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pubs.acs.org/acscatalysis

Performance of DFT+U Approaches in the Study of Catalytic
Materials
Marçal Capdevila-Cortada,† Zbigniew Łodziana,‡ and Núria López*,†
†

Institute of Chemical Research of Catalonia (ICIQ), The Barcelona Institute of Science and Technology, Av. Països Catalans, 16,
43007 Tarragona, Spain
‡
The Henryk Niewodniczanski Institute of Nuclear Physics (IFJ-PAN) Radzikowskiego 152, 31-342 Kraków, Poland

1. INTRODUCTION
Catalytic materials are complex systems that contain diﬀerent
units, typically a carrier and an active phase. For many
reactions, redox-active materials constitute the active phase.
Active catalytic phases based on metal oxides include, but are
not limited to, silver, copper, vanadium, molybdenum, iron,
cobalt, and titanium oxides, and most notably ceria.1,2 This
results in variable occupations of the d or f states of the cations.
Therefore, the proper energy alignment of these states is
mandatory for the adequate description of the chemistry
responsible for the catalytic processes. In particular, all the
challenges related to energy-harvesting and storage technologies are intrinsically linked to the small energy diﬀerence
between the diﬀerent spin conﬁgurations, the easy transfer
between them, and the correct alignment of the energy states.
The most powerful approach to describe the electronic
structure is based on the use of density functional theory
(DFT). This theory has reached an excellent balance in the
accuracy−cost trade-oﬀ,3,4 and the computational codes are
nowadays benchmarked and stable.5 Comparison with experiments is then based on upgrading the results of the electronic
structure calculations through the use of transition-state theory,
further embedding these calculated parameters in microkinetic
models3,6 and kinetic Monte Carlo codes.7,8 However, the
intrinsic errors that appear at the DFT level get ampliﬁed in the
multiscale approaches and produce unrealistic rates for the
catalytic process of interest.
The disappointing failure of DFT in the identiﬁcation of the
electronic structure of NiO (a Mott insulator with crystal ﬁeld
splitting eﬀects, predicted to be a metal even at the GGA level
of theory)9 was one of the ﬁrst indications that the selfinteraction term (one electron interacts with itself), intrinsically
present in the formulation of DFT, was the main responsible
for this recurrent eﬀect. The self-interaction error aﬀects the
electronic structure, up to the point of being qualitatively wrong
(e.g., metals instead of semiconductors).10 The problem is
especially present in the systems with localized d or f orbitals,
where electron self-repulsion leads to wrong occupation
numbers and false alignment of the electronic levels, directly
inﬂuencing the calculated chemistry of these materials when
employed in the redox cycles. Although a variety of selfinteraction corrections have been proposed,11−15 the computational eﬃciency required for the description of catalytic
processes, together with numerical stability issues, currently
excludes the vast majority of these methods. In practice, the
simplest concept of the Hubbard model, borrowed from
condensed matter physics, is commonly used.16 In this model,
the electrons localized on d or f orbitals are subject to an
© 2016 American Chemical Society

