Molecular Dynamics simulations

Lecture 04:

Pair Potentials andForce Calculations

Dr. Olli PakarinenUniversity of Helsinki

Fall 2012

Lecture notes based on notes by Dr. Jani Kotakoski, 2010

CONSTRUCTING A POTENTIAL FUNCTION

I In MD simulations, interactions between particles are typicallypresented with potential functions or functionals with freeparameters.

I The free parameters are fitted to experimental data or moreaccurate calculations.

I Another option is to directly use quantum mechanical models forcalculating the forces on-the-fly.

I This lecture, however, concentrates on classical analyticalpotentials in the form of pair potentials.

I In these models, we apply the Born-Oppenheimer approximationto simplify the interactions (electrons are assumed to be alwaysin the ground state), and the interaction strength depends onlyon the interatomic separation.

I As has already been pointed out, a typical way to construct ainter-atomic potential is to split it first in parts:

U(r) =∑

iU1(ri) +

∑i,j

U2(ri, rj) +∑i,j,k

U3(ri, rj , rk) + . . .

(1)

first term is external potential, second one is a two-particle terminvolving atoms i and j, and the third one is a three particle termfor i, j and k.

I The field quantity related to this scalar is the force on atom i:

fi = −∇riU(r) (2)

I In chemistry, it’s common to directly describe the force withoutdefining the corresponding potentials.

I Therefore, chemical interaction models are often referred to asforce fields, even when they are not constructed this way.

I Other names for analytic potentials are empirical potentialsor classical potentials.

I They are used in many fields – also outside MD simulations –with field-specific emphasis.

INTERACTION MODELS IN DIFFERENT FIELDS

I Chemistry; Accurate reaction rates and molecular energytransfer are the key points of interest→ small systems and shorttimes. The forces and integrators must be accurate.

I Materials Science; Solid state structures like surfaces, grainboundaries and interfaces are of interest as well as many-atomdynamics, like in crack motion. Systems can be large – N ≈ 106

– and the forces are relatively accurate (from quantummechanics). Simulation times below µs range.

I Statistical Mechanics; Correlated many body motion is the mainemphasis, especially liquids. Systems are larger and timeslonger than above→ simple interaction models.

I Biochemistry/–physics; The molecular structure and correlatedmotion (as in protein folding) require simple forces and longtimes.

ORIGIN OF ATOMIC INTERACTIONS

I As we know by now, the potential energy function is (typically) ofthis shape:

U(r)

r

Equilibrium

Attraction

Pauli repulsion

Purely repulsive

potential

'Typical'

attractive

potential

I At very small interatomicseparations Coulombicrepulsion between the nucleidominates.

I After this, Pauli rule andCoulombic interaction betweenthe electrons of the two atomsare dominating contributors inthe repulsion.

I Attraction between two atoms can be due to van der Waalsinteraction, Coulomb interaction (for an ionic system), covalentbonding (pairing of valence electrons) or metallic bonding(sharing of valence electrons).

DIFFERENT WAYS TO FORM BONDS

I Simplistic introduction by khanacademy can be found at YouTube(click http://is.gd/fEdrq for the video). Different kinds ofatomic interactions are explained below.

Ionic Bonding

I Charge transfer occurs between two atoms which have alarge difference in electronegativity: one of the atoms,typically a halogen, rips of an electron from the other one,typically an alkali metal.

I Due to the transferred electrons, one of the formed ionshas a positive charge whereas the other one becomesnegative.

I The ions attract each other via Coulombic interaction toform a compound, e.g. NaCl.

I Ionic bonding depends only on the distance between theatoms. Thus the angular relations play no role.

Covalent Bonding

I When two atoms have a similar electronegativity a, e.g. in thecase of atoms of the same element, they can share some of theelectrons.

I For example, in an oxygen dimer, O2, two oxygen atoms (with sixvalence electrons each) will share four electrons between theatoms so that each effectively has a filled outermost electronlevel.

I The atoms become bonded together since the shared electronshave a similar probability to localize around either one of theatoms.

I In contrast to ionic bonding, due to spatially localized electrons,covalent bond energetics depend on the angles between theatoms (when more than two atoms are bonded with covalentbonds).

aExactly the same electronegativity leads to non-polar covalent bonds, whereasdifferences in it lead to polar bonds.

Metallic Bonding

I In contrast to ionic and covalent bonds, a single metallic bonddoes not exist.

