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Physical Chemistry Chemical Physics
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COVER ARTICLEJenkins et al.Theory of gold on ceria
www.rsc.org/pccp Volume 13 | Number 1 | 7 January 2011 | Pages 1–348
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22 Phys. Chem. Chem. Phys., 2011, 13, 22–33 This journal is c the Owner Societies 2011
Cite this: Phys. Chem. Chem. Phys., 2011, 13, 22–33
Theory of gold on ceria
Changjun Zhang,aAngelos Michaelides
aand Stephen J. Jenkins*
b
Received 8th July 2010, Accepted 15th September 2010
DOI: 10.1039/c0cp01123a
The great promise of ceria-supported gold clusters as catalysts of the future for important
industrial processes, such as the water gas shift reaction, has prompted a flurry of activity aimed
at understanding the molecular-level details of their operation. Much of this activity has focused
on experimental and theoretical studies of the structure of perfect and defective ceria surfaces,
with and without gold clusters of various sizes. The complicated electronic structure of ceria,
particularly in its reduced form, means that at present it is highly challenging to carry out
accurate electronic structure simulations of such systems. To overcome the challenges, the
majority of recent theoretical studies have adopted a pragmatic and often controversial approach,
applying the so-called DFT + U technique. Here we will briefly discuss some recent studies of
Au on CeO2{111} that mainly use this methodology. We will show that considerable insight has
been obtained into these systems, particularly with regard to Au adsorbates and Au cluster
reactivity. We will also briefly discuss the need for improved electronic structure methods,
which would enable more rigorous and robust studies in the future.
1. Introductory remarks
Amongst the least reactive of all bulk metals, gold would, until
relatively recently, have seemed an odd choice of material to
investigate for its potential as a catalyst. Since the seminal
work of Haruta,1 however, it has become clear that gold
nanoparticles can show dramatic chemical activity in certain
contexts when suitably prepared, and this unexpected
observation has prompted concerted efforts both to explain
this phenomenon and to expand its range of application.2–19
Today, research into catalysis by gold is a thriving field,
driven not only by the economic attractions of the material
(historically one of the more abundant and cheaper of the
precious metals) but also by its highly desirable propensity to
catalyse reactions at relatively low temperatures. Various
theories have been put forward to explain the activity of
nanoparticulate gold, including those based on structural
a London Centre for Nanotechnology and Department of Chemistry,University College London, London WC1H 0AH
bDepartment of Chemistry, University of Cambridge, Lensfield Road,Cambridge, CB2 1EW, UK. E-mail: [email protected];Fax: +44 (0)1223 762829; Tel: +44 (0)1223 336502
Angelos Michaelides
Angelos Michaelides obtained aPhD in Theoretical Chemistry in2000 from Queen’s UniversityBelfast (supervised by Peijun Hu).He then moved to David King’sgroup in Cambridge and in 2002was awarded a junior researchfellowship from Gonville andCaius College. In 2004 he movedto the Fritz Haber Institute,Berlin, first as an Alexander vonHumboldt fellow and then later asa staff scientist, in MatthiasScheffler’s Theory Department.In 2006 he moved to UCL andbecame Professor in 2009.
Angelos’ current research (www.chem.ucl.ac.uk/ice) involvesthe application and development of electronic structure methodsto study catalytic and environmental interfaces.
Stephen J. Jenkins
Stephen Jenkins studied Theo-retical Physics at the Univer-sity of Exeter, completing hisPhD in 1995 (supervised by G.P.Srivastava and John Inkson).After a spell as a post-doc inExeter, he joined David King’sgroup in Cambridge in 1997.Awarded a Royal Society Univer-sity Research Fellowship in 2001,Stephen was appointed to a Uni-versity Lectureship in PhysicalChemistry in 2009, and succeedsDavid King as head of theCambridge Surface Chemistry
group. His current research (www-jenkins.ch.cam.ac.uk)includes the fundamentals of adsorption and catalysis, surfacestructure and chirality, and the surface and interface propertiesof spintronic materials.
PCCP Dynamic Article Links
www.rsc.org/pccp PERSPECTIVE
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This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13, 22–33 23
effects (the presence of step and kink sites), on electronic
effects (the presence of non-metallic and/or ionic sites), and
on support effects (either through the stabilisation of unusual
structural/electronic features, or through reactions occuring
across the particle/support interface itself). The reality may
well be that all such phenomena are important at some time or
another in varying proportions in different scenarios and for
different reactions.
One of the more exotic materials considered as a possible
support for nanoparticulate gold is cerium dioxide
(ceria, CeO2). A highly reducible oxide, ceria has found
applications as an oxygen buffer, as a catalyst, as a catalyst
support, and as a solid electrolyte.20 These uses typically rely
upon the easy creation and diffusion of oxygen vacancies within
the material. When the oxide is stoichiometric, the constituent
cerium atoms exist in the Ce4+ oxidation state, but in the
partially reduced form each missing lattice oxygen atom implies
the existence of two Ce3+ ions in sites close to (though not
necessarily immediately adjacent to21) the vacancy. The stoichio-
metric oxide is therefore characterised by a completely empty
f-band (located in an energy gap between the occupied O 2p
states and the unoccupied Ce 5d states) whilst the partially
reduced oxide features highly-localised partially-occupied
f orbitals split-off below the unoccupied f states (see Fig. 1).
A variety of experimental studies has provided evidence to
suggest that gold nanoparticles supported on ceria may be of
great interest from the catalytic viewpoint.22–34 These studies
in turn have prompted significant theoretical efforts to under-
stand the physical nature of the particle/support interaction in
this system, and its effect on catalytic activity.35–41 In this short
perspective article, we aim to provide a selective but
representative overview of progress in this direction over
recent years, highlighting not only the key areas in which
consensus appears to be emerging, but also what we believe to
be some of the most pertinent open questions that remain.
2. Theoretical approaches
Electronic structure simulations of nanoparticles on oxide
surfaces are challenging for a number of reasons. First, there
is the inevitable difficulty of describing finite clusters attached
to extended surfaces. This implies that the structures are
complex and system sizes rather large, and that neither
localised nor periodic basis functions are without significant
drawbacks. Second, and more challenging fundamentally, is
the question of identifying a suitable electronic structure
method with the requisite accuracy to describe the properties
of interest. Of the many electronic structure techniques that
have been developed, density functional theory (DFT) is the
most popular and robust theoretical approach currently
available for solving the electronic structures of solid surfaces.