additional on-site Coulomb interaction term, U, and site
exchange term, J. They can be perceived as penalty functions
forcing electron occupancy of particular orbitals that, for
instance, improve the prediction of band gaps. When doing so,
the electrons are more localized instead of being shared by
several atoms,17 thus breaking symmetries in local conﬁgurations and generating polarons that for some materials can be
mobile at low temperatures.18 The additional advantage of the
DFT+U methodology is the easy implementation of energy
derivatives (i.e., forces), crucial to study chemical reactions on
surfaces.
Although the physical meaning of the U parameter might
seem straightforward, it has been usually selected in an arbitrary
manner,19 ﬁtting it to a particular problem in such a way that
the experimental properties are correctly reproduced. Typical
examples of such an approach are ﬁttings to the band gap20 or
to the thermodynamics of a particular process.21 However,
ﬁtting to one property does not ensure that the others are
correctly reproduced, and this property dependence illustrates
the lack of universality of DFT+U approaches, contrarily to the
formal DFT formulation by Hohenberg, Kohn, and Sham.22
Alternative methodologies, focusing on the improvement of
the exchange correlation functionals, have been reviewed
recently by Paier.23 The so-called hybrid functionals are
based on the combination of DFT with methodologies that
contain a contribution from the exact (Hartree−Fock-like)
exchange, which can at least partially mitigate the problem of
electron self-interaction. Such functionals come in a variety of
ﬂavors depending on the particular fraction of the exact
exchange pair selected (e.g., B3LYP, PBE0, or M05-2X), and
sometimes even the fraction of the exact contribution can be
taken as a variable (e.g., HSE03 or HSE06). The most common
hybrid functionals for periodic boundary conditions calculations are HSE03, HSE06, and PBE0. Alternatively, the
random phase approximation (RPA)24−26 method has recently
emerged as a suitable tool to improve the results from DFT and
more sophisticated approaches taking Taylor expansions of the
correlation kernel have recently been put forward.27 Still, these
approaches are computationally very demanding and cannot be
routinely employed for models of real catalysts, except for
obtaining more reliable, benchmark-like, single-point energy
values.
The aim of the present viewpoint is to illustrate how DFT+U
methodologies have been employed in the study of catalytic
materials and how we are using the approach in a black box
Received: July 8, 2016
Published: November 7, 2016
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the speciﬁc formulation of this term results in the particular
implementation of the method: for example, the around mean
ﬁeld (AFM)37 or the fully localized limit (FLL)36 methods, the
latter being more widely implemented and used.
The occupation number of localized orbitals is computed by
the projection of the Kohn−Sham states on the localized basis
set. The implementation of this projection is an important
ingredient of the DFT+U method. For the plane wave basis set
representation used for slab models in heterogeneous catalysis,
the local basis sets are related to the pseudopotentials. Some
simpliﬁcations are usually included in the implementation to
allow eﬃcient computations of energy derivatives, like forces on
atoms. In general, the rotationally invariant implementations
ensure that the occupation numbers are independent with
respect to the localized atomic orbital basis set rotations.38 The
electron−electron interaction integrals of the Hubbard term
can be factorized with respect to angular and radial
contributions. Such factorizations involve Slater integrals Fk of
the radial part of the atomic wave functions. For d electrons F0,
F2, and F4 are required (F6 for f states). Within this approach,
eﬀective Coulomb and exchange terms can be computed as U =
F0 and J = (F2 + F4)/14. In practice, simpliﬁed formulations are
used, where only the lowest Slater integrals F0 are considered.
In this formulation, it is customary to introduce an eﬀective
Coulomb interaction Ueff = U − J that incorporates the
exchange correction J. This simpliﬁed DFT+U scheme can be
written as39,40

manner. To this end, we have taken an example that has been
extensively analyzed in our group.1,28,29 We aim at warning
potential users about the associated errors that are nonsystematic and aﬀect adsorption and reaction steps in diﬀerent
manners, depending on the chemical process taking place on
the surface (acid−base vs redox). The origin of the
discrepancies to hybrid methods and potential ways to mitigate
the problems are discussed.

2. COMPUTATIONAL TECHNIQUES
Even though DFT can be perceived as the major computational
tool for the description and prediction of heterogeneous
catalytic processes, it is not free from important shortcomings.
As the exact exchange correlation (xc) functional is unknown,
the approximate expressions for xc fail to capture the groundstate properties of the systems where the localization of
electrons is present. This is related to diﬃculties in recovering
the many-body interactions in the electron gas.10,30
In heterogeneous catalysis, the active materials often present
localized electrons (strongly correlated), that is, transition
metal oxides. Their chemical properties are governed by the
properties of the valence electrons. These electrons are
localized on particular d or f orbitals, and the approximate xc
functionals tend to delocalize them, overstabilizing metallic
ground states. As a result, DFT might predict metallic
properties of catalysts that are insulators, as mentioned before
for NiO.9
This intrinsic problem of DFT is related to the unphysical
self-interaction of electrons, that is, the charge density of an
individual electron interacting with itself. This repulsive
interaction induces a disproportionate delocalization of the
wave functions. From this perspective, the explicit introduction
of the Hartree−Fock approach (the Fock exchange term is free
of self-interaction) is often used to improve the accuracy of
DFT via the so-called hybrid functionals.23,31 This usually leads
to insulating ground states, although this approach is still based
on single-electron description. Still, better treatment of these
eﬀects requires the improved description of electron interaction
many-body terms that have been formulated in the dynamical
mean ﬁeld theory32 or reduced density matrix functional
theory.33 All these approaches signiﬁcantly improve the
prediction of the ground-state properties of bulk materials;
however, they are computationally much more intensive than
standard DFT. Consequently, at present, they are not practical
for the description of complex surfaces or catalytic processes.
DFT+U methods, on the other hand, have only minor
computational cost compared to standard DFT. The
fundamentals of the Hubbard model state that the electronic
correlations associated with a few localized orbitals are
responsible for the complex electronic structure. These
methods are based on model Hamiltonians: the Hubbard
model.16,34−36 Within this approach, the electrons in the
studied system are divided into two categories: (i) the strongly
correlated ones, which are usually localized on atomic d or f
orbitals, are those subject to Hubbard treatment; and (ii) the
remaining electrons that are treated in the standard DFT
manner.
Then the energy of the system can be written as EDFT+U =
EDFT + EHubbard[n] − EDC[n], where EHubbard is related to the
term of the Hubbard Hamiltonian for localized orbitals, EDC is
the double counting term, and EDFT is the total energy of
standard DFT approach within the LDA or GGA approximation. The double counting term is not uniquely deﬁned, and