I Instead, metallic bonding is collective in nature and alwaysinvolves a group of atoms.

I A metal is characterized by its electric conductivity which is dueto delocalization of electrons in the material.

I Metallic bonding is a result of the attraction between thedelocalized electrons and ions which are embedded in theelectron cloud (or, free electron gas) of the the delocalizedvalence electrons.

Dipole Interactions (van der Waals)

I For noble gas atoms, the attraction arises from dipole interactionwhich is due to the perturbed charge densities of the two atomsinvolved in the bonding.

I This interaction is weak, but still forms the basis for noble gasdimers and solids at low temperatures.

CHARACTERISTIC BONDING FOR ELEMENTS

DIFFERENT INTERATOMIC POTENTIALS

Pair potentials: U(r) = U0 +∑

i,j U2(ri, rj)

Pair functionals: U[F , r] =∑

i,j U2(ri, rj) +∑

i F(∑

j g2(ri, rj))

Cluster potentials: U(r) = U0 +∑

i,j U2(ri, rj)+∑i,j,k U3(ri, rj , rk)

Cluster functionals: U[F , r] =∑

i,j U2(ri, rj)+

∑i F(∑

j g2(ri, rj),∑

j,k g3(ri, rj , rk))

Real potentials are often combinations of the above.

IDEALIZED PAIR POTENTIALS

I Idealistic potentials can serve as the first approximation.

Hard Sphere Potential

UHS(r) ={ ∞, r < σ

0, r > σ(3)

I First ever MD potential

I Billiard-ball physics

I Works for packing problems

Square Well Potential

USW(r) =

∞, r < σ1−ε, σ1 6 r < σ20, r > σ2

(4)

Soft Sphere Potentials

USS(r) = ε[σ

r

]ν(5)

ν = 1 ν = 12

MORE REALISTIC PAIR POTENTIALS

Lennard-JonesI Lennard-Jones potential [Proc. R. Soc. Lond. A 106 (1924) 463] is

perhaps the best known pair potential which can be used inrealistic simulations (although only for certain specificstructures).

I It was developed to describe dipole interactions. Let’s have alook at how to figure out a functional from for such a potential[following Kittel, Introduction to Solid State Physics, 8th ed., Wiley, p. 53].

I When we have a system of two identical inert gas atoms, whatholds them together to form a dimer?

I If the charge distributions of the atoms would be rigid, thecohesive energy would be zero.

I However, the atoms induce dipole moments in each other, andthe induced moments cause an attractive interaction.

I As a model, let’s consider two oscillators, which are separatedby distance R.

I Each oscillator has charges of ±e with separations x1 and x2.The particles oscillate along the x axis.

+ - + -x2 x1

R

I Let p1 and p2 denote themomenta.

I The force constant is C.

I The Hamiltonian of the unperturbated system is

H0 =1

2mp21 +

12Cx2

1 +1

2mp22 +

12Cx2

2 . (6)

I We assume frequence ω0 for the strongest optical absorptionline of the atom. Thus C = mω2

0. Uncoupled energy is2× 1

2 hω0.

I Coulomb interaction energy of the two oscillators becomes

H1 =e2

R +e2

R + x1 − x2−

e2

R + x1−

e2

R − x2. (7)

I In the approximation |x1|, |x2| << R this translates to

H1 ≈ −2e2x1x2

R3 . (8)

I The total Hamiltonian H0 +H1 can be diagonalized by thenormal mode transformation

xs ≡1√2(x1 + x2); xa ≡

1√2(x1 − x2), (9)

I or

x1 =1√2(xs + xa); x2 =

1√2(xs − xa). (10)

I Similarly, the momenta associated with the normal modes are

p1 ≡1√2(ps + pa); p2 ≡

1√2(ps − pa). (11)

I H = H0 +H1 becomes

H =

[1

2m p2s +

12

(C −

2e2

R3

)x2

s

]+

[1

2m p2a +

12

(C +

2e2

R3

)x2

a

]. (12)

I From H, the frequencies of the two coupled oscillators can befound

ω =

[(C ± 2e2

R3

)/m]1/2

=ω0

[1± 1

2

(2e2

CR3

)−

18

(2e2

CR3

)+ . . .

](13)

I The zero point energy is 12 h(ωs +ωa), which means that

∆U =12

h(∆ωs + ∆ωa) = − hω0 ·18

(2e2

CR3

)2

= −AR6 . (14)

I So, the attractive interaction caused by the induced dipolemoment (i.e., van der Waals dispersion interaction or Londoninteraction) obeys the scaling law ∝ r−6.

