Theoretical Approaches to Excited-State-Related Phenomena in

Review pubs.acs.org/CR

Theoretical Approaches to Excited-State-Related Phenomena in Oxide Surfaces Carmen Sousa,† Sergio Tosoni,†,‡ and Francesc Illas*,† †

Chem. Rev. 2013.113:4456-4495. Downloaded from pubs.acs.org by KAOHSIUNG MEDICAL UNIV on 08/27/18. For personal use only.

Departament de Química Física and Institut de Química Teòrica i Computacional (IQTCUB), Universitat de Barcelona, C/Martí i Franquès 1, 08028 Barcelona, Spain ‡ Departamento de Química, Universidad de Las Palmas de Gran Canaria, Campus Universitario de Tafira, 35017 Las Palmas de Gran Canaria, Spain 5.3. Photodesorption 6. Excited States at Oxide Surfaces Related to Photocatalysis 7. Excited States at Oxide Surfaces Related to DyeSensitized Solar Cells 8. Concluding Remarks and Perspectives Author Information Corresponding Author Notes Biographies Acknowledgments List of Acronyms References

CONTENTS 1. Introduction 2. Survey of Electronic Structure Methods for Ground and Excited States 2.1. Quantum Mechanics and Wave Functions 2.2. From Quantum Mechanics to Quantum Chemistry 2.2.1. Wave-Function-Based Methods in Quantum Chemistry 2.2.2. Density-Functional-Theory-Based Methods in Quantum Chemistry 2.3. Describing Excited States 2.3.1. Configuration Interaction and Related Methods 2.3.2. Time-Dependent Density Functional Theory 2.4. Excited States from Band Structure and Quasiparticle Excitation Calculations 3. Modeling Oxide Surfaces 3.1. Embedded Cluster Model Approach 3.1.1. Defining the Quantum Region 3.1.2. Adding Static Short-Range Interactions 3.1.3. Adding Long-Range Electrostatic Interactions 3.1.4. Adding Long-Range Polarization 3.1.5. Case of Covalent Oxides 3.2. Periodic Slab Model Approach 3.3. Oxide Nanoparticle Surfaces 4. Excited States in Clean Oxide Surfaces 4.1. Stoichiometric Surfaces 4.2. Defective Surfaces 4.3. Nanostructures 5. Excited States Related to Adsorbates at Oxide Surfaces 5.1. Metal Atoms and Metal Clusters 5.2. Probe Molecules © 2012 American Chemical Society

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1. INTRODUCTION To the general public, metal oxides are surely synonymous with corrosion and therefore considered mostly as undesirable materials. To chemists and to materials science specialists, however, metal oxides and other solid oxides in a very broad sense are interesting materials with a wide range of properties which ultimately define their possible technological applications. Depending on the composition, oxides exhibit different physical properties.1 Hence, these materials can be strongly insulating or even exhibit metallic conductivity. Often they have very high melting points with concomitant refractory properties. Some oxides exhibit extreme hardness, which makes them suitable as abrasive agents. From a chemical point of view, oxides might behave as very inert materials or, on the contrary, display a complex reactivity which is of paramount importance in catalysis.2 Precisely because of their relationship to catalysis, there has been an enormous interest in applying surface science techniques to these materials, although this is not so simple because their insulating character complicates the use of electrons as surface probes.3 Oxide thin films supported on metal surfaces4−7 overcome this problem and permit application of standard surface science techniques and have therefore been proposed as appropriate models of oxide surfaces and as catalyst models.8 Recent advances in this area are discussed in detail in this special issue. Oxide thin films are also of interest in many relevant technologies such as corrosion protection of metals by passive films,9 ferroelectric ultrathin

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Special Issue: 2013 Surface Chemistry of Oxides Received: June 7, 2012 Published: November 29, 2012 4456

