1.10

Interatomic Potential Development

G. J. Ackland
University of Edinburgh, Edinburgh, UK

ß 2012 Elsevier Ltd. All rights reserved.

1.10.1

Introduction

268

1.10.2
1.10.3
1.10.4
1.10.4.1
1.10.4.2
1.10.4.2.1
1.10.4.2.2
1.10.4.2.3
1.10.4.2.4
1.10.4.2.5
1.10.4.2.6
1.10.5
1.10.5.1
1.10.5.2
1.10.5.3
1.10.6

1.10.6.1
1.10.6.2
1.10.6.3
1.10.7
1.10.7.1
1.10.7.2
1.10.7.3
1.10.7.4
1.10.7.5
1.10.8
1.10.8.1
1.10.8.2
1.10.8.3
1.10.8.4
1.10.9
1.10.10
1.10.10.1
1.10.10.2
1.10.10.3
1.10.11
1.10.12
1.10.12.1
1.10.12.2
1.10.12.2.1
1.10.12.2.2
1.10.12.3

Basics
Hard Spheres and Binary Collision Approximation
Pair Potentials

LJ Phase Diagram
Necessary Results with Pair Potentials
Outward surface relaxation
Melting points
Vacancy formation energy
Cauchy pressure
High-pressure phases
Short ranged
From Quantum Theory to Potentials
Free Electron Theory
Nearly Free Electron Theory
Embedded Atom Methods and Density Functional Theory
Many-Body Potentials and Tight-Binding Theory
Energy of a Part-Filled Band
The Moments Theorem
Key Points
Properties of Glue Models
Crystal Structure
Surface Relaxation
Cauchy Relations
Vacancy Formation
Alloys
Two-Band Potentials
Fitting the s–d Band Model
Magnetic Potentials
Nonlocal Magnetism
Three-Body Interactions
Modified Embedded Atom Method
Potentials for Nonmetals
Covalent Potentials

Molecular Force Fields
Ionic Potentials
Short-Range Interactions
Parameterization
Effective Pair Potentials and EAM Gauge Transformation
Example: Parameterization for Steel
FeCr
FeC
Austenitic Steel

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268

Interatomic Potential Development


1.10.13
1.10.13.1
1.10.14
References

Analyzing a Million Coordinates
Useful Concepts Without True Physical Meaning
Summary

Abbreviations
BCA
bcc
DFT
DOS
EAM
fcc
FS
GGA
hcp
LDA
LJ
MD
MEAM
NFE

Binary collision approximation
Body-centered cubic
Density functional theory
Density of electronic states
Embedded atom method

Face-centered cubic
Finnis–Sinclair
Generalized gradient approximation
for exchange and correlation
Hexagonal closed packed
Local density approximation for
exchange and correlation
Lennard Jones
Molecular dynamics
Modified embedded atom method
Nearly free electron

Symbols
Ec Cohesive energy
KB Boltzmann constant
KF Fermi wavevector

1.10.1 Introduction
Nuclear materials are subject to irradiation, and their
behavior is therefore not that of thermodynamic
equilibrium. To describe the behavior that leads to
radiation damage at a fundamental level, one must
follow the trajectories of the atoms. Since millions of
atoms may be involved in a single event, this must be
done by numerical simulation, either molecular
dynamics (MD) or kinetic Monte Carlo. For either
of these, a description of the energy is needed: this is
the interatomic potential.
Now that accurate quantum mechanical force calculations are available, one might ask whether there is still
a role for atomistic potentials. In practice, ab initio calculations are currently limited to a few hundred atoms

and a few picoseconds, or a few thousands at T ¼ 0 K.
With MD and interatomic potentials, one can run calculations of millions of atoms for nanoseconds. With

289
289
290
290

kinetic Monte Carlo calculation, timescales extend for
years. These methods still provide the only method of
atomistic simulation in these regimes. The issue is not
whether they are still necessary; rather it is which of
their predictions are correct.
Before embarking on any simulation, it is essential
to consider whether the potential contains enough of
the right physics to describe the problem at hand. For
example, in studying copper, one might ask about the
following:
 Color (total electronic structure)
 Conductivity (electronic structure around Fermi
level)
 Crystal structure (ground state electronic structure)
 Freezing mechanism (bond formation and crystal
structure)
 Dislocation dynamics (stacking fault energy)
 Surface structures (bond breaking energies)
 Interaction with primary radiation (short-range
atom–atom interactions)
 Phonon spectrum (curvature of potential)
 Corrosion (chemical reaction)

In each case, a totally different aspect of physics
is required. An essential issue for empirical potentials
is transferability: can potentials fitted to one set of
properties describe another? In general they cannot,
so one has to be careful to use a potential in the type
of application that it was intended for.
For many physical problems such as dislocation
dynamics, swelling, fracture, segregation, and phase
transitions, much of the physics is dominated by
geometry. Here, finite size effects are more important
than accurate energetics and empirical potentials find
their home.

1.10.2 Basics
Materials relevant to reactors are held together by
electrons. An interatomic potential expresses energy
in terms of atomic positions: the electronic and magnetic degrees of freedom are integrated out.
Most empirical potentials are derived on the basis of
some approximation to quantum mechanical energies.
If they are subsequently used in MD of solids, then what


Interatomic Potential Development

is actually used are forces: the derivative of the energy.
For near-harmonic solids, it is actually the second
derivative of the energy that governs the behavior.
A dilemma: Does the primary term covering energetics also dominate the second derivative of the
energy? To take an extreme example, an equation
which calculates the energy of a solid exactly for all

configurations to 0.1% is: E ¼ mc 2 : most of the
energy is in the rest mass of the atoms. But this is
patently useless for calculating condensed matter
properties. We encounter the same problem in a less
extreme form in metals: should we concentrate the
energy gained in delocalizing the electrons to form
the metal, or is treating perturbations around the
metallic state more useful? In general, the issue is
‘What is the reference state.’ Most potentials implicitly assume that the free atom is the reference state.

1.10.3 Hard Spheres and Binary
Collision Approximation
The simplest atomistic model is the binary collision
approximation (BCA), which simulates cascades as a
series of atomic collisions. The ‘potential’ is then
basically a hard sphere, with collisions either elastic
or inelastic and the possibility of adding friction to
describe an ion’s progress through an electron field.
There is no binding energy, so the condensed phase is
stabilized due to confinement by the boundary conditions. This is a poor approximation, since the structures and energies of the equilibrium crystal and
defects do not correspond to real material. However,
it allows for quick calculation and the scaling of defect
production with energy of the primary knock-on atom.
The best-known code for this type of calculation is
probably Oak Ridge’s MARLOWE.1

1.10.4 Pair Potentials
Pairwise potentials are the next level in complexity
beyond BCA. They allow soft interactions between
particles and simultaneous interaction between many

atoms. Pair potentials always have some parameters
that relate to a particular material, requiring fitting to
experimental data. This immediately introduces the
question of whether the reference state should be free
atoms (e.g., in argon), free ions (e.g., NaCl), or ions
embedded in an electron gas (e.g., metals).
The two classic pair potentials used for modeling
are the Lennard Jones (LJ) and Morse potential. Each

269

consists of a short-range repulsion and a long-range
attraction and has two adjustable parameters.
sÀ12 sÀ6 !
À
VLJ ¼ e
r
r
VMorse ¼ e½2eÀ2ar À eÀar Š
ÀsÁÀ6
cohesion is on the basis of van der Waals
The r
interactions, while the eÀar is motivated by a screenedCoulomb potential. The repulsive terms were invented
ad hoc.
Because they have only two parameters, all simulations using just LJ or Morse are equivalent, the
values of e, s, or a , which simply rescale energy
and length. Hence, these potentials cannot be fitted
to other properties of particular materials.
Both LJ and Morse potentials stabilize closepacked crystal structures, and both have unphysically
low basal stacking fault energies. Equivalently, the

energy difference between fcc and hcp is smaller
than for any real material. For materials modeling,
this introduces a problem that the (110) dislocation
structure is split into two essentially independent
partials. For radiation damage, this means that configurations such as stacking fault tetrahedra are overstabilized, and unreasonably large numbers of
stacking faults can be generated in cascades, fracture,
or deformation. If the energy scale is set by the cohesive energy of a transition metal, then the vacancy and
interstitial formation energy tend to be far too large; if
the vacancy energy is fitted, then the cohesive energy
is too small.
For binary systems, these potentials can stabilize a
huge range of crystal structures, even without explicit
temperature effects; some progress has been made
to delineate these, but it is far from complete.2 Once
one moves to N-species systems, there are e and s
parameters for each combination of particles, that is,
N(N þ 1) parameters. Now it is possible to fit to properties of different materials, and the rapid increase
in parameters illustrates the combinatorial problem
in defining potentials for multicomponent systems.
1.10.4.1

LJ Phase Diagram

In practice, pair potentials are cut off at a certain
range, which can have a surprising effect on stability
as shown in Figure 1
While the LJ fluid is very well studied, the finite
temperature crystal structure has only recently been
resolved. The problem is that the fcc and hcp structures
are extremely close in energy (see Figure 1(b)), so the

entropy must be calculated extremely accurately.


270

Interatomic Potential Development

400
500
600
700
800
900
1000

300

200

0.5

100

0

ps 3/e

fcc
0.4


kT/e

0.3
hcp
Harmonic N = 6
Harmonic N = 123
Harmonic+fit N = 123
LS-NPT N = 63
LS-NVT N = 63
LS-NPT N = 123
Salsburg and Huckaby
3

0.2

2

3

rc/s

4

5

6

Figure 1 Energy difference between hcp and fcc for the
Lennard-Jones potential at 0 K, as a function of cutoff (rc )
with either simple truncation or with the potential shifted

to remove the energy discontinuity at the cutoff. Without
truncation the difference is 0.0008695 e, with hcp more stable.

