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Lecture notes on MMM

39

Chapter 3:Interatomic Potential

3.1 Introduction to interatomic potentials

In Molecular Dynamics (MD) and Monte Carlo (MC) simulation we must have some

rules, which governs the interactions between the atoms. In classical simulation these

rules are expressed in terms of potential functions. Interatomic potential describes how

the atoms interact with each other.For example, in case of ionic solids the main source of

cohesion is Columbic attraction force. Ions with the opposite charge attract each other. In

the absence of any repulsive force all the atoms will collapse into one point. But this does

not happen in real case so there must be some repulsive force, which prevents the atoms

to collapse into one point. There are two types of repulsive forces in ionic solids. One is

due to Columbic interaction. Ions with the same charge repulse ease other. Also when the

ions comes very close to each other a strong repulsive force due to Paulis exclusion

principle acts which is a very strong function of distance between the ions. There is one

more kind of interaction between the ions is van der Waals interaction. Which is the

weakest interaction among all. Similarly in case of metals the main source of cohesion is

free electrons. In case of metallic solids it is assumed that there is a pool of free electrons

and positive ions are submerged in the pool. So more the density of free electrons

stronger will be the attractive force. Similar to the ionic case a strong repulsive force, due

to Paulis exclusion principle, acts when the atoms comes very close to each other. The

potential function U(r1, r2, , rN) describes how the potential energy of a system of N

atoms depends on the coordinates of the atoms, r1, r2, , rN. In classical MD or MC

simulation electrons comes nowhere in the calculations. It is assumed the electrons adjust

to new atomic positions much faster than the motion of the atomic nuclei (Born-

Oppenheimer approximation). The behavior of the electrons is approximated in thepotential itself.


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3.2 How to obtain the potential function for a particular system?

3.2.1 Empirical potential function: A functional form is assumed for the

potential function and then we fit the parameters to reproduce a set of experimental data

like lattice parameter, elastic constants, thermal expansion coefficient etc. This gives

empirical potential functions (e.g. Lennard-Jones, Morse, Born-Mayer). These potentials

have no direct physical basis but still widely used in many types of problems because of

their simplicity, computationally less expensive and reasonable accuracy.

3.2.2 Semi-empirical Potentials:The electronic wave function can be calculated

for fixed atomic positions. But for a system consist of many atoms this type of calculation

is very difficult. Different approximations are used and analytic semiempirical potentials

are derived from quantum-mechanical arguments (e.g. Embedded Atom Method (EAM)

by Foiles, Baskes, and Daw, Glue Model by Ercolessi et al., bondorder potentials by

Tersoff and Brenner, etc.). These are also called many body potential because the

potential depends on the density of surrounding atoms. These potentials are more

complex and computationally more expensive but provide good description of different

types of materials like metals and ceramics.

3.2.3 Potentials from Ab-initio calculations: Direct electronic-structure

(quantum-mechanics-based) calculations of forces can be performed during so-called ab-

initio MD simulation (e.g., Carr-Parrinello method using plane-wave psuedopotentials).

Potentials obtained from ab-initio method are most accurate because these are obtained

from electronic calculation using first principle and number of assumptions or

experimental parameters used is negligible. For more detailed and elaborated analysis

these types of potentials are used.

3.3 The Born-Oppenheimer approximation

In the MD and MC methods potential functions are used to describe the interactionamong the atoms. But in actual case atoms consist of a nucleus and electrons around it

and electrons play a vital role in defining the interactions and bonds between the atoms.

In the potential functions interaction of electrons with nuclei are not considered. Is the


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use of the potential function is justified. Hamiltonian of the real material is defined by the

presence of nuclei and electrons and can be written as follows:

+

++=i i

2i

2

ij ji

2ji

2

i i

2i

rR

eZ

2

1

rr

e

2

1

RR

eZZ

2

1

2m

p

2M

PH

3.3.1

where Pi, Mi, Ri and Zi are momentum, mass, coordinates and atomic number of the

nucleus i and p, m, r and e are momentum, mass, coordinates and charge on electron .

A Schrdinger equation H=E should be solve to get the total wavefunction (Ri, r),

which tells everything about the system. But this is impossible for any system of practical

use. In 1923 Born and Oppenheimer shown that the electrons (me=5.510-4 amu) are

much lighter than nuclei and are moving much faster. So it can be safely assumed that the

nucleus is fixed with respect to electrons and the total wavefunction is factorized as:

(Ri, r) = (Ri) (r;Ri) 3.3.2

where (Ri) describe the nuclei, and (r; Ri) depends parametrically on Ri and describes

electrons. The problem can be reformulated in terms of two separate Schrdinger

equations:

Hel(r;Ri) = U(Ri) (r;Ri) 3.3.3

Hi(Ri) = E (Ri) 3.3.4

Where

+

+=i ii

2i

2

ij ji

ji2

elrR

eZ

rr

e

2

1

RR

ZZ

2

1

m2

pH 3.3.5

and

)U(R2M

PH i

i i

ii += 3.3.6

the equation 3.3.3 for the electronic problem gives eigenvalue of the energy U(Ri) that

depends parametrically on the coordinates of the nuclei, Ri. After determining U(Ri) we

put it in equation 3.3.4, which describe the motion of nuclei. Equation 3.3.4 does notinclude any electronic degrees of freedom, all electronic effects are incorporated in U(Ri)

that is called interatomic potential.