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film capacitors,10 and tunneling magnetoresistance sensors.11 Moreover, oxide thin films have their own specific and often unusal properties, such as charge flow between the metal and the oxide depending on the adsorbate.12 Nevertheless, one should not forget that most of the interesting chemical reactions involving oxide surfaces take place in solution, the presence of the solvent adding an extra amount of complexity and a rich chemistry.13 Oxides dominate the composition of the earth crust and, hence, are also at the focus of geochemistry.14,15 Because of the exceptionally wide variety of physical and chemical properties, oxides are increasingly emerging as technically important materials which are attractive in several applications such as photovoltaic devices, gas sensors, microelectronics, heterogeneous catalysis, and anticorrosion devices. Some properties of interest are ultimately determined by the bulk electronic structure of the oxides, even if the relationship between electronic structure and properties is not always fully understood. This is precisely the case for the so-called highcritical-temperature superconductivity16 exhibited by some copper oxides such as La2CuO4 or YBa2Cu3O6 when they are conveniently doped, resulting in nonstoichiometric compounds such as La2−xBaxCuO4 or YBa2Cu3O6+x. The degree of doping x influences remarkably the conductivity, which thus introduces a rather large degree of complexity to the microscopic studies, of either experimental or theoretical character. Before doping, these amazing materials are typical antiferromagnetic insulators. After doping, they become superconductors (i.e., the electric resistance becomes zero) below a certain critical temperature, Tc, which is rather large compared to the corresponding Tc of pure metals and other conventional superconductors. However, whereas the superconductivity exhibited by conventional superconductors is well understood through the theory developed by Bardeen, Cooper, and Schrieffer (BCS theory), a general theory for high-Tc superconductivity is still missing.17−19 Nevertheless, there are indications that the critical temperature is related to the magnetic order of the parent (undoped) compound, which is, in turn, related to the electronic structure and, more precisely, to the energy difference between different electronic states.20,21 Even if superconductivity is a property not directly related to chemistry, it is included here to show the importance of excited states of oxides, in general, and of oxide surfaces, in particular, in various fields from solid-state physics to materials science to chemistry. In the context of the present review, more addressed toward a chemical audience, the attention is focalized on two much more appropriate examples of the relevance of excited states in oxides: photocatalysis and dye-sensitized solar cells. Excited states in oxides and oxide surfaces, however, are relevant to many other basic processes as we will discuss at length later on. According to a recent paper by Herrmann,22 photocatalysis was initiated by Doerffler and Hauffe23,24 nearly 50 years ago, although it is generally accepted that the field bloomed after the English translation in Nature25 of the original work of Fujishima, Honda, and Kikuchi published in 1969.26 Photocatalysis has a tremendous impact on environmental27 and green28,29 chemistry, fine chemicals fabrication,30 hydrogen production,31−33 and other advanced oxidation-related processes. The basics of photocatalysis are rather simple: the absorption of photons of energy higher than or equal to the band gap of the photocatalyst, which is usually a semiconductor such as TiO2, SnO2, or ZnO in the form of a finely divided

powder, generates photoelectrons and photoholes. These, in turn, are able to reduce and oxidize adsorbed molecules, respectively. However, the detailed description of the photocatalytic process is much more complicated and has been the subject of several extensive reviews.34−37 Among many other processes, the electron−hole recombination rate or deactivation by charge trapping plays a crucial role. A recent review by Henderson describes all these processes from a surface-science point of view.38 The growing demand of green materials and devices being able to exploit renewable energy sources for chemicophysical processes has driven the efforts of many research groups toward the search of photocatalysts showing high activity under sunlight. This is not the case for excellent UV photocatalysts such as rutile or anatase polymorphs of TiO2 due to the rather high band gap that these materials exhibit (close to 3 eV). This has prompted chemists and material scientists to search for modifications of TiO2 which could exhibit photocatalytic activity under visible light. Doping seemed to be one of the obvious options, and indeed, doping with N has been found to be among the best alternatives.39 Recently, a two-dimensional phase of undoped TiO2 with a reduced band gap has been reported40 which indeed opens new and interesting possibilities. The type of crystal surface is another crucial aspect, and recent advances in synthesis have led to anatase nanoparticles with a majority of (001) reactive facets.41 From a theoretical point of view, which is the main focus here, studying the electronic structure of the various polymorphs of bulk TiO2 constitutes no doubt a first necessary approach. However, a more detailed study of the excited states of various surfaces, either extended or in nanoparticles, for the pure or doped material, is necessary to make real progress in the field and to better interpret the huge amount of experimental data existing for these surfaces.42 In the present review we aim to present the state of the art in the theoretical description of excited states in oxide surfaces, hoping to contribute to triggering the necessary theoretical studies on photocatalysis by the different forms of TiO2.43 There is no need to say that similar arguments apply to other semiconducting oxides of interest in photocatalysis such as ZnO, SrTiO3, or WO3, although, undoubtedly, TiO2 is the most studied material in this field. Solar cells44 constitute another excellent example of the importance of excited states in oxide surfaces and on the need for accurate theoretical studies on these materials. Solar cells are based on the photovoltaic effect, i.e., the creation of a voltage or electric current in a material after exposure to light. This physical phenomenon was first observed by Becquerel in 1839, but its application to solar cells was first carried out by the Bell laboratories back in 1954 using a Si semiconductor. The photovoltaic effect implies a light-induced electronic transition from the valence band to the conduction bands of the photovoltaic material. Note that this is different from the photoelectric effect, where incident photons of sufficient energy ionize the material’s surface. The reason to mention solar cells in this review comes from the fact that one of the most promising technologies to replace the Si-based solar cell is the dye-sensitized solar cell,45−52 first introduced by O’Regan and Grätzel,53 with proven efficiencies close to 10−12%. In fact, Han et al. have recently presented a new design with a certified efficiency of 11.4%.54 The main components of a dye-sensitized solar cell are an electrolyte containing a redox pair, an organic or metalloorganic dye, and some form of mesoporous TiO2 or 4457