I This is the principal attractive force in crystals of inert gases andmany organic molecules.

I However, we still need a repulsive part for the interaction.

I When two atoms (A & B) are brought together, at some pointtheir charge densities will start to overlap.

I With increasing overlap, the potential energy will becomepositive due to Pauli exclusion principle.

I This principle says that two electrons can not have equalquantum numbers, which would happen if some of the electronsfrom atom B would occupy states in atom A (or the other wayaround).

I Therefore, those electrons must be excited to higher energylevels, which leads to increase in the overall energy of thesystem.

I The analytical problem becomes overly complicated for thiscase, but fitting to experimental data has shown that a repulsivepotential of the form B/R12 describes it well enough whileallowing easy computations (2× 6 = 12 . . . ).

U(r)

r

Pauli repulsion

Equilibrium

Dipole-dipole interaction

I Adding the terms up gives thefamiliar functional form:

ULJ(r) = 4ε[(σ

r

)12−(σ

r

)6]

.

(15)

I LJ works reasonably well for noble gases close to equilibrium.However, due to U(r) ∝ r−12 it fails badly at very closeinteratomic separations (true behavior is U(r) ∝ e−r/r).

Morse PotentialI Pair potentials work reasonably for simple metals (e.g., Na, Mg,

Al and fcc/hcp metals). An often used one is the Morse potential[Phys. Rev. 34 (1930) 57].

I Also used in the mdmorse code.

I Functional form is

U(r) =∑

ij

{De−2α(rij−r0) − 2De−α(rij−r0)

}. (16)

-3

-2

-1

0

1

2

3

4

2 2.5 3 3.5 4 4.5 5

En

erg

y (e

V)

r (Å)

RAU

I Developed for solving theSchrödinger equation since it has ananalytical solution with this functionalform.

I Efficient to evaluate and decaysfaster than LJ→ less problems withcutoff.

I Many parametrizations exist (see,e.g., [Phys. Rev. 114 (1959) 687]).

ON PAIR POTENTIALS

I The “good” ones work for those materials which have aclose-packed structure as the ground state (usually fcc or hcp).

I However, people have tried some tricks with pair potentials toextend their usage to covalent materials.

I Diamond lattice: open structure, four nearest neighbours, veryfar from close packed.

I Still, it is actually possible to makediamond stable locally with a pairpotential, but this will become ratherpathological (Mazzone potential for Si,[Phys. Stat. Sol (b) 165 (1991) 395.]):

I Note, many of these tricks introducepathological problems to thepotentials. Extreme care is advised ifone ever needs to use any of suchpotentials.

I Other pair potentials include:I Buckingham potential:

UBuck(r) = A exp(−r/ρ) − C/r6. (17)

This potential tries to solve the problem of LJ at small separations,but remains finite at small interatomic distances.

I Born-Huggins-Meyer potential (developed for alkali halides):

UBHM(r) = A exp(−B(r − σ)) − C/r6 − D/r8. (18)

FORCE CALCULATION FROM U(r)I For a pure pair potential Uij , the force acting on atom i due to

atom j is

fij = −∇rij U2(rij) = +∇rji U2(rij) = −fji

= −

[∂U2∂xij

x +∂U2∂yij

y +∂U2∂zij

z]

, (19)

rij = ri − rj , xij = xi − xj , (20)

∂U∂xij

=dUdr

∂rij∂xij

,∂rij∂xij

=xijrij

(21)

PRACTICAL IMPLEMENTATION ON FORCE CALCULATION

do i=1,Ndo j=1,N

if (i==j) cyclerijx = rx(j)-rx(i)rijy = ry(j)-ry(i)rijz = rz(j)-rz(i)rijsq = rijx**2+rijy**2+rijz**2rij = sqrt(rijsq)if (rij < rcut) then

V = (Potential energy per atom)/2dVdr = ..derivative of potential energy with respect to its only

argument r..a = -dVdr/m/2.0 ! Unit transformations may be needed. Note the factor 1/2!ax(i) = ax(i)-rijx/rij*a ! The application on bothax(j) = ax(j)+rijx/rij*a ! i and j ensures thatay(i) = ay(i)-rijy/rij*a ! Newton’s third law isay(j) = ay(j)+rijy/rij*a ! fulfilledaz(i) = az(i)-rijz/rij*aaz(j) = az(j)+rijz/rij*a

endifenddo

enddo

I Above, we had the N2 loops since neighborlist was omitted forbrevity.