This has been done by Jackson using ‘latticeswitch Monte Carlo3 (see Figure 2). The equivalent
phase diagram for the Morse potential remains
unsolved. The LJ potential has been used extensively
for fcc materials, and it still comes as a surprise to
many researchers that fcc is not the ground state.
1.10.4.2 Necessary Results with Pair
Potentials
Apart from the specific difficulties with Morse and LJ
potentials, there are other general difficulties that are
common to all pair potentials, which make them
unsuitable for radiation damage studies.
Expanding the energy as a sum of pairwise interactions introduces some constraints on what data can
be fitted, even in principle. It is important to distinguish this problem from a situation where a particular
parameterization does not reproduce a feature of a
material. There are many features of real materials
that cannot be reproduced by pair potential whatever
the functional form or parameterization used.
1.10.4.2.1 Outward surface relaxation

For a single-minimum pair potential, the nearest neighbors repel one another, while longer ranged neighbors
attract. When a surface is formed, more long-range
bonds are cut than short-range bonds, so there is an
overall additional repulsion. Hence, the surface layer is
pushed outward. But in almost all metals, the surface
atoms relax toward the bulk, because the bonds at
the surface are strengthened. Similarly, pair potentials


0.1

0

1

1.5

2
rs 3

Figure 2 Pressure versus temperature phase diagram
for the crystalline region of the Lennard-Jones system in
reduced units where p is pressure and r is density. The
equilibrium density is at rs3 ¼ 1:0915. Filled squares are the
harmonic free energy integrated to the thermodynamic limit
from Salsburg, Z. W.; Huckaby, D. A. J. Comput. Phys.
1971, 7, 489–502. All other points are from lattice-switch
Monte Carlo simulations with N atoms, lines showing the
phase boundary deduced from the Clausius–Clapeyron
equation, from Jackson, A. N. Ph. D. Thesis, University of
Edinburgh, 2001; Jackson, A. N.; Bruce, A. D.; Ackland, G. J.
Phys. Rev. E 2002, 65, 036710.

give too large a ratio of surface to cohesive energy,
again consistent with the failure to describe the
strengthening of the surface bonds.
1.10.4.2.2 Melting points

With LJ, the relation between cohesive energy and

melting is Ec =kB Tm % 13, other pair potentials being
similar. Real metals are relatively easier to melt, with
values around 30. One can fit the numerical value of
the e parameter to the melting point, and accept the
discrepancy as a poor description of the free atom.
1.10.4.2.3 Vacancy formation energy

For a pair-potential, removing an atom from the
lattice involves breaking bonds. The cohesive energy
of a lattice comes from adding the energies of those
bonds. Hence, the cohesive energy is equal to the
vacancy formation energy, aside from a small difference from relaxation of the atoms around the vacancy.


Interatomic Potential Development

In real metals, the vacancy formation energy is typically one-third of the cohesive energy, the discrepancy
coming yet again from the strengthening of bonds to
undercoordinated atoms.
1.10.4.2.4 Cauchy pressure

Pairwise potentials constrain possible values of the
elastic constants. Most notably, it is the ‘Cauchy’
relation which relates C12 –C66 . In a pairwise potential, these are given by the second derivative of the
energy with respect to strain, which are most easily
treated by regarding the potential as a function of r 2
rather than r ; whence for a pair potential V ðr 2 Þ, it
follows:
2 X 00 2 2 2
V ðrij Þxij yij

C12 ¼ C66 ¼
O ij
where i; j run over all atoms and O is the volume
of the system.
In metals, this relation is strongly violated (e.g., in
gold, C12 ¼ 157GPa; C44 ¼ C66 ¼ 42GPa).

and simplify. Quantum mechanics can be expressed
in any basis set, so there are several possible starting
points for such a theory. Thus, a picture based on
atomic orbitals (i.e., tight binding) or plane waves (i.e.,
free electrons) can be equally valid: for potential
development, the important aspect is whether these
methods allow for intuitive simplification.
When a potential form is deduced from quantum
theory, approximations are made along the way. An
aspect often overlooked is that the effects of terms
neglected by those approximations are not absent in
the final fitted potential. Rather they are incorporated
in an averaged (and usually wrong) way, as a distortion of the remaining terms. Thus, it is not sensible to
add the missing physics back in without reparameterizing the whole potential.
1.10.5.1

Free Electron Theory

For a free electron gas with Fermi wavevector kF, the
energy U of volume O is5
U¼

1.10.4.2.5 High-pressure phases


Many materials change their coordination on pressurization (e.g., iron from bcc (8) to hcp (12)) and
some on heating (e.g., tin, from fourfold to sixfold).
This suggests that the energy is relatively insensitive
to coordination – for pair potentials, it is proportional. These problems suggest that a potential has
to address the fact that electrons in solids are not
uniquely associated with one particular atom, whether
the bonding be covalent or metallic. Ultimately, bonding comes from lowering the energy of the electrons,
and the number of electrons per atom does not change
even if the coordination does.

271

h"2 kF5
O
10p2 me

This contribution to the energy of the condensed phase
generates no interatomic force since U is independent
of the atomic positions. However, its contribution is
significant: metallic cohesive energy and bulk moduli
are correct to within an order of magnitude. Consideration of this term gives some justification for ignoring
the cohesive energy and bulk modulus in fitting a
potential, and fitting shear moduli, vacancy, or surface
energies instead. The discrepancy is absorbed by a
putative free electron contribution which does not
contribute to the interatomic atomic forces in a constant volume ensemble calculation.

1.10.4.2.6 Short ranged


It is worth noting that some properties that are
claimed to be deficiencies of pair potentials are actually associated with short range. So, for example, the
diamond structure cannot be stabilized by nearneighbor potentials, but a longer ranged interaction
can stabilize this, and the other complex crystal structures observed in sp-bonded elements.4

1.10.5 From Quantum Theory to
Potentials
To understand how best to write the functional form
for an interatomic potential, we need to go back to
quantum mechanics, extract the dominant features,

1.10.5.2

Nearly Free Electron Theory

In nearly free electron (NFE) theory, the effects of
the atoms are included via a weak ‘pseudopotential.’
The interatomic forces arise from the response of the
electron gas to this perturbation. To examine the
appropriate form for an interatomic potential, we
consider a simple weak, local pseudopotential V0 ðr Þ.
The total potential actually seen at ri due to atoms at
rj will be as follows:
X
V0 ðrij Þ þ W ðr Þ
V ðri Þ ¼
j

where W ðr Þ describes how the electrons interact
with one another. Given the dielectric constant,



272

Interatomic Potential Development

we can estimate W in reciprocal space using linear
response theory:
W ðqÞ ¼ V0 ðqÞ=eðqÞ
where eðqÞ is the dielectric function. In Thomas
Fermi theory,
eðqÞ ¼ 1 þ ð4kF =pa0 q 2 Þ
where a0 is the Bohr radius. A more accurate
approach due to Lindhard:



4kF
1 1 À x2 1 þ x
ln j
þ
j
eðqÞ ¼ 1 þ
pa0 q 2
4x
2
1Àx
where x ¼ q=2kF accounts for the reduced screening
at high q, and r0 is the mean electron density.
From this screened interaction, it is possible to

obtain volume-dependent real space potentials.6
The contributions to the total energy are as
follows:
 The free electron gas (including exchange and
correlation)
 The perturbation to the free electron band
structure
 Electrostatic energy (ion–ion, electron–electron,
ion–electron)
 Core corrections (from treating the atoms as
pseudopotentials)

 Soft phonon instabilities are an extreme case of the
Kohn anomaly. They arise when the lowering of
energy is so large that the phonon excitation has
negative energy. In this case, the phonon ‘freezes
in,’ and the material undergoes a phase transformation to a lower symmetry phase.
 Quasicrystals are an example where the atoms
arrange themselves to fit the Friedel oscillation.
This gives well-defined Bragg Peaks for scattering
in reciprocal space, and includes those at 2kF but
no periodic repetition in real space.
 Charge density waves refer to the buildup of
charge at the periodicity of the Friedel oscillation.
 ‘Brillouin Zone–Fermi surface interaction’ is yet
another name for essentially the same phenomenon, a tendency for free materials from structures
which respect the preferred 2kF periodicity for the
ions – which puts 2kF at the surface.
 ‘Fermi surface nesting’ is yet another example of
the phenomenon. It occurs for complicated crystal

structures and/or many electron metals. Here,
structures that have two planes of Fermi surface
separated by 2kF are favored, and the wavevector q
is said to be ‘nested’ between the two.
 Hume-Rothery phases are alloys that have ideal
composition to allow atoms to exploit the Friedel
oscillation.

This arises from the singularity in the Lindhard
function at q ¼ 2kF : physically, periodic lattice perturbations at twice the Fermi vector have the largest
perturbative effect on the energy. The effect of
Friedel oscillations is to favor structures where the
atoms are arranged with this preferred wavelength. It
gives rise to numerous effects.

NFE pseudopotentials enabled the successful prediction of the crystal structures of the sp3 elements. It is
tempting to use this model for ‘empirical’ potential
simulation, using the effective pseudopotential core
radius and the electron density as fitting parameters;
indeed such linear-response pair potentials do an
excellent job of describing the crystal structures of
sp elements.
For MD, however, there are difficulties: the electron density cannot be assumed constant across a free
surface and the elastic constants (which depend on
the bulk term) do not correspond to long-wavelength
phonons (which do not depend on the bulk term).
Since most MD calculations of interest in radiation
damage involve defects (voids, surfaces), phonons,
and long-range elastic strains, NFE pseudopotentials
have not seen much use in this area. They may be

appropriate for future work on liquid metals (sodium,
potassium, NaK alloys).
The key results from NFE theory are the following:

 Kohn anomalies in the phonon spectrum are particular phonons with anomalously low frequency.
The wavevector of these phonons is such as to
match the Friedel oscillation.

 The cohesive energy of a NFE system comes primarily from a volume-dependent free electron gas
and depends only mildly on the interatomic pair
potential.

In this model, interatomic pair potential terms arise
only from the band structure and the electrostatic
energy (the difference between the Ewald sum and a
jellium model) and give a minor contribution to the
total cohesive energy. However, these terms are
totally responsible for the crystal structure.
A key concept emerging from representing the
Lindhard screening in real space is the idea of a
‘Friedel oscillation’ in the long-range potential:
V ðr Þ /

cos 2kF r
ð2kF r Þ3


Interatomic Potential Development

 The pair potential is density dependent: structures

at the same density must be compared to determine the minimum energy structure.
 The pair potential has a long-ranged, oscillatory
tail.
 These potentials work well for understanding
crystal structure stability, but not for simulating
defects where there is a big change in electron
density.
 The reference state is a free electron gas: description of free atoms is totally inadequate.