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on NO with the high-spin coupling of the unpaired electrons of the Ni2+ cation leads to various possibilities, including Ni2+ reductions to Ni+. The result of the interaction is hard to predict without describing in detail the various possible electronic states, and for an extended surface, this is far from being straightforward.71,72 Therefore, except for simple oxide surfaces interacting with closed-shell molecules, the proper prediction of the electronic ground state may be a challenging task. Moreover, charge transfer between the adsorbate and the oxide surface often occurs and, at variance with the situation corresponding to metallic surfaces where the electrons are usually delocalized in the conduction band, leads to different, usually nearly degenerate, electronic states. Here, we will also review these apparently simple cases and will also describe the attempts to model more complicated systems such as those involved in photocatalysis and dye-sensitized solar cells. Finally, we must state that the field is really enormous, and the present review presents an overview of the most important topics.

another semiconducting oxide. For instance, dye-sensitized solar cells based on ZnO nanowires have been reported. Hence, dye-sensitized solar cells present all elements of interest discussed in the present review: they imply an oxide surface and involve excited states and charge transfer processes between the adsorbed dye and the oxide surface. We will review the theoretical attempts to study such a complex system, focusing especially on studies based on advanced electronic structure methods. Hopefully, a full understanding of the microscopic processes involved in these devices can help to engineer better systems and to overcome some of the problems55 encountered in the search for more efficient dyesensitized solar cells, which, according to recent advances reported in the literature, is very possible.56 From the previous discussion, it is clear that a precise knowledge of excited states of oxide surfaces, either naked or with adsorbed species, is important to fully understand the microscopic mechanism behind technologically important processes such as photocatalysis or those involved in dyesensitized solar cells, both of which indeed have a strong social impact. However, these are not the only processes at oxide surfaces where excited states play a role and where input from theory is needed. The case of oxygen vacancies in nonreducible oxides such as MgO is a clear example. The removal of an oxygen atom results in one or two electrons trapped in the cavity, which leads to a rich chemistry57 and peculiar spectral features.58 The removal of the oxygen atom can take place in the bulk of the material or on its surface, giving rise to different spectroscopic signatures, often difficult to assign; theoretical models are therefore necessary for a complete understanding of these systems. This is at variance with the situation encountered in reducible oxides such as titanium oxide, zirconium oxide, vanadium oxides, or cerium oxide where the creation of oxygen vacancies usually results in the appearance of states in the gap corresponding to the presence of reduced cations. A complete review of the ground-state properties of these oxides in their stoichiometric and reduced forms has been provided by Ganduglia-Pirovano et al.59 The interaction of metal atoms and clusters with oxide surfaces is also a process not as simple as can be anticipated. In the case of noble-metal atoms or clusters deposited on nonreducible oxides such as MgO, the ground state is, in principle, quite straightforward and normally of closed-shell or non spin-polarized type60−66 or with spin polarization being due to the presence of adsorbates.67,68 However, this is not always the case as illustrated by the recent work of Moseler et al.69 on supported Pd clusters showing that Pd13 supported on MgO has a quintet state. This suggests that one needs to explore various electronic states to find the ground state, which is in line with previous results for Pd clusters supported in MgO, some of which were found to exhibit a magnetic moment.70 In the case of transition-metal atoms with unfilled d-shells, the interplay between Hund’s rule, favoring the atomic state of highest multiplicity for the groundstate electron configuration, competes with the chemical bond with the surface, which tends to quench the magnetic moment of the adsorbed atom or cluster. The result is not straightforward, and accurate theoretical studies are required to provide a physically meaningful answer. A similar situation can also be encountered when metal clusters are deposited above oxide surfaces and even when an open-shell molecule such as NO is adsorbed on a magnetic oxide such as NiO. The interaction takes place above a Ni cation with a local 3d8configuration, and the coupling of the unpaired electron

2. SURVEY OF ELECTRONIC STRUCTURE METHODS FOR GROUND AND EXCITED STATES In this section, we provide a unified point of view of the different theoretical methods used in the study of electronic structure and focus especially on those allowing the treatment of excited states. Details of the methods outlined in this section can be found in specialized references, such as the excellent recent review by Huang and Carter,73 monographs,74,75 and textbooks.76 Here we only aim to provide the common basis behind these methods and illustrate how they rely on the principles of quantum mechanics. The final goal of this section is to supply some hints to the reader concerning when and how these methods can be applied to problems involving excited states in oxide surfaces. Nevertheless, readers with a strong theoretical background are referred directly to section 2.3 or to section 3. Nowadays, the methods of electronic structure can be classified into two large families. On one hand, one finds ab initio methods of computation of electronic wave functions and, on the other hand, methods based on modern density functional theory. In the forthcoming discussion we attempt to focus mainly on the mathematical foundation and physical significance rather than on technical aspects of computer implementation. Nevertheless, it is important to point out that the main goal of the methods of quantum chemistry is to explore potential energy surfaces; this is normally done within the Born−Oppenheimer approximation, which consists of decoupling the (slow) motion of the nuclei from the (fast) motion of the electrons. This is possible because of the difference of several orders of magnitude in terms of mass between electrons and nuclei, which makes any coupling in the motion irrelevant, as long as the calculation of chemical properties is involved. Hence, nuclei are usually considered as classical objects, and their dynamics is studied by means of the Newtonian equations of motion, while electrons are quantum objects, and the description of electronic motion in an atom, a molecule, or a crystal refers to the principles of quantum mechanics. Clearly, both nuclear and electronic degrees of freedom carry along relevant information concerning the chemical properties of any system. Most importantly, in the Born−Oppenheimer approximation it can be shown that nuclei move in the potential field generated by the electrons, which obviously depends on the nuclear configuration. The electronic structure provides information about chemical bonds, magnetic 4458