I However, even if we would use the neighborlist:do i=1,N

neighboursi=neighbourlist(startofneighbourlist)do jj=1,nneighboursi

we still would take each interaction into account twice, which iswaste of resources.

I To overcome this, we can adjust the loop todo i=1,N-1

do j=i+1,N

either when implementing the neighbor list or when calculating theforces.

I Note that this doesn’t work for all potentials.

FORCES FOR A THREE-BODY POTENTIAL

I For a pair potential, Uij = Uji since the potential only depends onrij = |rij | = |rji|. This simplifies the force calculation.

I In the case of a many-body potential, things are more difficult,since Uij 6= Uji.

I When we have both two-body terms Uij = U2(ri, rj) andthree-body terms Uijk = U3(ri, rj , rk), the force (on atom i)becomes

fi = −∇i

∑j(Uij + Uji) +

∑j,k

Uijk

= −

∑j(∇iUij +∇iUji) +

∑j,k∇iUijk

(22)

I Often the only three-body dependency is implemented through acosine term: U3(ri, rj , cos θijk).

I When this is the case, one can utilize the following equalities:

cos θijk =rij · rikrijrik

⇒ (23)

∇i cos θijk = ∇i

(rij · rikrij ik

)= . . .

=

[cos θijk

r2ij

−1

rijririj

]+

[cos θijk

r2ik

−1

rijrik

](24)

thus, there’s no need to actually evaluate the cos function, whichis computationally expensive.

I Also, depending on the potential, there may be symmetrieswhich can be used to reduce calculations.

TESTING THE CODE

I A practical way to test the implementation of force and potentialcalculation is to:

1 Calculate U @ T = 0 K, and compare with an analytical solutionfor some simple system (e.g., dimer or perfect crystal).

2 Simulate a two-atom system by starting with a very smallinter-atomic distance (with energy much higher than the bindingenergy, e.g. 1000 eV). Use a small enough time step (so thatdecreasing further won’t change the result) and check the kineticenergies of the atoms when the system has exploded. You shouldhave Kend = Ustart.

3 Third check would be numerical derivation of the potential energy:

∂U(vr)∂s = lim

h→0

U(r + hs) − U(r)h = ∇U(r) · s = −f(r) · s.

PROBLEMS WITH PAIR POTENTIALS

I Pair potentials have some inherent shortcomings, of which someare listed below:

I For example, for cubic materials, they by default satisfy the Cauchyrelation (c12 = c44) which is violated in most transition metals andsemiconductors.

Cauchy relation: A set of six relations between the compliance constantsof a solid which should be satisfied provided the forces between atoms inthe solid depend only on the distances between them and act along thelines joining them, and provided that each atom is a center of symmetry inthe lattice.

I Vacancy formation energies are overestimated and mostly as largeas cohesive energies due to poor description of environmentalchanges and the lattice relaxations around defects and surfaces.

I The ground state is always a close-packed structure (fcc or hcp).

I Pair potentials don’t take into account environmental changes.Instead, they predict that a two-atom bond is always equally strongno matter what happens around it. This is almost never true.

I Because of the lacking dependency of the environment, pairpotentials are also poor with surfaces (in addition to vacancies).

Example: Vacancy Formation Energy

I A simple estimate of vacancy formation energies can becalculated as

Ef = Etot(N, vac) − Etot(N, perfect). (25)

I Assuming a nearest-neighbor pair potential,E/bond = U(rnn) = φ, no relaxation, fcc structure (12neighbors):

Etot(N, vac) = 12 [(N − 12) 12φ+ 12 (12 − 1)φ] = 6(N − 1)φ

(26)

Etot(N, perfect) = 12N12φ = 6Nφ (27)

⇒ Ef = −6φ = −Ecoh. (28)

I However, ab initio calculations give [Solid State Physics: Advances in

Research and Applications, 43 (1990) 1]:

Element −Ecoh (eV) Ef (eV)V 5.31 2.1 ± 0.2

Nb 7.57 2.6 ± 0.3W 8.90 4.0 ± 0.2

I This comparison neglects the effect of relaxation, but in simplemetals it’s not likely to have an effect larger than ∼ 1 eV.

I Better, and more computationally expensive, potentials will bepresented during the coming lectures.