1.10.5.3 Embedded Atom Methods and
Density Functional Theory
In the density functional theory (DFT), the electronic energy of a system can be written as a functional of its electron density:
U ¼ F ½rðrފ
7

The embedded atom model (EAM) postulates that
in a metal, where electrostatic screening is good, one
might approximate this nonlocal functional by a local
function. And furthermore, that the change in energy
due to adding a proton to the system could be treated
by perturbation theory (i.e., no change in r). Hence,
the energy associated with the hydrogen atom would
depend only on the electron density that would exist
at that point r in the absence of the hydrogen.
UHðrÞ ¼ FH ðrðrÞÞ
The idea can be extended further, where one considers the energy of any atom ‘embedded’ in the
effective medium of all the others.8 Now, the energy
of each (ith) atom in the system is written in the
same form,
Ui ¼ Fi ðrðri ÞÞ

To this is added the interionic potential energy,
which in the presence of screening, they took as a
short-ranged pairwise interaction. This gives an
expression for the total energy of a metallic system:
X
X
Fi ðrðri ÞÞ þ
V ðrij Þ
Utot ¼
i

ij

To make the model practicable, it is assumed that r
can be evaluated
P as a sum of atomic densities fðr Þ, that
is, rðri Þ ¼ j fðrij Þ and that F and V are unknown
functions which could be fitted to empirical data. The
‘modified’ EAM incorporates screening of f and additional contributions to r from many-body terms.

273

1.10.6 Many-Body Potentials and
Tight-Binding Theory
1.10.6.1

Energy of a Part-Filled Band

An alternate starting point to defining potentials is
tight-binding theory. As this already has localized

orbitals, it gives a more intuitive path from quantum
mechanics to potentials. Consider a band with a density of electronic states (DOS) n(E) from which the
cohesive energy becomes
ð Ef
U ¼ ðE À E0 ÞnðEÞ dE
where E0 is the energy of the free atom, which to a
first approximation lies at the center of the band.
For example, a rectangular d-band describing both
spin states and containing N electrons, width W has
nðEÞ ¼ 10=W , and EF ¼ W ðN À 5Þ=10 þ E0 whence
(Figure 3),
N ð10 À N Þ
W
½1Š
U ¼À
10
This gives parabolic behavior for a range of energyrelated properties across the transition metal group,
such as melting point, bulk modulus, and Wigner–
Seitz radius. For a single material, the cohesion is
proportional to the bandwidth. Even for more complex band shapes, the width is the key factor in
determining the energy.
The width of the band can be related to its second
moment9 here:
ð
m2 ¼ ðE À E0 Þ2 nðEÞ dE
¼

ð E0 þW =2
E0 ÀW =2


10ðE À E0 Þ2 =W dE

¼ 5W 2 =6

½2Š

To build a band in tight-binding theory, we set up a
matrix of onsite and hopping integrals (Figure 4). For
a simple s-band ignoring overlap,
n(E)

W

E0

EF
E

Figure 3 Density of states for a simple rectangular band
model.


274

Interatomic Potential Development

Hopping integral h(r)
h

h


s

s

Onsite term s

s

h

h

0

0

h

s

h

0

0

h

h


s

h

h

0

0

h

s

h

0

0

h

h

s

1. hhhhh
2. hhshh


Figure 4 Matrix of onsite and hopping integrals for a
planar five-atom cluster – in tight binding this gives five
eigenstates, each of which contributes one level to the
‘density of states’: five delta functions. In an infinite solid,
the matrix and number of eigenstates become infinite, so
the density of states becomes continuous. Of course, tricks
then have to be employed to avoid diagonalizing the
matrix directly.
Figure 5 Dashed and dotted lines show two of the chains
of five hops which contribute to the fifth moment of the
tight-binding density of states.

S ¼ hÈi jVi jÈi i
hðrij Þ ¼ hÈi jVi jÈj i
The electron eigenenergies come from diagonalizing
this matrix (there are, of course, cleverer ways to
do this than brute force). Typically, we can use
them to create a density of states, n(E), which can be
used to determine cohesive energy (as above).
The width of this band depends on the off-diagonal
terms (in the limit of h ¼ 0, the band is a delta
function). One can proceed by fitting S and h, or
move to a further level of abstraction.
1.10.6.2

The Moments Theorem

A remarkable result by Ducastelle and CyrotLackmann10 relates the tight-binding local density
of states to the local topology. If we describe the
density of states in terms of its moments where the

pth moment is defined by
ð1
E p nðEÞ dE
mp ¼
À1

and recall that by definition
X
nðEÞ ¼
dðE À Ei Þ
i

i

i

where H is the Hamiltonian matrix written on the
basis of the eigenvectors. But, the trace of a matrix is
invariant with respect to a unitary transformation,
that is, change of basis vectors to atomic orbitals i.
Therefore,

X
X
½H p Šii 
mðiÞ
p
i

i

ðiÞ

A sum of local moments of the density of states mp .
These diagonal terms of H p are given by the sum of
all chains of length p of the form Hij Hjk Hkl . . . Hni .
These in turn can be calculated from the local topology: a prerequisite for an empirical potential. They
consist of all chains of hops along bonds between
atoms which start and finish at i (e.g., see Figure 5).
By counting the number of such chains, we can build
up the local density of states.
Unfortunately, algorithms for rebuilding DOS and
deducing the energy using higher moments tend to
converge rather slowly, the best being the recursion
method.11
The zeroth moment simply tells us how many
states there are.
The first moment tells us where the band center is.
Taking the band center as the zero of energy, the
second moment is as follows:
X
X
ðiÞ
½Hij Hji Š ¼
hðrij Þ2
½3Š
m2 ¼ ½H 2 Šii ¼
j

where i labels the eigenvalues, we get
ð1

X
X p
Ep
dðE À Ei Þ dE ¼
Ei ¼ Tr½H p Š
mp ¼
À1

mp ¼ Tr½H p Š ¼

j

where h is a two-center hopping integral, which can
therefore be written as a pairwise potential.
This result, that the second moment of the tightbinding density of states can be written as a sum of
pair potentials, provides the theoretical underpinning
for the Finnis–Sinclair (FS) potentials. Referring
back to the rectangular band model, we can take the
ðiÞ
second moment of the local density of states m2 as a
measure of the bandwidth.


Interatomic Potential Development

This gives the relationship between cohesive
energy, bandwidth, and number of neighbors ðzi Þ. In
qffiffiffiffiffiffiffi
ðiÞ
the simplest form Wi / m2

pffiffiffi
N ð10 À N Þ
Wi / À z
½4Š
20
that is, the band energy is proportional to the square root of
the number of neighbors.
Note that this is only a part of the total energy
due to valence bonding. There is also an electrostatic interaction between the ions and an exclusionprinciple repulsion due to nonorthogonality of the
atomic orbitals – it turns out that both of these can be
written as a pairwise potential V ðr Þ.
The moments principle was laid out in the late
1960s.12 To make a potential, the squared hopping
integral is replaced by an empirical pair potential
fðrij Þ, which also accounts for the prefactor in eqn
[4] and the exact relation between bandwidth and
second moment. Once the pairwise potential V ðrij Þ
is added, these potentials have come to be known as
FS potentials.13
X
X
X sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ecoh ¼
V ðrij Þ À
fðrij Þ
½5Š
Ui ¼ À

i


ij

j

where V and f are fitting functions.

275

Further work14 showed that the square root law
held for bands of any shape provided that there was
no charge transfer between local DOS and that the
Fermi energy in the system was fixed. For bcc, atoms
in the second neighbor shell are fairly close, and are
normally assumed to have a nonzero hopping integral.
Notice that the first three moments only contain
information about the distances to the shells of atoms
within the range of the hopping integral. Therefore, a
third-moment model with near neighbor hopping
could not differentiate between hcp and fcc (in fact,
only the fifth moment differentiates these in a nearneighbor hopping model!). This led Pettifor to consider a bond energy rather than a band energy, and
relate it to Coulson’s definition of chemical bond
orders in molecules.15 Generalizing this concept leads
to a systematic way of going beyond second moments
and generating bond order potentials.
One can investigate the second-moment hypothesis
by looking at the density of states of a typical transition
metal, niobium, calculated by ab initio pseudopotential
plane wave method, Figure 6, and comparing it with
the density of states at extremely high pressure. The
similarity is striking: as the material is compressed,

the band broadens but the structure with five peaks
remains unchanged. The s-band is displaced slightly
to higher energies at high pressure, but still provides a

Density of states (electrons eV–1)

2

1.5

1

0.5

0
-5

0

5

10

15

Energy (eV)
Figure 6 Density of states for bcc Nb. Dotted blue line is at ambient pressure, and solid red line is for 32% reduction in
volume. The Fermi energy is set to zero in each case.



276

Interatomic Potential Development

low, flat background, which extends from slightly
below the d-band to several electron volts above.

Table 1
Neighbor distances in fcc, hcp (c=a ¼
and bcc, in units of the nearest neighbor separation

pffiffiffiffiffiffiffiffi
8=3),

Structure

1.10.6.3

Key Points

fcc

 In a second-moment approximation, the cohesive
(bond) energy is proportional to the square root of
the coordination.
 Other contributions to the energy can be written as
pairwise potentials.

1(12)


bcc

1(8)

hcp

1(12)

pffiffiffi
2ð6Þ
qffiffi
4
3 (6)
pffiffiffi
2 (6)

pffiffiffi
3 (24)
pffiffiffi
2 (12)
qffiffi
8
3 (2)

2(12)
qffiffiffiffi
11
3 (24)
pffiffiffi
3 (18)


pffiffiffi
5ð24Þ
pffiffiffi
3 (8)
qffiffiffiffi
11
3 (6)

2(6)

The number of neighbors
pffiffiffiffi at that distance is given in brackets. For
fcc the shells fall at N for all integers up to 30 except one. As fcc
and hcp structures have identical numbers and ranges of first and
second neighbors, glue or pair potentials can only distinguish them
via long-range interactions.

