Pair Potentials and Force Calculations

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1 Molecular Dynamics simulations Lecture 04: Pair Potentials and Force Calculations Dr. Olli Pakarinen University of Helsinki Fall 01 Lecture notes based on notes by Dr. Jani Kotakoski, 010 CONSTRUCTING A POTENTIAL FUNCTION In MD simulations, interactions between particles are typically presented with potential functions or functionals with free parameters. The free parameters are fitted to experimental data or more accurate calculations. Another option is to directly use quantum mechanical models for calculating the forces on-the-fly. This lecture, however, concentrates on classical analytical potentials in the form of pair potentials. In these models, we apply the Born-Oppenheimer approximation to simplify the interactions (electrons are assumed to be always in the ground state), and the interaction strength depends only on the interatomic separation.

2 As has already been pointed out, a typical way to construct a inter-atomic potential is to split it first in parts: U(r) = i U 1 (r i ) + i,j U (r i, r j ) + i,j,k U 3 (r i, r j, r k ) +... (1) first term is external potential, second one is a two-particle term involving atoms i and j, and the third one is a three particle term for i, j and k. The field quantity related to this scalar is the force on atom i: f i = ri U(r) () In chemistry, it s common to directly describe the force without defining the corresponding potentials. Therefore, chemical interaction models are often referred to as force fields, even when they are not constructed this way. Other names for analytic potentials are empirical potentials or classical potentials. They are used in many fields also outside MD simulations with field-specific emphasis.

3 INTERACTION MODELS IN DIFFERENT FIELDS Chemistry; Accurate reaction rates and molecular energy transfer are the key points of interest small systems and short times. The forces and integrators must be accurate. Materials Science; Solid state structures like surfaces, grain boundaries and interfaces are of interest as well as many-atom dynamics, like in crack motion. Systems can be large N 10 6 and the forces are relatively accurate (from quantum mechanics). Simulation times below µs range. Statistical Mechanics; Correlated many body motion is the main emphasis, especially liquids. Systems are larger and times longer than above simple interaction models. Biochemistry/ physics; The molecular structure and correlated motion (as in protein folding) require simple forces and long times. ORIGIN OF ATOMIC INTERACTIONS As we know by now, the potential energy function is (typically) of this shape: U(r) Pauli repulsion Attraction Equilibrium r Purely repulsive potential 'Typical' attractive potential At very small interatomic separations Coulombic repulsion between the nuclei dominates. After this, Pauli rule and Coulombic interaction between the electrons of the two atoms are dominating contributors in the repulsion. Attraction between two atoms can be due to van der Waals interaction, Coulomb interaction (for an ionic system), covalent bonding (pairing of valence electrons) or metallic bonding (sharing of valence electrons).

4 DIFFERENT WAYS TO FORM BONDS Simplistic introduction by khanacademy can be found at YouTube (click for the video). Different kinds of atomic interactions are explained below. Ionic Bonding Charge transfer occurs between two atoms which have a large difference in electronegativity: one of the atoms, typically a halogen, rips of an electron from the other one, typically an alkali metal. Due to the transferred electrons, one of the formed ions has a positive charge whereas the other one becomes negative. The ions attract each other via Coulombic interaction to form a compound, e.g. NaCl. Ionic bonding depends only on the distance between the atoms. Thus the angular relations play no role. Covalent Bonding When two atoms have a similar electronegativity a, e.g. in the case of atoms of the same element, they can share some of the electrons. For example, in an oxygen dimer, O, two oxygen atoms (with six valence electrons each) will share four electrons between the atoms so that each effectively has a filled outermost electron level. The atoms become bonded together since the shared electrons have a similar probability to localize around either one of the atoms. In contrast to ionic bonding, due to spatially localized electrons, covalent bond energetics depend on the angles between the atoms (when more than two atoms are bonded with covalent bonds). a Exactly the same electronegativity leads to non-polar covalent bonds, whereas differences in it lead to polar bonds.