E DFT + U = E DFT +

1
U ∑ Tr[n I (1 − n I )]
2
I

(1)

where I stands for the site of the ions, and nI corresponds to the
density matrices projected on localized atoms. Unless otherwise
noted, this is the method discussed throughout this work. The
value of U can be then obtained by diﬀerent methods.35,41−48
For the plane wave codes, one of the most extended
methodologies is the constrained DFT approach that follows
the linear response (LR) theory, presented by Cococcioni and
Gironcoli.40 The idea behind this methodology is to evaluate
the eﬀect of the perturbation of the bare and screened density
response functions once a perturbation is applied to the local
projector. To ensure that the values are meaningful, these
authors indicated that the perturbation needs to be performed
in supercells with increasing size, to guarantee that the values
are not aﬀected by periodic boundary conditions.40 It has to be
noted that the U value shows an important dependence on the
functional of choice (LDA vs GGA), the pseudopotential, the
choice of the ﬁtting properties (if any), or projection operators.
Indeed, important diﬀerences have been found, and thus, the U
term shall better be reparameterized for each computational
setup.49−52 It has been proposed that the use of a unique U
value for a metal ion in diﬀerent environments could lead to
inaccuracies when the local electronic structure changes.41 All
these procedures to obtain the Hubbard U are particularly
suited for bulk systems. For the slab models representing
catalysts, the U values taken from the bulk are typically used.
However, the changing local environment of the surface atoms
during a chemical reaction is a potential source of errors.
An exhaustive analysis employing this methodology for
cerium compounds was performed by Lu and Liu.41 These
authors illustrated that adequate U values for Ce atoms in
diﬀerent conﬁgurations ranging from isolated atoms and ions,
to small oxide compounds or extended CeO2 bulk and surface
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Table 1. Most Typical U Values Suggested for Several Metal Oxides Using GGA+U Approachesa
Mn+

U (eV)

Ce3+/Ce4+
Ce3+/Ce4+
Ce4+
Ce3+
Ce4+
Ce4+
Ce3+

2.0
2.0−3.0
2.0−3.0
4.5
5.0
5.13
6.70

ﬁtting experimental properties
ﬁtting experimental properties
ﬁtting experimental properties
linear response
band gap states
linear response
linear response

Kresse20
Illas19
Fabris61
Fabris51
Watson62
Liu41
Liu41

Ti3+/Ti4+
Ti4+
Ti4+
Ti4+
Ti4+
Ti3+
Ti4+
Ti4+
Ti4+

2.0−3.0
3.0
3.0
3.4
4.2
4.35
4.76
4.95b
10.0

ﬁtting experimental properties and band gap states
ﬁtting experimental properties
ﬁtting hybrid−dft calculations
linear response
band gap states
ﬁtting experimental properties
ﬁtting experimental properties
linear response
ﬁtting experimental properties

Metiu63
Nolan64
Selloni65
Mattioli66
Watson67
Wolverton52
Wolverton52
Kitchin68
Dupuis69

Co2+/Co3+
Co2+/Co3+
Co2+
Co3+
Co2+
Co4+
Co2+
Co3+

3.3
3.52
3.75
4.26
4.4
4.77
4.0−5.0
6.7

ﬁtting experimental
ﬁtting experimental
ﬁtting experimental
ﬁtting experimental
linear response
ﬁtting experimental
ﬁtting experimental
linear response

properties
properties
properties
properties

Ceder70
Nørskov71
Wolverton52
Wolverton52
Selloni72
Wolverton52
Doublet73
Selloni72

Mn4+
Mn2+
Mn4+
Mn4+
Mn3+
Mn4+

1.0−1.6
2.98
3.19
3.0−4.0
4.54
6.63

ﬁtting experimental
ﬁtting experimental
ﬁtting experimental
ﬁtting experimental
ﬁtting experimental
linear response

properties
properties
properties
properties and hybrid-dft calculations
properties