1.10.7 Properties of Glue Models
The embedded atom and FS potentials fall into a
general class of potentials of the form:
"
#
X
X
X
Ui ¼
Fi
fðrij Þ þ
V ðrij Þ

i

j

j

with a many-body cohesive part and a two-body
repulsion. Both fðrij Þ and V ðrij Þ are short ranged,
so MD with these potentials is at worst only as costly
as a simple pair potential (computer time is proportional to number of particles).
These models are sometimes referred to as glue
potentials,16 the many-body F term being thought of
as describing how strongly an atom is held by the
electron ‘glue’ provided by its environment.
The pragmatic approach to fitting in all glue
schemes is to regard the pair potential as repulsive
at short-range with long-range Friedel oscillations.
Compared with most pair potential approaches, this
is unusual in that the repulsive term is longer ranged
than the cohesive one.
1.10.7.1

Crystal Structure

According to the tight-binding theory on which the
FS potentials are based, the relative stability of bcc
and fcc is determined by moments above the second,
which in turn relate to three center and higher hops.
These third and higher moments effects are explicitly
absent in second-moment models, and so by implication, the correct physics of phase stability is not

contained in them. There is no such clear result in
the derivation of the EAM; however, since the forms
are so similar, the same problem is implicit.
In glue models, energy is lowered by atoms having
as many neighbors as possible; thus, fcc, hcp, and bcc
crystal structures (and their alloy analogs) are normally stable (see Table 1); bcc is normally stable in
potentials when the attractive region is broad enough

to include 14 neighbors, fcc/hcp are stable for narrower attractive regions in which only the eight nearest bcc-neighbors contribute significantly to the
bonding. Indeed, without second neighbor interactions, bcc is mechanically unstable to Bain-type shear
distortion. The fcc–hcp energy difference is related
to the stacking fault energy: it is common to see MD
simulations with too small an hcp–fcc energy difference producing unphysically many stacking faults
and over widely separated partial dislocations.
Phase transitions are observed in some potentials.
As free energy calculations are complicated and
time consuming,17 it is impractical to use them
directly in fitting – one would require the differential of the free energy with respect to the potential
parameters, and this could only be obtained numerically. Consequently, most potentials are only fitted
to reproduce the zero temperature crystal structure,
and high-temperature phase stability is unknown for
the majority of published potentials. One counterexample is in metals such as Ti and Zr, where the
bcc structure is mechanically unstable with respect
to hcp, but becomes dynamically stabilized at high
temperatures. Here, the transition temperature is
directly related to a single analytic quantity: the
energy difference between the phases. Although
about half of this difference comes from electronic
entropy,18 which suggests a temperature-dependent
potential, phase transition calculations have been

explicitly included in some recent fits.19 The case
of iron is also anomalous, as the phase transition is
related to changes in the magnetic structure.
1.10.7.2

Surface Relaxation

Glue models atoms seek to have as many neighbors as
possible; therefore, when a material is cleaved, the
surface atoms tend to relax inward toward the bulk to
increase cohesion. This effect also arises because of


Interatomic Potential Development

the longer range of the repulsive part of the potential:
at a surface, the further-away atoms are absent. This
is in contrast to pair potentials and in agreement with
real materials.
1.10.7.3

Cauchy Relations

The functional form of the glue model places fewer
restrictions on the elastic constants of materials than
pair potentials do; for example, the Cauchy pressure
for a cubic metal is as follows20:
"
#2
d2 F ðxÞ X 0 2 2

C12 À C66 ¼
f ðr Þxj
dx 2
j
If the ‘embedding function’ F (minus square root in
FS case) has positive curvature, the Cauchy pressure
must be positive, as it is for most metals. A minority of
metals have negative Cauchy pressure. It is debatable
whether this indicates negative curvature of the embedding function, or a breakdown of the glue model.
There are also some Cauchy-style constraints on
the third-order elastic constants. But in general, ‘glue’
type models can fit the full anisotropic linear elasticity of a crystal structure.
1.10.7.4

Vacancy Formation

In a near-neighbor second-moment model for fcc,
breaking one of twelve bonds reduces the cohesive
energy of each
atom ffi adjacent to the vacancy by a
pffiffiffiffiffiffiffiffiffiffiffi
factor of ð1 À 11=12Þ ¼ 4:25%. Other glue models
give a similar result. Meanwhile, the pairwise (repulsive) energy is reduced by a full 1=12 ¼ 8:3%.
Thus, energy cost to form a vacancy is lower in
glue-type models than in pairwise ones. For actual
parameterizations, it tends to be less than half the
cohesive energy.
1.10.7.5

277


the atom i
In the EAM, the function Fi depends on P
being embedded, while the charge density j fj ðrij Þ
into which it is embedded depends on the species and
position of neighboring atoms. By contrast for FS potentials, the function F is a given (square root), while fðrij Þ
is the squared hopping integral, which depends on
both atoms. There is no obvious way to relate this
heteroatomic hopping integral to the homoatomic
ones, but a practical approach is to take a geometric
mean21: one might expect this form from considering
overlap of exponential tails of wavefunctions.

1.10.8 Two-Band Potentials
In the second-moment approximation to tight binding, the cohesive energy is proportional to the square
root of the bandwidth, which can be approximated as
a sum of pairwise potentials representing squared
hopping integrals. Assuming atomic charge neutrality, this argument can be extended to all band occupancies and shapes22 (Figure 7).
The computational simplicity of FS and EAM follows from the formal division of the energy into a sum
of energies per atom, which can in turn be evaluated
locally. Within tight binding, we should consider a
local density of states projected onto each atom. The
preceding discussion of FS potentials concentrates
solely on the d-electron binding, which dominates
transition metals. However, good potentials are difficult to make for elements early in d-series (e.g., Sc, Ti)
where the s-band plays a bigger role. An extension to
the second-moment model, which keeps the idea of

DOS


Alloys

To make alloy potentials in the glue formalism, one
needs to consider both repulsive and cohesive terms.
Thinking of the repulsive part as the NFE pair
potential, it becomes clear that the long-range behavior depends on the Fermi energy. This is composition
dependent – the number of valence electrons is critical, so it cannot be directly related to the individual
elements. The short-ranged part should reflect the
core radii and can be taken from the elements.
Despite this obvious flaw, in practice, the pairwise
part is usually concentration-independent and is
refitted for the ‘cross’ heterospecies interaction.

Ef
E

DOS

Ef
E
Figure 7 In second-moment tight binding, the band
shape is assumed constant at all atoms, the effect of
changing environments being a broadening of the band.


278

Interatomic Potential Development

locality and pairwise functions, is to consider two

separate bands, for example, s and d.
This was first considered for the alkali and alkaline earth metals, where s-electrons dominate. These
appear at first glance to be close-packed metals,
forming fcc, hcp, or bcc structures at ambient pressures. However, compared with transition metals,
they are easily compressible, and at high pressures
adopt more complex ‘open’ structures (with smaller
interatomic distances). The simple picture of the
physics here is of a transfer of electrons from an s- to
a d-band, the d-band being more compact but higher
in energy. Hence, at the price of increasing their
energy (U ), atoms can reduce their volumes (V ).
Since the stable structure at 0 K is determined by
minimum enthalpy, H ¼ U þ PV at high pressure, this
sÀd transfer becomes energetically favorable. The
net result is a metal–metal phase transformation
characterized by a large reduction in volume and
often also in conductivity, since the s-band is free
electron like while the d-band is more localized.
Two-bands potentials capture this transition, which
is driven by electronic effects, even though the crystal
structure itself is not the primary order parameter.
Materials such as cerium have isostructural transitions. It was thought for many years that Cs also had
such a transition, but this has recently been shown to
be incorrect,23 and the two-band model was originally designed with this misapprehension in mind.24
For systems in which electrons change, from an s-type
orbital to a d-type orbital as the sample is pressurized,
one considers two rectangular bands of widths W1 and
W2 as shown in Figure 8 with widths evaluated using

eqn [3]. The bond energy of an atom may be written as

the sum of the bond energies of the two bands on that
atom as in eqn [4], and a third term giving the energy of
promotion from band 1 to band 2 (see eqn [8]):
X Wi1
ni1 ðni1 À N1 Þ
Ubond ¼
2N1
i
Wi2
þ
ni2 ðni2 À N2 Þ þ Eprom
½6Š
2N2
where N1 and N2 are the capacities of the bands (2 and 10
for s and d respectively) and ni1 and ni2 are the occupation of each band localized on the ith atom.
For an ion with total charge T, assuming charge
neutrality,
ni1 þ ni2 ¼ T

½7Š

The difference between the energies of the band centers a1 and a2 is assumed to be fixed. The values of a
correspond to the appropriate energy levels in the
isolated atom. Thus, a2 À a1 is the excitation energy
from one level to another. For alkali and alkaline earth
metals, the free atom occupies only s-orbitals; the
promotion energy term is therefore simply
Eprom ¼ n2 ða2 À a1 Þ ¼ n2 E0

½8Š


where E0 ¼ a2 À a1 .
Thus, the band energy can be written as a function
of ni1 , ni2 , and the bandwidths (evaluated at each
atom as a sum of pair potentials, within the secondmoment approximation). Defining,
i ¼ ni1 À ni2

½9Š

and using eqn [7], we can write as follows:

D(E)
N/W1 + N/W2

d-band

N/W1
s-band
a1 − W1/2

a1

Ef

a2 − W2/2

a1 + W1/2

E


a2 a + W /2
2
2

Figure 8 Schematic picture of density of electronic states in rectangular two-band model. Shaded region shows those
energy states actually occupied.