However, DFT with standard exchange–correlation func-
tionals (e.g. generalised gradient functionals, such as PW9142
or PBE43) is far from a panacea and suffers from a number of
well-known deficiencies. The one most relevant to the present
topic of gold on ceria is an inability to correctly treat the
strongly correlated nature of the cerium f electrons.
The problem of strongly correlated electrons is easily
explained, though much less easily remedied. Essentially, one
should recall that the electron–electron interactions included
within standard DFT exchange–correlation functionals are
only mean-field approximations to what are, in reality, a set
of complex dynamical phenomena. In most situations, this
approach works remarkably well, allowing orbitals to be
calculated within an independent particle picture. In contrast,
however, when an orbital is very highly localised (as occurs, of
course, for f orbitals in the cerium atom) the Coulomb
repulsion is sufficient to provide a strong disincentive towards
instantaneous double occupancy of that orbital. That is, the
probability of a specific electron being found in such an orbital
at any given moment is strongly modified by whether another
electron is or is not already there at the time: the motion of
electrons then becomes ‘‘strongly correlated’’. Effects of this
nature cannot be captured by a pure mean-field approach,
since this assumes that electrons can be treated as independent
particles, so standard DFT exchange–correlation functionals
are doomed to failure; a classic example of such a situation is
to be found in NiO, where DFT incorrectly predicts a metallic
bandstructure and must be augmented in some more or less
sophisticated manner in order to achieve a physically reason-
able result.44
Although alternative explictly correlated electronic structure
methods for solids have recently begun to emerge, e.g. the
Random Phase Approximation45 and QuantumMonte Carlo,46
a pragmatic ad hoc correction known as the ‘‘Hubbard U’’ has
Fig. 1 Top: Schematic representations of the stoichiometric (left) and
reduced (right) CeO2{111} surface. White atoms are cerium, red atoms
oxygen, and the oxygen vacancy is circled. Bottom: Electronic struc-
ture of the stoichiometric CeO2{111} surface (left) and the reduced
surface (right) with occupied states shaded and unoccupied states
unshaded. The density of states for the reduced surface is shown
spin-resolved. The Fermi levels are set at zero and the units of the
energy (y) axis are in eV.
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24 Phys. Chem. Chem. Phys., 2011, 13, 22–33 This journal is c the Owner Societies 2011
much more commonly been applied in practical calculations.47
Its application is, however, not free from controversy, as it
includes within DFT an additional potential, given the symbol U,
which effectively penalises double occupancy of highly
localised orbitals. In doing so, one steps outside the bounds
of ‘‘pure’’ DFT, but experience shows that the results of
DFT + U calculations can be considered reliable, so long as
the value ofU is appropriately chosen.97 To further complicate
matters, reports in the literature often cite an effective
parameter, Ueff, defined as the difference between the spheri-
cally averaged value of U and the screened exchange inter-
action parameter, J. In addition, the Hubbard correction is
typically applied only to a subset of orbitals (i.e. the most
strongly localised ones) which in this case corresponds to the
Ce 4f electrons.
The issue of choosing the value of Ueff thus assumes
considerable importance in assessing calculations reported in
the literature for ceria. Values employed for cerium oxides
typically range between 3–6 eV. One should note that the
DFT+ U approach is generally implemented in the context of
either the local density approximation (LDA + U) or the
generalised gradient approximation (GGA + U);48,49 the
latter may itself be sub-divided according to the particular
flavour of GGA functional used, which in the present context
has tended to be of either the PW91 or the PBE variety. On the
basis of studies for bulk CeO2 and bulk Ce2O3,50–53 various
authors have favoured Ueff values ranging between 5–6 eV for
LDA + U and between 3–5 eV for GGA + U.21,53–58 How
well any of these conclusions may safely be carried over to
surface calculations, however, has proved to be a point of
significant contention, as will be seen below. Moreover, it has
been pointed out (e.g. ref. 52) that there probably exists no
single value of Ueff that is optimal for calculation of allmaterial
properties, which is to say that a value of Ueff that yields
excellent results for one property (e.g. lattice constant) may
yield poor results for another (e.g. band gap) and vice versa.
3. The nature of the clean ceria surface with and
without defects
The most stable surface of ceria is the CeO2{111} facet,57–60
and it is this that has been the subject of most study to date.
This surface exhibits three-fold rotational symmetry and may
be thought of in terms of stacked O–Ce–O trilayers (Fig. 1).
Removal of a single oxygen atom at the surface leaves three
undercoordinated cerium atoms to accommodate the two
resulting excess electrons (3Ce4+ - 2Ce3+ + Ce4+). An
early confirmation of the necessity to account for strong
correlation effects in any theoretical assault on this system
may, however, be gleaned from the fact that standard DFT
calculations incorrectly predict delocalisation of these excess
electrons across all three undercoordinated cerium atoms
(3Ce4+ - 3Ce3.33+). Correct localisation is obtained when
Ueff values of around 5 eV are employed.21,56–58,61,62 The
energy cost for removal of an oxygen atom (i.e. creation of a
surface oxygen vacancy) has been calculated by Zhang et al.63
using GGA + U with Ueff = 5 eV to be 2.23 eV; in contrast,
the creation of a cerium vacancy has been calculated in the
same work to require an energy of 4.67 eV.
Removal of a single subsurface oxygen atom leads to an
electronic structure almost indistinguishable from that of a
single surface oxygen vacancy.58,62,63 Their energy difference is
on the order of 0.1 eV, which is very small compared to
the vacancy formation energy of well over 2.0 eV. These
theoretical conclusions are in good agreement with the elegant
high-resolution scanning tunneling microscopy work of Esch
et al.,64 which showed that the two types of defect (i.e. surface
oxygen vacancy and subsurface oxygen vacancy) are present
with almost equivalent concentrations. In addition to the
single vacancies, Esch et al.64 further discovered that upon
prolonged annealing vacancy clusters appear, with linear
vacancy clusters being dominant and triangular vacancy
clusters being the next most abundant. Interestingly, those
vacancy clusters expose exclusively Ce3+ ions. Because
electrons left behind by the released oxygen atoms are
localised at adjacent Ce ions, these observations suggest that
electron localisation plays an important role in shaping the
vacancy clusters.64,65
In order to address this issue, Zhang et al.62 carried out a
systematic DFT + U study on a series of vacancy clusters.