Schrdinger equation is replaced with Newtonian equation in classical MD simulation.

This replacement is justified for most of the cases except for lightest atoms.


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3.4 Characteristics of potential

If we look into the literature will find different kinds of potential available for simulation

of different materials. Many things govern our choice of potential. Firstly, what type of

materials we are going to simulate? Second, what will be the size of our simulation

domain? Thirdly, what type of accuracy we want for our simulation. Fourthly, what types

of computational resources are available for simulation? Finally, up to what time we have

to simulate the problem (in molecular dynamics simulations). Typical time of these

problems vary form hundreds of femto-second (1x10-15 second) to few microsecond.

Potentials can be appropriate or inappropriate for a given problem. Following

characteristics should be considered while selecting a potential.

1.Accuracy potential is able to reproduce properties of interest as closely as possible.

High accuracy is typically required in Computational Chemistry. Computational domain

is typically very small in this case.

2. Transferability potential can be used to study a variety of properties for which it

was not fit. It is the most important characteristic of potential for empirical potential.

Empirical potential is fit using one set of parameters obtained from different experiments.

Transferability requires that the same potential must be able to reproduce other set of

parameters, which were not used to fit the potential.

3. Computational speed - calculations are fast with simple potentials. Most of empirical

potentials have simple form hence they are used extensively in different types of

problems. Computational speed is often critical in Materials Science where processes

have a collective character and big systems should be simulated for long times.

3.5 Pair potentials vs. many-body potentials

The total energy of a system of N atoms with interaction described by an empirical

potential can be expanded in a many-body expansion:

+++= > >> i ij jk

kji2i

jij

i2i

i1N21 )r,r,r(U)r,r(U)r(U)r,,r,rU( 3.5.1

where,1. U1 one-body term, due to an external field or boundary conditions (wall of a

container).


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2. U2 two-body term, or pair potential. The interaction of any pair of atoms

depends only on their spacing and is not effected by the presence of other atoms.

3. U3 three-body term arises when the interaction of a pair of atoms is modified by

the presence of a third atom.

Based on this expansion, we can loosely separate potentials into two classes:

1. Pair potentials (only U2 is present) and

2. Many-body potentials (U3 and higher terms are included).

In most of materials interaction of two atoms is affected by the presence of its neighbors.

In many body potentials we accounted the presence of its neighbors in the calculations

while in case of two body potentials we treat the interaction between the two atoms or

ions independently. Pair potentials are used for simulation of ionic solids and ideal gas.

Pair potential are computationally less expensive hence it is also used for other materialslike metals for simulating the general class effect with reasonable accuracy. MBP are

computationally more expensive but describes the materials like metals more elaborately.

We will also consider examples of many-potentials in which multi-body effects are

included implicitly, through an environment dependence of two-body terms

3.6 Short review of potentials used in MD and MC

Pair Potentials (hard spheres, Lennard-Jones, Morse) can be used :1. for inert gases, intermolecular van der Waals interaction in organic materials;

2. for investigation of general classes of effects (material non-specific).

The total potential energy of the system of N atoms interacting via pair potential is:

ijiji ij

ij2N21 rrrwhere)r(U)r,,r,rU(

== >

3.6.1

Commonly used examples of pair potentials:3.6.1 Hard/soft spheres potential It is the simplest potential without any

cohesive interaction i.e. there is no attractive force present. A very strong repulsive forceact when the spacing between the two atoms reaches below certain cutoff distance. It is

useful in theoretical investigations of some idealized problems. But it is difficult to

handle singularity in simulation. So we use a soft sphere model in which the

transformation of potential is gradual. Still it is not a realistic model for real materials.

We have very limited use of these potentials for real problems.


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potentialspherehardrrfor0

rrfor)U(r

0ij

0ij

ij

>

= 3.6.2

potentialspheresoft-r

r)U(r

n

0

ijij

= 3.6.3

3.6.2 Ionic potential In ionic solids the main source of cohesion is Coulomb

interaction of charges attraction.

ij

ji

ijr

qq)U(r = 3.6.4

These are very strong forces and have long range of influence i.e. the force dies out very

slowly. As we will see later that we can not use simply a cutoff radius and determine the

sum of total energy of forces as in case of other potentials. We have to give some specialtreatment to obtain the summation of the potential. Ions like O-- have a tendency to

polarize and form dipoles in presence of electric field. These dipoles also play an

important role in cohesive energy of the system. So the columbic potential is often added

to other functional forms to account polarization.

3.6.3 Lennard-Jones potential From computational point of view this potential

is very efficient. It describes the van der Waals interaction in inert gases and molecular

systems very accurately. Lennard-Jones potential can also be used to study the solid and

liquids. It is often used to model general effects (drop collision with surface, study of the

behavior of crack in fracture mechanics problems etc.) with very large system rather than

properties of a specific material.