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nucleus−electron potential (V̂ Ne), and the electron−electron repulsion (V̂ ee). In this way the first and third terms of the electronic Hamiltonian do depend on electronic coordinates only. In the framework of density functional theory (vide infra), the second term is referred to as the external potential. Another important remark is that the electronic Hamiltonian contains one- and two-body operators only whereas |Ψi⟩ contains information about the N electrons in the system. Therefore, one can obtain the total energy by making use of the one- and two-particle density matrices only, which are obtained by integrating over all coordinates of N − 1 and N − 2 electrons, respectively.80,81 The one- and two-particle density matrices can be obtained from the N-electronic wave function. However, density functional theory provides a practical alternative. Both wave-function- and density-functional-based methods will be described in the forthcoming subsections. Here, we just mention that solving eq 2 for |Ψi⟩ is equivalent to finding the family of {cki} coefficients in eq 1, and this in turn requires knowledge of the N-electron basis. This can for instance be obtained from the eigenstates of the Hamiltonian of noninteracting electrons, which are generally known because the Schrödinger equation, eq 2, for one-electron systems can be analytically solved. In the case of one-electron atoms or ions, the resulting one-electron states are the well-known atomic orbitals. The general form of an N-electron state is a superposition of all possible vector states that can be constructed by choosing a particular one-electron state for each electron in the system. This is generally known as a configuration interaction (CI) wave function. However, to fulfill the Pauli principle, the states describing an N-electron system must be antisymmetric, which leads to the well-known concept of Slater determinants. Thus, the CI wave function is a linear combination of all Slater determinants that can be constructed by distributing the N electrons in the one-electron states included in the, in principle, infinite, one-electron basis in all possible ways. For a twoelectron system, this takes the following form:

properties, and electronic transitions. These aspects as well will be briefly reviewed in the next subsection. The Born−Oppenheimer approximation is implemented in geometry optimization and transition-state location algorithms where the electronic structure is relaxed at a fixed nuclear configuration, the gradient of the energy with respect to the nuclear motion is calculated, the nuclei are moved to reduce the gradient, and the electronic structure is recalculated on the new structure. By exploring the multidimensional potential energy surface generated by the motions of the nuclei, it is possible to identify and characterize the stationary points corresponding to minima and transition states. The minima correspond to thermodynamically stable structures at a temperature of 0 K, whose relative energies identify the relative stability of conformers (in the case of molecules) or polymorphs (in the case of solid structures). Transition states are relevant in studies of catalysis, where they permit evaluation of the kinetic barrier of a reaction. Eventually, the second derivative of the energy with respect to the nuclear motion (Hessian matrix) may also be calculated. The analysis of the eigenvalues of the Hessian matrix allows one to ensure that the resulting structure corresponds to a true minimum or to a transition state in the corresponding potential energy surface. The eigenvalues also provide the normal modes, and hence, it is possible to simulate the vibrational properties of the system, as well as to characterize the thermodynamic stability of the system at any temperature and pressure, exploiting the stationary-state theory and the statistical thermodynamics. Energy derivatives are not trivial, and some methods offer special technical advantages when gradients or higher order derivatives are to be computed. In addition, the proper interpretation of electronic spectra requires simultaneously handling several electronic states and not only the ground state. The choice of a particular computational method must contemplate the problem to be solved and in any case is a compromise between accuracy and feasibility. 2.1. Quantum Mechanics and Wave Functions

We start our review of methods of quantum chemistry by recalling the fundamental aspects of quantum mechanics that are relevant to this purpose; for a deeper insight, see refs 77−79. The postulates of quantum mechanics establish that a pure state of a given system is described by a state or ket vector, |Ψi⟩, belonging to a mathematically well-defined vector space, i.e., the Hilbert space denoted by H. A complete description of the Hilbert space is beyond the scope of the present review; we will just mention that H is a vector space defined on the complex numbers set, it has the inner or scalar product defined, it has an infinite dimension, and it is complete. This last property is especially important and means that, for any vector |Ψi⟩ ∈ H, there exists a family of, in principle, infinite and complex numbers, {cki}, and an infinite family of vectors, {|Φk⟩}, such that |Ψ⟩ i =