Cockayne74
Wolverton52
Wolverton52
Kresse54
Wolverton52
Kitchin68

Fe3+
Fe3+
Fe2+/Fe3+
Fe2+/Fe3+
Fe2+
Fe3+

3.0
3.0
3.61
3.7
4.04
4.09

ﬁtting
ﬁtting
ﬁtting
ﬁtting
ﬁtting
ﬁtting

experimental
experimental
experimental
experimental
experimental
experimental

properties
properties
properties
properties
properties
properties

Rollman75
Morgan76
Łodziana77
De Leeuw78
Wolverton52
Wolverton52

Mo6+
Mo6+
Mo6+
Mo4+/Mo6+

2.0
6.0
6.3
8.6

ﬁtting
ﬁtting
ﬁtting
ﬁtting

experimental properties
hybrid−dft calculations
hybrid−dft calculations
experimental properties

method

reference

CeOx

TiOx

CoOx

properties
properties

MnOx

FeOx

MoOx
Metiu79
Asta80
Willock81
Bell56

Commonly ﬁtted properties are the band gap, the lattice parameter, the bulk modulus, or the ΔH of formation of the oxide. The band gap states
method refers to properly localize these states within the band gap. The linear response method refers to the approach introduced by Cococcioni and
Gironcoli.40. bDiﬀerent functionals, pseudopotentials, and TiO2 polymorphs lead to diﬀerent values of U.59
a

and nearest neighbors interactions are needed for Ce in its
metallic form.53
An extensive compilation of U values for Co, Fe, Ni, Cr, Mn,
V, and Ti ions on diﬀerent oxide and ﬂuoride environments has
been presented by Aykol and Wolverton on the basis of the
experimental ΔH of formation ﬁtting approach.52 These
authors aimed at reproducing accurate thermochemistry,54,55
making special emphasis on the role of the oxidation state and
ligand contributions. They observed, for instance, that the U
term is not transferable when going from oxygen to ﬂuoride

models, presented some marked characteristics: (i) the ion
charge does not signiﬁcantly aﬀect the value of U when
properly screened by counterions (U values for Ce atoms, Ce in
Ce3HxO7 clusters, or CeO2 were ranging in the 4.1−5.2 eV
range); and (ii) when ions are isolated the values are much
larger (close to 15 and 18 eV for Ce2.5+ and Ce3.5+,
respectively). Thus, the environment and its polarizability
turn out to be crucial to screen the Coulomb interactions on Ce
4f orbitals eﬀectively. This observation agrees with Cococcioni
and Gironcoli,40 who found that to obtain Ce 4f values, on-site
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Figure 1. (a) Adsorption of formaldehyde (A1) on the CeO2(111) regular surface and ﬁrst hydrogen stripping to CHO and OH (I1). (b) Reaction
scheme for this elementary step. (c) Overlaid structures of TS1 for the diﬀerent U values. Red stands for oxygen, yellow for cerium, black carbon,
and white hydrogen.

bulk positions. Using the same setup, screened hybrid HSE06
calculations86 have been performed and employed as reference
to illustrate the dependence of the reactivity with the diﬀerent
values of U. The iron oxide slab contains 18 atomic layers and a
k-point sampling of 5 × 3 × 1 was employed.

ligands, and is larger when the charge of the cation increases
(although exceptions are identiﬁed). Similarly, Bell and coworkers,56 when investigating the reduction energies for
transition (Ti, V, Mo) and rare earth (Ce) metal oxides,
observed that the value of U had to be adjusted for each
reaction. The same authors in the investigation of the reducible
transition metal oxides, basically Bi2Mo3O12 for hydrogen
abstraction from propene, observed that both the barrier and
the ﬁnal state energy depend on the U parameter used for
Mo.57 When compared to their reference functional, M06-L,58
the thermodynamic and the kinetic parameters were not
reproduced with a single U value. In a similar study that focused
on the structure of diﬀerent TiO2 polymorphs, Kitchin et al.
realized that the U value obtained by linear response for the Ti
d orbitals depends on the functional and pseudopotential
employed, and the particular crystal structure.59 Similar
conclusions were obtained by Greeley et al. for several
transition-metal (hydroxy)oxides, where the dependence of
the enthalpy of formation with the U term varies signiﬁcantly
when van der Waals eﬀects were considered.60
In Table 1 we present a summary of the most common U
values employed together with the GGA formulation. These
values can range from 2.0 to 10.0 eV for TiOx or from 2.0 to 8.6
eV for MoOx, depending on a variety of factors.
In the present work, we have employed PBE+U with variable
U values in order to unravel the robustness of the diﬀerent
approaches to obtain U and how the values aﬀect the catalytic
properties of the materials. To this end, we have used the VASP
code82,83 with the PBE functional,84 PAW pseudopotentials,85
and plane waves with a cutoﬀ energy of 500 eV for the valence
electrons (5s, 5p, 4f, 6s for Ce atoms, 3d, 4s for Fe, 2s, 2p for O
and C, and 1s for H). The lattice parameter was optimized
using a dense Γ-centered 7 × 7 × 7 k-point mesh that leads to a
value of 5.497 Å, in good agreement with the experimental aexp
of 5.411 Å. The (111) surface, the most stable termination of
ceria, was modeled as a p(2 × 2) supercell with periodically
repeated slabs consisting of three O−Ce−O layers (nine single
layers) separated by 15 Å of vacuum space, which was
optimized using a Γ-centered (3 × 3 × 1) k-point mesh. The
ﬁve outermost single layers and the adsorbate(s) were allowed
to relax, whereas the rest of the atoms were kept ﬁxed to their

