

2M

dV

dt

"#$

%&'

2

)(pe) p)V (MD20)

First two terms are, of course, just the kinetic and potential energy of the system of

particles. The last two terms have the same form and correspond to kinetic and

potential energy associated with the volume. Following the standard Lagrangian

method2the equations of motion (in the vectorial form) are

d L2mi

di

dt

"

#$%

&'(

)**

+

,--

dt=.grad

i

Ep =LF

i (MD21)

where i = 1,2, .., N and Fi is the force acting on atom i given as !grad

i

Ep

. After

differentiation and use of the relationship L=V1/3

, equations of motion for the

particles are

d2

i

dt2

=

Fi

m iV1/3

" 2

3V

d

i

dt

#

$%

&

'(

dV

dt

(MD22)

Using the Lagrangian (MD20) the equation of motion for the volume V is

Md 2V

dt 2=p !p e (MD23)

where p is the instantaneous pressure equal to pi

i=1

N

! , where pi are the atomic level

pressures given by equation (G34).

2 In the Lagrangian mechanics the Lagrangian isL

(1

!

1

! ) E kin

#Epot and it is a

function of generalized coordinatesiand generalized velocities ! . 3N equations of motion for

the generalized coordinatesiof the corresponding system of N particles are then

d

dt

!

!"i

#

$%&

'()* !

!"i

# = 0


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Equations of motion (MD22) and (MD23) are then solved for bothiand V using a

molecular dynamics algorithm. For example, when employing the Verlet algorithm

with velocities, we regard the right sides of equations of motion as generalized forces

and obtain according to (MD5)

!i (J + 1)"t( ) =! i J"t( ) +"t !!i J"t( )+ "t( )

2

2miV1/ 3 J"t( )

Fi J"t( )

#1

3"t( )

2!!

i J"t( )

!V J"t( )V J"t( )

V (J +1)"t( ) =V J"t( ) +"t!V J"t( ) +"t( )

2

2Mp J"t( )#pe( )

(MD24)

The corresponding rates !

i and !V are then calculated analogously, following

equation (MD6), as

!

ii ( J + 1)"t( ) 1+

"t

3

!V (J +1)"t( )V (J +1)"t( )

#

$

%%

&

'

((=

"t

2

Fi (J +1)"t( )

miV1/3 (J + 1)"t( )

+

Fi J"t( )

miV1/3 J"t( )

#

$

%%

&

'

((

+!

ii J"t( ) 1)

"t

3

!V J"t( )V J"t( )

#

$

%%

&

'

((

!V (J + 1)"t( ) = !V J"t( ) +"t

2M

p (J + 1)"t( )+p J"t( ))2pe#$ &'

(MD25)

However, when calculating the pressure p (J +1)!t( ) that enters the equation for !V ,

we need to know the velocities in J+1 iteration, i. e. !

i(J + 1)"t( ) . Yet, according to

(MD25) this can only be done if !V (J +1)!t( ) is already known. In order to resolvethis inconsistency we introduce approximate velocities

!!i

(J +1)#t( ) =

i(J + 1)#t( ) $

iJ#t( )

#t (MD26)

and evaluate the pressure p (J +1)!t( ) using these velocities.

There is no rigorously defined criterion for the choice of the effective mass of the

block, M, and it has to be determined by trial and error. To keep the impact of the

choice of M insignificant this must be as small as possible. The usual procedure is to

change M during calculation so that it becomes smaller and smaller. When V


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converges to the value for which p = pethen !V! 0 and all the terms containing M inthe above equations converge to zero.

MD CALCULATIONS AT CONSTANT STRESS TENSOR

(Parrinello-Rahman algorithm)

Quantities conserved:Temperature, T,total number of particles, N, and the average

stress tensor, !"# , that includes both hydrostatic and shear components. This is a

canonical isothermal-isostress ensemble (N, !"# , T)

The procedure employed is a generalization of the previous case of constant pressure.

First we regard the block of particles as a parallelepiped (not cube) defined by three

vectors a !($=1, 2, 3) that are represented by the 3x3 matrix with elements a !"

, as

shown in Fig. 2. The volume of this block is ! = a1 " (a 2 # a3 ) . We then introduce a

scaling of the position vectors of the particles, ri , by defining them as linear

combinations of vectors a !

ri = a!

