Comprehensive nuclear materials 1 08 ab initio electronic structure calculations for nuclear materials

1.08 Ab Initio Electronic Structure Calculations for
Nuclear Materials
J.-P. Crocombette and F. Willaime
Commissariat a` l’Energie Atomique, DEN, Service de Recherches de Me´tallurgie Physique, Gif-sur-Yvette, France

ß 2012 Elsevier Ltd. All rights reserved.

1.08.1
1.08.2
1.08.2.1
1.08.2.2
1.08.2.2.1
1.08.2.2.2
1.08.2.3
1.08.2.3.1
1.08.2.3.2
1.08.2.3.3
1.08.3
1.08.3.1
1.08.3.1.1
1.08.3.1.2
1.08.3.2
1.08.3.2.1
1.08.3.2.2
1.08.3.2.3
1.08.3.2.4
1.08.3.2.5
1.08.3.3
1.08.3.3.1
1.08.3.3.2
1.08.3.4

1.08.4
1.08.4.1
1.08.4.1.1
1.08.4.1.2
1.08.4.1.3
1.08.4.2
1.08.4.2.1
1.08.4.2.2
1.08.4.2.3
1.08.4.2.4
1.08.4.2.5
1.08.4.3
1.08.5
1.08.5.1
1.08.5.1.1
1.08.5.1.2
1.08.5.1.3
1.08.5.1.4
1.08.5.1.5
1.08.5.2

Introduction
Methodologies and Tools
Theoretical Background
Codes
Basis sets
Pseudoization schemes
Ab Initio Calculations in Practice
Output
Cell sizes and corresponding CPU times

Choices to make
Fields of Application
Perfect Crystal
Bulk properties
Input for thermodynamic models
Defects
Self-defects
Hetero-defects
Point defect assemblies
Kinetic models
Extended defects
Ab Initio for Irradiation
Threshold displacement energies
Electronic stopping power
Ab Initio and Empirical Potentials
Metals and Alloys
Pure Iron and Other bcc Metals
Self-interstitials and self-interstitial clusters in Fe and other bcc metals
Vacancy and vacancy clusters in Fe and other bcc metals
Finite temperature effects on defect energetics
Beyond Pure Iron
helium–vacancy clusters in iron and other bcc metals
From pure iron to steels: the role of carbon
Interaction of point defects with alloying elements or impurities in iron
From dilute to concentrated alloys: the case of Fe–Cr
Point defects in hcp-Zr
Dislocations
Insulators
Silicon Carbide
Point defects

Defect kinetics
Defect complexes
Impurities
Extended defects
Uranium Oxide

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Ab Initio Electronic Structure Calculations for Nuclear Materials

1.08.5.2.1
1.08.5.2.2

1.08.5.2.3
1.08.5.2.4
1.08.6
References

Bulk electronic structure
Point defects
Oxygen clusters
Impurities
Conclusion

Abbreviations
bcc
CTL
DFT
DLTS
EPR
fcc
FLAPW
FP
GGA
LDA
LSD
LVM
PAW
PL
RPV
SIA
SQS
TD-DFT

Body-centered cubic
Charge transition levels
Density functional theory
Deep level transient spectroscopy
Electron paramagnetic resonance
Face-centered cubic
Full potential linearized augmented
plane waves
Fission products
Generalized gradient approximation
Local density approximation
Local spin density approximation
Local vibrational modes
Projector augmented waves
Photo-luminescence
Reactor pressure vessel
Self-interstitial atom
Special quasi-random structures
Time dependent density functional theory

1.08.1 Introduction
Electronic structure calculations did not start with
the so-called ab initio calculations or in recent years.
The underlying basics date back to the 1930s with an
understanding of the quantum nature of bonding in
solids, the Hartree and Fock approximations, and the
Bloch theorem. A lot was understood of the electronic structure and bonding in nuclear materials
using semiempirical electronic structure calculations,
for example, tight binding calculations.1 The importance of these somewhat historical calculations should

not be overlooked. However, in the following sections,
we focus on ‘ab initio’ calculations, that is, density
functional theory (DFT) calculations. One must
acknowledge that ‘ab initio calculations’ is a rather
vague expression that may have different meanings
depending on the community. In the present chapter
we use it, as most people in the materials science
community do, as a synonym for DFT calculations.

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The popularity of these methods stems from the fact
that, as we shall see, they provide quantitative results
on many properties of solids without any adjustable
parameters, though conceptual and technical difficulties subsist that should be kept in mind. The presentation is divided as follows. Methodologies and tools
are briefly presented in the first section. The next
two sections focus on some examples of ab initio results
on metals and alloys on one hand and insulating
materials on the other.

1.08.2 Methodologies and Tools
1.08.2.1

Theoretical Background

In the following a very basic summary of the DFT is
given. The reader is referred to specialized textbooks2–4
for further reading and mathematical details. Electronic
structure calculations aim primarily at finding the
ground state of an assembly of interacting nuclei and
electrons, the former being treated classically and the
latter needing a quantum treatment. The theoretical
foundations of DFT were set in the 1960s by the
works of Hohenberg and Kohn. They proved that the
determination of the ground-state wave function of
the electrons in a system (a function of 3N variables
if the system contains N electrons) can be replaced by
the determination of the ground-state electronic density (a function of only three variables). Kohn and Sham
then introduced a trick in which the density is expressed
as the sum of squared single particle wave functions,
these single particles being fictitious noninteracting
electrons. In the process, an assembly of interacting
electrons has been replaced by an assembly of fictitious
noninteracting particles, thus greatly easing the calculations. The electronic interactions are gathered in a
one-electron term called ‘the exchange and correlation
potential,’ which derives from an exchange and correlation functional of the total electronic density. One
finally obtains a set of one-electron Schro¨dinger equations, whose terms depend on the electronic density,
thus introducing a self-consistency loop.
No exact formulation exists for this exchange
and correlation functional, so one has to resort to


Ab Initio Electronic Structure Calculations for Nuclear Materials

approximations. The simplest one is the local density

approximation (LDA). In this approximation, the density of exchange and correlation energy at a given
point depends only on the value of the electronic
density at this point. Different expressions exist for
this dependence, so there are various LDA functionals. Another class of functionals pertains to the
generalized gradient approximation (GGA), which
introduces in the exchange and correlation energy
an additional term depending on the local gradient
of the electronic density. These two classes of functionals can be referred to as the standard ones. Most of
the ab initio calculations in materials science are performed with such functionals. Recently effort has been
put into the development of a new kind of functional,
the so-called hybrid functionals, which include some
part of exact exchange in their expression. Such functionals, which have been used for years in chemistry,
have begun to be used in the nuclear materials
context, though they usually involve much more
time-consuming calculations. One of their interests
is that they give a better description of the properties
of insulating materials.
We finish this very brief theoretical introduction
by mentioning the concepts of k-point sampling and
pseudoization.
In the community of nuclear materials, most
calculations are done for periodic systems, that is,
one considers a cell periodically repeated in space.
Bloch theorem then ensures that the electronic wave
functions should be determined only in the irreducible Brillouin zone, which is in practice sampled with
a limited number of so-called k points. A fine sampling is especially important for metallic systems.
Most ab initio calculations use pseudopotentials.
Pseudoization is based on the assumption that it is
possible to separate the electronic levels in valence
orbitals and core orbitals. Core electrons are supposed to be tightly bound to their nucleus with

their states unaffected by the chemical environment.
In contrast, valence electrons fully participate in the
bonding. One then first considers in the calculation
that only the valence electrons are modified while the
core electrons are frozen. Second, the true interaction
between the valence electrons and the ion made
of the nucleus and core electrons is replaced by a
softer pseudopotential of interaction, which greatly
decreases the calculation burden. Various pseudoization schemes exist (see Section 1.08.2.2.2).
Beyond ground-state properties, other theoretical
developments allow the ab initio calculations of additional features. Detailing these developments is

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beyond the scope of this text; let us just mention
among others time-dependent DFT for electron
dynamics, GW calculations for the calculation of
electronic excitation spectra, density functional perturbation theory for phonon calculations, and other
second derivatives of the energy.
1.08.2.2

Codes

Ab initio calculations rely on the use of dedicated
codes. Such codes are rather large (a few hundred
thousand lines), and their development is a heavy
task that usually involves several developers. An
easy, though oversimplified, way to categorize codes
is to classify them in terms of speed on one hand and
accuracy on the other. The optimum speed for the

desired accuracy is of course one of the goals of the
code developers (together with the addition of new
features). Codes can primarily be distinguished by
their pseudoization scheme and the type of their
basis set. We will not describe many other numerical
or programming differences, even though they can
influence the accuracy and speed of the codes.
The possible choices in terms of basis sets and
pseudoization are discussed in the following paragraphs. Pseudoization scheme and basis set are intricate as some bases do not need pseudoization and
some pseudoizations presently exist only for specific
basis sets. These methodological choices intrinsically
lead to accurate but heavy, or conversely fast but
approximate, calculations. We also mention some
codes, though we have no claim to completeness on
that matter. Furthermore, we do not comment on the
accuracy and speed of the codes themselves as the
developing teams are making continuous efforts to
improve their codes, which make such comments
inappropriate and rapidly outdated.
1.08.2.2.1 Basis sets

For what concerns the basis sets we briefly present
plane wave codes, codes with atomic-like localized
basis sets, and all-electron codes.
All-electron codes involve no pseudoization
scheme as all electrons are treated explicitly, though
not always on the same footing. In these codes, a
spatial distinction between spheres close to the nuclei
and interstitial regions is introduced. Wave functions
are expressed in a rather complex basis set made of

different functions for the spheres and the interstitial
regions. In the spheres, spherical harmonics associated with some kind of radial functions (usually
Bessel functions) are used, while in the interstitial