Interatomic Potential Development

X 
T
À i ðWi1 À Wi2 Þ À ðWi1 þ Wi2 Þ
4
4
i


2
2
 þ T Wi1 Wi2
þ
þ i
8
N1 N2
 T Wi1 Wi2
þ i
À
N1
N2

4
T À i
E0
½10Š
þ
2
Although this expression looks unwieldy, it is computationally efficient, requiring only two sums of pair
potentials for Wi and a minimization at each site
independently with respect to i , which can be done
analytically.
In addition to the bonding term, a pairwise repulsion between the ions, which is primarily due to
the screened ionic charge and orthogonalization of
the valence electrons, is added. In general, this pair
potential should be a function of i and j . But to
maintain locality, one has to write this pairwise contribution to the energy in the intuitive form, as the
sum of two terms, one from each ‘band,’ proportional
to the number of electrons in that band:
Ubond ¼

V ðrij Þ ¼ ðni1 þ nj 1 ÞV1 ðrij Þ þ ðni2 þ nj 2 ÞV2 ðrij Þ

½11Š

We rearrange this to give the energy as a sum
over atoms:
"
#
X
X
X

Upair ¼
ni1
V1 ðrij Þ þ ni2
V2 ðrij Þ
½12Š
i

j 6¼i

j 6¼i

The total energy is now simply
Utot ¼ Upair þ Ubond

½13Š

This depends on i , which takes whatever values to
minimize the energy, s
@Utot
¼0
@ i

½14Š

explicitly for i0 independently at each atom, with the
constraint that ji j cannot be greater than the total
number of electrons T per atom. The fixed capacities
of the bands (N1 and N2 ) can also prohibit the realization of i0 . It is therefore necessary to limit the
values which i may have to those where i0 does
not imply negative band occupation.

The expressions for i involves only constants and
sums of pair potentials, and can be evaluated independently at each atom at a similar computational
cost to a standard many-body type potential.

279

The variational property expressed in eqn [14] can
be exploited to derive the force on the ith atom:
dUtot
fi ¼ À
dri
@Utot
@Utot @
¼À
j À
@ri 
@ dri
@Utot
¼À
j
@ri 

½15Š

Hence, the force is simply the derivative of the
energy at fixed . Basically, this is the Hellmann–
Feynman theorem25 which arises here because  is
essentially a single parameter representation of the
electronic structure.
This result means that, like the energy, the force

can be evaluated by summing pairwise potentials.
Hence, the two-band second-moment model is well
suited for large-scale MD. The force derivation itself
is somewhat tedious, and the reader is referred to the
original papers. There is no Hellman–Feynman type
simplification for the second derivative, so analytic
expressions for the elastic constants in two-band
models are long ranged and complicated. Consequently, elastic constants are best evaluated numerically.
1.10.8.1

Fitting the s–d Band Model

To make a usable potential, the functional forms of f
and V must be chosen. Although this is somewhat arbitrary, the physical picture of hopping integral and
screened ion–ion potential suggests that both should
be short ranged, continuous, and reasonably smooth.
Popular choices are cubic splines, power series, and
Slater orbitals. The promotion energy E0 is simply that
required to promote an electron from the s level into the
d level of an isolated atom. The band capacities are
Ns ¼ 2, Nd ¼ 10 and the total number of electrons per
atom depends on the element, for example, in Cs T ¼ 1.
In the first application, parameters were fitted to
the energy–volume relations for bcc and fcc cesium
and the transition pressure between phases Cs-II
and Cs-III.
Figure 9 shows the energy–volume curves for the
fcc and bcc structures calculated using the model.
At ambient conditions for Cs, there are no d-electrons,
so the fitting process is just like a normal FS potential.

This determines the s-band parameters, and the d-band
parameters are then fitted to the high-pressure phase
data, where both s- and d-electrons contribute.
Although an isostructural phase transition is likely
to be accompanied by instability of the bulk modulus,
there may also be a precursor shear instability. Thus,


Interatomic Potential Development

1.0
0.5
0.0
-0.5

9.0
fcc lattice parameter (Å)/
cohesive energy (eV per atom)

h

280

-1.0

Energy per atom (eV)

1.5
fcc
bcc

4.3 GPa

1.0
0.5
0.0
-0.5

8.0
7.0
6.0
5.0
4.0
3.0
2.0
1.0
0.0

-1.0
-1.5

0

50

100
150
Volume per atom (Å3)

200


Figure 9 Top: Variation of i with compression, showing
s!d transfer in model cesium (T ¼ 1). Bottom: An energy–
volume curve for the two-band potential. The minimum lies
at –1.3163 eV and 115.2 A˚3 per atom. The gradient of the
straight dash–dotted line is the experimental Cs-II–Cs-III
transition pressure. In reality, cesium also has a bcc–fcc
phase transition at 2.3 GPa; however, first principles
calculations show that these two structures are almost
degenerate in energy at 0 K.

the mechanism may involve shearing rather than
isostructural collapse, particularly if a continuous
interface between the two phases exists, as in a shockwave. With the two-band model, the transition is first
order, the volume collapse occurring before the bulk
modulus becomes negative in the unstable region.
Although the shear and tetragonal shear decrease in
the unstable region, neither actually goes negative.
The two-band model is applicable to transition
metals, but since the d-band is occupied at all pressures, electron transfer is continuous and there is no
phase transition. This makes the empirical division of
the energy into s and d components challenging.
However, once appropriately scaled for ionic charge
and number of electrons T, in principle, the method
could be extended to alloys with noninteger T. The
results of such an extrapolation are extraordinarily
good (Figure 10), considering that there is no fitting
to any material other than Cs. The extrapolation
breaks down at high Z where the amount of sp hybridization is not fully captured in the parameterization.
As with FS, no information about band shape is
included, and so the sequence of crystal structures

cannot be reproduced.
While the extrapolated potentials do not represent
the optimal parameterization for specific transition

1

2

3

4 5 6 7 8 9
Number of sd electrons

10 11

Figure 10 Extrapolation of two-band model fitted to Cs to
various {6s5d}-band materials, with the lengths being
scaled according to the Fermi vector (Z À1=3 ) and the
energies by (Z 1=2 ). Lattice parameters are shown by the
dashed line (calculated) and squares (experiment), and the
cohesive energies by the solid line (calculated) and circles
(fcc at experimental density).

metals, the recovery of the trends across the group
lends weight to the idea that the two-band model
correctly reproduces the physics of this series.
The s–d two-band approach has also been applied
with considerable success by considering the s-band as
an alloying band.26 This has been applied to the FeCr
system, which we discuss in more detail later.

1.10.8.2

Magnetic Potentials

The two-band approach can be applied to magnetic
materials, where the bands spin up and spin down
bands have the same capacity (N ¼ Nd" ¼ Nd# ¼ 5). If
in addition we assume that the bands have the same
width and shape (see, e.g., Figure 11), there is a
remarkable collapse of the model onto the singleband EAM form, with a modified embedding function.
The formalism here extends to the two-band
model, but the physics is analogous to other magnetic
potentials.27 For simplicity, consider a rectangular
d-band of full width W centered on E0 . The bond
energy for a single spin-up band relative to the free
atom is given by


 " 
ð Ef ¼ Z=N À 1 W
"
" Z
2
À 1 W =2
NE=W dE ¼ Z
U ¼
N
ÀW =2
½16Š
where Z" is the occupation of the band and uparrows

denote ‘spin up.’


Interatomic Potential Development

281

2.5

Density of states

2

1.5

1

0.5

0
-8

-6

-4

-2
Energy (eV)

0


2

Figure 11 bcc iron density of states for majority and minority spin bands from ab initio spin-dependent GGA
pseudopotential calculations with 4913 k ¼ points, adjusted so that the Fermi energy lies at the zero of energy. The two-band
model assumes that the bands have the same shape and width, but are displaced in energy relative to each other: the figure
shows this to be reasonable.

To describe the ferromagnetic case, it is assumed
that there are two independent d-bands corresponding
to opposite spins, and that these can be projected onto
an atom to form a local density of states. For a free
atom atomic case, Hund’s rules determine a high-spin
case (e.g., S ¼ 2 for iron), and there is an energy U x
associated with transferring an electron to a lower spin
state. In the solid, the simplest method is to set U x to
be proportional to the spin with the coefficient of
proportionality being an adjustable parameter, E0 .
U x ¼ ÀE0 jZ" À Z# j

½17Š

Defining the spin, S ¼ jZ" À Z# j and assuming charge
neutrality (T ¼ ðZ" þ Z# Þ), the two-d-band binding
energy on a site i is then as follows:
Ui ¼ Ui" þ Ui# þ Uix ¼ þ

Wi 2
ðT þ Si2 Þ À TWi =2 À E0 Si
4N

½18Š

Differentiating this equation about Si gives us the
optimal value for the magnetization of a given atom
of Si ¼ 2NE0 =Wi , and the many-body energy of an
atom with T ¼ 6, N ¼ 5 (suppressing the i label) as
U ¼ À6W =5 À 5E02 =W E0 =Wi
¼ À2W =5 À 4E0 E0 =Wi ! 0:4

0:4
½19Š

where neither band is allowed to have occupancy
more than 5 or less than 0. For a material with
T d-electrons (where T > 5), transfer of electrons
between the spin bands becomes advantageous for
W > 10E0 =ð10 À T Þ. For smaller W, the spin " band
is full and the energy is simply proportional to the
bandwidth of the # band as in the FS model. Similar
cases apply to the T < 5 case when the minority band
may be empty.
There has been some controversy about the
expression for U x . In the two-band model, this is a
promotion energy from the minority spin band to the
majority. In the atomic case, it is the energy to violate
Hund’s rule, and the implicit reference state is the
high-spin atom. Electron transfer is bound by the
number of electron, so the function has discontinuous
slope at Si ¼ 0:4. By contrast, the approach of
Dudarev and coworkers27 uses a Stoner model for

the spin energy, which introduces quadratic and
quartic terms in U x ðSi Þ. In that case, the implicit
reference state is the nonmagnetic solid, and any
value of Si is acceptable.
Within the second-moment model, the bandwidth
W is given by the square root of r, the sum of the
squares of the hopping integral. Applying this, and
the usual pairwise repulsion V ðr Þ, gives an expression for the two-band energy