=

6

ij

12

ij

ijrr

4)U(r

3.6.5

3.6.4 Morse potential It is similar to Lennard-Jones but is a more bonding-typepotential and is more suitable for cases when attractive interaction comes from the

formation of a chemical bond. This potential can be used for materials, which we

encounter in daily life like metals. It is proposed by P. M. Morse, Phys. Rev. 34, 57,

1930. It was a popular potential for simulation of metals that have fcc and hcp structures.

A fit for many metals is given by Girifalco and Weizer, Phys. Rev. 114, 687, 1959.


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=

)r-(r-e2

)r-(r2-e)U(r 0ij0ijij

3.6.6

3.6.5 6-exp (Buckingham) potential exp term (Born-Mayer) provides a better

description of strong repulsion due to the overlap of the closed shell electron clouds,

which is important in simulation of bombardment by energetic atoms or ions, etc.

6ij

BMijij rB

Rr-eA)U(r = 3.6.7

repulsion. Most of the potentials like Columbic potential and EAM potential (which

is a many body potential) use exp potential to describe the strong repulsive

3.7 Lennard Jones potential

3.7.1 IntroductionThe expression for Lennard Jones potential can be given by the following equation.

=

6

ij

12

ij

ij rr4)U(r

3.7.1

where rij is the distance between the atoms under consideration and and are material

specific constants. Values of constants and are determined by fitting the potential

for particular material. The potential is plotted below which describes different zones of

the potential like dipole-dipole attraction, equilibrium and Pauli repulsion. X-axis showsthe distance between the two atoms while y-axis represent the potential energy.


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distance between two atoms r

PotentialEnergy

4 6 8 10-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

Pauli Repulsion

Equilibrium

Dipole-dipole attraction

Figure 1 plot of Lennard-Jones pair potential for Ar

When the spacing between the atoms is very large a week dipole-dipole attraction acts

between them. As the spacing between the atoms decreases the attractive force increases.

The term ~1/rij6, dominating at large distance, constitute the attractive part and describes

the cohesion to the system. A 1/r6 attraction describes van der Waals dispersion forces

(dipole-dipole interactions due to fluctuating dipoles). These are rather weak interactions,

which however are responsible for bonding in closed-shell systems, such as inert gases.

A strong repulsive force due to overlapping of close shell electrons counteracts the

attractive force before reach the equilibrium position. At equilibrium position resultant

force between the atoms is zero and potential energy is minimum at this position. If the

spacing between the atoms further decreases the repulsive force increases very rapidly.

The term ~1/rij12 represents the repulsion between atoms when they are brought close to

each other. Its physical origin is related to the Pauli principle: when the electronic cloudssurrounding the atoms starts to overlap, the energy of the system increases abruptly. The

exponent 12 was chosen on a practical basis: Lennard-Jones potential is particularly easy

to compute. In fact, on physical grounds an exponential behavior would be more

appropriate. Exponential term for repulsion (Born-Mayer potential) is typically used in

simulations where high-energy inter-atomic collisions are involved.


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3.7.2 Application of L-J potential:

1. The potential provides a good description of van der Waals interaction in inert gases

and molecular systems (Ar, Kr, CH4, O2, H2, C2H4, etc.). Parameters are given in [J.Chem. Phys. 104 8627 (1996)]. Parameters for inert gases can be also found in Ashcroft-

Mermin textbook. For example,

Ar ( = 0.0104 eV, = 3.40 )

Ne ( = 0.0031 eV, = 2.74 )

Kr ( = 0.0140 eV, = 3.65 )

Xe ( = 0.020 eV, = 3.98 ).

2. The main reason for popularity of Lennard-Jones potential is that it is used in

simulations when the objective is to model a general class of effects and the only

requirement is to have a physically reasonable potential.

Many studies on Lennard-Jones solids, liquids, surfaces, and clusters have been

performed. It is the potential of choice in studies when the focus is on fundamental

issues, rather than on properties of a specific material.

3.7.3 Derivation of the force for pair potential

In MD simulation we need forces that are acting on the atoms. The forces are given by

the gradient of the potential energy surface (the force on atom i is a vector pointing in the

direction of the steepest decent of the potential energy):

)r,,r,rU(rF N21ii

= 3.7.2

ir

operates on the position ri of atom i. Any change in the total potential energy that

results from a displacement of atom i contributes to the force acting on atom i.

For a pair potential:)r(U)r,,r,rU(

i ijij2N21

>

=

3.7.3

Where 222ij )()()(r jijijiji zzyyxxrr ++==

The force on atom i is:


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==ij

iji

N21ii)U(r

r)r,,r,rU(rF

3.7.4

ij

ij

ij i

ij

i

ij

i

ij

ijij iii dr

)dU(r

z

rz

y

ry

x

rx)U(r

zz

yy

xx

+

+

=

+

+

=

=

=

++=

ijij

ij

ij

ij ij

ji

ij

ij

ij ij

ij

ij

ij

ij

ij fdr

)dU(r

r

rr

dr

)dU(r

r

zz

r

yy

r

xx

3.7.5

for Lennard-Jones Potential:

=

612

4)(ijij

ijrr

rU

3.7.6

+= 7

612

6124

)(

ijijij

ij

rrdr

rd

=

6

8

6

21)(

24ijij ij

ji

irr

rrF

3.7.7

3.7.4 Potential cut-off

The potential functions like L-J have an infinite range of interaction i.e. whatever is the

value of r there is a corresponding value of potential. However as we increase value of rafter a certain value of r value of potential energy corresponding to this r is so small that

we can ignore the value for a practical problem. In practice a cutoff radius Rc is

established and interactions between atoms separated by more than Rc are ignored. There

are two reasons for this:

1. If we dont take a cutoff radius then the number of pair interactions grows as N 2.

Example: Consider system of 3000 atoms. If we dont tank a cutoff radius then the total

number of interactions (for force evaluation) in the simulation will be N2/2 = 45 lakhs.