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Ab Initio Electronic Structure Calculations for Nuclear Materials

regions wave functions are decomposed in plane
waves. All electron codes are very computationally
demanding but provide very accurate results. As an
example one can mention the Wien2k5 code, which
implements the FLAPW (full potential linearized
augmented plane wave) formalism.6
At the other end of the spectrum are the codes
using localized basis sets. The wave functions are
then expressed as combinations of atomic-like orbitals. This choice of basis allows the calculations to be
quite fast since the basis set size is quite small (typically, 10–20 functions per atom). The exact determination of the correct basis set, however, is a rather
complicated task. Indeed, for each occupied valence
orbital one should choose the number of associated
radial z basis functions with possibly an empty polarization orbital. The shape of each of these basis
functions should be determined for each atomic
type present in the calculations. Such codes usually
involve a norm-conserving scheme for pseudoization
(see the next section) though nothing forbids the use
of more advanced schemes. Among this family of
codes, SIESTA7,8 is often used in nuclear material
studies.
Finally, many important codes use plane waves as

their basis set.9 This choice is based on the ease of
performing fast Fourier transform between direct and
reciprocal space, which allows rather fast calculations. However, dealing with plane waves means
using pseudopotentials of some kind as plane waves
are inappropriate for describing the fast oscillation of
the wave functions close to the nuclei. Thanks to
pseudopotentials, the number of plane waves is typically reduced to 100 per atom.
Finally, we should mention that other basis sets
exist, for instance Gaussians as in the eponymous
chemistry code10 and wavelets in the BigDft
project,11 but their use is at present rather limited
in the nuclear materials community.
1.08.2.2.2 Pseudoization schemes

As explained above, pseudoization schemes are especially relevant for plane wave codes. All pseudoization schemes are obtained by calculations on isolated
atoms or ions. The real potential experienced by the
valence electrons is replaced by a pseudopotential
coming from mathematical manipulations. A good
pseudopotential should have two apparently contradictory qualities. First, it should be soft, meaning that
the wave function oscillations should be smoothened
as much as possible. For a plane wave basis set, this
means that the number of plane waves needed to

represent the wave functions is kept minimal. Second,
it should be transferable, which means that it should
correctly represent the real interactions of valence
electrons with the core in any kind of chemical environment, that is, in any kind of bonding (metallic,
covalent, ionic), with all possible ionic charges or
covalent configurations conceivable for the element
under consideration. The generation of pseudopotentials is a rather complicated task, but nowadays

libraries of pseudopotentials exist and pseudopotentials are freely available for almost any element,
though not with all the pseudoization schemes.
One can basically distinguish norm-conserving
pseudopotentials, ultrasoft pseudopotentials, and PAW
formalism. Norm-conserving pseudopotentials were
the first ones designed for ab initio calculations.12
They involve the replacement of the real valence
wave function by a smooth wave function of equal
norm, hence their name. Such pseudopotentials are
rather easy to generate, and several libraries exist
with all elements of the periodic table. They are reasonably accurate although they are still rather hard,
and so they are less and less used in plane wave codes
but are still used with atomic-like basis sets. Ultrasoft
pseudopotentials13 remove the constraint of norm
equality between the real and pseudowave functions.
They are thus much softer though less easy to generate than norm-conserving ones. The Projector
Augmented Wave14 formalism is a complex pseudoization scheme close in spirit to the ultrasoft scheme but
it allows the reconstruction of the real electronic density and the real wave functions with all their oscillations, and for this reason this method can be considered
an all-electron method. When correctly generated,
PAW atomic data are very soft and quite transferable.
Libraries of ultrasoft pseudopotentials or PAWatomic
data exist, but they are generally either incomplete or
not freely available.
Plane wave codes in use in the nuclear materials
community include VASP15 with ultrasoft pseudopotentials and PAW formalism, Quantum-Espresso16
with norm-conserving and ultrasoft pseudopotentials
and PAW formalism, and ABINIT17 with normconserving pseudopotentials and PAW formalism.
Note that for a specific pseudoization scheme
many different pseudopotentials can exist for a
given element. Even if they were built using the

same valence orbitals, pseudopotentials can differ by
many numerical choices (e.g., the various matching
radii) that enter the pseudoization process.
We present in the following a series of practical
choices to be made when one wants to perform


Ab Initio Electronic Structure Calculations for Nuclear Materials

ab initio calculations. But the first and certainly most
important of these choices is that of the ab initio
code itself as different codes have different speeds,
accuracies, numerical methods, features, input files,
and so on, and so it proves quite difficult to change
codes in the middle of a study. Furthermore, one
observes that most people are reluctant to change
their usual code as the investment required to fully
master the use of a code is far from negligible (not to
mention the one to master what is in the code).
1.08.2.3

Ab Initio Calculations in Practice

In this paragraph, we try to give some indication of
what can be done with an ab initio code and how it is
done in practice. The calculation starts with the positioning of atoms of given types in a calculation cell of
a certain shape. That would be all if the calculations
were truly ab initio. Unfortunately, a few more pieces
of information should be passed to the code; the
most important ones are described in the final section.

The first section introduces the basic outputs of
the code, and the second one deals with the possible
cell sizes and the associated CPU times.
1.08.2.3.1 Output

We describe in this section the output of ab initio
calculations in general terms. The possible applications in the nuclear materials field are given below.
The basic output of a standard ab initio calculation is
the complete description of the electronic ground
state for the considered atomic configuration. From
this, one can extract electronic as well as energetic
information.
On the electronic side, one has access to the electronic density of states, which will indicate whether
the material is metallic, semiconducting, or insulating
(or at least what the code predicts it to be), its possible magnetic structure, and so on. Additional calculations are able to provide additional information on
the electronic excitation spectra: optical absorption,
X-ray spectra, and so on.
On the energetic side, the main output is the total
energy of the system for the given atomic configuration. Most codes are also able to calculate the forces
acting on the ions as well as the stress tensor acting on
the cell. Knowing these forces and stress, it is possible
to chain ground-state calculations to perform various
calculations:
 Atomic relaxations to the local minimum for the
atomic positions.

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 From the relaxed positions (where forces are zero),
one can calculate second derivatives of the energy

to deduce, among other things, the phonon spectrum. This can be done either directly, by the socalled frozen phonon approach, or by first-order
perturbation theory (if such feature is implemented in the code). In this last case, the third-order
derivative of the energy (Raman spectrum, phonon
lifetimes) can also be computed.
 Starting from two relaxed configurations close in
space, one can calculate the energetic path in space
joining these two configurations, thus allowing the
calculation of saddle points.
 The integration of the forces in a Molecular
Dynamics scheme leads to so-called ab initio
molecular dynamics (see Chapter 1.09, Molecular Dynamics). Car–Parrinello molecular dynamics18 calculations, which pertain to this class of
calculations, introduce fictitious dynamics on
the electrons to solve the minimization problem
on the electrons simultaneously with the real
ion dynamics.

1.08.2.3.2 Cell sizes and corresponding
CPU times

The calculation time of ab initio calculations varies –
to first order – as the cube of the number of atoms or
equivalently of electrons (the famous N3 dependence)
in the cell. If a fine k-point sampling is needed, this
dependence is reduced to N2 as the number of k
points decreases in inverse proportion with the size
of the cell. On the other hand, the number of selfconsistent cycles needed to reach convergence tends
to increase with N. Anyway, the variation of calculation time with the size of the cell is huge and thus
strongly limits the number of atoms and also the cell
size that can be considered. On one hand, calculations
on the unit cell of simple crystalline materials (with a

small number of atoms per unit cell) are fast and can
easily be performed on a common laptop. On the
other hand, when larger simulation cells are needed,
the calculations quickly become more demanding.
The present upper limit in the number of atoms
that can be considered is of the order of a few
hundreds. The exact limit of course depends on the
code and also on the number of electrons per atoms
and other technicalities (number of basis functions, k
points, available computer power, etc.), so it is not
possible to state it precisely. Considering such large
cells leads anyway to very heavy calculations in
which the use of parallel versions of the codes is


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Ab Initio Electronic Structure Calculations for Nuclear Materials

almost mandatory. Various parallelization schemes
are possible: on k points, fast Fourier transform,
bands, spins; the parallelization schemes actually
available depend on the code.
The situation gets even worse when one notes that
a relaxation roughly involves at least ten ground-state
calculations, a saddle point calculation needs about
ten complete relaxations, and that each molecular
dynamics simulation time step (of about 1 fs) needs
a complete ground-state calculation. Overall, one can
understand that the CPU time needed to complete an

ab initio study (which most of the time involves various starting geometry) may amount up to hundreds
of thousands or millions of CPU hours.
1.08.2.3.3 Choices to make

Whatever the system considered and the code used,
one needs to provide more inputs than just the atomic
positions and types. Most codes suggest some values
for these inputs. However, their tuning may still be
necessary as default values may very well be suited
for some supposedly standard situations and irrelevant for others. Blind use of ab initio codes may thus
lead to disappointing errors. Indeed, not all these
choices are trivial, so mistakes can be hard to notice
for the beginner. Choices are usually made out of
experience, after considering some test cases needing
small calculation time.
One can distinguish between choices that should
be made only once at the beginning of a study and
calculation parameters that can be tuned calculation
by calculation. The main unchangeable choices are
the exchange and correlation functional and the
pseudopotentials or PAW atomic data for the various
atomic types in the calculation.
First, one has to choose the flavor of the exchange
and correlation functional that will be used to
describe the electronic interactions. Most of the
time one chooses either an LDA or a GGA functional. Trends are known about the behavior of these
functionals: LDA calculations tend to overestimate
the bonding and underestimate the bond length in
bulk materials, the opposite for GGA. However,
things can become tricky when one deals with defects

as energy differences (between defect-containing and
defect-free cells) are involved. For insulating materials or materials with correlated electrons, the choice
of the exchange and correlation functional is even
more difficult (see Section 1.08.5).
The second and more definitive choice is the one
of the pseudopotential. We do not mean here the
choice of the pseudoization scheme but the choice

of the pseudopotential itself. Indeed, calculated energies vary greatly with the chosen pseudopotential, so
energy differences that are thermodynamically or
kinetically relevant are meaningless if the various
calculations are performed with different pseudopotentials. The determination of the shape of the atomic
basis set in the case of localized bases is also of
importance, and it is close in spirit to the choice of
the pseudopotential except that much less basis sets
than pseudopotentials are available.
More technical inputs include
 the k-point sampling. The larger the number of
k points to sample the Brillouin zone, the more
accurate the results but the heavier the calculations will be. This is especially true for metallic
systems that need fine sampling of the Brillouin
zone, but convergence with respect to the number
of k points can be accelerated by the introduction
of a smearing of the occupations of electronic
levels close to the Fermi energy. The shape and
width of this smearing function is then an additional parameter.19
 the number of plane waves (obviously for plane wave
codes but also for some other codes that also use
FFT). Once again the larger the number of plane
waves, the more accurate and heavier the calculation.

 the convergence criteria. The two major convergence criteria are the one for the self-consistent
loop of the calculation of the ground-state electronic
wave functions and the one to signal the convergence
of a relaxation calculation (with some threshold
depending on the forces acting on the atoms).