POTENTIALS AT CUTOFF

I Since we have to limit the interaction to a certain distance rc fromeach atom, this means that we have a discontinuity in the U(r).

I Discontinuity in the potential leads to a jump in the force at rc,which makes the physics of a simulation questionable.

U(r)

r

rcut

I So, we need to have a potentialwhich has a continuous firstderivative.

I To achieve this at rc, we mustsmoothly drive the potential to zerowithin a certain distance ranger ∈ [rc, rc + ∆r].

I In many modern potentials, thepotential is defined with a proper firstderivative at rc.

Driving a Lennard-Jones potential to zero

I As we saw, Lennard-Jones is defined as

U(r)

r

Pauli repulsion

Equilibrium

Dipole-dipole interactionULJ(r) = 4ε

[(σr

)12−(σ

r

)6]

. (29)

I There are (at least) two ways how to drive this potential to zero:

I Shift and tilt the potential to get U(r) and U ′(r) continuous at rc.

I Use a third order polynomial for r ∈ [rc, rc + ∆r].

I Shift-and-tilt requires playing around with the actual potentialequation:

U(r) = ULJ(r) − (r − rc)U′LJ(rc) − ULJ(rc). (30)

I Now U(rc) = 0 as is U ′(rc) = 0.

I However, since we are changing the equation, the fitting must beredone with the new equation.

U(r)

r

shift-and-tilt

ULJ

P(r)

I Or, as said, we can solve constantsfor the polynomialP(r) = ar3 + br3 + cr + d forr ∈ [rc, rc + ∆r] with conditions:

P(rc) = ULJ(rc)

P ′(rc) = U ′LJ(rc)

P(rc + ∆rc) = 0P ′(rc + ∆rc) = 0

(31)

I Note that even smooth-looking potential functions can haveproblematic forces (even if continuous).

REPULSIVE POTENTIALS

I Another part where we may need to replace the original potentialis at the shortest interatomic distances.

I In most analytic potentials the repulsive part simply allowshaving the potential minimum at correct place.

I It also gives a proper description of the near-equilibrium systemwhen the interatomic distance is somewhat smaller than theideal dimer distance.

I However, as we saw for pair potentials, they usually give wrongdescription of the energy when the atoms are very close to eachother.

I This typically happens in ion irradiation or nuclear physics, orwhen other energetic phenomena are involved.

I Therefore, there is a need to use a repulsive potential at theseshort distances – fitted smoothly to the equilibrium potential.

I The usual equation for a repulsive potential is

U(r) = Z1Z2e2

4πε0r Φ(ra ) (32)

where Φ(x) is a screening function and a = a(Z1, Z2) is ascreening length.

I At very short distances, Φ = 1 and the potential reduces toCoulomb interaction, whereas at longer distances Φ ≈ 0.

I The most used repulsive potential is the ZBL universalpotential a.

I Within this model, an approximation is given for the screeningfunction Φ.

a[Ziegler, Biersack and Littmark, The Stopping and Range of Ions in Matter(Pergamon, New York, 1985)]

I From fits to a massive data set for dimers, the authors came upwith a screening function of the form

Φ(x) =4∑

i=1aie−bix (33)

with parameters for ai: 0.1818, 0.5099, 0.2802 and 0.02817 andfor bi: 3.2, 0.9423, 0.4029 and 0.2016.

I The accuracy of the potential is expected to be in the range of5 − 10%.

I Another way of obtaining a repulsive potential is to carry outelectronic structure calculations for dimers with different lengthsor – even better – for different bond lengths inside solids.

I Fitting is done similarly to the running to zero at cutoff, e.g. witha fitted polynomial.

SUMMARY

I Different scientific disciplines have different requirements for theaccuracy of potential energy functions and forces.

I The simplest interaction model is provided by pair potentials,which are simple functions of only one variable (interatomicdistance r) and a few fitted parameters.

I Interatomic potentials are typically constructed of separateattractive and repulsive parts. Attraction depends on thematerial, repulsion arises from the Pauli rule and Coulombicrepulsion.

I Pair potentials have some serious drawbacks, but can be usedfor some metallic systems and to describe van der Waalsinteractions between noble gas atoms.

I Because forces are direct derivatives of the potentials, U musthave a continuous first derivative for all r. This poses a problemat cutoff, since U must be smoothly driven to zero. Also, most ofthe equilibrium potentials have wrong scaling at very short r sothat they have to be modified.