While using a cutoff of 8-10 A we can reduce a number of interacting neighbors for each

atom to ~50 and we will have to evaluate force only ~50N = 150 thousand times for the

simulation. So with the introduction of a cutoff radius we can reduce the computational

requirement many times.


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2. The size of the system that can be simulated is finite. As we will see later with the help

of the finite size of the domain we evaluate the properties of the bulk materials using

periodic boundary conditions. And we will also see that these periodic boundary

conditions will require a cutoff radius.

But this introduction of cutoff radius also introduces some errors in the calculation as

explained below.


PotentialEnergy

4 6 8 10-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

Pauli Repulsion

Equilibrium


Jump in potential

Cutoff Radius

Figure 2 Plot of Lennard-Jones potential for Ar with the introduction of the cutoff radius

A simple truncation of the potential creates a jump in the potential at the cutoff distance.

This can spoil the energy conservation (i.e. energy of the system will not be remain

constant) or lead to unphysical behavior in simulations of the effects where contribution

of far-away molecules is important (surface tension, stacking faults, etc.). To avoid this

potential can be shifted to get potential energy zero at cutoff distance.

>

=cij

cijcij

ij RrRrRUrUrU

0)()()( 3.7.8

In the sifted potential value of potential at cutoff radius is subtracted from the potential

when the distance rij is less than or equal to cutoff radius andits value is zero when the


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distance rij is greater than cutoff radius. This arrangement smooth out the problem of

variation of energy.


PotentialEnergy

4 6 8 10-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

original potential

shifted potential

Pauli Repulsion

Equilibrium


Shiftd potential

origional potential

But for shifted potentials forces, which are derivative of potential energy, can have a

jump at the cutoff. To avoid this, a smooth transition function that brings potential to zero

can be added.

+

=

1973.8,1504,Phys.Rev.AJ.FordandstoddardS.D.byfunctionoff-cut

6122612612

47364)(ccc

ij

ccijij

ijRRR

r

RRrrrU

3.7.9

with cutoff function

0)( =cRU and 0)(

==

cijij

ij

Rrdr

rdU 3.7.10

In any case, physical quantities (cohesive energy, total pressure etc.) are affected by the

truncation and most modern potentials for real materials are designed with a cutoff radius

in mind, and go to zero at Rc together with several first derivatives of the potential

function.


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3.7.5 Limitations of pair potentials

1. Presence of other atoms affects the strength of bond between two atoms. It is called

many body effects and also known as environment dependence of bonds. Many body

effects are important in real materials. Pair potentials do not have environmental

dependence (e.g. atom in the bulk is too similar to the atom on the surface or near a

defect site). In reality, the strength of the individual bonds should decrease as the local

environment becomes too crowded due to the Paulis principle, but pair potentials do not

depend on the environment and cannot account for this decrease. Thus modeling of real

materials cannot be done accurately using only pair potentials.

2. Most of the ceramic materials have covalent bond. In covalent bonds the two or more

atoms share the valence electrons to complete their outer shell. And this sharing in turn

give rise to the directional nature of the bond i.e. the bond angle between the atoms

remains constant. Pair potentials do not account for directional nature of the bond. In

the same fashion covalent contributions (d orbitals) of the transition metals cannot be

described using pair potential. Pair potentials work better for metals in which cohesionis

provided bys andp electrons.

3. Energy required to remove an atom from its equilibrium position and thus form a

vacancy is called vacancy formation energy. The vacancy formation energy, Ev, is

significantly overestimated by pair potentials. Pair potential gives the estimation of Ev

equal to the cohesive energy per atom in the system Ec. In reality vacancy formation

energy is less then the cohesive energy per atom as in case of Cu Ev = 0.33 Ec and for Au

Ev = 0.25 Ec.

4. The ratio between the cohesive energy and the melting temperature, Tm, is

underestimated by as much as 2-3 times. Metals have some extra cohesion that is less

effective than pair potential in keeping the system in the crystalline state.

5. For cubic crystals (bcc, fcc) there exist a relation between the elastic constants (C12 =

C44), which is known as Cauchy relation. Ionic materials follow Cauchy relation, but


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metals show significant deviation from the Cauchy relation. Pair potentials do not

describe the deviations from the Cauchy relation for elastic constants in cubic crystals in

case of metals. (C12 = C44 or G = 3/5 B)

3.7.6 Relationship between pair potential and elastic constants

The bonding energy density of the system of N atoms interacting via pair potential is:

=ij

ij

a

b )U(r2N

1E 3.7.11

where a is the average volume per atom.