1.08.3 Fields of Application
Ab initio calculations can be applied to almost any
solid once the limitations in cell sizes and number of
atoms are taken into account. Among the materials
of nuclear interest that have been studied one can cite
the following: metals, particularly iron, tungsten,
zirconium, and plutonium; alloys, especially iron
alloys (FeCr, FeC to tackle steel, etc.); models of
fuel materials, UO2, U–PuO2, and uranium carbides;
structural carbides (SiC, TiC, B4C, etc.); waste materials (zircon, pyrochlores, apatites, etc.).
In this section, we rapidly expose the types of
studies that can be done with ab initio calculations.
The last two sections on metallic alloys and insulating materials will allow us to go into detail for some
specific cases.


Ab Initio Electronic Structure Calculations for Nuclear Materials

1.08.3.1

Perfect Crystal

1.08.3.1.1 Bulk properties

Dealing with perfect crystals, ab initio calculations
provide information about the crystallographic and
electronic structure of the perfect material. The
properties of usual materials, such as standard metals,
band insulators, or semi-conductors, are basically
well reproduced, though some problems remain, especially for nonconductors (see Section 1.08.5.1 on
SiC). However, difficulties arise when one wishes to
tackle the properties of highly correlated materials
such as uranium oxide (Section 1.08.5.2). For instance, no ab initio code, whatever the complexity
and refinements, is able to correctly predict the fact
that plutonium is nonmagnetic. In such situations, the
nature of the chemical bonding is still poorly understood, so the correct physical ingredients are probably not present in today’s codes. These especially
difficult cases should not mask the very impressive
precision of the results obtained for the crystal structure, cohesive energy, atomic vibrations, and so on of
less difficult materials.
1.08.3.1.2 Input for thermodynamic models

The information on bulk materials can be gathered in
thermodynamical models. Most ab initio calculations
are performed at zero temperature. Even with this
restriction, they can be used for thermodynamical
studies. First, ab initio calculations enable one to
consider phases that are not accessible to experiments. It is thus possible to compare the relative
stability of various (real or fictitious) structures for
a given composition and pressure.
Considering alloys, it is possible to calculate the
cohesive energy of various crystallographic arrangements. Solid solutions can also be modeled by so-called
special quasi-random structures (SQS).20 Beyond a
simple comparison of the energies of the various structures, when a common underlying crystalline network
exists for all the considered phases, the information

about the cohesive energies can be used to parameterize rigid lattice inter-atomic interaction models (i.e.,
pair, triplet, etc., interactions) that can be used to perform computational thermodynamics (see Chapter
1.17, Computational Thermodynamics: Application
to Nuclear Materials). These interactions can then be
used in mean field or Monte-Carlo simulations to
predict phase stabilities at nonzero temperature.21
As examples of this kind of studies one can cite the
determination of solubility limits (e.g., Zr and Sc in
aluminum22) and the exploration of details of the

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phase diagrams (e.g., the inversion of stability in the
iron-rich side of the Fe–Cr diagram23).
Directly considering nonzero temperature in
ab initio simulation is also possible, though more
difficult. First, one can calculate for a given composition and structure the electronic and vibrational
entropy (through the phonon spectrum), which leads
to the variation in heat capacity with temperature.
Nontrivial thermodynamic integrations can then be
used to calculate the relative stability of various structures at nonzero temperature. Second, one can perform
ab initio molecular dynamics simulations to model
finite temperature properties (e.g., thermal expansion).
1.08.3.2

Defects

Point defects are of course very important in a nuclear
complex as they are created either by irradiation or
by accommodation of impurities (e.g., fission products (FP)) (see Chapter 1.02, Fundamental Point

Defect Properties in Ceramics and Chapter 1.03,
Radiation-Induced Effects on Microstructure).
More generally, they have a tremendous role in the
kinetic properties of the materials. It is therefore not
surprising if countless ab initio studies exist on point
defects in nuclear materials. Most of them are based
on a supercell approach in which the unit cell of the
perfect crystal is periodically repeated up to the largest possible simulation box. A point defect is then
introduced, and the structure is allowed to relax. By
difference with the defect-free structure, one can calculate the formation energy of the defect that drives its
equilibrium concentration. Some care must be taken in
writing this difference as the number and types of
atoms should be preserved in the process. Point defects
are also the perfect object for the saddle point calculations that give the energy that drives their kinetic
properties. Ab initio permits accurate calculation of
these energies and also consideration of (for insulating
materials) the various possible charge states of the
defects. They have shown that the properties of defects
can vary greatly with their charge states.
Many different kinds of defects can be considered.
A list of possible defects follows with the characteristic
associated thermodynamical and kinetic energies.
1.08.3.2.1 Self-defects

Vacancies and interstitials, with the associated formation energy driving their concentration and migration energy driving their displacement in the solid;
the sum of these two energies is the activation energy
for diffusion at equilibrium. For such simple defects,


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Ab Initio Electronic Structure Calculations for Nuclear Materials

In the nuclear context, such defects can be fission
products in a fuel material, actinide atoms in a waste
material, helium gases in structural materials, and
so on; ab initio gives access to the solution energy of
these impurities, which allows one to determine their
most favored positions in the crystal: interstitial
position, substitution for host atoms, and so on. The
kinetic energies of migration of interstitial impurities
are accessible as well as the kinetic barrier for the
extraction of an impurity from a vacancy site.

ab initio molecular dynamics. We are aware of studies
in GaN27 and silicon carbides.28,29 The procedure is
the same as that with empirical potentials: one initiates a series of cascades of low but increasing energy
and follows the displacement of the accelerated
atom. The threshold energy is reached as soon as
the atom does not return to its initial position at
the end of the cascade. Such calculations are very
promising as empirical potentials are usually imprecise for the orders of energies and interatomic
distances at stake in threshold energies. However,
they should be done with care as most pseudopotentials and basis sets are designed to work for
moderate interatomic distances, and bringing two
atoms too close to each other may lead to spurious
results unless the pseudopotentials are specifically
designed.

1.08.3.2.3 Point defect assemblies

1.08.3.3.2 Electronic stopping power

it is possible to go beyond the 0 K energies and to
access the free energies of formation and migrations
by calculating the vibrational spectra in the presence
of the defect in the stable position and at the saddle
point (see Section 1.08.4.2.3).
1.08.3.2.2 Hetero-defects

In this class, one can include the calculation of interstitial assemblies as well as the complexes built with
impurities and vacancies. One then has access to the
binding of monoatomic defects to the complexes,24
possibly with the associated kinetic energy barriers.
1.08.3.2.4 Kinetic models

As for perfect crystals, the information obtained by
ab initio calculations can be gathered and integrated in
larger scale modeling, especially, kinetic models. Many
kinetic Monte-Carlo models were thus parameterized
with ab initio calculations (see e.g., the works on pure
iron25 or FeCu26and Chapter 1.14, Kinetic Monte
Carlo Simulations of Irradiation Effects).
1.08.3.2.5 Extended defects

Even if the cell sizes accessible by ab initio calculations
are small, it is possible to deal with some extended
defects. Calculations then often need some tricks to
accommodate the extended defect in the small cells.
Some examples are given in the next section on studies on dislocations.

1.08.3.3

Ab Initio for Irradiation

Irradiation damage, especially cascade modeling, is
usually preferentially dealt by larger scale methods
such as molecular dynamics with empirical potentials
rather than ab initio calculations. However, recently
ab initio studies that directly tackle irradiation processes have appeared.
1.08.3.3.1 Threshold displacement energies

First, the increase in computer power has allowed the
calculations of threshold displacement energies by

Second, recent studies have been published in the
ab initio calculations of the electronic stopping power
for high-velocity atoms or ions. The framework
best suited to address this issue is time-dependent
DFT (TD-DFT). Two kinds of TD-DFT have been
applied to stopping power studies so far.
The first approach relies on the linear response
of the system to the charged particle. The key quantity here is the density–density response function
that measures how the electronic density of the solid
reacts to a change in the external charge density.
This observable is usually represented in reciprocal
space and frequency, so it can be confronted directly
with energy loss measurements. The density–
density response function describes the possible
excitations of the solid that channel an energy transfer from the irradiating particle to the solid. Most
noticeably the (imaginary part of the) function

vanishes for an energy lower than the band gap
and shows a peak around the plasma frequency.
Integrating this function over momentum and
energy transfers, one obtains the electronic stopping
power. Campillo, Pitarke, Eguiluz, and Garcia have
implemented this approach and applied to some
simple solids, such as aluminum or silicon.30–32
They showed that there is little difference between
the usual approximations of TD-DFT: the random
phase approximation, which means basically no
exchange correlation included, or adiabatic LDA,
which means that the exchange correlation is local
in space and instantaneous in time. The influence
of the band structure of the solid accounts for
noticeable deviations from the homogeneous electron gas model.