!=1

3

" i! ri$ = a!$#i!!=1

3

"%

&'(

)*and in the matrix form r

i = h

i (MD27)

where his the matrix with elements a!

" . This introduces the scaled vectors of particle

positionsiwith components ! i

". Since we want to achieve that the cell varies in

. . .

a a

Fig. 2. Parallelepiped defined by three vectors a !($=1, 2, 3).


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time during the MD calculation we regard the vectors a !, or alternatively the matrix

h , as dynamical variables and we have to introduce additional equations of motion

for them. This is attained by using the following Lagrangian proposed by Parrinello

and Rahman (Phys. Rev. Lett. 45, 2384, 1980; J. Appl. Phys. 52, 7158, 1981)

=

1

2

mi

!

i

TG

!

i( )i=1

N

" #Ep (r1,r2,...,rN ,V)+1

2

MTr( !hT !h) (MD28)

where the letter T denotes the transpose of a matrix. M is again a formally defined

'effective mass of the block'; in this formulation it actually has dimensions of the

mass.

The corresponding generalized equations of motion, which can again be solved using

one of the molecular dynamics algorithms, are

m i

d h!

i( )dt

=

Fi"

mi h

T

( )"1

!

h

T

h

!

i (MD29)

M!!h = # (MD30)

The instantaneous stress tensor = i

i=1

N

" , where i are the atomic level stresses

given by equation (G33). The matrix = (a2" a

3,a

3" a

1,a

1" a

2) describes the size

and orientation of the cell faces.

If only pressure is considered then the cell can be taken as cubic and a !"= L#!" so

that h = LI , where Iis the unit matrix, and. hTh=L2I . Taking L3= V and inserting

these quantities into equation (MD29) yields equation (MD22). However, equation

(MD30) does not lead to (MD23) since the dynamical variables and the effective

mass describing the cell have been defined differently.


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PHYSICAL INTERPRETATION OF MD CALCULATIONS

USING CONCEPTS OF STATISTICAL MECHANICS

The analyses of the atomic structures found in MD simulations can be carried out

using the same methods as those described in the section on General aspects of

computer modeling. This means we can employ various graphical methods, radial

distribution function, Voronoi polyhedra, atomic level stresses. All these quantitiesare now time dependent and therefore always their time averages have to be

evaluated. However, while MD calculations represent time variation of the system

studied, because of the ergodic theorem many quantities commonly used in the

statistical mechanics can be determined and used in calculations of physical

properties of the system

Fluctuations

In the thermodynamic equilibrium the fluctuation of a physical quantity A is defined

as the root mean square (RMS) deviation

A2! A

2

(MD31)

Fig. 3. Schematic picture of fluctuations of a quantity A

time

A

Fluctuations


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17

Following (MD11) and using the ergodic theorem

A = lim !"#( )1

! A

2(t)0

!

$ dt =1

M A

2(J%t)J=1

M

& (MD32)

where !t is the time step,Mthe total number of MD steps and #the total time of theMD run (after equilibration). A is given by equation (11.2).

Physical quantities obtainable from fluctuations

Specific heat at constant volume:CV =!E!T

"#$

%&'V

.

E is the energy of the system and it is identical with the Hamiltonian given byequation (G2), i. e. it includes both the kinetic and potential energy. It follows fromthe theory of fluctuations and statistical mechanics that CV is related to the

fluctuation of the energy3

CV =E2! E

2

kBT2

(MD33)

3If the distribution function f(Q) in (MD12) is the Boltzmann distribution then the average value of

the energy of the corresponding system is

E =E exp !E k

BT( )dQ"

exp !E kBT( )dQ"

where the integration is over the whole phase space of the system. Differentiation of E with

respect to the temperature T yields

! E!T

"#$

%&'

V

=1

kB

T2

E2

exp (E kBT( )dQ)exp (E k

BT( )dQ)

( Eexp (E kBT( )dQ)exp (E k

BT( )dQ)

"#$ %

&'

2

*+,,

-.//

which corresponds to

! E

!T

#$

&'V

=

1

kBT

2E

2 ( E 2[ ]


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18

When evaluating CV using the fluctuations of energythe corresponding MD

calculation must be done at constant temperature, number of particles and, of

course, volume. Thus (MD33) cannot be used when the calculation is done for

microcanonical ensemble when the energy of the system is fixed but temperature

changes. Similarly it cannot be used when the volume is allowed to change, for

example, when dealing with an isolated cluster with free surfaces.