In equilibrium, when the force acting on any particle is zero, the structure must be stable

with respect to the application of a small homogeneous strain tensor . Then the

displacement vector for each interatomic distance, rij is uij = aij

where aij is

undeformed value of rij (uij = rij - aij). The elastic energy is expanded into a Taylor series

with respect to small displacements:

+

+

+=

ij

ij

ij

ij

ij

ij

ij

ij

ij

ij

a

b uurr

)U(ru

r

)U(r)U(r

NE

0

2

0

0 2

1

2

1 3.7.12

where the evaluation is at the undeformed values of rij and summation is implied byrepeated indices. Using uij

= aij we can rewrite this equation as

+++= CAEE bb2

1)0()( 3.7.13

where

=

ij

ij

ij

ij

a r

rU

NA

a

)(

2

1

0

3.7.14

is tensor of internal stress. First invariant of this tensor (Axx + Ayy + Azz) is the pressure.

In equilibrium A = 0, and

=

ij

ijij

ijij

ij

a rr

rU

NC

aa

)(

2

1

0

2

3.7.15

for pair potential


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ij

ij

ij

ij

ij

ij

dr

rdU

r

r

r

rU )()(

=

3.7.16

ij

ij

ijij

ij

ijij

ij

ij

ijij

ijij

ij

rd

rdU

rrd

rdU

rrd

rUd

r

rr

rr

rU )(1)(1)()(2

2

2

2

+

=

3.7.17

+

=

ij ij

ij

ijij

ij

ijij

ij

ija rd

rdU

rrd

rdU

rrd

rUd

rNC

ijijijijijij2

2

2aa

)(1aaaa

)(1)(1

2

1 3.7.18

C is symmetric with respect to all changes of indices and using Voigt notation (C11 =

C1111, C12 = C1122, C44 = C2323, C66 = C1212 etc.) we have Cauchy relation C12/C44=1

satisfied.

The Cauchy relation is often satisfied for van der Waals solids and ionic crystals. It is

never valid for metals (e.g. C12/C44 is 1.5 for Cu, 1.9 for Ag, 3.7 for Au). This means thatfor van der Waals and ionic solids the total energy may be reasonably well described by

the pair potential approximation. But for metals pair interaction may be used to represent

only part of the total energy.


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3.8 Embedded-atom and related methods for metallic systems

As we discussed above, pair potentials cannot provide an adequate description of metallic

systems. An alternative simple but rather realistic approach to the description of bondingin metallic systems is based on the concept of local densitythat is considered as the key

variable. This allows one to account for the dependence of the strength of individual

bonds on the local environment, which is especially important for simulation of surfaces

and defects. In metals the main source of cohesion is free electrons.

Many methods that have been proposed since early 1980s have different names (e.g.

embedded-atom method - EAM, effective medium theory, Finnis-Sinclair potential, the

glue model, corrected effective medium potential - CEM, etc.) and are based on different

physical arguments (e.g. tight-binding model, effective-medium theory), but result in a

similar expression for the total energy of the system of N atoms:

=+==ij

ijji

ij

ijijiii

i

itot rfrFEEE )()(2

1)( 3.8.1

Interpretation and functional form ofF,f, and depend on a particular method. From the

point of view of effective medium theory or the embedded-atom method, the energy of

the atom i is determined by the local electron density at the position of the atom and the

functionfdescribes the contribution to the electronic density at the site of the atom i from

all atomsj. The sum over function fis therefore a measure oflocal electron density i.

The embedding energyFis the energy associated with placing an atom in the electron

environment described by . The pair-potential term describes two-body contribution.

The general form of the potential can be considered as a generalization of the basic idea


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of the Density Functional Theory the local electron density can be used to calculate the

energy.

In addition to having different physical interpretations, the different methods differ in the

way function are determined. Some authors derive functions and parameters from first

principles calculations, others guess the functions and fit parameters to experimental

data Results are usually rather similar.

The main advantage of these methods over pair potentials is the ability to describe the

variation of the bond strength with coordination. Increase of coordination decreases the

strength of each of the individual bonds and increases the bond length.

+

=

ij

ijij

i

ij

ijjii rrfFE )(2

1)(

3.8.2

In order to use this potential in MD simulation we need to find the forces:

ij

ji

ijij

rr

ij

ijrr

i

j

j

ijrr

j

i

i

ij

ijij

ij

jjiiii

ii

toti

i

r

rr

r

r

r

rfF

r

rfF

rFFr

Er

Er

F

)()()()()()(

)()()(

+

+

=

++===

=====

3.8.3

Only inter-particles distances rij are needed to calculate energy and forces the

calculation is nearly as simple and efficient as with pair potentials.

The EAM potential can be called an environment-dependent pair potential. The lack

of explicit 3-body terms makes it challenging to design potentials for metals where

covalent effects are important.