Ab Initio Electronic Structure Calculations for Nuclear Materials

The second approach is more straightforward
conceptually but more cumbersome technically. It
proposes to simply monitor the slowing down of the
charged irradiated particle in a large box in real space
and real time. The response of the solid is hence not
limited to the linear response: all orders are automatically included. However, the drawback is the size of
the simulation box, which should be large enough to
prevent interaction between the periodic images. Following this approach, Pruneda and coworkers33 calculated the stopping power in a large band gap
insulator, lithium fluoride, for small velocities of the
impinging particle. In the small velocity regime, the
nonlinear terms in the response are shown to be

important.
Unfortunately, whatever the implementation of
TD-DFT in use, the calculations always rely on very
crude approximations for the exchange-correlation
effects. The true exchange-correlation kernel
(the second derivative of the exchange-correlation
energy with respect to the density) is in principle
nonlocal (it is indeed long ranged) and has memory.
The use of novel approximations of the kernel was
recently introduced by Barriga-Carrasco but for
homogeneous electron gas only.34,35
1.08.3.4

Ab Initio and Empirical Potentials

Ab initio calculations are often compared to and
sometimes confused with empirical potential calculations. We will now try to clarify the differences
between these two approaches and highlight their
point of contacts. The main difference is of course
that ab initio calculations deal with atomic and electronic degrees of freedom. Empirical potentials
depend only on the relative positions of the considered atoms and ions. They do not explicitly consider
electrons. Thus, roughly speaking, ab initio calculations deal with electronic structure and give access to
good energetics, whereas empirical potentials are not
concerned with electrons and give approximate
energetics but allow much larger scale calculations
(in space and time).
Going into some details, we have shown that
ab initio gives access to very diverse phenomena.
Some can be modeled with empirical potentials,
at least partly; others are completely outside the

scope of such potentials.
In the latter category, one will find the phenomena
that are really related to the electronic structure
itself. For instance, the calculations of electronic
excitations (e.g., optical or X-ray spectra) are conceptually impossible with empirical potentials. In the

231

same way, for insulating materials, the calculation of
the relative stability of various charge states of a given
defect is impossible with empirical potentials.
Other phenomena that are intrinsically electronic
in nature can be very crudely accounted for in empirical potentials. The electronic stopping power of
an accelerated particle is an example. As indicated
above, it can be calculated ab initio. Conversely, from
the empirical potential perspective one can add an
ad hoc slowing term to the dynamics of fast moving particles in solids whose intensity has to be established by fitting experimental (or ab initio) data.
In a related way, some forms of empirical potentials
rely on electronic information; for instance, the
Finnis–Sinclair36 or Rosato et al.37 forms. In the
same spirit, a recent empirical potential has been
designed to reproduce the local ferromagnetic order
of iron.38 However, this potential assumes a tendency
for ferromagnetic order, while ab initio calculation can
(in principle) predict what the magnetic order will be.
Therefore, ab initio is very often used as a way to
get accurate energies for a given atomic arrangement.
This is the case for the formation and migration
energies of defects, the vibration spectra, and so on.
These phenomena are conceptually within reach of

empirical potentials (except the ones that reincorporate electronic degrees of freedom such as charged
defects). Ab initio is then just a way to get proper and
quantitative energetics. Their results are often used
as reference for fitting empirical potentials. However,
the fit of a correct empirical remains a tremendous
task especially with the complex forms of potentials
nowadays and when one wants to correctly predict
subtle, out of equilibrium, properties.
Finally, one should always keep in mind that
cohesion in solids is quantum in nature, so classical
interatomic potentials dealing only with atoms or
ions can never fully reproduce all the aspects of
bonding in a material.

1.08.4 Metals and Alloys
The vast majority of DFT calculations on radiation
defects in metallic materials have been performed in
body-centered cubic (bcc) iron-based materials, for
obvious application reasons of ferritic steels but also
because of the more severe shortcoming of predictions based only on empirical potentials. A number
of accurate estimates of energies of formation and
migration of self-interstitial and vacancy defects
as well as small defect clusters and solute-vacancy
or solute-interstitial complexes have been obtained.


Ab Initio Electronic Structure Calculations for Nuclear Materials

DFT calculations have been intensively used to
predict atomistic defect configurations and also transition pathways. An overview of these results is presented below, complete with examples in other bcc

transition metals, in particular tungsten, as well as
hcp-Zr. These examples illustrate how DFT data
have changed the more or less admitted energy landscape of these defects and also how they are used to
improve empirical potentials. In the final part of this
chapter, a brief overview of typical works on dislocations (in iron) is presented.
1.08.4.1

55
Fe

50
45
P-bcc
(PW)

40
35
E (mRy)

232

30

P-bcc
(LSD)

25

Pure Iron and Other bcc Metals

Ferritic steels are an important class of nuclear materials, which include reactor pressure vessel (RPV)
steels and high chromium steels for elevated temperature structural and cladding materials in fast reactors
and fusion reactors, see Chapter 4.03, Ferritic Steels
and Advanced Ferritic–Martensitic Steels. From a
basic science point of view, the modeling of these
materials starts with that of pure iron, in the ferromagnetic bcc structure. Iron presents several difficulties for DFT calculations. First, being a three
dimensional (3D) metal, it requires rather large basis
sets in plane wave calculations. Second, the calculations
need to be spin polarized, to account for magnetism,
and this at least doubles the calculation time. But most
of all, it is a case where the choice of the exchangecorrelation functional has a dramatic effect on bulk
properties. The standard LDA incorrectly predicts
the paramagnetic face-centered cubic (fcc) structure
to be more stable than the ferromagnetic bcc structure.
The correct ground state is recovered using gradient
corrected functionals,39 as illustrated in Figure 1.
Finally, it was pointed out that pseudopotentials tend
to overestimate the magnetic energy in iron,40 and
therefore, some pseudopotentials suffer from a lack of
transferability for some properties. In practice, however, in the large set of the results obtained over the last
decade for defect calculations in iron, a quite remarkable agreement is obtained between the various
computational approaches. With a few exceptions,
they are indeed quite independent on the form of the
GGA functional, the basis set (plane wave or localized),
and the pseudopotential or the use of PAWapproaches.
1.08.4.1.1 Self-interstitials and selfinterstitial clusters in Fe and other bcc metals

The structure and migration mechanism of selfinterstitials in iron is a very good illustrative example
of the impact of DFT calculations on radiation defect

P-fcc
(PW)

20

F-bcc
(LSD)

15
10
5

P-fcc
(LSD)

F-bcc
(PW)

0
2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75
s arbitrary units (a.u.)
Figure 1 Calculated total energy of paramagnetic (P) bcc
and fcc and ferromagnetic (F) bcc iron as a function of
Wigner–Seitz radius (s). The dotted curve corresponds to
the local spin density (LSD) approximation, and the solid
curve corresponds to the GGA functional proposed by
Perdew and Wang in 1986 (PW). The curves are displaced
in energy so that the minima for F bcc coincide. Energies
are in Ry (1 Ry ¼ 13.6057 eV) and distances in bohr
(1 bohr ¼ 0.5292 A˚). Reproduced from Derlet, P. M.;

Dudarev, S. L. Prog. Mater. Sci. 2007, 52, 299–318.

studies. Progress in methods, codes, and computer
performance made this archetype of radiation
defects accessible to DFT calculations in the early
2000s, since total energy differences between simulation cells of 128þ1 atoms could then be obtained
with a sufficient accuracy. In 2001, Domain and
Becquart reported that, in agreement with the
experiment, the h110i dumbbell was the most stable
structure.41 Quite unexpectedly, the h111i dumbbell
was predicted to be $0.7 eV higher in energy, at
variance with empirical potential results that predicted a much smaller energy difference. DFT calculations performed in other bcc metals revealed
that this is a peculiarity of Fe,42 as illustrated
in Figure 2, and magnetism was proposed to be
the origin of the energy increase in the h111i dumbbell in Fe. The important consequence of this result
in Fe, which has been confirmed repeatedly since


Ab Initio Electronic Structure Calculations for Nuclear Materials

Defect energy relative to the <111> (eV)

2.5
2
1.5

2.5
V
Nb
Ta

Fe

2

Cr
Mo
W
Fe

1.5

1

1

0.5

0.5

0

0
-0.5

-0.5
-1
<111>

233

<110>

Tetra.

<100>

-1
Octa. <111>

SIA configuration

<110>

Tetra.

<100>

Octa.

SIA configuration

Figure 2 Formation energies of several basic SIA configurations calculated for bcc transition metals of group 5B (left)
and group 6B (right), taken from Nguyen-Manh et al.42 Data for bcc Fe are taken from Fu et al.43 Reproduced from
Nguyen-Manh, D.; Horsfiels, A. P.; Dudarev, S. L. Phys. Rev. B 2006, 73, 020101.

then, is that it excludes the SIA migration to
occur by long 1D glides of the h111i dumbbell
followed by on-site rotations of the h110i dumbbell,
as predicted previously from empirical potential
MD simulations. Moreover, DFT investigation of

the migration mechanism yielded a quantitative
agreement with the experiment for the energy
of the Johnson translation–rotation mechanism (see
Figure 3), namely $0.3 eV.43
These DFT calculations were followed by a very
successful example of synergy between DFT and
empirical potentials. The DFT values of interstitial
formation energies in various configurations and
interatomic forces in a liquid model have indeed
been included in the database for a fit of EAM type
potentials by Mendelev et al.45 This approach has
resulted in a new generation of improved empirical
potentials, albeit still with some limitations. When
considering SIA clusters made of parallel dumbbells,
the Mendelev potential agrees with DFT for predicting a crossover as a function of cluster size from the
h110i to the h111i orientation between 4 and 6 SIA

clusters.44 However, discrepancies are found when
considering nonparallel configurations.46 More precisely, new configurations of small SIA clusters were
observed in MD simulations performed at high temperature with the Mendelev potential. The energy
of the new di-interstitial cluster, made of a triangle of
atoms sharing one site (see Figure 4), is even lower
than that of the parallel configuration within DFT
but higher by 0.3 eV with the Mendelev potential (see
also Section 1.08.4.3 on dislocations). The new triand quadri-interstitial clusters, with a ring structure
(see Figure 4), are one of the few examples in which a
significant discrepancy is found between various
DFT approaches. Calculations with the most accurate description of the ionic cores predict that the
new tri-interstitial configuration is slightly more stable than the parallel configuration, whereas more
approximate ones predict that it is 0.7 eV higher.

The first category includes calculations in the PAW
approach, performed using either the VASP code or
the PWSCF code and also ultrasoft pseudopotential
calculations. The second one includes calculations


234

Ab Initio Electronic Structure Calculations for Nuclear Materials
0.8
DFT-GGA
Mendelev
Ackland

Energy barrier (eV)

0.6

0.4

0.2

0.0
[011]

[110]

[111] Crowd.

[111]

Figure 3 Left: Johnson translation–rotation mechanism of the h110i dumbbell; white and black spheres indicate the initial
and final positions of the atoms, respectively. Reproduced from Fu, C. C.; Willaime, F.; Ordejon, P. Phys. Rev. Lett. 2004,
92, 175503. Right: Comparison between the DFT-GGA result and two EAM potentials for the energy barriers of the
Johnson mechanism and the h110i to h111i transformation. Reproduced from Willaime, F.; Fu, C. C.; Marinica, M. C.;
Torre, J. D.; Nucl. Instrum. Meth. Phys. Res. B 2005, 228, 92.

