Transcript Potentials

ELECTRONIC STRUCTURE OF MATERIALS
From reality to simulation and back
A roundtrip ticket
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
v(r)  [e2a(rr0 )  2ea(rr0 ) ]
• 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.
dE
0
dr r
0
dP
d2 E
B  V
V
dV V
dV 2
0
V0
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
V(R)   v(| ri  rj |)
i j
• 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:
C6   d  A ( ) B ( )
• dipole-induced-dipole (C6/r6)
– Short-range effects-repulsion caused by cores: exp(-r/c)


Fit for a Born (1923) potential
Zi Z j
A
V (R) 

1
| ri  rj | | ri  rj |n
•Attractive charge-charge interaction
•Repulsive part determined by atom core.
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=1500eV 8.87
• Now we need a check. The “bulk modulus”.
– We get 4.35 x 1011 dy/cm2
• You get to what you fit!
experiment is 2.52 x 1011 dy/cm2

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
1
4
(ra)
v2 (r)  (B /r – A)e
for r<a
 /(rij a) /(rik a)
v 3(r)   e
[cosijk  1/3]2
i, j,k
• Minimum at
rk
109o
ri
i
rj
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).
V (R)   F(i )    (rij )
atoms

pairs
Embedding function
electron density
pair potential
• Good for spherically, closed-packed, symmetric atoms: FCC Cu, Al, Pb
• Not so good for BCC.
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 about relaxation and other monkey-tricks
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.
Go ahead on “real” systems
N
E
N
D
LLLLL
N
E
O
D
N
E
O
M
D
M
N
O
E
D
NN
M
E
O
D
E
M
O
L
D
O
M
E
M
D
N
L
O
D
M
M EN
0,30
0,25
Energy (eV)
0,20
0,15
0,10
0,05
0,00
-0,05
1,0
1,5
2,0
2,5
Distance (A)
3,0
3,5
4,0
[210]
<110>
GB
The interatomic potential used in
the simulation is based on the
Embedded Atom Method (EAM).
For additional simulations we used
Ab initio method in combination with
the usual copper pseudopotential.
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