Metallic Bonding In contrast to ionic and covalent bonds, a single metallic bond does not exist. Instead, metallic bonding is collective in nature and always involves a group of atoms.

5 Metallic Bonding In contrast to ionic and covalent bonds, a single metallic bond does not exist. Instead, metallic bonding is collective in nature and always involves a group of atoms. A metal is characterized by its electric conductivity which is due to delocalization of electrons in the material. Metallic bonding is a result of the attraction between the delocalized electrons and ions which are embedded in the electron cloud (or, free electron gas) of the the delocalized valence electrons. Dipole Interactions (van der Waals) For noble gas atoms, the attraction arises from dipole interaction which is due to the perturbed charge densities of the two atoms involved in the bonding. This interaction is weak, but still forms the basis for noble gas dimers and solids at low temperatures. CHARACTERISTIC BONDING FOR ELEMENTS

6 DIFFERENT INTERATOMIC POTENTIALS Pair potentials: U(r) = U 0 + i,j U (r i, r j ) Pair functionals: U[F, r] = i,j U (r i, r j ) + ( ) i F j g (r i, r j ) Cluster potentials: U(r) = U 0 + i,j U (r i, r j )+ i,j,k U 3(r i, r j, r k ) Cluster functionals: U[F, r] = i,j U (r i, r j )+ ( i F j g (r i, r j ), ) j,k g 3(r i, r j, r k ) Real potentials are often combinations of the above. IDEALIZED PAIR POTENTIALS Idealistic potentials can serve as the first approximation. Hard Sphere Potential U HS (r) = {, r < σ 0, r σ (3) First ever MD potential Billiard-ball physics Works for packing problems

7 Square Well Potential U SW (r) =, r < σ 1 ε, σ 1 r < σ 0, r σ (4) Soft Sphere Potentials [ σ ] ν U SS (r) = ε (5) r ν = 1 ν = 1

8 MORE REALISTIC PAIR POTENTIALS Lennard-Jones Lennard-Jones potential [Proc. R. Soc. Lond. A 106 (194) 463] is perhaps the best known pair potential which can be used in realistic simulations (although only for certain specific structures). It was developed to describe dipole interactions. Let s have a look at how to figure out a functional from for such a potential [following Kittel, Introduction to Solid State Physics, 8 th ed., Wiley, p. 53]. When we have a system of two identical inert gas atoms, what holds them together to form a dimer? If the charge distributions of the atoms would be rigid, the cohesive energy would be zero. However, the atoms induce dipole moments in each other, and the induced moments cause an attractive interaction. As a model, let s consider two oscillators, which are separated by distance R. Each oscillator has charges of ±e with separations x 1 and x. The particles oscillate along the x axis. + x x R Let p 1 and p denote the momenta. The force constant is C. The Hamiltonian of the unperturbated system is H 0 = 1 m p Cx m p + 1 Cx. (6) We assume frequence ω 0 for the strongest optical absorption line of the atom. Thus C = mω 0. Uncoupled energy is 1 hω 0.

9 Coulomb interaction energy of the two oscillators becomes H 1 = e R + e R + x 1 x e R + x 1 In the approximation x 1, x << R this translates to e R x. (7) H 1 e x 1 x R 3. (8) The total Hamiltonian H 0 + H 1 can be diagonalized by the normal mode transformation x s 1 (x 1 + x ); x a 1 (x 1 x ), (9) or x 1 = 1 (x s + x a ); x = 1 (x s x a ). (10) Similarly, the momenta associated with the normal modes are p 1 1 (p s + p a ); p 1 (p s p a ). (11) H = H 0 + H 1 becomes H = [ 1 m p s + 1 (C e R 3 ) ] [ 1 xs + m p a + 1 ) ] (C + e R 3 xa. (1) From H, the frequencies of the two coupled oscillators can be found ω = ) ] 1/ [ [(C ± e R 3 /m = ω 0 1 ± 1 ( ) e CR 3 1 ( ) ] e 8 CR (13) The zero point energy is 1 h(ω s + ω a ), which means that U = 1 h( ω s + ω a ) = hω 0 1 ( ) e 8 CR 3 = A R 6. (14)