A correlation between the coordination number of each Ce3+
ion and the energy of its f states was identified: specifically, the
higher the coordination number, the less stable the Ce3+ ion
would be. Because the Ce ions neighboring the vacancies have
lower coordination numbers than those that are further away
from the vacancies, this correlation explains well the electronic
features of the vacancy clusters. However, Zhang et al.62
predicted that a triangular surface vacancy cluster is actually
more stable than the linear vacancy clusters, which disagrees
with the experimental observations. Conesa66 also used
DFT + U to study the defects containing two and three
vacancies. Whilst the author identified a similar correlation
between the coordination number of Ce3+ ions and their
energies, he also found the electronic configuration with the
lowest energy is not that which contains the Ce3+ ions located
as immediate neighbors to the vacancies, although the energy
differences between them are small (less than 0.1 eV per
vacancy). Because the reduction of Ce ions leads to displace-
ments of neighboring O atoms, such electronic configurations
as predicted in Conesa’s work would produce different O
structural features than are observed in the STM images.
These apparent disagreements suggest a need for further
experiments and calculations on vacancy clusters. Another
important issue deserving of further study is the mobility of
oxygen vacancies. Experimentally, whilst Namai et al.
reported the defect mobility at room temperature,67,68
Esch et al.64 suggested that the diffusion of oxygen vacancies
requires temperature higher than 400 1C. In theory, energy
barriers for vacancy diffusion have been computed with
interatomic potential approaches69 and DFT methods,70,71
and the obtained values vary in the range of B0.5–1.1 eV.
Nolan et al.72 employed the DFT+ Umethod to calculate the
diffusion for a single vacancy in a (2�2�2) unit cell (CeO1.96)
and obtained an energy barrier of 0.53 eV. Substituting the
calculated barrier in the classical transition-state theory
expression yields a rate of less than 10�8 ps�1 at 300 K (assuming
a standard pre-exponential factor of 1013), which suggests very
low mobility for the vacancy at room temperature.
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This journal is c the Owner Societies 2011 Phys. Chem. Chem. Phys., 2011, 13, 22–33 25
4. Gold adatoms on stoichiometric ceria surfaces
The first theoretical calculation to address the adsorption of
gold on ceria was that of Liu et al.,35 in which single adatoms
were considered on both the stoichiometric and the partially
reduced CeO2{111} surface. Significantly, the adsorption
energy was found to be rather larger when the adatom
occupied an oxygen vacancy site (1.86 eV) than when it sat
atop an oxygen atom on the ideal surface (1.26 eV) pointing
towards a possible role for such vacancies as nucleation sites
for larger clusters. Furthermore, these calculations also
indicated that the charge state of gold adatoms could be
profoundly altered by their local environment. On the
stoichiometric surface, the calculations indicated that the 6s
orbital of the gold atom became depopulated in favour of an
adjacent cerium atom upon adsorption (Ce4+ - Ce3+); the
adatom itself acquired a net positive charge of +0.35|e|98
(see Fig. 2) and was found to retain no net spin, in sharp
contrast to an isolated gold atom for which the 6s orbital hosts
a single unpaired spin.
It is worth mentioning that these early calculations for gold
on ceria were not carried out using the DFT + U technique
that underlies more recent work. Neither, however, were they
carried out without due regard to the issue of strong correla-
tion. In fact, an alternative method of achieving localisation of
f electrons was employed, via a technical adjustment to the
self-consistency procedure adopted during iterative solution of
the Kohn–Sham equations of DFT, perhaps best described as
‘‘pruned density mixing’’.99 Nevertheless, it is certainly fair to
say that the ‘‘pruned density mixing’’ technique is not trivial to
apply, and that careful adjustment of parameters was required
to achieve convergence in its one application to date.35 The
DFT + U method suffers from the ambiguity noted above,
regarding the correct choice for Ueff, but does benefit from
being by now rather standard, implying both that DFT+ U is
readily available as an option in many modern DFT codes, but
also that calculations performed using this method are more
straightforwardly comparable with others in the literature
obtained by similar means; convergence is also routinely
achieved without the extensive optimisation efforts necessary
in application of the ‘‘pruned density mixing’’ approach. For
those reasons alone, the present authors have adopted the
GGA + U methodology in all of their joint publications on
this system.
Our recent GGA + U (Ueff = 5 eV) results55 for individual
gold adatoms on the CeO2{111} surface are, indeed, in
qualitative agreement with the earlier work, albeit the adatom
charge (+0.17|e|) and adsorption energy (0.96 eV) for the atop
site are both rather smaller than before. Upon investigating
alternative adsorption sites, however, we discovered that a
position bridging between two oxygen atoms was, in fact,
preferred (adsorption energy 1.17 eV, adatom charge
+0.32|e|); interestingly, the unpaired spin resulting from
depopulation of the gold 6s orbital was found to be located
not on the nearest cerium atom, but instead upon the third
nearest55 (see Fig. 3). A very similar configuration (both in
terms of energy and electronic structure) was reported
subsequently by Hernandez et al.,73 using a near-identical
computational set-up, although they additionally point out
the existence of multiple, slightly less stable, electronic local
minima associated with some of the adsorption sites they
considered. Camellone and Fabris74 also concur with the same
basic picture, using Ueff = 4.5 eV.
Castellani et al.,75 however, adopt a mixed methodology
whereby LDA + U (Ueff = 5 eV) is used to obtain optimised
geometries, and then a single-point GGA + U (Ueff = 3 eV)
calculation is performed to obtain energies, charges and spin
moments. These choices are motivated by evidence from bulk
ceria compiled by Loschen et al.,51 but in light of other work
on the clean reduced surface57,62 it is questionable whether the
relatively low value of Ueff used in the GGA + U calculations
will correctly reproduce the localisation of charge on cerium f
orbitals. Certainly, the results for gold adsorption presented
by Castellani et al.75 are in marked disagreement with those of
Fig. 2 Electronic structure of a single Au adatom on the stoichio-
metric CeO2{111} surface (atop site). In (a) the top panel shows the
density of states (DOS) for the clean surface, similar to the bottom left
panel of Fig. 1. The lower panel in (a) shows the spin-polarised DOS
that arises from adsorption of Au in an atop site, highlighting the
existence of a single occupied f-orbital (occ-f). In (b) the blue isosurface
shows net spin associated with this occupied f-orbital, localised on one
of the Ce ions neighbouring the Au adatom. Reprinted with permission
from Liu et al.35 Copyright (2005), American Physical Society.
Fig. 3 Electronic structure of a single Au adatom on the stoichio-
metric CeO2{111} surface (bridge site). In the left panel, the total
density of states (DOS) and atom-resolved projected density of states
(PDOS) of the Au adsorbed on the bridge site, the latter of which are
rescaled so that characteristic features can be seen more clearly. The
Ce and O atoms in the PDOS refer to the reduced Ce (i.e. Ce3+) and
the O between Ce3+ and the Au atom, respectively. In the right panel
is depicted the three-dimensional isosurface plot of the spin
density (blue areas) of the system. Reprinted with permission from
Zhang et al.55 Copyright (2008), American Chemical Society.