282

Interatomic Potential Development

Ui ¼

X
j

À B=

V ðrij Þ À

pffiffiffiffiffi
rj

pffiffiffiffiffi
rj H ð2W À 5E0 Þ

À 4E0 H ð5E0 À 2W Þ


½20Š

where H is the Heaviside step function, B is a constant, and the zero of energy corresponds to the
nonmagnetic atom.
Note that this form does not explicitly include S,
and that it has the
pffiffiffiEAM form with an embedding
function FI ðxÞ ¼ x ð1 À B=xÞ.
Although this model incorporates magnetism and
provides a way to calculate the magnetic moment at
each site, it is possible to use it without actually
calculating S. The additional many-body repulsive
term is similar to, for example, the many-body
potential method of Mendelev et al. It is also interesting that in the original FS paper, it was not possible to fit the properties of the magnetic elements Fe
and Cr; an extra term was added ad hoc. Later parameterizations of FS potentials for iron with a pure
square root for F have not exactly reproduced the
elastic constants.28
The implication of this work is that for secondmoment type models, there should be a one-toone relation between the local density r and the
magnetic moment. Figure 12 shows this relation
for two parameterizations, r from Dudarev–Derlet
and Mendelev et al. and the magnetic moment calculated with spin-dependent DFT projected onto
atoms. It shows that there are two cases. For atoms
associated with local defects, the density varies quite
sharply with r, while for the crystal under pressure
the variation is slower. It is noteworthy that the same
broad features are present in both potentials, even
though the Mendelev et al. potential was fitted without consideration of magnetic properties, albeit with
a FS-type embedding function. This suggests that
the magnetic effects were unwittingly captured in

the fitting process.
1.10.8.3

Nonlocal Magnetism

The two-band model projects the magnetism onto
each atom. It does not properly describe magnetic
interactions, so it cannot distinguish between ferro,
para, and antiferromagnetism. In order to do so, we
need to include an interionic exchange term.
Pauli repulsion arises from electron eigenstates
being orthogonal. While its nature on a single atom
is complex, its interatomic effects can be modeled as

a pairwise effect of repulsion between electrons of
similar spins. The secondary effect of magnetization
is that there are more electrons in one band than in
another, and more same-spin electron pairs to repel
one another, and so the repulsion between those
bands is enhanced.
Conceptually, this can be captured in two pairwise
effects, the standard nonmagnetic screened-Coulomb
repulsion of the ions plus the core–core repulsion,
and an additional Si dependent term arising from
Pauli repulsion between like-spin electrons.
V ðrij Þ ¼ V0 ðrij Þ þ ðSi" Sj" þ Si# Sj# ÞVm ðrij Þ

½21Š

Note that an antiferromagnetic state with Si þ Sj ¼ 0

would have a lower repulsive energy. In the tightbinding picture, this would be compensated by a
much reduced hopping integral, and hence, lower W.
If we insist on Si > 0, then we suppress these solutions
and can model ferromagnetic or diamagnetic iron.
Also, as with DFT-GGA/LDA, the spin is Ising-like.
At the time of writing, no good parameterization
of this type of potential exists. The difficulty is that
determining the spins Si is a nonlocal process: the
optimal value of the spin on site i depends on the spin
at site j. The only practical way to proceed appears to
be to treat the spins as dynamical variables, in which
case it is probably better to treat them as noncollinear
Heisenberg moments.

1.10.8.4

Three-Body Interactions

It is worth noting that the ‘glue’ type potentials
cannot be expanded in a sum of two-body, threebody, four-body, etc. terms. Three-body terms enter
into the free electron picture through nonlinear
response of the electron gas, and into the tightbinding picture in the fourth moment description
and beyond.
At constant second moment, increasing the fourth
and higher even moments of the DOS tends to lead to
a bimodal distribution. Bimodal distributions will be
favored by materials with half-filled bands. Thus,
three-body terms are likely to be important in structures with few small-membered rings of atoms, and
hence, small low moments. The bcc structure is a
borderline example of this, but the classic is the diamond structure. Diamond has no rings of less than six

atoms, resulting in a strongly bimodal DOS. This
bimodal structure in the tight-binding representation
is also interpreted as bonding and antibonding states in
a covalent picture.


Interatomic Potential Development

283

Magnetic moments in iron versus r
Dudarev/Derlet potential case2
3
bcc Fe perfect crystal
Vacancy
100 sia
110 sia
111 sia
Octa
Tetra

m /mB

2

1

0

0


0.5

1

2

1.5
r

(a)

2.5

3

Magnetic moments in iron versus r
Ackland/Mendelev potential
3
bcc Fe perfect crystal
Vacancy
100 sia
110 sia
111 sia
Octa
Tetra

m /mB

2


1

0

20

30

40

(b)

50

60

70

r

Figure 12 Relationship between ab initio calculated magnetic moment per atom, and r from (a) Dudarev–Derlet magnetic
potential and (b) Mendelev et al.

Interatomic potentials for carbon and silicon
fall into this category. However, once a band gap is
opened, the Fermi energy and perfect screening
are lost, and the rigid band approximation is less
appropriate.


1.10.9 Modified Embedded Atom
Method
The modified embedded atom method (MEAM) is
an empirical extension of EAM by Baskes, which


284

Interatomic Potential Development

includes angular forces. As in the EAM, there are
pairwise repulsions and an embedding function. In
the EAM, the ri is interpreted as a linear supposition
of species-dependent spherically averaged atomic electron densities (here designated by fðr Þ); in MEAM ri
is augmented by angular terms. The spherically symmetric partial electron density rð0Þ is the same as the
electron density in the EAM:
ð0Þ

ri

¼

X

fð0Þ ðrij Þ

j

where the sum is over all atoms j, not including
the atom at the specific site of interest i. The angular

contributions to the density are similar to spherical
harmonics: they are given by similar formulas weighted
by the x; y; and z components of the distances between
atoms (labeled by a; b; g):
2
32
X
X
a
ij
4
ðrð1Þ Þ2 ¼
fð1Þ ðrij Þ 5
rij
a
j
2
32
2
32
X
X
X
a
b
1
ij ij
4
ðrð2Þ Þ2 ¼
fð2Þ ðrij Þ 2 5 À 4

fð2Þ ðrij Þ5
3
r
ij
j
j
a;b
2
32
2
32
X X
X
aij bij gij
1
ð3Þ 2
ð3Þ
ð2Þ
4
5 À 4
ðr Þ ¼
f ðrij Þ
f ðrij Þ5
3 j
rij3
j
a;b;g

The fðlÞ are so-called ‘atomic electron densities,’
which decrease with distance from the site of interest,

and the a; b; and g summations are each over the three
coordinate directions (x, y, z). The functional forms for
the partial electron densities were chosen to be translationally and rotationally invariant and are equal to
zero for crystals with inversion symmetry about all
atomic sites. Although the terms are related to powers
of the cosine of the angle between groups of three
atoms, there is no explicit evaluation of angles, and
all the information required to evaluate the MEAM is
available in standard MD codes. Typically, atomic
electron densities are assumed
to decrease
Â
à exponentially, that is, fðlÞ ðRÞ ¼ exp Àb ðlÞ ðR=re À 1Þ where the
decay lengths (re and b ðlÞ ) are constants.
While there is no derivation of the MEAM from
electronic structure, it also introduced the physically
reasonable idea of many-body screening, which is missing in pair-functional forms such as EAM. Thus, fðRij Þ
is reduced by a screening factor determined by the other
atoms k forming three-body triplets with i and j : primarily those lying between i and j. This eliminates the need
for an explicit cut-off in the ranges of V ðr Þ and fðlÞ ðr Þ.

For close-packed materials, the improvement of
MEAM over standard EAM is marginal; the angular
terms come out to be small. For sp-bonded materials, a
large three-body term can stabilize tetrahedrally coordinated structures, but since the physics arises from
preferred 109 angles rather than preferred fourfold
coordination, it suffers problems similar to Stillinger–
Weber type potentials (discussed below). Very high
angular components enable one to fit the complex
phases of lanthanides and actinides. It is tempting to

attribute this to the correct capture of the f-electron
physics, although the additional functional freedom may
play a role in enabling fits to low symmetry structures.

1.10.10 Potentials for Nonmetals
While much of the work on structural materials has
concentrated on metals, there are important issues
involving nonmetallics for coatings, corrosion, and
fuel. In this section, we review other types of potentials.
1.10.10.1

Covalent Potentials

Empirical potentials for covalent materials have been
much less successful than for metals. As with the NFE
pair potentials, the bulk of the energy is contained in
the covalent bond, and potentials which well describe
distortions from fourfold coordination tend to fail
when applied to other bond situations, such as surfaces, high pressure phases, or liquids.
A commonly used example, the Stillinger–Weber
potential,29 is written as

2
X
XX
1
V ðrij Þ þ
F ðrij ÞF ðrik Þ À cos yijk
Ui ¼
3

j
j
k
where V ðrij Þ and F ðrij Þ are short-ranged pair potentials. The form of the three-body term, with its minimum at 109 , ensures the stability of a tetrahedrally
bonded network. The stacking fault energy (equivalently, the difference between cubic and hexagonal
diamond) is zero, so the ground state crystal structure
is not unique; however, this is not far from correct,
and hexagonal diamond can be found in carbon
and silicon. Moreover, it has little effect in many
simulations since the large kinetic barrier against
cubic–hexagonal phase transitions prevents them
occurring in simulations. Solidification and recovery
from cascade damage are counterexamples.
Stillinger–Weber works well for fourfoldcoordinated amorphous networks, and vibrational
properties of the diamond structures. It gives too low