∑ cki|Φk⟩

|Ψk⟩ =

(3)

the generalization to N electrons being straightforward. The only remaining point concerns the representation of the oneelectron states in a given basis. The usual approach consists of using the position representation which leads to the usual definition of the wave function in terms of electron coordinates, x⃗, thus finding the key one-to-one correspondence between quantum vector states and quantum wave functions. This oneto-one correspondence extends to the eigenvalue equations Ĥ |Ψk⟩ = Ek |Ψk⟩ ↔ Ĥ Ψk(x ⃗) = Ek Ψk(x ⃗)

(4)

which, as commented above, can be exactly solved for oneelectron systems, providing a rather simple way to obtain oneelectron basis sets. From this one-electron basis set, the Nelectron basis set can be constructed as described above. There are several possible choices for the one-electron basis sets which can be used to construct the N-electron basis. In nonrelativistic quantum mechanics, one-electron wave functions are always written as the (tensor) product of space, r,⃗ and spin, ω, parts. For the space part, different choices are possible,82−84 such as hydrogen-like orbitals, Slater-type orbitals (STOs), Gaussian-type orbitals (GTOs), plane waves (PWs), or other numerical representations.

(1)

Now, recall that the goal of any quantum chemical method is to obtain approximate solutions to the eigenvalue problem defined by the Schrödinger time-independent equation Ĥ |Ψ⟩ i = Ei|Ψ⟩ i

∑ cij ,k det|ΦΦ⟩ i j

(2)

where Ĥ = T̂ + V̂ NeV̂ ee is the usual electronic Hamiltonian hermitic operator in the Born−Oppenheimer approximation and includes the kinetic energy of the electrons (T̂ ), the 4459


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2.2. From Quantum Mechanics to Quantum Chemistry

molecular orbitals are used as a one-electron basis to construct the Slater determinants. However, use of atomic orbitals leads to the valence bond (VB) theory81 initially proposed by Heitler and London, whereas use of molecular orbitals leads, of course, to the molecular orbital configuration interaction (MO-CI) theory,76 the rigorous generalization of Mulliken’s initial molecular orbital theory. The VB wave functions are relatively easy to interpret85−87 because the different Slater determinants can be represented as resonant forms and usually only the valence space is considered within a minimal basis description. The reason for the use of such a limited space is that these VB wave functions are difficult to compute because of the use of a nonorthogonal basis set.88 On the other hand, the MOs are usually taken as orthonormal, a choice that permits very large CI expansions to be carried out and extended basis sets,89−93 with new and efficient algorithms still being actively investigated.94−97 Notice that, once a finite set of “2m” atomic/molecular spin orbitals is given, the dimension of the N-electron space and, hence, of the N-electron subspace is also fixed. This is because with N electrons and “m” spin orbitals, m > N, the number of Slater determinants that can be constructed is

To solve the eigenvalue problem outlined by eq 2 or, equivalently, to obtain the set of coefficients defining the wave function in eq 3 requires dealing with complicated differential equations. However, eq 2 can be transformed to a simpler matrix eigenvalue problem, leading to the well-known set of secular equations

∑ ⟨Φk|Ĥ |Φj⟩cji = Eicki ∀ k

(5)

Equivalently, defining ⟨Φk⟩|Ĥ |Φl = Hkl, one has

∑ Hkjcji = Eicki ∀ k

(6)

or in matrix representation

HC = CE

(7)

which is the general matrix equation to be solved in any quantum chemical problem provided an orthonormal set is used. Transforming the Schrödinger equation to the matrix form given by eq 6 or 7 reduces the problem of solving differential equations to a simpler eigenvalue problem, although one needs to avoid the, in principle, infinite dimension of the corresponding matrices. This is achieved by projecting eq 6 into a subspace S of finite dimension defined by the appropriate projection operator:

PŜ =

∑ |Φk⟩⟨Φk| k∈S

⎛ 2m ⎞ dim FCI = ⎜ ⎟ or more precisely ⎝N⎠ ⎛ m ⎞⎛ m ⎞ dim FCI = ⎜ ⎟⎜ N ⎟ ⎝ Nα ⎠⎝ β ⎠

(8)

if the system contains Nα and Nβ electrons with α and β spins, respectively. The dimension of the FCI problem (dim FCI) grows so fast that practical computations can be carried out only for systems with a small number of electrons. Therefore, the FCI method is often used to calibrate more approximate methods.98,99 The simplest N-electron wave function that can be imagined consists of a single Slater determinant. In this case, there is no eigenvalue problem and the energy is computed as an expectation value. Of course, constraining the wave function to just one Slater determinant largely reduces the variational degrees of freedom of the wave function. In fact, the energy is uniquely defined by the one-electron basis used to construct this particular Slater determinant. The one-electron basis set can be chosen in such a way that the energy expectation value is the lowest possible, i.e., variationally. In practice one usually chooses