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26 Phys. Chem. Chem. Phys., 2011, 13, 22–33 This journal is c the Owner Societies 2011
Zhang et al.,55 Hernandez et al.73 and Camellone and Fabris,74
both regarding the location and the charge state of the
adatom; they conclude that it sits atop an oxygen atom and
remains essentially neutral. Use of the same mixed methodo-
logy by Branda et al.76 yields a different preferred adsorption
site for Au, bridging between an O and a Ce atom, but again
indicates a neutral charge state; results in line with previous
work55,73,74 are obtained when a purely GGA+U (Ueff = 5 eV)
approach is adopted.76 Our view is that the low Ueff values
suggested by Loschen et al.51 are intended to provide a
balanced description of both CeO2 and Ce2O3, whereas a
better representation of CeO2 alone may be achieved with
higher Ueff values. Furthermore, the lower lattice constant
achieved through use of LDA + U in the structural calcula-
tion would, we believe, tend to favour unphysical delocalisa-
tion of charge between neighbouring cerium atoms, owing to
their closer proximity. Alternatively, one might say that whilst
charge localisation is most strongly controlled by the choice of
Ueff, an important secondary effect arises due to the influence
of strain (relative to the prevailing calculated lattice constant)
on the stability of the large Ce3+ cation versus the smaller
Ce4+ cation (i.e. compressive strain disfavours the reduction
of individual ions, Ce4+ - Ce3+, implied by localisation).77 It
should nevertheless be granted, of course, that the lower lattice
constant obtained with LDA + U lies closer to the experi-
mental value than does that calculated with GGA + U.
A valuable analysis has now been provided by Branda
et al.,78 addressing precisely the subtle interplay between
lattice constant and the chosen Ueff value in determining
the charge transfer between adatom and surface. They
demonstrate that, for gold adsorbed atop an oxygen atom,
GGA + U (Ueff = 3 eV) consistently predicts zero or minimal
charge transfer at lattice constants in the range 5.35–5.55 eV,
whereas GGA + U (Ueff = 5 eV) consistently predicts full
charge transfer at lattice constants above about 5.40 eV
(and very little charge transfer at lattice constants below). If
one accepts the larger Ueff value and any reasonable lattice
constant, the unequivocal conclusion is that significant charge
transfer occurs. If, on the other hand, one prefers the smaller
Ueff value, the result is a less clearcut preference for a lack of
charge transfer, since metastable solutions with some degree of
charge transfer lie within a few tens of meV from the ground
state. Branda et al.78 avoid drawing a firm conclusion over
which picture is correct, but we feel it is important to note that
their results obtained at smaller (essentially experimental)
lattice constants reveal an unphysical (albeit small) delocalisa-
tion of charge amongst cerium f orbitals, whereas those
obtained at larger lattice constants do not.
Out of interest, we have now performed GGA + U
(Ueff = 3 eV) and GGA + U (Ueff = 5 eV) calculations for
the clean reduced CeO2{111} surface, holding the lattice
constant fixed at the LDA + U (Ueff = 5 eV) value reported
by Castellani et al.75 (i.e. 5.39 A). In the former case, we find
unphysical delocalisation of charge amongst three cerium
atoms, rather than the physically correct solution showing
localisation on just two. Only with the larger Ueff value do we
obtain properly localised solutions, and even then only when
the initial guess for the electron distribution is carefully
chosen. It seems clear, therefore, that use of the lower lattice
constant (5.39 A) produces unphysical results unless the higher
Ueff value is used for the GGA+ U calculations, and risks error
even then. Since we and others have also previously reported57,62
that GGA+U (Ueff = 3 eV) produces similarly spurious results
for oxygen vacancies even at larger lattice constants, it seems to
us on balance that the most consistent choice is to use GGA+U
(Ueff E 5 eV) at its own theoretically obtained (albeit somewhat
too large) lattice constant. Such a choice for CeO2{111}/Au
leads to adsorption at the bridge site, with full charge transfer
from the gold adatom to a single cerium atom, as described by
Zhang et al.,55 Hernandez et al.,73 Camellone and Fabris,74 and
Branda et al.76
As a final comment on the subject of adatom adsorption on
the stoichiometric surface, we note that Branda et al.78 report
the results of hybrid functional (HSE79) calculations
(as opposed to DFT + U) which favour a solution with no
charge transfer over a second solution with strong charge
transfer, but only by a very small margin. Given the uncer-
tainties inherent in hybrid functional DFT, its application on
ceria systems is not free of problems. Indeed, as Da Silva
et al.80 and Kullgren et al.81 have found, whilst hybrid
functionals such as PBE0,82 HSE79 or B3LYP83 give a reason-
ably good prediction on bulk CeO2 and Ce2O3 systems, the
overall description of the electronic properties (particularly band
gaps) can actually be substantially worse than obtained when
using LDA + U or GGA + U with adjustable U parameters.
5. Gold adatoms on defective ceria surfaces
Notwithstanding the complexities of gold adsorption on stoichio-
metric surfaces, it is highly likely that adsorption at defect sites
will play a major (possibly even dominant) role under
Table 1 Calculated adsorption energies for a single gold atom on thestoichiometric (Stoich.) and defective (O-Vac. and Ce-Vac.)CeO2{111} surfaces. The ‘‘PW91+’’ calculations employed the‘‘pruned density mixing’’ technique. For the various DFT + Ucalculations, we adopt the notation of Branda et al.,78 whereby‘‘Ln’’ indicates LDA + U with Ueff = n eV, and ‘‘Gm’’ indicatesGGA+U withUeff =m eV. All the GGA-based calculations used thePW91 functional, except for those reported by Camellone & Fabris74
which employed the PBE functional. The notation ‘‘L5-G3’’ impliesthat geometry optimisation was done wholly within the L5 approach(and at the L5 lattice constant) followed by a single-point calculationfor adsorption energy and electronic structure carried out within theG3 approach. For the stoichiometric surface, results marked with asuperscript dagger correspond to adsorption atop an O atom, thosemarked with a double-dagger are for adsorption in a Ce–O bridge site,and the remaining plain entries are for adsorption in the O–O bridgesite. In all cases at the O-Vac. and Ce-Vac. surfaces, the adatom bondsdirectly into the vacancy site.
Functional Stoich. O-Vac. Ce-Vac.