<011>
<101>
<101>

<011>

<110>

<110>

Figure 4 New low-energy configurations of SIA clusters in Fe, which revealed discrepancies between DFT and
empirical potentials and between various approximations within DFT. Reproduced from Terentyev, D. A.; Klaver, T. P. C.;
Olsson, P.; Marinica, M. C.; Willaime, F.; Domain, C.; Malerba, L. Phys. Rev. Lett. 2008, 100, 145503.

with less transferable ultrasoft pseudopotentials with
VASP and norm-conserving pseudopotentials with
SIESTA.46 Such a discrepancy is not common in
defect calculations in metals. Further investigations
are required to understand more precisely its origin,
in particular the possible role of magnetism.
The structures of the most stable SIA clusters in
Fe, and more generally of their energy landscape,
remain an open question. One would ideally need

to combine DFT calculations with methods for exploring the energy surface, such as the Dimer47 or
ART48 methods. Such a combination is possible in
principle, and it has indeed been used for defects in
semiconductors,49 but due to computer limitations this
is not the case yet in Fe. The alternative is to develop
new empirical potentials in better agreement with

DFT energies in particular for these new structures,
to perform the Dimer or ART calculations with these
potentials, and to validate the main features of the
energy landscape thus obtained by DFT calculations.
To summarize, the energy landscape of interstitial
type defects has been revisited in the last decade
driven by DFT calculations, in synergy with empirical potential calculations.
1.08.4.1.2 Vacancy and vacancy clusters
in Fe and other bcc metals

DFT has some limitations in predicting accurate
vacancy formation energies in transition metals. The
exceptional agreement with the experiment obtained
initially within DFT-LDA50 was later shown to result
from a cancellation between two effects. First, the


Ab Initio Electronic Structure Calculations for Nuclear Materials

Ackland et al.94
Mendelev et al.45
DFT-GGA

1.2
Migration energy (eV)

structural relaxation, which was neglected by Korhonen
et al.50 is now known to significantly reduce the vacancy
formation energy, in particular in bcc metals.51 Second,
due to limitations of exchange-correlation functionals
at surfaces, DFT-LDA tends to underestimate the
vacancy formation energy. This discrepancy is even
larger within DFT-GGA, and it increases with the
number of valence electrons. It is therefore rather
small for early transition metals (Ti, Zr, Hf,), but it is
estimated to be as large as 0.2 eV in LDA and 0.5 eV in
GGA-PW1 for late transition metals (Ni, Pd, Pt).52
However, the effect is much weaker for migration energies.52 A new functional, AM05, has been proposed to
cope with this limitation.53
Less spectacular effects are expected in vacancytype defects than in interstitial-type defects when
going from empirical potentials to DFT calculations.
The discussion on vacancy-type defects in Fe will be
restricted to the results obtained within DFT-GGA,
due to the superiority of this functional for bulk properties. For pure Fe, DFT-GGA vacancy formation and
migration energies are in the range of 1.93–2.23 eVand
0.59–0.71 eV.41,43,54 These values are in agreement
with experimental estimates at low temperatures in
ultrapure iron, namely 2.0 Æ 0.2 eV and 0. 55 eV,
respectively. These values can be reproduced by
empirical potentials when included in the fit, but one
discrepancy remains with DFT concerning the shape
of the migration barrier. It is indeed clearly a single
hump in DFT25 and usually a double hump with

empirical potentials.
Concerning vacancy clusters, the structures predicted by empirical potentials, namely compact structures, were confirmed by DFT calculations, but
there are discrepancies in the migration energies. In
both cases, the most stable divacancy is the nextnearest-neighbor configuration, with a binding energy of 0.2–0.3 eV.25,55,56 The migration can occur by
two different two-step processes, with an intermediate configuration that is either nearest neighbor or
fourth nearest neighbor.56 A quite unexpected result
of DFT calculations was the prediction of rather low
migration energies for the tri- and quadrivacancies,
namely 0.35 and 0.48 eV.25 Depending on the potential, this phenomenon is either not reproduced or only
partly reproduced (see Figure 5).57
Stronger deviations from empirical potential predictions for divacancies are observed in DFT calculations performed in other bcc metals. The most
dramatic case is that of tungsten, where the nextnearest-neighbor interaction is strongly repulsive
(0.5 eV) and the nearest-neighbor interaction is

235

1.0
0.8
0.6
0.4
0.2
0.0

V

V2

V3

V4

V5

Figure 5 Migration energies of vacancy clusters in Fe,
as a function of cluster size. Reproduced from Fu, C. C.;
Willaime, F. (2004) Unpublished.

vanishing.58 This result does not explain why voids
are formed in tungsten under irradiation.
1.08.4.1.3 Finite temperature effects on
defect energetics

The properties of radiation defects at high temperature may change due to three possible contributions to
the free energy: electronic, magnetic, and vibrational.
These three effects can be well modeled in bulk bcc
iron,59 but they are more challenging for defects. The
electronic contribution, which exists only in metals,
arises due to changes in the density of states close to
the Fermi level. The electronic entropy difference
between, for example, two configurations is, to first
order, proportional to the temperature, T, and the
change in density of states at the Fermi level. This
electronic effect is straightforward to take into account
in DFT calculations. It was shown in tungsten to
decrease the activation free energy for self-diffusion
by up to 0.4 eV close to the melting temperature. Thus,
although this effect is relatively small in general, it
cannot be neglected at high temperature.
The magnetic contribution is important in iron.
Spin fluctuations were shown to be the origin of the

strong softening of the C0 elastic constant observed as
the temperature increases up to the aÀg transition
temperature,60 and it drives, for instance, the temperature dependence of relative abundance of <100> and
<111> interstitial loops formed under irradiation.61
It is also known to have a small effect on vacancy
properties, but to the authors’ knowledge there is presently no tractable method to predict this effect for
point defects quantitatively from DFT calculations.
This is probably one of the important challenges in
the field.


236

Ab Initio Electronic Structure Calculations for Nuclear Materials

Finally, vibrational entropy effects can in principle be obtained either in the quasi-harmonic
approximation from phonon frequency calculations
or directly from first-principles molecular dynamics. There are very few examples of such calculations in the literature. The vibrational modes of
vacancies and self-interstitials in iron have been
investigated by DFT calculations, and their formation entropies have been estimated.62 As illustrated
recently in Mo, it is also possible to calculate the
temperature dependence of the vacancy formation
enthalpy, from DFT molecular dynamics simulations, including anharmonic effects, as well as the
defect jump frequency, going beyond the transition
state approximation.63
1.08.4.2

Beyond Pure Iron

1.08.4.2.1 helium–vacancy clusters in iron

and other bcc metals

Irradiation of metals by neutrons produces, besides
point defects, rare gases by transmutation reactions.
Helium is a major concern since it has a very low
solubility in metals, see Chapter 1.06, The Effects of
Helium in Irradiated Structural Alloys. It is deeply
trapped by vacancies, and helium–vacancy clustering
can ultimately lead to bubble formation and void
swelling. At variance with empirical potential predictions, DFT calculations showed that interstitial
helium is unambiguously located on tetrahedral
sites, not only in iron, but also in all other bcc
metals.64–67 An improved Fe–He pair potential was
then obtained by fitting to the DFT results.68 DFT
(b)

calculations account for the very fast migration of
interstitial helium as well as for its deep trapping to
vacancies, although not as deep as predicted by
empirical potentials. Note that an unexpected effect
was observed in Vanadium, where the helium atom in
a vacancy is found to be off centered.66 The energy
landscape of the helium–divacancy complex also
revealed unexpected configurations, in particular,
for the lowest energy configuration where the helium atom is located halfway between two nearestneighbor vacancies (see Figure 6). More generally, a
systematic study of the energetics of all small HenVm
clusters in iron was performed,64 giving very useful
data for the kinetic modeling of helium–vacancy clustering and dissociation.69 Quite interestingly, interstitial He atoms are found to attract one another, even
in the absence of vacancies.24,64 This clustering of
helium atoms may then yield the emission of selfinterstitials. Finally, the interaction of helium with

self-interstitials is, as expected, much weaker but
also attractive.24
Similar studies have been performed on small
helium–vacancy clusters in tungsten58,70 and also on
the behavior of hydrogen in iron and tungsten. It
should be noted that at low temperature, quantum
effects must be taken into account in the migration
properties in particular for hydrogen.71
1.08.4.2.2 From pure iron to steels:
the role of carbon

In steels, the presence of carbon, even though its
concentration is very low, considerably affects defect properties because of the strong carbon-defect
(d)

(a)

(e)

Energy (eV)

1.2

(f)

(c)

0.8

0.4

0.0
3 nn

1 nn
Reaction coordinate

2 nn

2 nn

Figure 6 Schematic representation of the energy landscape of the HeV2 complex. Reproduced from Fu, C. C.;
Willaime, F. Phys. Rev. B 2005, 72, 064117.


Ab Initio Electronic Structure Calculations for Nuclear Materials

interaction. DFT calculations reproduce the wellknown fact that carbon is located in octahedral sites,
and they also confirm the strong attraction between
interstitial carbon and a monovacancy, with a binding
energy of about 0.5 eV.72–74 This strong attraction is
the origin of the confusing discrepancy between the
vacancy migration energy in ultrapure iron, $0.6 eV,
and the effective vacancy migration energy in iron
with carbon or in steels, that is, $1.1 eV, which corresponds to first order to the sum of the vacancy
migration energy and the carbon-vacancy binding
energy.74 More interestingly, DFT calculations predict that the complex formed by a vacancy and
two carbon atoms, VC2, is extremely stable, due to
the formation of a strong covalent bond between the
carbon atoms. The VC–C binding energy is indeed

close to 1 eV,72–74 and VC2 complexes are expected to
play a very important role.
The interaction between carbon and selfinterstitials is also attractive but weaker. In agreement
with experiments,75 DFT calculations confirmed a
binding energy of $0.2 eV76 and predict, at variance with initial empirical potential results, that the
nearest-neighbor configurations are repulsive and
that the most attractive configuration is that shown
in Figure 7. This shortcoming of empirical potentials
was overcome recently with an improved potential
derived taking into account information from the
electronic structure.77
The strong interaction of carbon with vacancies
also affects the energetics of helium–vacancy clusters,
and it is important to take this into account to reproduce, for example, thermal helium desorption experiments performed in iron.78
Similar calculations have been performed with
nitrogen.72
1.08.4.2.3 Interaction of point defects with
alloying elements or impurities in iron

The diffusion of point defects produced by irradiation may induce fluxes of solutes, for example, toward

237

or away from defect sinks, depending on the defect–
solute interactions. DFT is again a very powerful tool
to predict such interactions, which can then be used
in kinetic models. This approach is also useful in the
absence of irradiation, and a very interesting example
has been obtained in the simulation of the first stages
of the coherent precipitation of copper in bcc–Fe.