3. DISCUSSION
3.1. Formaldehyde Decomposition on CeO2(111).
Cerium oxide is among the most interesting oxides from a
chemical point of view. The coexistence of acid−base, redox,
and oxygen storage material properties makes it an outstanding
ingredient in catalytic formulations where it can act as active
phase, cooperative or synergetic catalytic phase, or just as
support.28,87 The hydrophobicity of the surface can also be
tuned by the reduction degree.88,89 Applications centered on its
oxidative capacity have been preeminent: ceria is then usually
present in combination with metal phases, although recently it
has been shown to be active as a single phase in the Deacon
and HBr oxidation reactions,90,91 hydrogenations,92−95 and
selective C−C bond breaking.96 Shape-directed synthesis
protocols enable the synthesis of tailored ceria nanoparticles,97,98 in particular, nanocubes exposing {100} facets.
The latter, which are polar and present two main
reconstructions,99,100 exhibit unique catalytic activity.93
Many of these reactions combine acid−base and redox
elementary steps. In the latter, one of the electrons from
reactants and/or products can be accommodated at the surface
cations. Therefore, the correct identiﬁcation of the energy gain/
loss by the reductive nature of the cations on the surface
constitutes a major issue. To illustrate the eﬀect, we have taken
as an example the dissociation of formaldehyde on the ceria
(111) surface, for which detailed temperature program
desorption experiments exist.101−103 The elementary step for
dehydrogenation of formaldehyde is found in Figure 1.
The most common values of U for LDA and PBE functionals
are 5.3 and 4.5 eV,51 which have been extensively used in the
literature. The results of the reaction energy proﬁles obtained
with PBE+U for diﬀerent values of U are intriguing (Figure 1).
Formaldehyde adsorption remains constant, as it does not
involve any surface reduction, and thus, all points (CH2O + *
→ CH2O*) lie in the same energy. In contrast, the elementary
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step for the decomposition of formaldehyde (CH2O* →
CHO* + H*) is always exothermic, but the energy span
between the small and large U values is more than 2 eV. The
reaction is more exothermic as the U increases. A similar
behavior was observed by Bennett and Jones for CO+NO
reaction on the ceria surface.104 More importantly, the energy
of the transition state also presents a wide range of values that
expands about 1 eV. Considering that the computed activation
energy is introduced in the exponential in the transition-state
theory, this means that at room temperature the diﬀerence in
the kinetic coeﬃcient for small and large values of U on Ce can
account for several orders of magnitude.
A clearer view of the process can be noticed when the
adsorption-, transition-, and ﬁnal-state energies are included in
the same plot, see Figure 2. The ﬁgure clearly illustrates that the
balance between thermodynamics and kinetics is broken. The
kinetics of the system are wrong for several U values, and the
energy of the transition state with respect to the gas-phase
reference changes from positive to negative depending on the

strength of the eﬀective U parameter. In Figure 2, the problem
of electronic structure and how it is dealt with in DFT+U
approaches becomes clear. While formaldehyde adsorption is a
constant and thus is not aﬀected by changes in the U
parameter, the following step in the decomposition is strongly
aﬀected. For small U values, the electrons are still delocalized,
and thus, the values for the transition and ﬁnal states are rather
constant. For higher values of U, the adsorption remains
constant, but the transition and ﬁnal states reduce their energies
according to a linear dependence, yet with diﬀerent slopes.
At small U values, electrons are not localized even in reduced
surface conditions, and thus, for U values lower than a certain
threshold (that depends on the material), all the energy
parameters for adsorption, transition, and thermodynamics of
the elementary steps are practically constant. Instead, at
medium to large U values, both the transition and ﬁnal states
are stabilized, but the size of this stabilization is doubled for the
ﬁnal state. This means that the thermodynamics and the
kinetics are no longer synchronous, and the choice of U implies
an error in one or the other. Comparison with experiments for
formaldehyde decomposition sets an experimental constraint to
the transition-state energy, as it should be larger than zero to
explain that desorption is preferred when compared to
dehydrogenation.102,103 This only happens for U values below
5.5 eV, thus constituting the upper bound to this parameter.
The inﬂuence of the U parameter on the direct reducibility of
the surface has also been assessed (Figure 3). This reducibility,

Figure 3. Reduction energy (Ered, red) and electron localization
(turquoise) as a function of the Hubbard U parameter. The horizontal
solid line accounts for the screened hybrid HSE06 Ered result. The
background color indicates when the electron is properly localized in
the Ce center (turquoise).

