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Interatomic Potentials
• Before we can start a simulation, we need the model!
• Interactions between atoms, molecules,… are determined by quantum mechanics:– Schrödinger Equation + Born-Oppenheimer (BO) approximation
– BO: Because electrons T is so much higher (1eV=10,000 K) than true T and they move so fast, we can get rid of electrons and consider interaction of nuclei in an effective potential “surface.” V(R).
– Approach does not work during chemical reactions.
• Crucial since V(R) determines the quality of result.
• But we don’t know V(R).– Semi-empirical approach: make a good guess and use experimental
data to fix it up
– Quantum chemistry approach: works in a real space.
– Ab initio approach: it works really excellent but…
Semi-empirical potentials
• Assume a functional form, e.g. 2-body form.• Find some data: theory + experiment• Use theory + simulation to fit form to data.
• What data? – Atom-atom scattering in gas phase– Virial coefficients, transport in gas phase– Low-T properties of the solid, cohesive energy, lattice constant, bulk
modulus.– Melting temperature, critical point, triple point, surface tension,….
• Interpolation versus extrapolation. • Are results predictive?
Lennard-Jones potential V(R) = i<jv(ri-rj) v(r) = 4[(/r)12- (/r)6]
= minimum
= wall of potential
Reduced units:– Energy in – Lengths in
Good model for rare gas atoms
Phase diagram is universal!(for rare gas systems)
.
Morse potential
• Like Lennard-Jones
• Repulsion is more realistic-but attraction less so.
• Minimum neighbor position at r0
• Minimum energy is • Extra parameter “a” can be used to fit a third property:
lattice constant, bulk modulus and cohesive energy.
v(r) [e 2a(r r0) 2e a(r r0)]
dE
dr r00 B V
dP
dV V0
Vd2E
dV 2V0
Various Other Potentials
a) simplest: Hard-sphere
b) Hard-sphere, square-well
c) Coulomb (long-ranged) for plasmas
d) 1/r12 potential (short-ranged)
Atom-atom potentials
• Total potential is the sum of atom-atom pair potentials
• Assumes molecule is rigid, in non-degenerate ground state, interaction is weak so the internal structure is weakly affected. Geometry (steric effect) is important.
• Perturbation theory as rij >> core radius– Electrostatic effects: multipole expansion (if molecules are
charged or have a permanent dipole, …)
– Induction effects (by a charge on a neutral atom)
– Dispersion effects:
• dipole-induced-dipole (C6/r6)
– Short-range effects-repulsion caused by cores: exp(-r/c)
V (R) v(| ri rj |)i j
C6 d A( )B()
Fit for a Born (1923) potential
EXAMPLE: NaCl
• Obviously Zi=1
• Use cohesive energy and lattice constant (at T=0) to determine A and n:• EB=ea/d + er/dn
dEB/dr= –ea/d2 + ner/dn-1
=0
=> n=8.87 A=1500eVǺ8.87
• Now we need a check. The “bulk modulus”. – We get 4.35 x 1011 dy/cm2 experiment is 2.52 x 1011 dy/cm2
• You get to what you fit!
•Attractive charge-charge interaction
•Repulsive part determined by atom core.
V (R) ZiZ j
| ri rj |1
A
| ri rj |n
Silicon potential• Solid silicon is NOT well described by a pair potential.• Tetrahedral bonding structure caused by the partially filled
p-shell: sp3 hybrids (s+px+py+pz , s-px+py+pz , s+px-py+pz , s+px+py-pz)
• Stiff, short-ranged potential caused by localized electrons.
• Stillinger-Weber (1985) potential fit to:Lattice constant,cohesive energy, melting point, structure of liquid Si
for r<a
• Minimum at 109o
ri
rk
rj
i
v2(r) (B /r4 – A)e(r a) 1
v3(r) i, j,k e
/(rij a)/(rik a)[cosijk 1/3]2
Metallic potentials
• Have a inner core + valence electrons
• Valence electrons are delocalized. Pair potentials do not work very well. Strength of bonds decreases as density increases because of Pauli principle.
• EXAMPLE: at a surface, LJ potential predicts expansion but metals contract
• Embedded Atom Method (EAM) or glue models better.Daw and Baskes, PRB 29, 6443 (1984).
Embedding function electron density pair potential
• Good for spherically, closed-packed, symmetric atoms: FCC Cu, Al, Pb
• Not so good for BCC.
V (R) atoms F(i)
pairs (rij )
Problems with potentials
• Difficult problem because potential is highly dimensional function. Arises from QM so it is not a simple function.
• Procedure: fit data relevant to the system you need to simulate, with similar densities and local environment.
• Use other experiments to test potential if possible.
• Do quantum chemical (SCF or DFT) calculations of clusters. Be aware that these may not be accurate enough.
• No empirical potentials work very well in an inhomogenous environment.
• This is the main problem with atom-scale simulations--they really are only suggestive since the potential may not be correct. Universality helps.
Some tests
-Lattice constant
-Bulk modulus
-Cohesive energy
-Vacancy formation energy
-Property of an impurity
What are the forces?
• Common examples are Lennard-Jones (6-12 potential), Coulomb, embedded atom potentials.
• They are only good for simple materials.
• The ab initio philosophy is that potentials are to be determined directly from quantum mechanics as needed.
• But computer power is not yet adequate, in general. • But nearing in the future (for some problems).
• A powerful approach is to use simulations at the quantum level to determine parameters at the classical level.
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GB
[210]
<110>
The interatomic potential used in the simulation is based on the Embedded Atom Method (EAM). For additional simulations we usedAb initio method in combination withthe usual copper pseudopotential.