DFT calculations predicted that the vacancyformation energy in metastable bcc–Cu (which is
not known experimentally since bulk Cu is fcc) is
0.9 eV, that is, much smaller than that in bcc iron,
namely 2.1 eV. This leads to strong trapping of vacancies by the Cu precipitates. As a result, precipitates
containing up to several tens of copper atoms are
quite surprisingly predicted to be much more mobile
than individual copper atoms in the iron matrix.26
Another very illustrative example is given by the
study of atomic transport via interstitials in dilute
Fe–P alloys. DFT results indeed predict that Fe–P
mixed dumbbells are highly mobile but that they can
be deeply trapped by a substitutional P atom.79
A systematic study of the interaction of monovacancies and self-interstitials with all transition-metal
solutes has been reported recently (see Figure 8).80
1.08.4.2.4 From dilute to concentrated alloys:
the case of Fe–Cr

In the approach described earlier, which considers
low concentrations of solutes and defects, the number
of independent configurations is rather small, and
they can be easily taken into account in kinetics
model. The situation is much more complex when
considering Fe–Cr with Cr concentration in the
range 10–20%. Nevertheless, first results have been
obtained by considering the interaction of defects
with one or two Cr atoms in the Fe matrix.81 These
data could ideally be used to fit an improved empirical potential, but the Fe–Cr system is rather difficult
to model because of the strong interplay between
magnetic and chemical interactions. This is also
clearly one of the challenges in the field.

Figure 7 Structure of carbon–vacancy and carbon–self-interstitial complexes in iron, predicted from DFT calculations.
Reproduced from Fu, C. C.; Meslin, E.; Barbu, A.; Willaime, F.; Oison, V. In Theory, Modeling and Numerical Simulation of
Multi-Physics Materials Behavior, 2008; Vol. 139, pp 157–164, 168.


238

Ab Initio Electronic Structure Calculations for Nuclear Materials

Blochl14
Zunger et al.20 and Becquart and Domain55
Djurabekova et al.56
Fu and Willaime57 and Becquart and Domain58

ce

0

nn

-0.6

34

di

st
an

-0.3

1

2

E bvac-3d (eV)

0.3

Ti

V

Cr

Mn

Fe

Co

Ni

Cu

Zr

Nb

Mo

Tc

Ru

Rh

Pd

Ag

Hf

Ta

W

Re

Os

Ir

Pt

Au

nc

e

0
n
23 n d
4 ista

-0.3
-0.6

1

E bvac-4d (eV)

0.3

nc

e

0
-0.3

12 nn
3 4 dis
ta

E bvac-5d (eV)

0.3

-0.6

Figure 8 DFT-GGA solute-vacancy binding energies in iron for 3D, 4D, and 5D elements for 1–5 nn relative positions.
Reprinted with permission from Olsson, P.; Klaver, T. P. C.; Domain, C. Phys. Rev. B 2010, 81, 054102. Copyright (2010) by
the American Physical Society.

1.08.4.2.5 Point defects in hcp-Zr

Point defects in hcp-Zr have also been studied via
DFT calculations. It was found in particular that the
vacancy migration energy is lower by $0.15 eV within
the basal plane than out of the basal plane.82 The
situation for the self-interstitial is quite complex,
since among the known configurations, at least three
configurations are found to have almost the same
formation energy (within 0.1 eV): the octahedral (O),
split dumbbell (S), and basal octahedral (BO)
configurations.83,84
1.08.4.3

Dislocations

The collective behavior of dislocations can be
described thanks to dislocation dynamics codes. In
order to reinforce the physical foundation, input data
such as mobility laws can be obtained from atomistic
calculations of individual dislocations. These defects
can now be investigated using more accurate ab initio
electronics structure methods. We exemplify these

studies by focusing in the following section on the
properties of dislocations in bcc metals and especially

iron. In these materials, dislocation properties are
known to be closely related to their core structure.
When dealing with dislocations, special care
should be taken in the positioning of the dislocations
and in the boundary conditions of the calculations.
For instance, considering h111i screw dislocations,
the two cell geometries proposed in the literature –
the cluster approach85 and the periodic array of
dislocation dipoles86 – have been thoroughly
compared.87 The calculations of dislocations are
extremely demanding as they can include up to 800
atoms, so studies usually use fast codes such as
SIESTA.8 The construction of simulation cells appropriate for such extended defects should be optimized
for cell sizes accessible to DFT calculations, and the
cell-size dependence of the energetics evidenced
in both the cluster approach and the dipole approach
for various cell and dipole vectors should be rationalized. The quadrupolar arrangement of dislocation
dipoles is most widely used for such calculations87
although the cluster approach with flexible boundary
conditions can be considered a reference method
when no energies are necessary (i.e., only structures).


Ab Initio Electronic Structure Calculations for Nuclear Materials

DFT calculations in bcc metals such as Mo, Ta, Fe,
and W85,87–91 predict a nondegenerate structure for

the core, as illustrated in Figure 9 using differential
displacement maps as proposed by Vitek.92 The edge
component reveals the existence of a significant core
dilatation effect in addition to the Volterra field,
which can be successfully accounted for by an anisotropic elasticity model.93
Thanks to good control of energy, it is also possible to obtain quantitative results on the Peierls potential; namely, the 2D energy landscape seen by a
straight screw dislocation as it moves perpendicular
to the Burgers vector. This is exemplified in the
following Figure 10(a), where a high symmetry direction of the Peierls potential is sampled: the line going
between two easy core positions along the glide direction, that is, the Peierls barrier. These calculations

239

were performed by simultaneously displacing the
two dislocations constituting the dislocation dipole
in the same direction and by using a constrained
relaxation method. In the same work, the behavior
of the Ackland–Mendelev potential for iron,45 which
gives the correct nondegenerate core structure unlike
most other potentials, has been tested against the
obtained DFT results. It appears that it compares
well with the DFT results for the g-surfaces, but
discrepancies exist on the deviation from anisotropic
elasticity of both edge and screw components and on
the Peierls potential. Indeed, the empirical potential
results do not predict any dilatation elastic field
exerted by the core. Besides, the Peierls barrier displayed by the Ackland–Mendelev potential yields a
camel hump shape, as illustrated in Figure 10(a),
and at the halfway position, the core spreads between

[110]

(a)

[111]

[112]

(b)

(c)

Figure 9 (a) Differential displacement map of the nondegenerate core structure of a <111> screw dislocation in Fe, as
obtained from SIESTA GGA. (b) Same as (a) after subtraction of the Volterra anisotropic elastic field and magnified by a factor
of 20. (c) Same as (b) for the displacement in the (111) plane (or edge component) and a magnification by a factor of 50.
Reproduced from Ventelon, L.; Willaime, F. J. Comput. Aided Mater. Des. 2007, 14, 85–94.
200

100

40
30
20

SIESTA GGA
Ackland
Ackland–Mendelev
Dudarev–Derlet

0

-100

10
0
0.0

(a)

SIESTA GGA
SIESTA LDA
Ackland–Mendelev

Energy (meV/b)

Energy barrier (meV/b)

50

0.2

0.4
0.6
Reaction coordinate

0.8

-200
0.0

1.0
(b)

0.2

0.4
0.6
Polarity

0.8

1.0

Figure 10 (a) Peierls barrier in Fe calculated with the Ackland–Mendelev potential45 and with SIESTA using the two
exchange-correlation functionals, LDA and GGA. Reproduced from Ventelon, L.; Willaime, F. J. Comput. Aided Mater. Des.
2007, 14, 85–94. (b) Dependence of the dislocation core energy with the modulus of its polarization calculated using SIESTA
and the three empirical potentials, namely, the Ackland,94 Ackland–Mendelev,45 and Dudarev–Derlet38 potentials.
Reproduced from Ventelon, L.; Willaime, F. Philos. Mag. 2010, 90, 1063–1074.


240

Ab Initio Electronic Structure Calculations for Nuclear Materials

two easy core positions, whereas it exhibits a single
hump barrier within DFT and a nearly hard-core
structure at halfway position. The effect of the
exchange-correlation functional within DFT appears
to be significant.87
More insight into the stability of the core

structure can be gained by looking at the response
of the polarization of the core, as represented in
Figure 10(b). In the Ackland–Mendelev and DFT
cases, these calculations confirm that the stable core
is completely unpolarized, and they prove that there
is no metastable polarized core.95
Finally, the methodology exists for calculating the
structure and formation and migration energies of
single kinks, but using it with DFT96 remains challenging because cells with about 1000 atoms are
needed, together with a high accuracy.

1.08.5 Insulators
From the atomistic and electronic structure point
of view, it is legitimate to distinguish between
electrically conducting materials on one hand and
insulating or semiconducting materials on the other.
Indeed, insulating materials exhibit specific behaviors,
especially for the point defects. Due to the existence of
a gap in the electronic density of states, the point
defects may be charged. There is recent evidence
that the properties of the point defects, especially
their kinetic properties, such as the migration energy,
depend a lot on their charge state. The charge of a
given point defect depends on the position of the
Fermi level within the band gap: a low lying Fermi
level (close to the valence band) favors positively
charged defects, whereas a Fermi level close to the
conduction band favors negatively charged defects.
The positions of the Fermi level corresponding to
transitions between charge states are called charge

transition levels (CTL). The correct determination
of these CTL allows the correct prediction of the
charge states of the defects, as piloted by Fermi
level position, that is, the doping conditions for the
material. Standard DFT methods fail to reproduce
accurately these CTL, and the research of more accurate methods is presently a very active field in the
electronic structure community, with major implications for microelectronic research as well as for nuclear
materials, especially in view of the aforementioned
variation of point defect kinetic properties with their
charge state. All these charge aspects of point defects
are completely out of range for empirical potentials.