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Chapter 4: Molecular Dynamics Simulations

4.1 Introduction

Molecular dynamics simulation is a technique for computing the equilibrium and

transport properties of a classical many-body system. In this context, the word classical

means that the nuclear motion of the constituent particles obeys the law of classical

mechanics. This is an excellent approximation for a wide range of materials. Molecular

dynamics simulations are in many respects very similar to real experiments. When we

perform a real experiment, we proceeded as follows. We prepare a sample of the material

that we wish to study. We connect this sample to a measuring instrument (e.g., a

thermometer, UTM, manometer), and we measure the property of interest during acertain time interval. If our measurements are subject to statistical noise then longer we

average the more accurate is our measurement becomes. In a Molecular Dynamics

simulation, we follow exactly the same approach. First, we prepare a sample: we select a

model system consisting of N particles and we solve Newtons equation of motion for

this system until the properties of the system no longer change with time. After

equilibration, we perform actual measurement. In fact, some of the most common

mistake that can be made when performing a computer experiment are very similar to the

mistakes that can be made in real experiments.(e.g. the sample is not prepared correctly,

the measurement is too short, or we do not measure what we think.)

4.2 Molecular Dynamics: A program

The best introduction to Molecular Dynamics simulations is to consider a simple

program. The program we consider is kept as simple as possible to illustrate a number of

important features of Molecular Dynamics simulations.

The program constructed as follows:

1. We read in the parameters that specify the condition of the run (e.g. initial

temperature, number of particle, density, time step).

2. We initialize the system (i.e. we select the initial positions and velocities).

3. We compute the forces on all particles.


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4. We integrate Newtons equations of motion. This step and the previous one make

up the core of the simulation. They are repeated until we have computed the time

evolution of the system for the desired length of time.

5. After completion of the central loop we compute and print the averages of the

measured quantities, and stop.

a flow chart for Molecular Dynamics algorithm isshown below:

no

Initialize coordinates,Temperature, velocity etc.

Compute force between the

Integrate the Newtonsequation of motion and find

new coordinate and velocities

System reached

Production runMeasurement of properties

progr am md simple MD program

cal l i ni t initializationt = 0do whi l e ( t . l t . t max) MD loop

cal l f orce( f , en) determine forcescal l i nt egr at e( f , en) integrate equation of motion

t = t + del tcal l sampl e sample averageend dostopend


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4.2.1 Initialization

To start the simulation, one must assign positions and velocities to all the particles in thesystem. The particle position should be chosen compatible with the structure that we are

aiming to simulate. The particle should not be placed in positions that result in anappreciable overlap of the atomic or molecular cores. In the present case, we put eachparticle on its lattice site and then we assign each velocity component a value that isdrawn from random distribution in the interval [-0.5, 0.5]. Subsequently, the velocitiesare shifted such that the total momentum is zero and the velocities are scaled to adjust themean kinetic energy to the desired value. In thermal equilibrium, the following relationshould hold:

,/2 mTkv B= 4.2.1

where vis the component of the velocity of a given particle. We can use this relation

to define an instantaneous temperature at time t, T(t):

=

N

i f

i

BN

tmvtTk

1

2, )()( 4.2.2

Algorithm for initialization of a Molecular Dynamics program

Subr out i ne i ni t Initialization of MD programsumv = 0sumv2 = 0do i = 1, npart

x( i ) = l at t i ce_pos( i ) place the particles on a latticev( i ) = ( r anf ( ) - 0. 5) give random velocitiessumv = sumv + v( i ) velocity of center of masssumv2= sumv2 + v( i ) **2 kinetic energy

enddosumv = sumv/ npar t velocity of center of mass

sumv2 = sumv2/ npar t mean square velocitiesf s = sqr t ( 3 * t emp/ sumv2) scale factor of velocitiesdo i = 1, npart set desired kinetic energy and set

v( i ) = ( v( i ) - sumv) * f s velocity of center of mass to zeroxm( i ) = x( i ) v( i ) * dt position at previous time step

enddoreturnstop


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The instantaneous temperature T(t) can be adjusted to match the desired temperature Tby

scaling all velocities with a factor (T/T(t))1/2. This initial setting of temperature is not

particularly critical, as the temperature will change during equilibration. The approximate

previous position can be obtained by the use of initial velocities byxm(i)= x(i) . v(i) . dt,

wherex is present position v is initial velocity and dtis time step for Molecular Dynamics

simulation.

4.2.2 Force Calculation

The calculation of forces on every particle is the most time consuming part of Molecular

Dynamics simulation. If we consider a model system with pair wise additive interactions

we have to consider the contribution to the force on particle i due to all its neighbors. If

we consider only interaction between a particle and the nearest image of another particle,

this implies that, for a system ofN particles, we must evaluate N (N -1)/2 pair

distances.

Algorithm for calculation of force

subr out i ne f or ce ( f , en) determine the forceen = 0 and energydo i = 1, npart

f ( i ) = 0 set force to zero

end dodo i = 1, npar t 1 loop over all pairdo j = i + 1, npart

xr = x( i ) x( j )xr = xr box*ni nt ( xr / box) periodic boundary conditionr 2 = xr **2i f ( r 2. l t . r c2) t hen test cutoff

r 2i = 1/ r 2r 6i = r 2i **3f f = 48*r2i *r6i *( r 6i - 0. 5) Lennard-Jones potentialf ( i )= f ( i ) + f f * xr update forces

f ( j ) = f ( j ) - f f * xren = en+4*r 6i *( r 6i - 1) - ecut update energy

endi fenddo

enddoreturn


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This implies that, if we use no tricks, the time needed for the evaluation of the forces

scales as N2. There exist different techniques to speed up the evaluation of both short

range and long-range forces in such a way that computing time scales as N, rather than

N2.