Specific heat at constant pressure:C p =!H!T

"#$

%&'p

.

H=E + pV is the enthalpy of the system and C pis related to its fluctuations

C p =H2 ! H

2

kBT 2

(MD34)

When evaluating C p using the fluctuations of enthalpythe corresponding MD

calculation must be done at constant temperature, pressure and number of

particles.

Importantly, both CV and Cp can be evaluated according to their definitions,

CV =!E!T

"#$

%&'V

and C p =!H!T

"#$

%&'p

if we determine by a molecular dynamics calculation

that employs the microcanonical ensemble the temperature dependence of the energy,

E, or the enthalpy, H, respectively. These quantities can then be differentiated withrespect to the temperature. These calculations have to be done at constant volume

when evaluating CV and at constant pressure when evaluating Cp. In the latter case,

if p = 0 the enthalpy is equal to the energy E.

Isothermal compressibility:!T ="1

V

#V#p

$%&

'()T

is related to the fluctuations of the volume, V

!T =V

2 " V 2

V k B T (MD35)

When evaluating !T

using the fluctuations of volume

the corresponding MD

calculation must be done at constant temperature, pressure and number of

particles.


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19

Time-Independent Correlations Cross Correlations of fluctuations

If the deviation of a quantity A from its average value is !A = A" A then the time

independent correlation between two quantities A and B is defined as

!A!B = A" A

( ) B " B

( )= AB " A B

(MD36)

Obviously, if A and B are independent quantities then AB = A B and

!A!B = 0. Following (MD11.2)

!A!B =1

MA(J!t)B(J!t)

J=1

M

" # 1M

A(J!t)J=1

M

"$%&'()

1

MB(J!t)

J=1

M

"$%&'()

(MD37)

where Mis the total number of MD steps and !t the time step.

Physical quantities obtainable from time-independent correlations

The thermal pressure coefficient: !V ="p"T

#$%

&'(V

is determined by the time-independentcorrelation between the potential energy, E p ,

and the pressure, p, using the relation

!E p!p = k BT2"V # $kB( ) (MD38)

where ! = N V is the density (J. S. Rowlinson: Liquids and liquid mixtures,

Butterworth: London, 1969). When evaluating !V using this correlation relationthecorresponding MD calculation must be done at constant volume, temperature and

number of particles.

Isobaric thermal expansion coefficient: !p =1

V

"V"T

#$%

&'(p

is determined by the time-independent correlation between the volume, V, and the

enthalpy, H

!V!H = kBT2V"p (MD39)

When evaluating !p using this correlation relationthe corresponding MD calculation

must be done at constant pressure, temperature and number of particles.

It should be noted that !T, the isothermal compressibility given by (MD35),

!Vand "

pobey the following relationship: !p ="T#V


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20

Tensor of elastic moduli

The tensor of elastic moduli C!"#$ relates the applied stress, !"#a

, and the induced

strain, !"#a

, in a linearly elastic material, i. e the Hooke's law !"#a=

C"#$%& $%a

$,%=1

3

' applies. This tensor may be determined from the time-independent correlation of the

stress tensor components

C!"#$ =%

k BT&'!"&' #$ (MD40.1)

where the stress tensor !"# is given by (G15). Since in equilibrium !"# = 0

C!"#$ =%

k BT&!"& #$ (MD40.2)

This relationship can then be used to determine the temperature dependence of elastic

moduli in an MD calculation. However, such calculation requires extremely long

MD runs that are often unattainable.

Time Dependent Correlations

Definition of correlations

Let A(t) and B(t) be two physical quantities of the system studied, i. e. quantities

averaged over all N particles constituting the system. These quantities generally vary

with time. The question asked is whether there is any relation between the value of

the quantity A at a time t1 and the value of the quamtity B at another time t2.

Mathematically, this is expressed by the time dependent correlation function defined

as

B(t2)A(t

1) =lim ! " #( )

1

! A(t

1+ $t )B(t

2+ $t )d $t

0

!