In the last two sections, we exemplify the research
on insulating materials by summarizing the available
results for two important insulating nuclear materials: silicon carbide and uranium dioxide.
Silicon carbide is an important candidate material
for fusion and fission applications. Even if it arguably
a less crucial material than UO2, we start with this
material as its electronic structure is simpler. UO2 is
obviously the basic model material for the nuclear
fuel of usual reactors.
1.08.5.1

Silicon Carbide

This brief survey exemplifies the kind of calculations
that can be performed on common insulating materials (as opposed to correlated ones such as UO2) in a
nuclear context. Specificities of insulating materials
when compared with metallic systems will clearly
appear, especially for what concerns the possible

charge states of the defects and the difficulties standard DFT calculations have in satisfactorily reproducing the quantities that govern them.
SiC exists in many different structures. Nuclear
applications are interested with the so-called b structure (3C–SiC), a zinc blende crystal cubic form. We
shall therefore focus on this structure, although many
additional calculations have been performed on other
structures of the hexagonal type, which are more of
interest for microelectronics applications.
Silicon carbide is a band insulator whose bulk
structural properties are well reproduced by usual
DFT calculations. The electronic structure of the
bulk material is also well reproduced except for the
usual underestimation of the band gap by DFT calculations. Indeed, the measured gap is 2.39 eV,97 whereas
standard DFT-LDA calculations give 1.30 eV.97
1.08.5.1.1 Point defects

The first DFT calculations of point defects in silicon
carbide,98 dating back to 1988, were burdened by
strong limitations in computing time. For this reason,
they were performed with relatively small supercells
(16 and 32 atoms), largely insufficient basis sets
(plane waves with energy up to 28 Ry), and further
approximations, namely for the relaxation of atomic
positions. Moreover, they were limited to high symmetry configurations. The results were only qualitative; however, it was already clear that vacancies and
antisites could be relatively abundant, at equilibrium,
with respect to interstitial defects. The authors dared
to approach some defect complexes and could predict
that antisite pairs and divacancies were bound.


Ab Initio Electronic Structure Calculations for Nuclear Materials

Vacancies were thoroughly studied at the turn of
the century.97,99–101 The most prominent result may
be the metastability of the silicon vacancy. Indeed,
following a suggestion coming from a self-consistent
DFT-based tight-binding calculation by Rauls and
coworkers,102 the electron paramagnetic resonance
(EPR) spectra of annealed samples of irradiated
SiC were measured103 and compared with calculated
hyperfine parameters. This showed that silicon
vacancies are metastable with regard to a carbon
vacancy–carbon antisite complex (VC–CSi); a fact
that has since been consistently confirmed by the
other calculations.
Interstitials were less studied than vacancies. One
should however mention a study104 devoted to carbon
and silicon in interstitials in silicon carbide. Beyond
these studies dedicated to one type of defect, very
complete and comprehensive work on both vacancies
and interstitials was also published. One should cite
Bernardini et al.105 devoted to the formation energies
of defects, while Bockstedte et al.106 goes further as it
also covers migration energetics of basic intrinsic
defects (vacancies, interstitials, antisites). It is worth
noting that in such covalent compounds there are
many possible atomic structures for defects as simple
as a monointerstitial and that all these structures must
be considered in the calculation (see Figure 11).
As examples, the results of these various studies on
what concerns formation energies and CTL of vacancies are summarized in the following tables.

One can see a general agreement in the formation
energies of the neutral defects, especially in the recent references. The small differences are related to
k-point sampling or cell size in the calculations.
Larger discrepancies appear between the various predicted CTL. They relate to the inaccuracy of standard
DFT calculations in treating empty or defect states.

Si

CTC

Front

A simple example relates directly to the underestimation of the band gap: the silicon interstitial (in the
ISi
TC configuration) in the neutral state shows up as
metallic in standard calculations, the defect states
lying inside the conduction band. This fact, on one
hand, calls for a better description of the exchangecorrelation potential for these configurations; on the
other, it makes the convergence with k points and
cell size very slow, as has recently been pointed
out.107 This drawback of standard DFT-LDA/GGA
supercell calculations is common to other defects in
SiC. Even when calculated defect states fall within the
band gap, their position inside it can be grossly miscalculated with standard DFT calculations.
The errors produced by standard DFT calculations for the CTL are well known nowadays. The
determination of an accurate method to calculate
these CTL is an active field of research with works
on advanced methods such as GW (e.g., the results on
SiO2108) or hybrid functionals.109 For what concerns
nuclear materials, and especially SiC, GW corrections and excitonic effects will allow further comparisons with experiments (Table 1).110

1.08.5.1.2 Defect kinetics

Before the aforementioned work by Bockstedte and
coworkers106 almost no work was devoted to migration properties of point defects in SiC. We should,
however, cite previous preliminary works by the
same group,111,112 a work on the mechanisms of
formation of antisite pairs,113 and a work on vacancy
migration published in 2003.114 The comprehensive
study of migration barriers in Bockstedte et al.106
showed, first of all, that vacancies have much higher
migration energies than those of interstitials: higher
than 3 eV for the former in the neutral state, around
1 eV for the latter (0.5 for IC, 1.4 for ISi). Another

C

CTSi

CHex

241

Csp<110>

Csp<100>

CspSi<100>

Side

Figure 11 Possible geometries for a carbon interstitial in cubic SiC. Reprinted with permission from Bockstedte, M.;
Mattausch, A.; Pankratov, O. Phys. Rev. B 2003, 68, 205201. Copyright (2003) by the American Physical Society.


242

Ab Initio Electronic Structure Calculations for Nuclear Materials

Table 1

Formation energies for vacancies and their charge transition levels according to various authors
VC

VSi

References

0

þ/0

þ/þþ

0

þ/0

0/À

À/ÀÀ

98
191
99
97
192
106
105

5.6
–
4.01
–
3.74
3.78
3.84

1.7
–
1.41
–
1.18
–
–

1.9
–
1.72
–
1.22

1.29
–

7.6
–
8.74
7.7
8.37
8.34
8.78

–
0.54
0.43
0.50
–
0.18
0.41

–
1.06
1.11
0.56
0.57
0.61
0.88

–
1.96
1.94

1.22
1.60
1.76
1.40

The values are for the 3C-polytype in silicon-rich conditions. Values are expressed in eV.

remarkable finding is the strong variation of the
migration energy with the charge state; indeed, the
migration energy for the carbon vacancy is raised by
almost 2 eV going from the neutral to the 2þ charged
state, whereas the silicon vacancy finds its migration
barrier reduced by 1 eV when its charge goes from
neutral to 2À. Interstitials are reported to have their
lowest migration barriers in the neutral state, except
for the ISi
TC configuration, which is expected to have
an almost zero energy barrier of migration in the
2þ and 3þ charge states. Such large changes in
the migration energies of defects with their charge
should induce tremendous variations in their kinetic
behavior under different charge states.
The energy barriers of recombinations of
close interstitial vacancy pairs have also been tackled.115–117 It appears that the energetic landscape for
the recombination of Frenkel pairs is extremely
complex. One should distinguish the regular recombination of an homo interstitial-vacancy pairs from
those of hetero interstitial-vacancy pairs, which
leads to the formation of an antisite. Recent works
tend to suggest that the latter may, in certain conditions, have a lower energy than the recombination
of a regular Frenkel pair. A kinetic bias for the

formation of antisites, preliminary to decomposition,
may thus be active in SiC under irradiation.118
Calculations of threshold displacement energies
from first-principles molecular dynamics29 have also
been reported. Their results show that this quantity is
strongly anisotropic, and they found average values
(38 eV for Si and 19 eV for C) that are in agreement
with currently accepted values (coming from experimental evaluations that are, however, largely dispersed). These calculations prove that available
CPU power is now large enough to calculate TDE
from ab initio molecular dynamics. This is good news
as empirical potentials are basically not reliable in the
prediction of TDE.

1.08.5.1.3 Defect complexes

Several defect complexes have been studied by
first-principles calculations in silicon carbide. The
identification of EPR signals, deep level transient
spectroscopy (DLTS), or photoluminescence (PL)
experiments based on calculated properties have
been attempted for some of them. Crucial to these
identifications is the reliability of the predictions of
charge transition levels (for the position of DLTS
peaks) and of annealing temperatures, through more
or less complicated mechanisms.
One of the first, and simplest, defect complex identified through comparison of theory and experiment
was the VC–CSi coming from the annealing of silicon
vacancies in 6H-SiC, as previously mentioned. More
complex antisite defects or antisite complexes119,120 as
well as divacancy complexes121–123 were called upon

for the attribution of PL or EPR peaks.
Various kinds of carbon clusters were studied in
detail theoretically.124–127 The cited works deal with
the stability, electrical properties, and local vibrational
modes (LVM) of several structures. It was shown that
the aggregation of carbon interstitials with carbon
antisites can lead to various bound configurations.
In particular, two, three, or even four carbon atoms
can substitute one silicon atom forming very stable
structures. The binding energy of these structures is
high: from 3.9 to 5 eV, according to the charge state, for
the (C2)Si, and further energy is gained when adding
further carbon atoms. Silicon clusters did not raise as
much interest as carbon ones; however, a recent
work107 deals with the stability and dynamics of such
silicon clusters (see Figure 12).
1.08.5.1.4 Impurities

The interest in SiC as a large band gap semiconductor for electronic applications has promoted works
on typical dopants. Most of the calculations focus on
hexagonal SiC, but one can reasonably assume that


Ab Initio Electronic Structure Calculations for Nuclear Materials

Ie2

Energy barrier (eV)

ITC + ITC

Ia2

Ie2

Ia2

Ib2

Ia2

Ic2

Ia2

Id2

ISisp<110> + I

243

Id2

3.0

3.0

2.5

2.5

2.0

2.0
Emigra(I)
1.5
Ebind(I2e)

1.5

Etrans(I2e-a)

1.0
0.5

Emigra(I2a)

Ebind(I2d)

Etrans(I2c-a)
in-plane

Erotation(I2a)

Eopen(I2a-c)

out-plane

Erotation (I2a)

1.0

0.5
0.0

0.0

Reaction coordinate
Figure 12 Energetic landscape of silicon mono- and di-interstitial in cubic SiC. Reproduced from Liao, T. (2009)
Unpublished.