Ered, has been taken as the energy diﬀerence between the p(2 ×
2) supercell with one extra electron added (where a
background charge is added to preserve charge neutrality)
and the bare p(2 × 2) surface, as considered in previous
work.1,28 The electron localization on a single cerium atom of
the topmost layer is also shown in Figure 3, for each value of U.
The interpretation of the dependence with the U term in the
redox steps and the diﬀerence between the kinetics and
thermodynamics can be traced back to the local structures and
the electronic states of these conﬁgurations. The diﬀerent
slopes observed for the reaction and transition-state energies
with the U parameter originate from the number of electrons
that are localized at the surface. The ratio between the slopes
gives the ratio between the electrons at the surface.

Figure 2. (Top) Activation energy (Ea, red) and reaction energy (ΔE,
blue) for the ﬁrst C−H bond breaking of formaldehyde on CeO2(111)
as a function of the U parameter. (Bottom) Adsorption energy of
formaldehyde (blue), transition step for ﬁrst hydrogen striping (red),
and ﬁnal state (green), all with respect to gas-phase formaldehyde and
the bare CeO2(111) surface, as a function of U. In both cases, the
horizontal lines account for the screened hybrid HSE06 results. PBE0
calculations provide nearly identical results as HSE06 (see text). For
the redox processes, dependence on the U parameter is observed when
electron localization in the Ce centers appears (turquoise background
color). Notably, the dependences of the kinetic and thermodynamic
terms are diﬀerent.
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The structurally consistent U suggested by Kulik and Marzari
is at the origin of the problem. These authors observed the
need for a smooth potential energy curve incorporating the
variations of U (structurally consistent U) when the signiﬁcant
changes in the hybridization and occupancy of a transitionmetal center occur along the reaction coordinate. When the
authors introduced a polyatomic molecular system, they
recognized that more than one geometric variable aﬀects the
occupation matrix of the d-states and thus the U term depends
on several variables, U(R). Other structurally consistent
approaches were also indicated in the literature, and the eﬀect
when changing the coordination by adsorption of diﬀerent Ocontaining species was reported for Ti.68,110 Kulik and Marzari
proposed CCSD(T) calculations as an appropriate benchmark
and suggested that the inclusion of a U(R) in the nudged elastic
band is “trivial”. However, ﬁve years after this comment, we are
not aware of such implementation. Unfortunately, although this
approach seems evident, it could only be applied to very small
systems (with one metal atom) that did not present a set of
close-lying states.
Indeed, the search of transition-state calculations with DFT
and plane waves codes has an important problem when dealing
with the electronic structure. The reason is that in the
algorithms to search for transition states, the ﬁles that describe
the electronic structure are taken to be the same (i.e., the U for
each of the conﬁgurations along the band is identical). As we
have seen before, the strength of the U parameter seems to
provide unreasonable values for the kinetics. We propose here a
methodology that can smooth the errors: (i) to identify acid−
base and redox steps; (ii) to determine of the most adequate U
value by comparison of the thermodynamics of the steps to a
reference calculation (ideally RPA or QMC, otherwise hybrid
DFT); (iii) to unravel the dependence of the thermodynamics
with the U (as in Figure 2); (iv) to elaborate on the kinetics an
accurate count of the electron localization at the transition state
and comparison to its corresponding ﬁnal state is needed, as
this gives the slope for the dependence of the transition state
with U; and (v) to rescale the U value for the transition state
according to the geometric perturbation between transition and
ﬁnal states. This procedure is more stable than employing
variable U for each initial image in the band, as the initial
guesses both in terms of electronic structure and geometry
might signiﬁcantly diﬀer from the ﬁnal solution. This is possible
because, at a diﬀerence from molecular calculations, many of
these steps correspond to bound initial and ﬁnal states.
3.2. Iron Oxides. Iron oxides are among the cheapest
catalysts and present a very versatile structure that accommodates diﬀerent oxygen stoichiometries. The structure of the
material presents certain diﬃculties in the adequate description
of its electronic structure that are linked to the metal−insulator
transition occurring in some of its phases (Verwey transition).77
The use of U, applied to the Fe 3d orbitals, is crucial to obtain a
proper orbital ordering to understand surface reconstructions in
a material as common as Fe3O4. In that case, the U value
employed was 3.61 eV. 77 For a completely diﬀerent
stoichiometry, in the study of water interacting with iron
oxide ﬁlms, the eﬀective U value was very similar, 3.0 eV. In
both cases the U values were ﬁtted for the PW91 functional
and, as also observed by Lu and Liu for ceria, the charge of the
cation is not the primary term aﬀecting the U value.41 To
analyze the extension of the problem, we have investigated the
same reaction (CH2O* + * → CHO* + H*) on Fe2O3. Again,
the observed patterns for adsorption, reaction energy for the