Liu et al.35 PW91+ 1.26w 1.86 —Chen et al.84 G5 0.88w 2.58 5.65Branda et al.78 G5 0.96w — —Zhang et al.55 G5 1.17 2.75 5.94Hernandez et al.73 G5 1.15 2.41 —Camellone & Fabris74 G4.5 1.18 2.29 5.68Branda et al.76 G5 1.15 — —Branda et al.76 L5 - G3 0.71ww — —Castellani et al.75 L5 - G3 0.66w — —Branda et al.78 L5 - G3 0.61w — —Branda et al.78 G5 0.96w — —
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catalytically relevant conditions. Certainly the adsorption
energy associated with depositing single gold adatoms into
either cerium or oxygen vacancy sites is agreed to be rather
greater than for the stoichiometric surface. Various calcula-
tions have, for instance, indicated a preference for adsorption
into an oxygen vacancy site relative to the stoichiometric
surface (see Table 1) the most reliable results being, we believe,
those presented recently by Zhang et al.55 (1.58 eV preference),
Hernandez et al.73 (1.26 eV preference) and Camellone and
Fabris74 (1.11 eV preference). The work of Chen et al.84
appears accurately to represent the structure and energetics
of the gold adatom in the oxygen vacancy site, but overstates
the relative preference for that site because the most stable site
for adsorption on the stoichiometric surface is not considered
in that work. The adsorption energy of 1.86 eV reported
by Liu et al.35 for the oxygen vacancy site now looks
conspicuously low in relation to these four GGA + U
calculations. All works that address explicitly the calculated
electronic structure,55,73,74 however, agree with the original
finding of Liu et al.35 that the direction of charge transfer is
reversed relative to adsorption on the stoichiometric surface;
one of the two reduced cerium atoms associated with the oxygen
vacancy donates an electron (2Ce3++Ce4+ - Ce3+ + 2Ce4+)
to the adatom, which gains a substantial net negative charge
in the vicinity of �0.60|e| due to filling of the 6s orbital
(see Table 2 and Fig. 4).
An alternative scenario involves the adsorption of gold into
cerium vacancies, the energetics of which were first considered
by Hu and co-workers84,85 who report an extremely high
adsorption energy of 5.65 eV in this site. Subsequent work
by Zhang et al.55 confirms a similarly high adsorption energy
(5.94 eV) and adds the observation that the substitutional gold
atom gains a very large positive charge of +1.23|e|. Camellone
and Fabris74 report an adsorption energy of 5.68 eV for such
a site. These high adsorption energies do not, however,
necessarily imply that cerium vacancies will dominate the
adsorption properties of gold, since the creation of a cerium
vacancy itself requires significant energy.
In order to address this issue, Zhang et al.63 extended their
work to include a first-principles thermodynamic analysis of
gold adsorption, allowing both oxygen and cerium vacancies
to compete with the stoichiometric surface for gold adatoms.
As mentioned in the section dealing with clean surfaces above,
the formation energy for a single cerium vacancy (4.67 eV) far
exceeds that of a single oxygen vacancy (2.23 eV) when
calculated within the GGA+ U (Ueff = 5 eV) methodology.63
Application of thermodynamic arguments then leads to the
conclusion that the free energy of an oxygen vacancy is much
less than that of a cerium vacancy, under conditions similar to
those found in, for example, the water gas shift (WGS)
reaction (400–800 K, with partial pressure ranging from
1 kPa to 100 kPa). Indeed, the free energy of the oxygen
vacancy is comparable with that of the stoichiometric surface
in the relevant range, whereas the cerium vacancy is unstable
by a margin of around 8 eV. Even composite defects including
both cerium and oxygen vacancies together are found to be
unstable by more than 2 eV. Accordingly, one expects that
oxygen vacancies will be abundant under these conditions,
whereas cerium vacancies ought to be extremely rare. The
inclusion of gold within the model makes only a quantitative
difference to the outcome under WGS conditions.63 Coupling
the creation of a cerium vacancy with gold adsorption into the
vacant site is strongly disfavoured, from a free energy
perspective, by around 7 eV relative to adsorption of gold
on the stoichiometric surface. And once again, composite
defects involving both cerium and oxygen vacancies together
are also disfavoured. Adsorption of gold into oxygen
vacancies is, however, still strongly favoured over adsorption
onto the stoichiometric surface, even when the cost of creating
the oxygen vacancy is included within the free energy.
In a rare investigation of the less stable {110} and {100}
surfaces of CeO2, Nolan et al.86 considered the effect of
substitutional gold on the formation of oxygen vacancies.
They found that occupancy of a Ce site with Au rendered
nearby lattice O thermodynamically unstable, implying that
substitutional gold would almost inevitably be accompanied
Table 2 Calculated charge states for a single gold atom adsorbed onthe stoichiometric (Stoich.) and defective (O-Vac. and Ce-Vac.)CeO2{111} surfaces. The key to the ‘‘Functional’’ column is explainedin the caption to Table 1. For the stoichiometric surface, resultsmarked with a superscript dagger correspond to adsorption atop anO atom, those marked with a double-dagger are for adsorption in aCe–O bridge site, and the remaining plain entries are for adsorption inthe O–O bridge site. In all cases at the O-Vac. and Ce-Vac. surfaces,the adatom bonds directly into the vacancy site
Functional Stoich. O-Vac. Ce-Vac.
Liu et al.35 PW91+ +0.35w �0.58 —Branda et al.78 G5 +0.32w — —Zhang et al.55 G5 +0.32 �0.62 +1.23Hernandez et al.73 G5 +0.34 �0.6 —Branda et al.76 G5 +0.33 — —Branda et al.76 L5 -G3 �0.06ww — —Castellani et al.75 L5 - G3 �0.02w — —Branda et al.78 L5 -G3 +0.05w — —Branda et al.78 G5 +0.32w — —
Fig. 4 Electronic structure of a single Au adatom adsorbed into an O
vacancy on the CeO2{111} surface. In (a) the left panel shows
schematically the structure and the formal charges of atoms surrounding
the bare vacancy, while the right panel shows the density of states
(DOS) projected onto the f-orbitals of the two Ce3+ ions. In (b) the left
panel shows the structure and formal charges after adsorption of Au
into the vacancy site, while the right panel shows the DOS projected
onto the f-orbitals of the single remaining Ce3+ ion and the s- and
d-orbitals of the Au adatom. Reprinted with permission from
Hernandez et al.73 Copyright (2009), Royal Society of Chemistry.