the results would not be very different in cubic SiC.
One can find calculations dealing with boron129,130
as an acceptor and nitrogen131,132 or phosphorus133 as
a donor. Other impurities were studied: transition
metals,134–136 oxygen,137 important for the behavior of
the SiO2/SiC interface, hydrogen,138–140 rare gases,141
and palladium.142 A systematic study of substitutional
impurities has recently appeared,143 which focuses on
the trends of carbon vs. silicon substitution according
to the position of species in the periodic table.
1.08.5.1.5 Extended defects

Another major subject, which has attracted much
interest for the hexagonal types of SiC, is related to
the electronic properties of extended defects, surfaces/interfaces, stacking faults, and dislocations.
The reason why extended defects have been mainly
studied in the hexagonal types of silicon carbide lies
in the fact that electronic properties of dislocations
and stacking faults are particularly important for
understanding the degradation of hexagonal SiC
devices144 and the remarkable enhancement of dislocation velocity under illumination in the hexagonal

phase.145 Nevertheless, some studies have been done
for cubic SiC on the electronic structure of stacking
faults146–151 and various types of dislocations.152–155
Obviously, a lot of work remains to be done on the
extended defects in b SiC.
1.08.5.2

Uranium Oxide

1.08.5.2.1 Bulk electronic structure

Due to its technological importance and the complexity of its electronic structure, uranium oxide
has become one of the test cases for beyond

LDA methods. Indeed, UO2 comes out as a metal
when its electronic structure is calculated with
LDA or GGA. This result has been found by
many authors using many different codes or numerical schemes (the primary calculation being the
work of Arko and coworkers156). The physical difficulty lies in the fact that UO2 is a Mott insulator.
f electrons are indeed localized on uranium atoms
and are not spread over the material as usual
valence electrons are.
The first correction that has been applied is the
LDAþU correction in which a Hubbard U term
acting between f electrons is added ‘by hand’ to the
Hamiltonian.157,158 This method allows the opening of an f–f gap.157 However, it suffers from the
existence of multiple minima in the calculations, so
the search for the real ground state is rather tricky
as the calculation is easily trapped in metastable
states.159

Hybrid functionals are another type of advanced
methods that are very often used nowadays in the
quantum chemistry community. Their principle is to
mix a part of Hartree–Fock exact exchange with a
DFT calculation; it has been applied to UO2 has been
made by Kudin et al.160 These methods are very
promising for solid-state nuclear materials. However,
the same problem of metastability as in LDAþU
exists for such hybrid functionals,161 and the computational load is much heavier than that in common or
LDAþU calculations. Recently, an alternative to
LDAþU has been proposed: the so-called local
hybrid functional for correlated electrons162 in
which the hybrid functional is applied only to the
problematic f electrons. An application on UO2 is
available.163


244

Ab Initio Electronic Structure Calculations for Nuclear Materials

1.08.5.2.2 Point defects

While UO2 comes out as a metal with LDA or GGA
DFT calculations, its structural properties are quite
well reproduced by these standard methods. Based
on this observation, some studies, using this standard
framework, have been published on point defects.164–166
The values obtained for the formation energies for the
composite defects (oxygen and uranium Frenkel pairs

and Schottky defect) compare well with experimental
estimates. However, as UO2 is predicted to be a metal
with such methods, it is impossible to consider the
charge state of the defects. More recent studies using
the þU correction have been published. Most of them
still focus on neutral defects.159,167–169 The discrepancies between the results obtained in these various
studies are larger than the spread usually observed
in ab initio calculations; for instance, the formation
energy of the oxygen Frenkel pair is found anywhere
between 2.6170 and 6.5 eV.159 This suggests some
hidden problem in the calculations, probably related
to the possible occurrence of metastable minima in
the calculations.
We are aware of only one study of the charge state of
point defects. This work,171 done within LDAþU,
predicts the following charge states: À4 charge for
uranium vacancy, À2 for oxygen interstitial, and
from þ2 to 0 for oxygen vacancy depending on the
position of the Fermi level. However, in this last
work, the formation energies of the composite defects
(Frenkel and Schottky) built from charged defects are
in not as good agreement with experiments as the ones
obtained with neutral defects. Together with the large
spread of values mentioned here, this underestimation
shows that the correct reproduction of point defects
in UO2 with ab initio seems not yet at hand.
Beyond the formation energy of isolated defects,
some studies focus on their migration. Gupta and

coworkers172 calculated the migration energy of the

oxygen vacancy (1.0 eV) and interstitial (1.1 eV). In
this last case, they found that the stable position
for a monointerstitial is a dumbbell configuration.
This point is refuted by others173 who calculated
the migration energy of oxygen mono- (0.81 eV)
and di-interstials (0.47 eV) and implemented this
information in a Kinetic Monte-Carlo model,
showing that the di-interstitial configuration, though
less abundant than the single interstitial, may play a
dominant role in oxygen diffusion in hyperstoichiometric oxide.
1.08.5.2.3 Oxygen clusters

Another point of interest beyond point defects is the
clustering of oxygen interstitials. Indeed, oxygen
interstitial clustering has been deduced from diffraction experiments174 many years ago. However, a
debate remains on the exact shape of such clusters.
Two configurations are contemplated: the so-called
Willis clusters174,175 or cubo-octahedral clusters that
have been observed by neutron diffraction in U4O9176
and U3O7.177 These clusters are made of 12 oxygen
and 8 uranium atoms and amount for 4 oxygen interstitials. An additional oxygen interstitial may reside
in the center of the cluster, forming a so-called filled
cube-octahedral cluster (with five interstitials).
Recent calculations have proved that Willis clusters are in fact unstable and transform upon relaxation into assemblies of three or four interstitials
surrounding a central vacancy cluster (Figure 13).178
The three interstitial–1 vacancy cluster has been
found independently by other authors173,180 who
refer to it as split di-interstitials. These clusters prove
in fact to have a formation energy higher than the
cube-octahedral cluster (Figure 14), especially the

filled one.178,179

O

O

OV
OЈ
U

U

OЈЈ

OV

OV

OЈЈ

(a)

(b)

Figure 13 Relaxation process of a Willis cluster of oxygen interstitials in UO2. Reproduced from Geng, H. Y.;
Chen, Y.; Kaneta, Y.; Kinoshita, M. Appl. Phys. Lett. 2008, 93.


Ab Initio Electronic Structure Calculations for Nuclear Materials

Figure 14 Cubo-octahedral cluster of oxygen interstitials
in UO2. Reprinted with permission from Geng, H. Y.; Chen,
Y.; Kaneta, Y.; Kinoshita, M. Phys. Rev. B 2008, 77,
180101(R).180 Copyright (2008) by the American Physical
Society.

1.08.5.2.4 Impurities

Lattice sites and solution energies of FP are of major
importance in fundamental studies of nuclear fuels,
see Chapter 2.20, Fission Product Chemistry in
Oxide Fuels. They pilot the dependence of the
behavior of FP on fuel stoichiometry and temperature as well as their possible release from the fuel in
the context of a direct storage of spent fuel. As experimental studies in this field are very difficult, ab initio
results are of great value.
In such studies, one considers the insertion of a
fission atom in interstitial or vacant sites of UO2.
A difficulty arises for the latter case.181 Indeed, one
then has to distinguish between the incorporation
energy, defined as the energy to incorporate the FP
in a preexisting vacancy site, and the solution energy,
which is the one relevant for full thermodynamical
equilibrium, in which the amount of available vacant
site is taken into account. One then adds to the incorporation energy the so-called apparent formation
energy, which is defined as the logarithm of the
vacancy concentration multiplied by the temperature.
Such apparent formation energies depend on the
stoichiometry of UO2þx. A positive (respectively, negative) solution energy then means that the FP is
insoluble (respectively, soluble) in UO2þx.
The first DFT study of the incorporation of a FP

in UO2 is the one by Petit et al.182 on krypton in the
late 1990s. It was performed within the LMTO-ASA
formalism, which could give only qualitative results.

245

Crocombette181 used more modern plane wave formalism to calculate the insertion of some FP (krypton, iodine, cesium, strontium, and helium) but
neglected atomic relaxation, which limits the accuracy of the results. Freyss et al.183 considered He
and Xe. All these calculations were performed with
standard LDA. More recent works always included
a þU correction. While the first papers dealt only
with interstitial and monovacancy sites, more recent
works may also consider divacancy or tri-vacancy sites
that often appear to be the most stable sites for FPs.
Many FPs have been recently considered. At the time
of writing one could find in the literature, beyond the
works already mentioned, calculations on helium,184
iodine,185 xenon,186 strontium,186 cesium,186–188 molybdenum,189 and zirconium.189 Yun et al.190 dealt with
helium and went beyond the solution energies as they
also considered migration and clustering energies.

1.08.6 Conclusion
It is hoped that the examples discussed above have
shown the tremendous interest of ab initio calculations for nuclear materials. Indeed, they allow the
qualitative and most of the time quantitative calculations of the basic energetic and kinetic properties that
have a major influence on the behavior of the materials at the atomic scale.
For metallic materials, the common theoretical
framework works quite well. One can thus nowadays
tackle objects of increasing complexity, for example,
assemblies of defects or dislocations. The main limit

for these materials is the severe restriction in possible
cell sizes.
Silicon carbide exemplifies the successes of ab initio
methods in modeling the properties of a band insulator of interest for the nuclear industry. However,
some difficulties remain, especially for what concerns
the correct prediction of CTL in these materials.
In actinide materials, the case of uranium oxide,
by far the most studied of the actinide compounds
of interest as a nuclear material, shows that a lot of
information can be obtained, for example, for the
solution energies of FP or the structure of oxygen
interstitial clusters. However, this information remains
only qualitative, due to the very complex electronic
structure of such actinide compounds with localized
f electrons. The solution for these difficulties with
insulating materials should come from the current
developments of hybrid functional or GW calculations, with the drawback that these advanced methods


246

Ab Initio Electronic Structure Calculations for Nuclear Materials

are at least one order of magnitude heavier than the
standard ones.
Ab initio methods have thus brought a lot of information for nuclear materials and will certainly continue to do so. Conversely, nuclear materials are a
very challenging field for the use of these ab initio
methods in many aspects: physical principles, numerical schemes, practical implementation, and so on.

Acknowledgments

The authors thank Drs Fabien Bruneval, Chu Chun
Fu, Guido Roma, and Lisa Ventelon for their valuable input.

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