When compared to more accurate screened hybrid
calculations (HSE06), the U values at which the agreement is
obtained vary for the kinetic and the thermodynamic
parameters (solid lines in Figure 2). This means that to
match the HSE06 results a value of 4.27 eV is required for the
thermodynamics of the process, but only of 3.33 eV for the
transition state. Similarly, hybrid PBE0 calculations provided
almost identical results (4.32 and 3.33 eV, respectively). The
calculation of Ered, Figure 3, where one electron was added to
the bare surface slab (comparable then with the previous
HCHO transition-state calculation), leads to a U value of 2.58
eV to reproduce the HSE06 result. Indeed, Kulik and
Marzari105 reported that the complete potential-energy surface
cannot be represented by a constant U value, through the study
of the dissociation of 4Φ FeO+ by internuclear separation.
Instead they proposed that the geometry plays a major role and
that, rather than being a parameter, U is a function of the local
geometry. It is worth stressing that although the HSE06
functional also contains certain degree of inaccuracy, it typically
outperforms LDA(+U) and GGA(+U), especially for the
description of semiconductor and insulator materials.23,106−109
Thus, hybrid functional results should not be considered as the
gold standard but rather as a higher rung in Jacob’s ladder of
DFT. Proper benchmark studies should be performed using
more accurate RPA or QMC calculations, but these are beyond
the scope of this work.
The origin of the diﬀerent U to ﬁt the HSE06 values thus can
be traced back to the local structures. For ceria, Lu and Liu41
identiﬁed the dependence of the U parameter on the geometry
of the environment: the larger the Ce−O distance, the larger
the oxygen screening loss, and thus, the larger the interaction of
the f electrons shall be (i.e., larger U). These authors illustrated
that for variations of 0.1 Å, the eﬀect on the U could be as large
as about 0.4 eV. We have compared the elastic modiﬁcations
both for the transition and ﬁnal states at a given U = 3.5 eV,
Figure 4.

Figure 4. (Top) Most relevant Ce−O distances in Å for the initial,
transition, and ﬁnal states of formaldehyde decomposition on CeO2 at
PBE+U (U = 3.5 eV) level. (Bottom) Electron localization in the
initial, transition, and ﬁnal states. Same color code as in Figure 1.

In contrast, acid−base steps do not present this problem. We
have performed the ﬁrst reaction in methanol decomposition
on ceria(110), in the same manner as presented before. The
adsorption, transition, and ﬁnal states (i.e., the thermodynamics
and kinetics) are completely stable and completely independent
of the U parameter. This seems to be a reasonable conclusion
because no surface atoms are reduced during the process.
However, it sets an internal contradiction to generate reaction
energy proﬁles that maintain the correct balance of the kinetics
and thermodynamics in complex reaction networks that contain
both acid−base and redox steps.
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ACS Catalysis
ﬁrst abstraction, and the corresponding transition-state energy
(Figure 5) resemble those in Figure 2. However, two

optimal value, as the local geometric perturbations induced by
single-atom conﬁgurations might be very large.
3.4. Other Materials Related to Energy Production
and Storage. The problems observed in understanding the
chemistry of ceria and iron oxides can be further extended to
other materials where an adequate band alignment is needed to
rationalize the chemical processes. Examples of such behavior
can be found in several studies,116−118 which illustrate that the
proper band alignment in titania is required for the photoelectrochemical properties to be well-balanced, particularly in
complex interfaces. The ﬁtting of U for the diﬀerent cations is
required. Still, some authors have proposed that the use of
linear response DFT+U in reactions, like the oxygen evolution
activity on rutiles, would only change the activity ordering of
diﬀerent materials.68 According to these authors, the weakening
of the adsorption energies (taken as descriptors) will not aﬀect
the relative ordering of reaction energies and, at most, the
variation will only aﬀect the top of the activity volcano. In view
of our results, larger diﬀerences shall be expected for doped
systems,119 particularly if chemical and electrochemical steps
are present in the reaction network.120
The problem is also prevalent for other compounds, like
cobalt oxide, with a wide range of stable compositions seem
particularly diﬃcult. Recent studies have employed the ﬁtted
values of U from Wang and Ceder,70 which were adapted to
reproduce thermodynamic properties (ΔH of formation of the
oxide), but still corrections were needed to obtain the proper
thermodynamics between the diﬀerent phases.121 Similar
problems arise in manganese oxides, for which site-dependent
values were proposed,54 and the most recent use of nickel oxohydroxides122−124 with varied valences of the Ni centers and
mixed Fe−Ni materials for energy-related catalysis125,126 are
foreseen. The implication of the U dependence will also aﬀect
other families of materials with applications in Li-ion batteries,
water splitting, or photovoltaics such as Prussian Blue
analogues,127,128 perovskites,129,130 or Li-ion battery cathodes.52