4.2.3Integrating the Equations of Motion

After computing all forces between the particles, we integrate Newtons equations of

motion. Algorithms have been designed to do this. Verlet algorithm is not only one the

simplest, but also usually the best.

The Taylor series expansion of the coordinate of the particle, around time t,

),(!32

)()()()( 43

2 tOrttmtfttvtrttr ++++=+ 4.2.3

similarly,

).(!32

)()()()( 4

32 tOr

tt

m

tfttvtrttr +

+= 4.2.4

Summing the two equations, we obtain

)(2

)()(2)()( 42 tOt

m

tftrttrttr ++=++

or2

2

)()()(2)( t

m

tfttrtrttr ++ 4.2.5

The estimate of the new position contains an error that is of the ordert4, where tis the

time step of our Molecular Dynamics scheme. Verlet algorithm does not use the velocity

Algorithm for integrating the equation of motionsubr out i ne i nt egr at e( f , en) integrate equation of motionsumv = 0

sumv2 = 0do i = 1, npar t MD loop

xx = 2 * x( i ) xm( i ) + del t **2 * f ( i ) Verlet algorithmvi = ( xx - xm( i ) ) / ( 2*del t ) velocitysumv = sumv + vi velocity of center of masssumv2 = sumv2 + vi **2 total kinetic energyxm( i ) = x( i ) update positions previous time stepx( i ) = xx update positions current time

end dot emp = sumv2/ ( 3*npar t ) instantaneous temperature

et ot = ( en + 0. 5*sumv2) / npar t total energy per particlereturnend


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to compute new position. The velocity can be derived from the knowledge of the

trajectory, using

)()(2)()( 3tOttvttrttr +=+

or

)(2

)()()( 2tO

t

ttrttrtv +

+

= 4.2.6

It is possible to cast Verlet algorithm in a form that uses positions and velocities

computed at equal times. This velocity Verlet algorithm looks like a Taylor series

expansion for the coordinates:

.2

)()()()( 2t

m

tfttvtrttr ++=+ 4.2.7

where the update of the velocities is given by

.2

)()()()( t

m

tfttftvttv

+++=+ 4.2.8

Note that, in this algorithm, we can compute the new velocities only after we have

computed the new positions and, from these the new forces. Now that we have computed

the new positions, we discard the positions at time (t - t). The current positions become

the old positions and the new positions become the current positions. After each time step

we compute the current temperature, the current potential energy and the total energy.

The total energy should be conserved. This completes the introduction to Molecular

Dynamics method.

4.5 Boundary Conditions

The length-scale of MD is limited a large fraction of atoms is therefore on the surface

or near the surface. This is not the case with most of the physical systems. How to

reproduce interaction of atoms in the MD computational cell with the surrounding

materials?

4.5.1 Free boundaries (or no boundaries):

this works for a molecule, a cluster or aerosol particle in vacuum. Free boundary can also

be appropriate for ultra-fast processes when the effect of boundary is not important due to

the short time scale of the involved processes, e.g. fast ion/atom bombardment, etc.


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Figure 3Free boundary condtion

4.5.2 Rigid Boundaries (atoms at the boundaries are fixed). In most

cases the rigid boundaries are unphysical and introduce artifacts into the simulation

results. Sometimes used in combination with other boundary conditions.

Figure 4 Rigid boundary condition

4.5.3. Periodic Boundary condition (eliminates surfaces the most popular

choice of boundary conditions) this boundary conditions are used to simulate processes

in a small part of a large system. all atoms in the computational cell (dark box) are

replicated throughout the space to form an infinite lattice. That is, if atoms in the


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computational cell have positions ri, the periodic boundary condition also produces

mirror images of the atoms at the positions defined as

Figure 5 Periodic boundary condition

cnbmalrr iimage

i

+++= , where a, b, c are vectors that correspond to the edges of the

box, l, m, n are any integers from - to +.

Each particle in the computational cell is interacting not only with other particles in the

computational box, but also with their images in the adjacent boxes. The choice of the

position of the original box (computational cell) has no effect on forces or behavior of the

systems.

Limitations of periodic boundary condition:

The size of the computational cell should be larger than 2Rcut, where Rcut is the cutoff

distance of the integration potential. In this case any atom i interact with only one image

of any atom j. it does not interact with its own image. This condition is called minimum

image criterion

The characteristic size of any structural feature in the system of interest or the

characteristic length-scale of any important effect should be smaller than the size of thecomputational cell. For example, low frequency parts of the phonon spectrum can be

affected, stress fields of the different images of the same dislocation can interact etc. to

check if there are any artifacts due to the size of the computational cell perform

simulation with different sizes and check if the result converge.

Calculation of distances between atoms with periodic boundary conditions:


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When the minimum image criterion is satisfied, a particle can interact only with the

closest image of any other particle.