Interatomic Potential Development

density (and coordination) for the liquid and highpressure phases, because it fails to reproduce the rebonding of atoms at the surface to remove dangling bonds.
An alternate approach to stabilizing diamond via
the 109 angle is to do so through its tetravalent
nature. The simplest type is the restricted bond pair
potential30:
X
X
Ui ¼
Aðrij Þ À
Bðrij Þ
j ¼1;4


j

where the attractive part of the potential is summed
over at most four neighbors (one per electron). This
formalism describes well the collapse of the network
under pressure or melting, but lacks shear rigidity
(only the repulsion of second neighbors provides
shear rigidity). There is also some ambiguity over
which four neighbors should be chosen, which
makes implementation difficult.
An embellishment on this is the bond-charge
model, in which the electrons in the bonds repel one
another. This adds a three-center term of the form
X
Cðrjk Þ
i

where j and k are bonded neighbors of i. This approach
avoids explicitly introducing the tetrahedral angle into
the potential. Note that although this term is associated with atom i (and is often interpreted as a bondbending term at i ), in the simplest form forces derived
from this term are independent of i.
The problem of defining ‘bonded neighbors’ can
be circumvented, in the spirit of the embedded atom
method, by having an embedding function that effectively cuts off after the bonding reaches four neighbors worth31 as in the Tersoff approach:
X
X
Aðrij Þ À
Bðrij Þ
Ui ¼

j

where
Bðrij Þ ¼ fðrij Þ

j

X

Gðrik ; rjk ; rij Þ

k

The bond ij is weakened by the presence of other
bonds ik and jk involving atoms i and j. The protetrahedral angular dependence is still necessary to
stabilize the structure, and further embellishment by
Brenner32 corrects for overbinding of radicals.
These potentials give a good description of the
liquid and amorphous state, and have become widely
used in many applications, in addition to elements such
as Si and C, as well as covalently bonded compounds
such as silicon carbide33 and tungsten carbide.34

1.10.10.2

285

Molecular Force Fields

Potentials based on bond stretching, bond bending,

and long-ranged Coulomb interactions are widely
used in molecular and organic systems. Chemists call
these potentials ‘force fields.’ They cannot describe
making and breaking chemical bonds, but by capturing
molecular shapes, they describe the structural and
dynamical properties of molecules well.
There are many commercial packages based on
these force fields, for example, CHARMM35 and
AMBER.36 They are primarily useful for simulating
molecular liquids and solvation, but have seen little
application in nuclear materials, on account of the
long-range Coulomb forces, which are costly to evaluate in large simulations.
1.10.10.3

Ionic Potentials

With no delocalized electrons, ionic materials should
be suitable for modeling with pair potentials. The
difficulty is that the Coulomb potential is long ranged.
This can be tackled by Ewald or fast multipole methods, but still scales badly with the number of atoms.
The simplest model is the rigid ion potential, where
charged (q) ions interact via long-range Coulomb
forces and short-ranged pairwise repulsions V ðr Þ.
X
qi qj
V ðrij Þ þ
U¼
4pe0 rij
ij
For example, a common form of the pair potential

in oxides consists of the combination of a (6-exp)
Buckingham form and the Coulomb potential:
U¼

X
ij

A expðÀarij Þ À b=rij6 þ

qi qj
4pe0 rij

where a and b are parameters and rij is the distance
between atoms i and j.
As with other potential, various adjustments are
needed in order to obtain reasonable forces at very
short distance; see for example, recent reviews of UO2.37
For nuclear applications, the most commonly studied material in the open literature is UO2, which is
widely used as a reactor fuel. It adopts a simple fluorite
structure with a large bandgap, which makes potentialfitting to get the correct crystal structure reasonably
straightforward. Early work fitted the potentials to
lattice parameter and compressibility, and later to
elastic constants and the dispersion relation. The elastic constants are c11 ¼ 395 GPa, c12 ¼ 121 GPa, and
c44 ¼ 64 GPa.38 As previously described, the Cauchy
relation generally applies to a pairwise potential,


286

Interatomic Potential Development


C12 ¼ C44 , which is seldom true experimentally for
oxides. However, the Cauchy relation is on the basis
of the assumption that all atoms are strained equally,
which is not the case for a crystal such as UO2 where
some atoms do not lie at centers of symmetry. Thus,
the violation of the Cauchy relation in UO2 can be
fitted by attributing it to internal motions of the atoms
away from their crystallographic positions. (The violation of the Cauchy relation is similar in oxides with
and without this effect, so it is debatable whether this
is the correct physical effect.)
The earlier potentials were based on the Coulomb
charge plus Buckingham described above; more
recent parameterizations include a Morse potential.
While this gives more degrees of freedom for fitting,
having two exponential short-range repulsions with
different exponents appears to be capturing the same
physics twice. Comparison of the parameters39 shows
that the prefactor for the U–U Buckingham repulsion
varies by ten orders of magnitude when fitted. Moreover, the original Catlow parameterization sets this
term to zero. This difference tells us that the small
U atoms seldom approach one another close enough
for this force to be significant. Even the ionic charges
vary between potentials by almost a factor of two,
with more recent potentials taking lower values.
Despite the huge disparity in parameters, the size of
cascades is similar and the recombination rate is high.
Polarizability is not incorporated in rigid ion potentials; they will always predict a high-frequency dielectric constant of 1, which is much smaller than typical
experimental values. The main consequence of this for
MD appears in the longitudinal optic phonon modes.

The solution is that ions themselves change in
response to environment. A standard model for this
is the shell model in which the valence electrons are
represented by a negatively charged shell, connected
to a positively charged nucleus by a spring. (Typically
this represents both atomic nucleus and tightly
bound electrons.) In a noncentrosymmetric environment (e.g., finite temperature), the shell center lies
away from the nuclear center, and the ion has a net
dipole moment – it is polarized.
X
X
qi qj
V ðrij Þ þ
þ
ki ðri À rishell Þ2
U¼
4pe
r
0 ij
i
ij
In this case, rij may refer to the separation between
nuclei i and j or the centers of the shells associated with
i and j. In MD, the shells have extremely low mass, and
are assumed to always relax to their equilibrium position: this is a manifestation of the Born–Oppenheimer
approximation used in DFT calculation.

Shell model potentials,40 which capture the dipole
polarizability of the oxygen molecules, were developed by Grimes and coworkers, and have been
through many extensions and reparameterizations

since then. Again, there have been many successful
parameterizations with wildly differing values for the
parameters; even the sign of the charge on the U core
and shell changes.41
A particular issue with ionic potentials is that of
charge conservation. A defect involving a missing ion
will lead to a finite charge. If the simulation is carried
out in a supercell with periodic boundary conditions,
this will introduce a formally infinite contribution
to the energy. The simple way to deal with this is to
ignore the long wavelength (k ¼ 0) term in the Ewald
sum. This, under the guise of a ‘neutralizing homogeneous background charge’ is routinely done in first
principles calculation. Alternately, a variable charge
approach can be used42 in which the extra charge is
added to adjacent atoms. The original approach then
involved minimizing the total energy with respect
to these additional charges, which is computationally
demanding. A promising new development is to limit
the range of the charge redistribution.43 While this
screening approximation is difficult to justify fully
in an insulator, it is very computationally efficient
or a system involving dilute charged impurities, and
appears to reproduce most known features of AlO.

1.10.11 Short-Range Interactions
In radiation damage simulations, atoms can come
much closer together than in any other application.
Interatomic force models are often parameterized on
data that ignore very short-ranged interactions, and
the physics of core wavefunction overlap is seldom

well described by extarpolation.
In radiation damage, we are normally dealing with
a case where two atoms come very close together, but
the density of the material does not change. By analogy with a free electron gas, we see that the energy
cost of compressing the valence wavefunctions is
important for high-pressure isotropic compression
but absent in the collision scenario. In the first
case, the ‘short range’ repulsion is a many-body
effect, while in the second case, it is primarily pairwise. Thus, we should not expect that fitting to
the high-pressure equation of state will give a good
representation of the forces in the initial stages of a
cascade, or even for interstitial defects. Indeed, when
there was no accurate measurement of interstitial


Interatomic Potential Development

formation energies, older potentials gave a huge range
of values based on extrapolation of near-equilibrium
data and uncertain partition of energy between pairwise
and many-body terms. Accurate values for these formation energies are now available from first principles
calculations and are incorporated in the fitting; therefore, a symptom of the problem is resolved.
At very short range, the ionic repulsive interaction
can be regarded as a screened-Coulombic interaction, and described by multiplying the Coulombic
repulsion between nuclei with a screening function
wðr =aÞ:
V ðr Þ ¼

Z1 Z2 e 2
wðr =aÞ

4pe0 r

where wðr Þ ! 1 when r ! 0 and Z1 and Z2 are the
nuclear charges, and a is the screening length. The
most popular parameterization of w is the Biersack–
Ziegler potential, which was constructed by fitting a
universal screening function to repulsions calculated
for many different atom pairs (Ziegler 1985). The
Biersack–Ziegler potential has the form
wðxÞ ¼ 0:1818eÀ3:2x þ 0:5099eÀ0:9423x
þ 0:2802eÀ0:4029x þ 0:02817eÀ0:2016x
where
0:8854a0
Z10:23 þ Z20:23
and x ¼ r =a and a0 ¼ 0:529 A˚ is the Bohr radius.
This potential must then be joined to the longer
ranged fitted potential. There are many ways to do
this, with no guiding physical principle except that
the potential should be as smooth as possible. Typical
implementations ensure that the potential and its first
few derivatives are continuous.
The short-range interactions arising from high
pressure come mainly from isotropic compression,
and should be fitted to the many-body part. To
achieve this division in glue models, the function
F ðrÞ should become repulsive at large r, but fðr Þ
should not become very large at small Rij .
a¼

1.10.12 Parameterization

Having deduced the functional form of the potential
from first principles, it remains to choose the fitting
functions and fit their parameters to empirical data.
Most papers simply state that ‘the potential was
parameterized by fitting to . . . .’ The reality is
different.