.This permits definition of the restriction of the Hamiltonian to S as ̂ Ŝ ĤS = PŜ HP

(9)

and, consequently, projection of the matrix in eq 7 on S, namely HSCS = CSES

(11)

(10)

which is the matrix equation to be solved in practice. We must point out that solving eq 10 is exactly equivalent to the linear variational method, and hence, the eigenvalues of eq 10 are upper bound to the exact eigenvalues in eq 2. The S subspace can be systematically increased, which permits consistent improvement of the precision of a computational result up to the required accuracy or to the computational limit. This approach is the basis of the ab initio CI methods and, more precisely, of the full configuration interaction (FCI) approach because all possible configurations within the S subspace are explicitly included in the CI wave function. The FCI leads to the exact solution in a given subspace. All ab initio wavefunction-based methods of quantum chemistry intend to approximate the FCI solution because, in practical computations, FCI can only be solved for rather small systems. Notice that the dimension of the FCI matrix in eq 10 grows as N!. This fact is a natural consequence of the procedure used to construct an appropriate basis, {|Φk⟩}, of N-electron states. 2.2.1. Wave-Function-Based Methods in Quantum Chemistry. The preceding discussion clarified that the best attainable approximation to the wave function and energy of a system of N electrons is given by the FCI approach. In fact, this is the exact solution in a given finite subspace, and it is independent of the N-electron basis used, provided that the considered basis sets expand the same subspace. This is a very important property because it means that the total energy and the final wave function are independent of whether atomic or

|Ψ0⟩ =

1 det|ΦΦ i j...ΦN ⟩ N!

(12)

with spin orbitals defined through the well-known MO-LCAO method originally designed by Roothaan.100 The MO-LCAO scheme permits work in a finite subspace from the very beginning. The variational problem reduces to finding the coefficients expanding the spin orbitals with the constraint that they remain orthonormal. This leads to a set of Euler equations, which in turn lead to the Hartree−Fock equations, finally giving the ϕk set, although in an iterative way because the Hartree− Fock equations depend on the orbitals themselves. This dependency arises from the fact that the HF equations are effective one-electron eigenvalue equations:

f ̂ ϕi = εiϕi 4460

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where f ̂ is the well-known one-electron Fock operator, the sum of the kinetic energy, nuclear attraction energy, and Coulomb and exchange effective potential operators. These effective potentials average the interaction with the rest of the electrons, which, of course, is given by the orbitals themselves and, hence, must be solved in a self-consistent way. Hence, HF is synonymous with the self-consistent field method. Solving eq 13 is not simple, and it is usually transformed to a matrix form by expanding the orbitals in a given basis, which leads to FA = SAE

converge very slowly or even diverge. However, the MPn methods have a special advantage over the truncated CI expansions. In the DCI and related methods, the relative weight of the different excitations differs from that in the exact FCI wave function because of the normalization of the DCI wave function. This normalization effect introduces spurious terms, and as a result, the energy of N-interacting molecules does not grow as N. This is the so-called size extensivity problem and is inherent to all truncated CIs. On the other hand, the MP series is size-consistent order by order. Successful attempts to render truncated CI expansions size extensive have been reported over the years,104 although the resulting methods are strongly related to the family of methods based on the cluster expansion of the wave function.105−107 The coupled cluster (CC) formulation of the wave function can be derived from FCI as in the case of the DCI method, although here the terms included are not selected by the degree of excitation with respect to the HF determinant only. The additional condition is that the different terms fulfill the socalled linked cluster theorem.105 The resulting system of equations is rather complicated and is not usually solved by diagonalization but rather by means of nonlinear techniques, and the result is not variational.108−110 The truncated CI and CC methods perform rather well when used to approximate the ground-state wave function. This is because the HF determinant provides an adequate zeroth-order approach. However, this is not necessarily the case when several excited states are to be studied. The logical extension of the truncated CI expansion is the so-called multireference CI (MRCI) approach, where excitations, usually single and double, for a set of reference determinants are explicitly considered;111,112 the method is referred to as the MR(SD)CI method and has been reviewed very recently.113 Energies and MRCI wave functions are obtained by solving the secular eq 10. Again the concept is quite simple, but solving the eigenvalue problem is not a simple task, and the different computational approaches involve very smart ideas and specialized codes coupled to vector, parallel, or vector−parallel processors. The implementation of Neese114 is particularly well suited to spectroscopyoriented problems and includes several interesting possibilities. For problems of chemical interest, the dimension of the MRCI matrices is so large that often a small block of eq 10 is diagonalized and the effect of the rest is taken up to second order by means of perturbation theory in different partitions. The reference space can be constructed by selecting either important determinants or important orbitals. The first idea is used in the CIPSI115−117 method, whereas the second one is the basis of the CASMP2118 and CASPT2119−122 methods, where CAS stands for complete active space. The active space is defined once a subset of orbitals is chosen, and it is complete because an FCI is performed within this restricted orbital space. Except for the simplest Hartree−Fock approach, the logic of the methods that we have discussed is based on solving the secular problem in a finite subspace defined by a chosen oneelectron orbital basis or on finding suitable approximations. In all cases the orbital set is fixed and, usually, obtained from a previous HF calculation. Then the contribution of each Slater determinant in eq 10 or each cluster amplitude in the CC methods is obtained variationally, as in the different CI methods, through perturbation theory, as in the MP series and related methods, or by mixed approaches, e.g., the CIPSI method. Nothing prevents one from using the variational method to optimize the orbital set and the configuration