10 So, the attractive interaction caused by the induced dipole moment (i.e., van der Waals dispersion interaction or London interaction) obeys the scaling law r 6. This is the principal attractive force in crystals of inert gases and many organic molecules. However, we still need a repulsive part for the interaction. When two atoms (A & B) are brought together, at some point their charge densities will start to overlap. With increasing overlap, the potential energy will become positive due to Pauli exclusion principle. This principle says that two electrons can not have equal quantum numbers, which would happen if some of the electrons from atom B would occupy states in atom A (or the other way around). Therefore, those electrons must be excited to higher energy levels, which leads to increase in the overall energy of the system. The analytical problem becomes overly complicated for this case, but fitting to experimental data has shown that a repulsive potential of the form B/R 1 describes it well enough while allowing easy computations ( 6 = 1... ). U(r) Pauli repulsion Dipole-dipole interaction Equilibrium r Adding the terms up gives the familiar functional form: [ (σ ) 1 ( σ ) ] 6 U LJ (r) = 4ε. r r (15) LJ works reasonably well for noble gases close to equilibrium. However, due to U(r) r 1 it fails badly at very close interatomic separations (true behavior is U(r) e r /r).

11 Morse Potential 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]. Also used in the mdmorse code. Functional form is U(r) = ij { } De α(r ij r 0 ) De α(r ij r 0 ). (16) Energy (ev) R A U r (Å) Developed for solving the Schrödinger equation since it has an analytical solution with this functional form. Efficient to evaluate and decays faster than LJ less problems with cutoff. Many parametrizations exist (see, e.g., [Phys. Rev. 114 (1959) 687]). ON PAIR POTENTIALS The good ones work for those materials which have a close-packed structure as the ground state (usually fcc or hcp). However, people have tried some tricks with pair potentials to extend their usage to covalent materials. Diamond lattice: open structure, four nearest neighbours, very far from close packed. Still, it is actually possible to make diamond stable locally with a pair potential, but this will become rather pathological (Mazzone potential for Si, [Phys. Stat. Sol (b) 165 (1991) 395.]): Note, many of these tricks introduce pathological problems to the potentials. Extreme care is advised if one ever needs to use any of such potentials.

Born-Huggins-Meyer potential (developed for alkali halides): U BHM (r) = A exp( B(r σ)) C/r 6 D/r 8.

12 Other pair potentials include: Buckingham potential: U Buck (r) = A exp( r/ρ) C/r 6. (17) This potential tries to solve the problem of LJ at small separations, but remains finite at small interatomic distances. Born-Huggins-Meyer potential (developed for alkali halides): U BHM (r) = A exp( B(r σ)) C/r 6 D/r 8. (18) FORCE CALCULATION FROM U(r) For a pure pair potential U ij, the force acting on atom i due to atom j is f ij = rij U (r ij ) = + rji U (r ij ) = f ji [ U = ˆx + U ŷ + U ] ẑ, (19) x ij y ij z ij r ij = r i r j, x ij = x i x j, (0) U = du x ij dr r ij x ij, r ij x ij = x ij r ij(1)

13 PRACTICAL IMPLEMENTATION ON FORCE CALCULATION do i=1,n do j=1,n if (i==j) cycle rijx = rx(j)-rx(i) rijy = ry(j)-ry(i) rijz = rz(j)-rz(i) rijsq = rijx**+rijy**+rijz** rij = sqrt(rijsq) if (rij < rcut) then V = (Potential energy per atom)/ dvdr =..derivative of potential energy with respect to its only argument r.. a = -dvdr/m/.0! Unit transformations may be needed. Note the factor 1/! ax(i) = ax(i)-rijx/rij*a! The application on both ax(j) = ax(j)+rijx/rij*a! i and j ensures that ay(i) = ay(i)-rijy/rij*a! Newton s third law is ay(j) = ay(j)+rijy/rij*a! fulfilled az(i) = az(i)-rijz/rij*a az(j) = az(j)+rijz/rij*a endif enddo enddo Above, we had the N loops since neighborlist was omitted for brevity. 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 is waste of resources. To overcome this, we can adjust the loop to do i=1,n-1 do j=i+1,n either when implementing the neighbor list or when calculating the forces. Note that this doesn t work for all potentials.