Figure 5. Adsorption energy of formaldehyde (blue), transition step
for ﬁrst hydrogen stripping (red), and ﬁnal state (green), all with
respect to gas-phase formaldehyde and the bare Fe2O3 surface, as a
function of U. The background color indicates when the electron is
properly localized in the Fe center (turquoise) and when spin-crossing
is observed (purple). Notably, the dependences of the kinetic and
thermodynamic terms are diﬀerent.

remarkable points emerge. First, the dependencies are diﬀerent
from those in Figure 2, where now the transition-state slope is
larger than that for the reaction energy. This behavior can be
traced back to the occupation of the d shell versus f shell in Fe
versus Ce. The former is by reduction more than half full, while
that of Ce is empty or with less than one electron. Second, at
high U values (6.5 eV, purple region of Figure 5), the electronic
state of the initial and ﬁnal states are not the same as for
intermediate values of U, and therefore, the corresponding
energies do not follow the trends. Yet, this highlights a problem
that is much more acute for less oxidized oxides, like FeO. Our
preliminary results for that surface show that spin-crossings can
occur at diﬀerent positions of the intrinsic reaction paths
connecting reactants and products, and thus the equivalent
results to those in Figure 2 and 5 will develop in families of
planes for each of the spin states considered.
3.3. Complex Interfaces and Single-Atom Catalysts.
The problem of diﬀerent U values might be also acute in the
case of isolated atoms or monomers. An example that parallels
our investigation on methanol conversion is the same reaction
on vanadyl ions and clusters supported on CeO2.111,112 These
atoms form a complex interface for which Ce 4f U values were
employed,111 but then the cooperative eﬀects (as there are two
diﬀerent centers that can be reduced competitively) are diﬃcult
to unravel. Therefore, identiﬁcation of the active vanadium
containing species might not be trivial.112
The case of the VOx/CeO2 materials shares some common
features with single-atom catalysts, SAC, proposed for isolated
atoms in oxide matrices. The term was coined by Zhang and
co-workers113 for the description of isolated Pt atoms
embedded in FeOx, which were tested in CO oxidation, but
follows similar principles to the Au/CeO 2 or Pt/CeO 2
chemistry reported by Flytzani-Stephanopoulos.114 In all
these cases, the theoretical simulations performed to understand this chemistry have been carried out with a U value
derived from the clean host material. For the Pt/FeOx oxide,
the authors took a U = 3 eV from hematite.115 However, as we
have shown, this approximation might really diverge from the

4. OUTLOOK
Computational techniques based on DFT have reached a level
of maturity that makes possible the accurate reproduction of
experiments and the prediction of reactivity up to levels that
seemed unreachable just a decade ago. The statement is
particularly true for metals and insulators, but for materials with
complex electronic structures, multiple spin conﬁgurations, and
diﬀerent degrees of electron localization, this is not so
straightforward. We have shown how the thermodynamics
and the kinetics for a simple reaction respond diﬀerently to the
external energy penalty included in the models to account for
proper representation of the band gap. The ultimate result is
that the dispersion leads to a complete disagreement between
the experimental observations and the computed values. In this
Viewpoint, we presented a revision of what needs to be done to
achieve a reasonable accuracy at a reasonable computational
cost, so as to improve the use of DFT for strongly correlated
systems. Most of the chemistry related to photochemistry,
energy extraction to suitable energy vectors, conversion, and
storage are based on the use of materials with complex
electronic structures, and thus, the results here are of
fundamental interest to the community.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: nlopez@iciq.es.
8376

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ORCID

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Marçal Capdevila-Cortada: 0000-0003-4391-6580
Núria López: 0000-0001-9150-5941
Notes

The authors declare no competing ﬁnancial interest.

■

ACKNOWLEDGMENTS
We thank MINECO (CTQ2015-68770-R) for ﬁnancial support
and BSC-RES for providing generous computer resources.
M.C.-C. also acknowledges MINECO for a “Juan de la Cierva
− Formación” fellowship (FJCI-2014-20568). Z.L. acknowledges NSC Poland support through project 2015/17/B/ST3/
02478.

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