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by an oxygen vacancy under near-equilibrium conditions. In
effect, therefore, these results indicate that substitution of an
isolated Au atom into a Ce vacancy is unstable relative to
adsorption of Au into a composite defect comprising both Ce
and O vacancy sites. Furthermore, the cost of creating the Ce
vacancy in the first place was not included in the analysis, so
the likely concentration of substitutional Au defects cannot be
directly addressed. We anticipate that, similar to the {111}
surface, such defects will be rare under conditions relevant to
WGS catalysis. Heavily oxygen-rich conditions would, of
course, tend to favour cerium vacancies over oxygen vacan-
cies, so under such circumstances it may be that substitutional
gold becomes important.
In the case of Au substituting Ce, there is another dimension
of complexity worthy of our attention. Substitution of Au,
with a lower valency (Au+ or Au3+), on a Ce4+ site could
result in an electronic hole localized on the neighbouring
oxygen ion(s). This phenomenon, also referred to as polaronic
localization, is quite common in many metal oxides87: despite
the existence of several equivalent oxygen sites surrounding a
metal ion defect in oxides, the hole-lattice coupling often
favours hole localization at a single oxygen, and thus breaks
the space group symmetry. This poses a challenge in theoretical
treatments, as standard DFT often fails to predict the correct
hole localization and the associated lattice distortion.88–90 The
origin of this problem has been tracked back to the inability of
standard DFT to cancel the electron self-interaction. There
have been some recent theoretical efforts in tackling this
problem, among which the DFT + U approach with the
Hubbard correction applied to the oxygen p orbitals has
shown some promising improvement.91–94 As the current
DFT + U studies of ceria-based systems often treat only the
Ce f orbitals with the Hubbard correction, it would be inter-
esting to see the effects on electronic structure, ionic structure
and energetics when the correction also applies on the oxygen
p orbitals.
Another category of surface defect worthy of serious
consideration is to be found in the form of step sites, but
these have been the subject of comparatively little study to
date. To our knowledge, only Castellani et al.75 have investi-
gated the binding of Au adatoms at step sites, working with a
model structure incorporating both convex and concave step
morphology. In most of the adsorption sites considered, these
authors find adsorption energies in the range 0.52–0.69 eV,
with a trivial degree of charge transfer between surface and
adsorbate. Although such results would seem distinctive
relative to the findings on planar surfaces reported by the
majority of recent works35,55,73,74 it should, however, be noted
that the particular methodology employed by Castellani et al.75
does in fact yield similarly weakly bound and neutral adsorp-
tion for the planar case. Indeed, the clearest message that may
be understood from this work is that Au adsorption in most of
the step-related adsorption sites is remarkably unperturbed
relative to the planar surface.75 We might therefore expect that
calculations performed using methodologies that provide
stronger binding and greater charge transfer on the planar
surface are likely also to do so on the stepped surface;
unfortunately such calculations have not, to our knowledge,
been performed to date. In just one adsorption site, involving
the concave step, is a significantly enhanced adsorption energy
of 2.36 eV reported by Castellani et al.,75 accompanied by a
charge transfer that leaves the Au adatom with a significant
positive charge of +0.32|e|.
6. Gold clusters on ceria
Although a great deal of attention has been lavished on the
adsorption of single gold adatoms on ceria surfaces
(see above) it is of course essential to ask whether these
isolated entities are likely to be plentiful under reaction
conditions. In general, we should think in terms of gold
clusters, whose size distribution will depend critically upon
both thermodynamic and kinetic aspects of the surface
preparation. Since the adsorption energies reported for gold
on the stoichiometric and reduced surfaces of ceria fall well
below the calculated cohesive energy of bulk gold, it is expected
that high temperatures will allow sintering, ultimately resulting
in relatively large gold particles. Low temperature preparation,
in contrast, may kinetically stabilise small particles, and this
may be further accentuated by low gold loading.
The earliest calculations to address gold clusters on ceria
were those of Liu et al.35 on CeO2{111}, where up to four
adatoms were considered (i.e. Au1, Au2, Au3 and Au4 clusters).
The first of these adatoms was assumed to occupy an oxygen
vacancy site, and the others to progressively occupy adjacent
sites lying nearly atop oxygen atoms. The adsorption energy
for each additional gold atom was in the range 1.84–2.45 eV,
which was noted to be much higher than the 1.26 eV calculated
in the same work for adsorption on the stoichiometric surface.
It was therefore concluded that, once the number of adsorbed
gold atoms exceeded the number of oxygen vacancies, clusters
would grow preferentially around these gold-filled vacancies,
rather than nucleating elsewhere on patches of stoichiometric
surface. Moreover, although the first gold atom in each cluster
was found to be negatively charged, consistent with its occu-
pation of an oxygen vacancy site, the additional gold atoms all
achieved a net positive charge, in the range +0.10–0.19|e|,
somewhat less than that associated with an isolated gold
adatom on the stoichiometric surface (+0.35|e|).
Sporadic attention has been paid to cluster morphology on
CeO2{111} in subsequent GGA+ U calculations, for example
a Au2 cluster anchored to an oxygen vacancy site was
considered by Camellone et al.74 (Ueff = 4.5 eV) and a Au3cluster anchored to a cerium vacancy site studied by
Chen et al.95 (Ueff = 5 eV). These latter authors also reported
calculations on a Au10 cluster, constructed by depositing a
cluster of assumed shape onto the stoichiometric surface,
performing molecular dynamics as a form of simulated
annealing, and then re-optimising; the stability of this cluster
against others of differing size and general shape was, how-
ever, not discussed.95
Gold clusters of up to 11 atoms were studied more system-
atically in very recent work by Zhang et al.96 based on the
GGA+ U (Ueff = 5 eV) methodology. Assuming the particles
to be anchored at a single oxygen vacancy, they built up a
sequence of clusters, one gold atom at a time, optimising each
structure before adding another atom and calculating the next
(see Fig. 5). Molecular dynamics calculations were also
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performed for some clusters, to test stability against un-
expected rearrangements. Both two- and three-dimensional
clusters were considered, and the latter was found to be
energetically favoured in general, with the exception of the
Au7 case. For the larger clusters, Au10 and Au11, it is possible
to discern the beginnings of face-centred cubic (fcc) and
hexagonal close-packed (hcp) stacking patterns, and it is
notable that these have very similar energies; the preference
for fcc stacking in bulk gold must emerge at rather larger
cluster sizes. This observation raises the fascinating prospect
that the smallest gold clusters (we speculate that this means
those having fewer than about 35 atoms) may exhibit fcc/hcp
polymorphism. This in turn may have catalytic consequences,
in terms of the number and type of active step and kink sites
exposed by the clusters. We believe that state-of-the-art
scanning transmission electron microscopy may now be
approaching sufficient resolution to confirm or refute these
intriguing structural predictions.