% (MD41)

In the thermodynamic equilibrium, the correlation may depend only on the differenceof the times: t2! t 1. Therefore, setting t1= 0 and writing t instead of t2, the time-

dependent correlation function is

B(t)A(0) =lim ! " #( )1

! A( $t )B(t + $t )d $t0

!

% (MD42.1)


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21

and in the framework of MD calculations

B(t)A(0) =1

MA(J!t)B(t +J!t)

J=1

M

" (MD42.2)

where M is the number of steps of the MD calculation. The short time limit of the

correlation function is:

lim(t!0) B(t)A(0)[ ] = A"B (MD43.1)

Since for t!" A and B cannot be correlated because the events are very far from

each other in time, the long time limit of the correlation function is:

lim(t ! ") B(t)A(0)[ ] = A # B (MD43.2)

Hence, if the correlation function approaches the product A ! B at a time t, it

means that no correlation exists at this time.

Autocorrelations

The autocorrelation function is a special case of time-dependent correlation function

when we set A = B

A(t)A(0) =lim ! " #( )1

! A(t + t')A(t' )dt'0

!

$ (MD44.1)

and in the framework of MD calculations

A(t)A(0) =1

MA(t + J!t)A(J! t)

J=1

M

" (MD44.2)

The physical meaning of the autocorrelation is that we ask how long the system

remembers that a quantity A had some value at a given time. The short time limit

of the autocorrelation functions is:

lim(t! 0) A(t)A(0) = A2

(MD45.1)

Obviously, as t! 0 the memory is completely preserved and there is a complete

correlation between the value at t = 0 and very short time after. On the other hand,

the correlation and thus the memory are completely lost at long times when t ! " .

The long time limit of the autocorrelation functions is


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22

lim(t ! ") A(t)A(0) = A 2

(MD45.2)

The autocorrelation function A(t)A(0) therefore varies between these two extreme

values and practically reaches A 2

after the correlationor relaxation timedefined as

!cor =2

A(t)A(0) " A 2( )dt

0

#

$

A2 " A

2 (MD46)

The reason for this definition is explained in Fig. 4, where the autocorrelation

function is shown schematically as a function of time. We define the correlation

time, !cor

, as a time when the autocorrelation function approaches closely A 2

. In

this case the area of the right-angled triangle marked by bold lines,

12

A2

! A2

( )"cor , is very close to that delineated by the autocorrelation functionand the two sides of the same triangle. This area is given by the integral

A(t)A(0) ! A2

( )dt0

"cor

# .

Since we assume that for time !cor

the autocorrelation function is very close to A 2

,

this integral is practically equal to A(t)A(0) ! A 2( )dt

0

"

# . Equating this integral to

the area of the right-angled triangle marked by bold lines, leads to the formula

(MD46) for the correlation time.

Autocorrelation

A

2

A2

Time

cor

Fig. 4. Schematic behavior of the autocorrelation function and evaluation of the

relaxation (correlation) time.


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23

Physical quantities obtainable from autocorrelations

In general, the time integrals of autocorrelation functions are related to mobility

coefficients defined as follows.

Let be a time dependent physical quantity (e. g. velocity, stress, flux etc.) with the

average value . Any local deviation, i. e. fluctuation, away from will tend to

dissipate towards the average value and in the linear approximation the rate of the

dissipation will be proportional to the magnitude of this deviation. Such dissipation

process will be described by the equation (Langevin equation)

d( " )

dtdissipation

=" #( " ) (MD47)

where & is the linear dissipation coefficient associated with the physical quantity .The term !"( ! ) represents the frictional force that opposes the change of . If

dissipates from some value

then it follows by integrating equation (MD47) that

the dissipation process decays exponentially and thus

= ( " )exp("#t) + (MD48)

The mobility related with this dissipation process is then defined as ! = 1/" and

according to the Green-Kubo equation it is determined by the time integral of the

autocorrelation of as follows4

! =1

2( t) (0) dt

0

"

# (MD49)

If (t ) =d(t) dt then the corresponding 'Einstein relation' is associated with

(MD49) that follows by integration by parts

! =

1

2

1

2lim " # $( )

1

"B(") % B(0)( )

2

(MD50)

In practice, #must be much larger than the correlation time for in order to evaluate

!

with sufficient precision.