14 FORCES FOR A THREE-BODY POTENTIAL For a pair potential, U ij = U ji since the potential only depends on r ij = r ij = r ji. This simplifies the force calculation. In the case of a many-body potential, things are more difficult, since U ij U ji. When we have both two-body terms U ij = U (r i, r j ) and three-body terms U ijk = U 3 (r i, r j, r k ), the force (on atom i) becomes f i = i j (U ij + U ji ) + j,k U ijk = j ( i U ij + i U ji ) + j,k i U ijk () Often the only three-body dependency is implemented through a cosine term: U 3 (r i, r j, cos θ ijk ). When this is the case, one can utilize the following equalities: cos θ ijk = r ij r ik (3) r ij r ik ( ) rij r ik i cos θ ijk = i =... r ij ik [ ] cos θ ijk = rij 1 [ cos θijk r ij + r ij r i rik 1 ] (4) r ij r ik thus, there s no need to actually evaluate the cos function, which is computationally expensive. Also, depending on the potential, there may be symmetries which can be used to reduce calculations.

15 TESTING THE CODE A practical way to test the implementation of force and potential calculation is to: 1 Calculate T = 0 K, and compare with an analytical solution for some simple system (e.g., dimer or perfect crystal). Simulate a two-atom system by starting with a very small inter-atomic distance (with energy much higher than the binding energy, e.g ev). Use a small enough time step (so that decreasing further won t change the result) and check the kinetic energies of the atoms when the system has exploded. You should have K end = U start. 3 Third check would be numerical derivation of the potential energy: U(vr) ŝ = lim h 0 U(r + hŝ) U(r) h = U(r) ŝ = f(r) ŝ. PROBLEMS WITH PAIR POTENTIALS Pair potentials have some inherent shortcomings, of which some are listed below: For example, for cubic materials, they by default satisfy the Cauchy relation (c 1 = c 44 ) which is violated in most transition metals and semiconductors. Cauchy relation: A set of six relations between the compliance constants of a solid which should be satisfied provided the forces between atoms in the solid depend only on the distances between them and act along the lines joining them, and provided that each atom is a center of symmetry in the lattice. Vacancy formation energies are overestimated and mostly as large as cohesive energies due to poor description of environmental changes and the lattice relaxations around defects and surfaces. The ground state is always a close-packed structure (fcc or hcp). Pair potentials don t take into account environmental changes. Instead, they predict that a two-atom bond is always equally strong no matter what happens around it. This is almost never true.

16 Because of the lacking dependency of the environment, pair potentials are also poor with surfaces (in addition to vacancies). Example: Vacancy Formation Energy A simple estimate of vacancy formation energies can be calculated as E f = E tot (N, vac) E tot (N, perfect). (5) Assuming a nearest-neighbor pair potential, E/bond = U(r nn ) = φ, no relaxation, fcc structure (1 neighbors): E tot (N, vac) = 1 [(N 1) 1φ + 1 (1 1) φ] = 6(N 1)φ (6) E tot (N, perfect) = 1 N1φ = 6Nφ (7) E f = 6φ = E coh. (8) However, ab initio calculations give [Solid State Physics: Advances in Research and Applications, 43 (1990) 1]: Element E coh (ev) E f (ev) V ± 0. Nb ± 0.3 W ± 0. This comparison neglects the effect of relaxation, but in simple metals it s not likely to have an effect larger than 1 ev. Better, and more computationally expensive, potentials will be presented during the coming lectures.