In addition to the structural issues raised by these cluster
studies, the work of Zhang et al.96 also confirmed that unusual
electronic properties were retained even for the largest clusters
considered. Specifically, although the net negative charge
residing on the gold atom sitting directly in the vacancy site
was found typically to be lessened by the addition of more
gold neighbours, the net positive charge found on other gold
atoms in direct contact with the oxide remained high. In the
hcp-stacked Au11 cluster, for example, the gold atom located
directly on top of the central oxygen vacancy retained a net
negative charge of�0.03|e| (cf.�0.62|e| for a single gold atom),
while six other gold atoms in the first layer of the cluster
displayed net positive charges averaging +0.17|e| per atom;
three further gold atoms in the second layer of the cluster
showed a net negative charge averaging �0.03|e|, and the
single gold atom in the third layer of the cluster had a net
negative charge of �0.12|e|. These results strongly suggest thatthe gold layer in immediate contact with the ceria surface will
display a significant positive charge, even for much larger clusters.
7. Reaction mechanisms
The first theoretical discussion of a reaction occurring on a
gold particle at a ceria surface addressed the WGS process
(i.e. CO + H2O - CO2 + H2) on a planar Au4 cluster
anchored at an oxygen vacancy on CeO2{111}.35 The first step
in the proposed mechanism involved adsorption of CO onto
one of the positively charged Au atoms clustered around the
negative one that sits in the vacancy site itself, with an
adsorption energy of 1.22 eV.35 This immediately highlights
one of the most important consequences of the Au charge
state, namely that it strongly affects adsorption energies.
Indeed, Liu et al.35 reported that CO adsorption energies on
single adatoms ranged from 0.09 eV in the case of negatively
charged Au located in an oxygen vacancy site, through to
2.37 eV in the case of positively charged Au located atop an
oxygen atom on the stoichiometric surface. The lesser positive
charges found for Au atoms in the basal layer of clusters are thus
likely to provide moderate adsorption energies, of which the
value quoted above for the Au4 cluster is probably fairly typical.
After adsorption of CO, the next stage in the mechanism
proposed by Liu et al.35 is partially dissociative adsorption of
H2O, calculated to be almost thermoneutral, and with an
activation barrier of 0.59 eV (see Fig. 6). The resulting OH
moiety can then react with CO to form COOH, via a transition
state activated by just 0.10 eV, releasing 0.37 eV at this step.
Fig. 5 Au clusters anchored to an O vacancy on CeO2{111}. Planar
two-dimensional clusters are denoted with a superscript (a). Clusters
forming a series leading eventually to hcp stacking are given a super-
scipt (b), and those forming a series leading eventually to fcc stacking
are highlighted with a superscript (c). Reprinted with permission from
Zhang et al.96 Copyright (2009), American Chemical Society.
Fig. 6 Reaction mechanism for the WGS reaction at a model Au4cluster anchored to an O vacancy on CeO2{111}. After initial adsorp-
tion of CO, subsequent steps involve dissociative adsorption of H2O,
reaction of the resulting OHmoiety with CO to form COOH, loss of H
from COOH to form gas-phase CO2, and finally recombinative
desorption of H2. Reprinted with permission from Liu et al.35
Copyright (2005), American Physical Society.
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A barrier of 1.08 eV must then be overcome in order to remove
H from COOH onto the cluster, in an almost thermoneutral
step that releases CO2 into the gas phase. And finally,
recombination of H adatoms to give gas-phase H2 requires
input of 0.90 eV. Since the adsorption of an H2O molecule
onto the cluster involves only a very small binding energy, it is
far from inevitable that there will a reasonable probability of
each one encountering a CO molecule on the same cluster
during its short adsorbed lifetime; clearly this will depend
sensitively upon the lifetime of adsorbed CO at the ambient
temperature. The observation that the adsorption energy of
CO (1.22 eV) exceeds the highest reaction barrier (1.08 eV) is
therefore highly relevant, implying that a favourable state of
affairs can be achieved through selection of an appropriate
temperature for the reaction, without recourse to prohibitive
pressures. The efficacy of gold clusters on ceria for the WGS
reaction is therefore explicitly linked to the unusual charge
state of its constituent atoms.35
More recent work, by Camellone and Fabris,74 confirms
through the use of GGA + U (Ueff = 4.5 eV) the qualitative
conclusions asserted by Liu et al.35 via ‘‘pruned density mixing’’
for the dependence of CO adsorption energy on the Au charge
state. Specifically, they report that adsorption of CO onto a
single Au adatom initially located in its favoured bridge site on
the stoichiometric CeO2{111} surface results in a spontaneous
relocation of the adatom into a neighbouring site atop an
oxygen atom, with an adsorption energy of 2.48 eV. On a Au2cluster, anchored at an oxygen vacancy site, the preferred
adsorption geometry features CO bound to the Au atom
furthest from the vacancy, and with a somewhat lower
adsorption energy of 1.60 eV, consistent with the assumption
that the Au atom in question has a lower positive charge than
that of a single adatom on the stoichiometric surface. The
negatively charged single Au adatoms found anchored into
oxygen vacancies are again reported as essentially inert
towards CO adsorption.74
Alternative pathways for the WGS reaction on CeO2{111}
have been investigated by Chen et al.,95 working with both a
Au3 cluster anchored at a cerium vacancy and with a Au10cluster located on the stoichiometric surface (GGA + U,
Ueff = 5 eV). They consider first a ‘‘redox’’ mechanism, in
which H2O dissociates stepwise at an oxygen vacancy site
resulting finally in filling of the vacancy with the O atom, and
creation of an H2 molecule on the cluster. The H2 molecule can
then desorb from the cluster, whilst CO is proposed to be
oxidised by an O atom from the lattice, thus regenerating an
oxygen vacancy adjacent to the cluster. Details of the CO
oxidation step itself are not reported, as the authors focus
instead upon the OH dissociation step, which they believe to
be kinetically limiting. To avoid the necessity for OH dissociation
involving only the oxide substrate or the gold cluster,
Chen et al.95 secondly consider a ‘‘formate’’ mechanism, in
which CO adsorbed on the cluster first abstracts H from OH
adsorbed at an oxide oxygen vacancy, then reacts with the
remaining O atom. The result is once again to fill the vacancy,
but this time by creating a formate moiety adsorbed adjacent
to the cluster; this in turn can decompose via various routes to
produce gas-phase CO2 and H2. The calculated activation
energy of formate production is, however, even higher than
those calculated for OH dissociation in the ‘‘redox’’ pathway.