According to the fluctuation-dissipation theorem the dissipation coefficient, &,

determining the rate of dissipation of fluctuations also determines, in the linear

4M. S. Green, J. Chem. Phys. 22, 398, 1954; R. Kubo, J. Phys. Soc. Japan 12, 570, 1957; Rep.

Prog. Phys. 29, 255, 1966; R. Kubo, M. Toda and N. Hashitsume, Statistical Physics II, Springer:Berlin, 1985).


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24

approximation, the dissipation rate if the deviation away from is induced by some

external action.

Self-diffusion coefficient

As first shown by Einstein when studying the Brownian motion, the self-diffusioncoefficient, D, is related to the mobility,

v, associated with the velocity of the

particles, according to the relation D = vkBT / m, where m is the mass of the

particles. Hence, if we identify with the vector of the velocity, v,then

v =

1

v2v! (t)v! (0)

!=1

3

"0

#

$ dt (MD51.1)

and the self-diffusion coefficient is

D = kBTv2 m

v! (t)v! (0)!=1

3

"0

#

$ dt = 13 v! (t)v! (0)!=13

"0

#

$ dt = 13 v(t )iv(0) dt0

#

$ (MD51.2)

since according to the equipartition theorem v2= 3k BT /m . Furthermore, the

relation (MD50) gives

D =lim ! " #( ) 1

6!r(!) $ r(0)[ ]

2 (MD52)

where r is the position vector. In practical calculations !must be much larger than

the correlation time for r .

Tensor of viscosity

In a linearly viscous material the tensor of viscosity,!"#$% , relates linearly the applied

stress,!"#a

,and induced strain rate, !"# ,according to the relation !"#a

= $"#%& '%&%,&=1

3

( .

This tensor is determined by the integral of the time dependent autocorrelation of the

stress tensor components

!"#$% = &o

kBT'"# (t)' $% (0)

0

(

) dt (MD53)

where the stress tensor !"# is given by equation (G15); !o is the average volume per

particle.


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In liquids, which are by definition isotropic, there are only two components of the

viscosity:

(i) The shear viscosity determined by the autocorrelation of shear stresses:

!s ="

o

kBT#$%(t )#$%(0)

0

&

' dt (MD54.1)

where !"# (! " # ) is a shear component of the stress tensor given by (G15).

(ii) The bulk viscosity determined by the autocorrelation of the hydrostatic

pressure:

!V ="okBT

p(t)p(0)0

#

$ dt . (MD54.2)

where p is 1/3Trace of the stress tensor.

Additional literature

D. Chandler: Introduction to Modern Statistical Mechanics, 1987, Oxford: Oxford

University Press.

M. P. Allen and D. J. Tildesley: Computer Simulation of Liquids, 1987, Oxford:

Oxford University Press.

J. P. Boon and S. Yip: Molecular Hydrodynamics, 1980, New York: McGraw-Hill;

also Dover Publications, 1991.

D. Frenkel and B. Smit: Understanding Molecular Simulations, Academic Press, New

York, 1996.


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APPENDIX - EQUIPARTITION THEOREM

The kinetic energy per degree of freedom is Ekin

df=

1

2mv2 and when the Boltzmann distribution

applies then the average kinetic energy per degree of freedom is

Ekin

df=

12mv

2exp !mv2 2kBT"

#$%&'dv

0

(

)

exp !mv2 2kBT"

#$%&'dv

0

(

) where kBis the Boltzmann constant.

We make the following substitution

mv2 2kBT= x2 ! v = x 2k

BT m

and thus

Ekin

df= k

BT

x2exp !x2"#

$%dx

0

&

'

exp !x2"#

$%dx

0

&

'

x2exp!x2"#$

%

&'dx

0

(

) = x xexp !x2"

#$

%

&'

*+,

-./dx

0

(

) = !x

exp !x2"#$

%&'

20

(

+1

2 exp!x2"

#$

%

&'dx

0

(

)

when integrating by parts.

Hence the average kinetic energy per degree of freedom is

Ekin

df=

kBT

2.

When there are N particles each with ! degrees of freedom then the average kinetic energy of the

assembly of these N particles is

E=N!k

BT

2

This is the equipartition theorem that represents for the system of N particles the definition of the

temperature

8. Molecular Dynamics

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