17 POTENTIALS AT CUTOFF Since we have to limit the interaction to a certain distance r c from each atom, this means that we have a discontinuity in the U(r). Discontinuity in the potential leads to a jump in the force at r c, which makes the physics of a simulation questionable. So, we need to have a potential which has a continuous first derivative. U(r) r cut To achieve this at r c, we must smoothly drive the potential to zero within a certain distance range r [r c, r c + r]. r In many modern potentials, the potential is defined with a proper first derivative at r c. Driving a Lennard-Jones potential to zero As we saw, Lennard-Jones is defined as U(r) Pauli repulsion Dipole-dipole interaction [ (σ ) 1 ( σ ) ] 6 U LJ (r) = 4ε. (9) r r Equilibrium r There are (at least) two ways how to drive this potential to zero: Shift and tilt the potential to get U(r) and U (r) continuous at r c. Use a third order polynomial for r [r c, r c + r].

18 Shift-and-tilt requires playing around with the actual potential equation: U(r) = U LJ (r) (r r c )U LJ(r c ) U LJ (r c ). (30) Now U(r c ) = 0 as is U (r c ) = 0. However, since we are changing the equation, the fitting must be redone with the new equation. U(r) shift-and-tilt Or, as said, we can solve constants for the polynomial P(r) = ar 3 + br 3 + cr + d for r [r c, r c + r] with conditions: U LJ r P(r) P(r c ) = U LJ (r c ) P (r c ) = U LJ (r c) P(r c + r c ) = 0 P (r c + r c ) = 0 (31) Note that even smooth-looking potential functions can have problematic forces (even if continuous).

19 REPULSIVE POTENTIALS Another part where we may need to replace the original potential is at the shortest interatomic distances. In most analytic potentials the repulsive part simply allows having the potential minimum at correct place. It also gives a proper description of the near-equilibrium system when the interatomic distance is somewhat smaller than the ideal dimer distance. However, as we saw for pair potentials, they usually give wrong description of the energy when the atoms are very close to each other. This typically happens in ion irradiation or nuclear physics, or when other energetic phenomena are involved. Therefore, there is a need to use a repulsive potential at these short distances fitted smoothly to the equilibrium potential. The usual equation for a repulsive potential is U(r) = Z 1Z e 4πε 0 r Φ( r a ) (3) where Φ(x) is a screening function and a = a(z 1, Z ) is a screening length. At very short distances, Φ = 1 and the potential reduces to Coulomb interaction, whereas at longer distances Φ 0. The most used repulsive potential is the ZBL universal potential a. Within this model, an approximation is given for the screening function Φ. a [Ziegler, Biersack and Littmark, The Stopping and Range of Ions in Matter (Pergamon, New York, 1985)]

20 From fits to a massive data set for dimers, the authors came up with a screening function of the form Φ(x) = 4 a i e b ix i=1 (33) with parameters for a i : , , 0.80 and and for b i : 3., 0.943, and The accuracy of the potential is expected to be in the range of 5 10%. Another way of obtaining a repulsive potential is to carry out electronic structure calculations for dimers with different lengths or even better for different bond lengths inside solids. Fitting is done similarly to the running to zero at cutoff, e.g. with a fitted polynomial. SUMMARY Different scientific disciplines have different requirements for the accuracy of potential energy functions and forces. The simplest interaction model is provided by pair potentials, which are simple functions of only one variable (interatomic distance r) and a few fitted parameters. Interatomic potentials are typically constructed of separate attractive and repulsive parts. Attraction depends on the material, repulsion arises from the Pauli rule and Coulombic repulsion. Pair potentials have some serious drawbacks, but can be used for some metallic systems and to describe van der Waals interactions between noble gas atoms. Because forces are direct derivatives of the potentials, U must have a continuous first derivative for all r. This poses a problem at cutoff, since U must be smoothly driven to zero. Also, most of the equilibrium potentials have wrong scaling at very short r so that they have to be modified.

Set the initial conditions r i. Update neighborlist. Get new forces F i

Set the initial conditions r i ( t 0 ), v i ( t 0 ) Update neighborlist Get new forces F i ( r i ) Solve the equations of motion numerically over time step Δt : r i ( t n ) r i ( t n + 1 ) v i ( t n )