Figure 6 Minimum image criteria

The closest image may or may not belong to the computational cell. Therefore, in the

code, if a particle j is beyond the range of interaction with particle i (Rij > Rcut), we have

to check the closest images. For example, an algorithm for checking the closest image is:

do i = 1, N ! loop over total number of atomsdo j = 1, nei ghbor ! loop up to number of neighbors

xr = x( i ) x( j )i f ( xr . gt . hbox) t hen ! half of the length of box = hbox

xr = xr box ! length of box = boxel sei f ( xr . l t . hbox) t hen

xr = xr + box

endi f. ! Other calculations.

end doend do

4.3 An Illustration

In this section we will look into the approach for a typical atomistic calculation problem.

Suppose we want to find out total energy and resultant force on an atom in a system of

Argon atoms having some initial positions with potential function for the Argon is given.

To simplify the problem let us consider a two-dimensional problem by assuming

that atoms can move only in x and y direction only.

Also we assume free boundary for the sake of simplicity.

We will use no cutoff radius in the calculation i.e. all atom will interact with all

other atoms.


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Let us assume that atoms are nicely sitting on the corners of a grid having equal spacing

in x and y direction. The potential energy of atom due to atom j can be obtained using the

following equation. (Eq. 3.7.1)

=

6

ij

12

ij

ij rr4)U(r

Figure 7 Arrangement of atoms in 2-D planefrom the above equation it is clear that if we have values of rij can determine the potential

energy of atom 1 due to j. we can obtain the total potential energy of atom 1 by taking

summation of all the U(r1j) over j. similarly we can determine the force due to atom j on

atom using the equation below

=

6

8

6

21)(

24ijij

ji

ijrr

rrF

the resultant force is the summation of the individual forces due all neighboring atoms.

parameters for potential of Argon are = 0.0104 eV, = 3.40

Suppose we want to evaluate total energy and total force on atom 1. First we have to find

the distance jr1 to determine potential energy as well as force due to atom j . The values

are tabulated as follows


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Atom no 1

interacts withR1j U(rij)

F(rij)in x-

direction

F(rij)in y-

direction

2 2.0 23.23 0.0 -142.4

3 4.0 -0.9722e-2 0.0 0.5782e-2

4 6.0 -0.1332e-2 0.0 0.1286e-2

5 2.0 23.23 -142.4 0.0

6 2.282 0.2523 -0.9478 -0.9478

7 4.472 -0.6482e-2 0.2958e-2 0.5917e-2

8 6.325 -0.9799e-3 0.2867e-3 0.8601e-3

9 4.000 -0.9772e-2 0.5782e-2 0.0

10 4.472 -0.6482e-2 0.5917e-2 0.2958e-2

11 5.657 -0.1869e-2 0.1332e-2 0.1332e-2

12 7.211 -0.4520e-3 0.2063e-3 0.3095e-3

13 6.000 -0.1332e-2 0.1286e-2 0.0

14 6.325 -0.9799e-3 0.8601e-3 0.2867e-3

15 7.211 -0.4520e-3 0.3095e-3 0.2063e-3

16 8.485 -0.1715e-3 0.8537e-4 0.8537e-4

total 46.679 -143.339 -143.339

It is clear from the table that total potential energy is positive and resultant force is not

zero, which shows that atom 1 is not at equilibrium position and the rest of the atoms

repels it collectively. It is also evident that the atoms beyond 4 A are pulling the atom 1

towards them i.e. an attraction force is acting between them. Also notice that energy due

to these atoms is negative. Atoms situated at less than or equal to 4 A are trying to push

away atom 1 i.e. repulsive force is acting and also notice that energy due to these atoms is

positive.

4.4 Ensembles

Ensemble is a set of all states, which a system can acquire with keeping some

macroscopic parameter constants. For example in constant energy and constant volume

ensemble (NVE) is a set of all states, which the system can acquire with keeping the

volume and total energy (KE + PE) constant. A typical MD simulation with periodic


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boundary condition is constant energy and constant volume ensemble (NVE), which is

also known as micro-canonical ensemble. Consider a many-body system with energy E

that is capable of occupying every quantum state. If we take an average of some property

of the system over all possible quantum states of is called an ensemble average of the

property. However, this is not the way one think about the average behavior of a system.

In most experiments we perform a series of measurements during a certain period of time

and then determine the average of these measurements. The idea behind Molecular

Dynamics is precisely that we can study the average behavior of the many-body particle

system simply by computing the natural time evolution of that system numerically and

averaging the quantity of interest over a sufficiently long period of time.

4.5 Computer ExperimentsAfter formulating and implementing the Molecular Dynamics program, we wish to use it

to measure the interesting properties of many body systems. These quantities can be

compared with the real experiments. Simplest among these are the thermodynamic

properties of the system under consideration, such as the temperature T, the pressureP,

and the heat capacity Cv. As mentioned earlier, the temperature is measured by

computing the average kinetic energy per degree of freedom. For a system offdegrees of

freedom, the temperature T is given by

f

KTkB

2= 4.5.1

whereKis kinetic energy.

There are several ways to measure the pressure of a classical N-body system. The most

common among these is based on the virial equation for pressure. For pair-wise

interactions, we can write

,).rf(r1

ijij

tgr-mmm-notes.pdf

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