The authors conclude that both pathways are limited by the
need to refill oxygen vacancies, since their view is that the
substrate is the proximal source of oxygen in the CO oxidation
step.95 This is a fundamental difference from the pathway
proposed by Liu et al.35 in which OH adsorbed on the cluster
supplied the necessary oxygen atom (without itself dissociating
prior to CO oxidation) and the substrate was not directly
involved other than to induce a positive charge state on the
gold cluster.
Camellone and Fabris,74 meanwhile, have proposed a
detailed mechanism for CO oxidation at a single Au adatom
on the stoichiometric surface (see Fig. 7). Such a mechanism
could constitute the CO oxidation step within the ‘‘redox’’
WGS pathway proposed by Chen et al.,95 or could indeed
stand alone in a process of CO oxidation by molecular O2.
After initial adsorption of CO (exothermic by 2.48 eV) the
molecule rotates towards the surface in a ‘‘spillover’’ step,
surmounting an activation barrier of 0.86 eV and achieving
eventually a metastable state in which the molecule bonds not
only to the Au adatom but also to one of the lattice oxygen
atoms. The conversion to this metastable state is endothermic
by 0.24 eV, and subsequent extraction of the lattice oxygen
atom to form desorbing CO2 requires an activation energy of
only another 0.24 eV. The Au adatom left behind by the
departing CO2 molecule migrates into the nascent oxygen
vacancy during this last step. As a means of CO oxidation,
therefore, this process is inherently self-limiting, since further
CO cannot adsorb on Au located in such a site. The authors
suggest, however, that clusters of Au atoms might allow
spillover to occur whilst being relatively more resistant to
deactivation.74 Such a situation might, for instance, allow
sufficient time for the vacancy to be refilled, either directly
by dissociation of molecular O2 in some manner, or via the
adsorption and dissociation of H2O as part of the ‘‘redox’’ or
‘‘formate’’ pathways of the WGS reaction.95
In addition, Camellone and Fabris74 report on a highly
exothermic (by 3.24 eV) and barrierless oxidation of CO to
CO2 by lattice oxygen when interacting with substitutional Au
located in a cerium vacancy (see Fig. 8). It is suggested that
molecular O2 can then adsorb in the oxygen vacancy created
Fig. 7 Reaction mechanism for CO oxidation at Au adsorbed on
stoichiometric CeO2{111}. After initial adsorption of CO on Au, the
spillover step leads to a metastable configuration in which the mole-
cule binds not only to Au but also to a lattice O atom. In the
subsequent oxidation step, this lattice O atom forms part of
the departing CO2 molecule, leaving an O vacancy into which the
Au atom migrates. Reprinted with permission from Camellone and
Fabris.74 Copyright (2009), American Chemical Society.
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by this first oxidation step, with an adsorption energy of
1.18 eV, and that a further CO molecule can then abstract
an oxygen atom from this molecule in a second oxidation step
leading to CO2 (exothermic by 2.76 eV and again barrierless).
The final state of the surface is identical to the initial state, so
the catalytic cycle is closed, with an overall release of around
3.59 eV per CO2 molecule formed (including the contribution
from the O2 adsorption energy). These results show substitu-
tional gold in cerium vacancies to be highly active towards CO
oxidation, although it remains uncertain whether such sites
typically exist in substantial concentration under realistic
reaction conditions.
8. Conclusions
In this brief perspective article we have discussed some of the
recent theoretical studies of relevance to gold at the surface of
ceria. In view of our aim to highlight only the key issues, we
have of course barely scratched the surface of the details to be
found within the wide literature on this topic. Inevitably, the
views we have put forward reflect primarily our own accumulated
experience of calculations on the gold/ceria system, rather
than a settled consensus accepted by all workers in the field.
We hope, however, that the current review will serve to point
readers in the direction of the original works, and references
therein, which will reveal both the great progress that has
recently been made (for example in predicting realistic adsorp-
tion geometries for small gold clusters, plausible mechanisms
for important catalytic reaction pathways, and the role of
different vacancies in modulating the properties of gold) and
the significant work that still remains to be done. Indeed, the
work carried out to date in this area has not been free from
controversies. Mainly these have focussed upon discussions
over what might be the best value of U to use, and the physical
implications of this choice in terms of adsorption structure
and energetics. Such discussions arise because, as is widely
recognised, DFT+ U is merely an ad hoc pragmatic approach
to a profound many-body problem. It is nevertheless the most
useful approach we currently have available in the light of
contemporary computational resources. We hope, however,
that more rigorous approaches will soon be applied to these
systems, resolving many of the intriguing controversies that
have stimulated so much active debate.
Acknowledgements
We warmly thank Ricardo Grau-Crespo and Franscesc Illas
for constructive criticism on a draft version of this work.
S.J.J. is grateful to The Royal Society for a University
Research Fellowship; C.Z. thanks the Isaac Newton Trust
for post-doctoral funding; and A.M.’s work is supported by
the EURYI scheme (www.esf.org/euryi), EPSRC and the
European Research Council.
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Fig. 8 Reaction mechanism for CO oxidation at Au substituted for
Ce on CeO2{111}. Step A involves abstraction of lattice O by CO,
forming gas-phase CO2. Step B covers the adsorption of molecular O2
into the O vacancy created in the previous step, including thermo-
neutral in situ rearrangement. Step C involves abstraction of O from
the adsorbed O2 molecule by CO, forming gas-phase CO2. The surface
at the end of Step C has returned to its condition at the start of Step A.
Energy is in eV. Reprinted with permission from Camellone and
Fabris.74 Copyright (2009), American Chemical Society.
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97 To reiterate, although the expression DFT + U is widely used, theDFT approach itself is formally exact and its failure to reproducethe behaviour of strongly correlated electronic systems is a failureof standard mean-field exchange–correlation functionals.
98 The charge on the atoms is typically estimated in such calculationswith a scheme such as Mulliken or Bader charge analysis.
99 In order to avoid problematic oscillations in the approach to self-consistency, it is standard practice in DFT to mix the electrondensity calculated at each new iteration with some contribution fromprevious iterations, prior to using this in formulating the potential tobe employed in calculating the next iteration. In essence, the work ofLiu et al.35 ‘‘switched off’’ this mixing procedure for the componentsof the density matrix most likely to correspond to the f orbitals(i.e. those corresponding to large reciprocal lattice vectors and/orthose whose magnitudes varied dramatically from one iteration tothe next, the latter criterion being continually reassessed dynamicallyduring the course of the calculation). The result is to ‘‘lock-in’’localised behaviour for the f electrons, whilst still benefitting fromimproved convergence properties for the s, p and d electrons.
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