287

Firstly, one must decide what functions to use for
the various terms. Here, one may be guided by the
physics (atomic charge density tails in EAM, square
root embedding in FS, Friedel oscillations), by the
anticipated usage (short-range potentials will speed
up MD, and discontinuities in derivatives may cause
spurious behavior), or simply by practicality (Can the
potential energy be differentiated to give forces?).
Secondly, one must decide what empirical data to
fit. Cohesive energies, elastic moduli, equilibrium
lattice parameters, and defect energies are common
choices. Accurate ab initio calculations can provide
further ‘empirical’ data, notably about relative structural stability, but now increasingly about point
defect properties. (It is, of course, possible to calculate all the fitting data from ab initio means. Potentials
fitted in this way are sometimes referred to as ab initio.
While this is pedantically true, the implication that
these potentials are ‘better’ than those fitted to experimental data is irritating.)
Ab initio MD can also give energy and forces for
many different configurations at high temperature.
Force matching44 to ab initio data is one of best ways
to produce huge amounts of fitting data. There have

been many attempts to fully automate this process,
but to date, none have produced reliably good potentials. This is in part because of the fact that although
MD only uses forces (differential of the potential),
many essential physical features (barrier heights,
structural stability, etc.) do depend on energy, which
in MD ultimately comes from integrating the forces.
Small systematic errors in the forces, which lead to
larger errors in the energies, can then cause major
errors in MD predictions. Furthermore, if the potential is being used for kinetic Monte Carlo, the forces
are irrelevant.
By using least squares fitting, all the data may be
incorporated in the fit, or some data may be fitted
exactly and others approximately. However, since the
main aim of a potential is transferability to different
cases, the stability of the fitting process should be
checked. The best way to proceed is to divide the
empirical data arbitrarily into groups for fitting and
control, to fit using only a part of the data, and then to
check the model against the control data. This process
can be done many times with different divisions of fitting
and control. Any parameter whose value is highly sensitive to this division should be treated with suspicion.
Structural stability is the most difficult thing to
check, since one simply has to check as many structures as possible. In addition to testing the ‘usual
suspects,’ fcc, bcc, hcp, A15, o-Ti, MD, or lattice


288

Interatomic Potential Development


dynamics can help to check for mechanical instability
of trial structures.
1.10.12.1 Effective Pair Potentials and EAM
Gauge Transformation
Although the various glue-type potentials attribute
different aspect of the physics to the N-body and pairwise terms in the potential, if one has complete freedom in choosing the functions for V ðr Þ, F ðrÞ, and f,
then it is possible to move energy between the two
terms. Johnson and Oh noted that the EAM potential
Ui ¼

X

Vij ðrij Þ þ Fi

hX

fj ðrij Þ

i

is invariant under a transformation
hX
i
hX
i
X
fj ðrij Þ ! Fi
fj ðrij Þ þ A
fj ðrij Þ
F"i

For an alloy,45
Vab ðr Þ ¼

1 fb ðr Þ
f ðr Þ
Va ðr Þ þ a Vb ðr Þ
2 fa ðr Þ
fb ðr Þ

!

Thus, it is possible to choose a ‘gauge’ for the potential, for example, by setting F 0 ðr0 Þ ¼ 0 for some reference density r. The advantage of the gauge
transformation is that it simplifies fitting the potential.
It eliminates terms in F 0 ðr0 Þ for pressure and elastic
moduli at the equilibrium volume: these terms are
nonlinear in the fitting parameters. Thus, the fitting
process can be done by linear algebra.
The downside of the gauge transformation is that
it destroys the physical intuition behind the form of
the many-body term. Moreover, the gauge is determined by a particular reference configuration, a simple concept for elements, but one which does not
transfer readily to alloys.
The FS potentials do not have this freedom, because
the function F is predefined as a square root. However,
they introduced the ‘effective pair potential’
pffiffiffiffiffi
Veff ðrij Þ ¼ V ðr Þ À fðr Þ= r0
where r0 is a reference configuration (typically the
equilibrium crystal structure). Many of the equilibrium
properties which they used for fitting depend only on
this quantity.

In addition to gauge transformation, MD depends
only on the derivative of the total energy. Energy can be
partitioned between atoms in any way one likes, without
changing the physical results. However, on-atom properties, such as the magnetic moment in magnetic

potentials, typically do depend on the partition of
energy between atoms. Such quantities do not have the
gauge-invariance property.
1.10.12.2
Steel

Example: Parameterization for

1.10.12.2.1 FeCr

Steel is of particular importance to radiation damage.
Stainless steel is based on FeCr alloys, which have
been observed by first principles calculation to
exhibit unusual energy of solution. For small Cr
concentrations, the energy of solution is negative;
however, once the concentration exceeds about
10%, it changes sign. Thus, the FeCr system has a
miscibility gap, but even at 0 K, there is a finite Cr
concentration in the Fe-rich region. The underlying
physics of this is that it is favorable for a Cr atom to
dissolve in ferromagnetic Fe, provided the Cr spin is
opposite to the Fe. Two adjacent Cr cannot be antiparallel to each other and to the Fe matrix. Thus,
nearby Cr atoms suffer magnetic frustration, which
leads to repulsion between Cr atoms in FeCr not seen
in pure Fe or pure Cr. Reproducing this effect in a

potential is a challenging problem.
In early work, EAM was regarded as being inappropriate for bcc metals (this turned out to be due to
the use of rapidly decaying functions). The original
FS functional form stabilized bcc elements, but
they were unable to obtain a good fit for the elastic
constants in Fe and Cr without introducing further
parameters.
The two-band model can be applied to the FeCr
system46 by assuming that the material can be treated
as ferromagnetic, and using s and d as the two bands.
They adopted the functional form of the interactions
from the iron potential by Ackland and Mendelev,
scaling the Cr electron density by the ratio of the
atomic numbers 24/26. The CrCr potential was
refitted to elastic and point defect properties. As the
previous Fe parameterization incorporated with effect
of s-electrons in a single embedding function, the socalled s-band density of this model in fact depends only
on the FeCr cross potential. It described the excess
energy of alloying by a many-body rather than pairwise
additive effect. By choosing values which favor Fe
atoms with a single Cr neighbor, this potential gives
the skew solubility. This is an ingenious solution:
magnetic frustration is essentially a 2 þ N-body effect.
Cr atoms repel when in an Fe rich ferromagnetic
environment; this is neatly captured by the longranged Slater orbital used for the s-electron. It is


Interatomic Potential Development

debatable whether this term is really capturing physics associated with the s-band.

A related approach47 created a potential in which
the embedding function depends directly on the local
Cr concentration. The skew embedding function
readily reproduced the phase diagram, which was
the intention of the work. However, the short-range
ordering and the Cr–Cr repulsion which appears to
underlie the physics of radiation damage are less well
reproduced.
1.10.12.2.2 FeC

Carbon dissolves readily in iron, producing a
strengthening effect that underlies all steel. The
physics of this is rather complex: the solution energy
is very high (6 eV), and carbon adopts an interstitial
position in bcc Fe with a barrier of 0.9 eV to migration.
It is attracted to tensile regions of the crystal and to
vacancies. It is repelled from compressive regions,
including interstitial atoms, although the asymmetry
of the interstitial means there are some tensile sites at
larger distances which are favorable. First principles
calculation also shows that the carbon forms covalently
bonded pairs in a vacancy site, and the energy gained
from the bond more than compensated for the reduced
space available to the second carbon atom. These
criteria prove rather demanding for parameterizing
FeC potentials, even though they only cover compositions with vey low carbon concentrations.
An early pair potential by Johnson48 proved
extremely successful, and it was only once first principles calculation revealed the repulsion between
C and interstitials that a major problem was revealed.
Although interstitial atoms are specific to radiation

damage applications, there is a strong implication
that the binding to other overcoordinated regions
such as dislocation cores may be wrong.
It appears to be very difficult to obtain the correct
bonding of carbon in all the cases above with smooth
EAM-type functions. Even in recent potentials,49
like those by Johnson, carbon binds chemically to
the interstitial.
There is a qualitative explanation for this. Electronic structure calculation50 shows that the electrons
pile up between the two nearest neighbors in the octahedral configuration, essentially forming two FeC
bonds. However, all the simple potentials described
above obtain similar bonding from all six neighbors,
stabilizing the octahedral site because the tetrahedral
site has only four neighbors. This approach favors
carbon bonding to highly coordinated defects, and
underlies the bonding to interstitials. An EAM

289

potential with a Tersoff–Brenner style saturation in
the C cohesion has addressed this problem.51 This is
tuned to saturate at two near neighbors, and so favors
the octahedral site but not overcoordination. As a consequence, it does not bind carbon to the interstitial.
1.10.12.3

Austenitic Steel

Few potentials exist for fcc iron. Calculating hightemperature phase transitions is a subtle process involving careful calculation of free energy differences,
which makes it difficult to incorporate in the fitting
process. Although the bcc–fcc transition has been

reported for one EAM iron potential,52 it is probably
fortuitous and has been disputed.53 In any case, it is at
far higher temperature than experimentally observed.
Worse, it is likely that magnetic entropy plays a
significant role,54 and the magnetic degrees of freedom are seldom included in potentials.
Some very recent progress has been made; an
analytic bond order potential53 shows bcc–fcc–bcc
transitions for iron and an MEAM parameterization
by Baskes successfully reproduces the bcc–fcc–bcc
phase transitions in iron on heating by using temperature-dependent parameters. It seems certain that the
challenge of austenitic steel will be receiving more
attention in the next few years.

1.10.13 Analyzing a Million
Coordinates
1.10.13.1 Useful Concepts Without True
Physical Meaning
For very large simulations, imaging is a problem, since
showing all atoms in a massive simulation is likely to
obscure the important regions. There are a number
of heuristic quantities arising from the interatomic
potential which can be used to pick out the atoms
associated with atypical configurations of interest.
 Most empirical potentials define the energy per
atom.
 The atomic level stress55
1 X a b
sab
f r À mi via vib
i ¼

2Oi j ij ij
where f is the force on atom i due to atom j which
determines the glass transition.56
 The concept of ‘local crystal structure’ can be used
to locate twin boundaries, phase transitions, etc. This
may be done by common neighbor analysis, or


290

Interatomic Potential Development

bespoke investigation of pair and angular distribution
functions to search for particular configurations.57
 The balance between pair and many-body energies in glue-type models, or more significantly the
various angular density functions of MEAM.
 The magnetization in ‘magnetic’ potentials.

15.

Although uniquely defined for a given potential, many
of these are not well-defined concepts in electronic
structure. Yet, they can be extremely useful in identifying the atoms in far-from-equilibrium environments.

19.

1.10.14 Summary

16.


17.
18.

20.

21.
22.
23.

Interatomic potential development is a continuing
challenge for materials modeling. They represent the
only way to perform MD, which in turn is crucial for
the nonequilibrium and off-lattice processes, which
dominate radiation damage. Despite best efforts, few
potentials can be reliably employed to predict quantitative energies beyond where they are fitted. Their
most useful role is to reveal processes and topologies
that might be of importance in real materials.

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