(14)

where A is the corresponding matrix grouping the coefficients defining the orbitals and S is the overlap matrix now appearing because the basis set orbitals are centered in different nuclei and, hence, are not orthogonal. The matrix in eq 14 is also solved iteratively, and the whole procedure is termed the HFSCF-LCAO method. An important remark here is that since a variational approach is used, the HF scheme is aimed at approximating the ground-state (of a given symmetry) wave function only. The resolution of the Hartree−Fock equations leads to 2m spin orbitals, but only N orbitals, i.e., the occupied orbitals, are needed to construct the HF determinant in eq 12. The rest of the spin orbitals, unoccupied in HF or virtual orbitals, can be used to construct additional Slater determinants. A systematic way to do it is substituting 1, 2, ..., N occupied spin orbitals with virtual orbitals, leading to Slater determinants with 1, 1, ..., N substitutions with respect to the HF determinant. The determinants thus constructed are usually referred to as single, double, ..., N excitations. Including all possible excitations leads to the FCI wave function. Clearly, the FCI wave function is invariant with respect to the orbital set chosen to construct the Slater determinants. However, using the Hartree−Fock orbitals has technical advantages because, at least for the ground-state wave function, the Hartree−Fock determinant contribution to the FCI wave function in eq 3 is by far the dominant term. The fact that the electronic Hamiltonian includes up to two-electron interactions suggests that double excitations would carry the most important weight in the FCI wave function; this is indeed found to be the case. Therefore, one may design an approximate wave function in which only the reference Hartree−Fock determinant plus the double-excited determinants are included. The result is called the double-excited CI (DCI) method and is routinely used in ab initio calculations. The practical computational details involved in DCI are not simple and will not be described here. Adding single excitations is important to describe some properties such as the dipole moment of CO76 and leads to the SDCI method. Extensions of SDCI by adding triple or quadruple excitations (SDTQCI) are also currently used, although the dimension of the problem grows very rapidly. Nevertheless, this type of method allows one to describe excited states in a rather accurate way and has long been used to interpret electronic spectra.101 The truncated CI methods described above are variational, and finding the energy expectation values requires the diagonalization of very large matrices. An alternative approach is to estimate the contribution of the excited determinants by using the Rayleigh−Schrödinger perturbation theory up to a given order. This is the basis of the widely used MP2, MP3, MP4, ... methods, which use a particular partition, the Möller− Plesset one, of the electronic Hamiltonian and an HF wave function as the zeroth-order starting point.102,103 A disadvantage of perturbation theory is that the perturbation series may 4461


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a nondegenerate ground state the one-electron density determines (up to an arbitrary constant) the external potential (the nucleus−electron potential) and hence the electronic Hamiltonian, the ground-state energy, and, indirectly, the electronic wave function. They have also proven that there is a universal functional, so that, for any N-electron system, the total energy is a functional of the density; i.e., given a density, there is a mathematical rule that permits the exact ground-state energy to be obtained. The resulting theoretical framework is nowadays referred to as density functional theory or simply DFT.80 Unfortunately, the universal functional is unknown, and one could think that, for practical purposes, DFT is useless. This view cannot be more incorrect, and DFT has become almost the standard choice for electronic structure calculations on molecules, clusters, surfaces, and solids, with thousands of papers published and new, improved, and more accurate functionals proposed almost every year. The practical use of DFT relies on a second theorem also proven by Hohenberg and Kohn. This is a variational theorem stating that the groundstate energy is an extreme (a minimum) for the exact density. Later, Kohn and Sham proposed a general framework which permits the practical use of DFT. In the Kohn−Sham formalism, one assumes that, for any real system of electrons, there is a fictitious system on N noninteracting electrons experiencing the real external potential and showing exactly the same electron density as the real system. This allows one to define a convenient reference of N one-electron systems as a Slater determinant. In this way DFT permits handling of discrete and periodic systems (see also the next section). The reference system allows a trial density to be obtained, but one still needs to compute the energy of the real system, and therefore, a model for the unknown functional is needed. To this purpose, the total energy is written as a combination of terms, all of them depending on the one-electron density only:

contribution at the same time. This is the basis of the multiconfigurational self-consistent field (MCSCF) methods, which are the logical extension of HF-SCF to a trial wave function made as a linear combination of Slater determinants.123 The mathematical problem is conceptually very similar to that of the HF-SCF approach, namely, finding an extreme of a function (the energy expectation value) with some constraints (orbital orthonormality). The technical problems encountered in MCSCF calculations were much more difficult to solve than those of the single-determinant particular case. One of the problems faced by the earlier MCSCF methods was the poor convergence of the numerical process and the criteria to select the Slater determinants entering into the MCSCF wave function. The first problem was solved by introducing quadratically convergent methods124,125 and the second one by substituting the determinant selection by an orbital selection and constructing the MCSCF wave function using the resulting CAS. The resulting MCSCF approach is known as CASSCF and turned out to be a highly efficient method.126−128 The CASSCF wave function is always precisely the zeroth-order wave function in the CASMP2 and CASPT2 methods and in CIPSI if desired. The CASSCF wave function has some special features which are worth mentioning. It is invariant with respect to rotations (linear combinations) among active orbitals. When the CAS contains all valence orbitals, the CASSCF wave function is equivalent to the wave function obtained by the spin-coupled valence bond method129−131 when all resonant forms involving valence orbitals are included. Before closing this section, we mention that, in practice, configuration-state functions (CSFs) are commonly used instead of Slater determinants. A CSF is simply a linear combination of determinants with coefficients fixed so as to have an eigenfunction of Ŝ2, the total square spin operator. The fixed coefficients are often obtained with the assistance of group theory.132 This choice ensures that truncated CIs are spin eigenfunctions and reduces the dimension of the secular problem. We end this short review on ab initio wave-function-based methods by noting that all of them can be applied to the embedded cluster model described in the next section whereas Hartree−Fock, MP2, and some sort of couple cluster expansion have been extended to account for periodic symmetry, although they remain applicable to the electronic ground state only.133−137 2.2.2. Density-Functional-Theory-Based Methods in Quantum Chemistry. The Schrödinger equationeq 2 or 4provides a way to obtain the N-electron wave function of the system, and the approximate methods described in the previous section permit reasonable approaches to these wave functions. From the approximate wave function the total energy can be obtained as an expectation value, and the different density matrices, in particular the one-particle density matrix, can be obtained in a straightforward way as ρ0 (X⃗ ) = N

E[ρ] = Ts[ρ] + Vext[r ] + VCoulomb[ρ] + VXC[ρ]

The first term is the kinetic energy of the noninteracting electrons, the second term accounts for the contribution of the external potential, and the third one corresponds to the Coulomb repulsion. Finally, the fourth term accounts for all the remaining effects, namely, the contribution to the kinetic energy due to the fact that electrons are interacting, the exchange part due to the Fermi character of the electrons, and the correlation contribution of the electron densities. Obviously, the success of DFT is strongly related to the ability to approximate the exchange and correlation energy (EXC) in a sufficiently accurate and, if possible, systematic way,139 although a number of fundamental challenges still remain, as discussed in the recent excellent review by Yang et al.140 In any case, eq 16 plus the Hohenberg−Kohn variational theorem permits modification of the density by optimizing the orbitals used to construct the Slater determinant from which the density of the reference system is constructed. This determinant is referred to as the Kohn−Sham determinant, and the orbitals defining it are usually termed Kohn−Sham orbitals. In addition, the Kohn−Sham orbitals can be expressed in a given basis set as in the case of the Hartree−Fock method commented above. When a CGTO basis set is chosen, one has the LCGTO-DF methods.141−143 The orbital variation must preserve the orthonormality of the orbitals in the Kohn−Sham determinant to hence maintain the number of electrons. The overall procedure is then very similar to the well-known

∫ Ψ*0 (X1⃗ , X⃗2 , ..., XN⃗ ) Ψ0(X1⃗ , X⃗2 , ..., XN⃗ ) d X⃗ ...d X⃗ 2

(16)

N

(15)

where the subindex zero indicates that this is the ground-state density arising from the ground-state wave function, Ψ0. The integration in eq 15 is carried out for the spin and space coordinates of all electrons but one. In 1964 Hohenberg and Kohn proved a theorem that states that, in certain conditions, the inverse of this proposal also holds.138 They proved that for 4462


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ized DFT methods suffer from the same drawbacks as the wellknown UHF formalism. This is especially important when attempting to study systems with many open shells as in isolated atoms or diradicals. The unrestricted Kohn−Sham determinant, in these cases, corresponds necessarily to a mixture of the different possible multiplets compatible with the electronic configuration, which results in spin contamination. Here one may argue that since DFT only makes use of the density provided by the Kohn−Sham determinant, which in turn represents the density of the noninteracting reference system, it is not necessary to be bothered with spin contamination. It has also been claimed that spin contamination in DFT is less severe than in UHF.152 However, one must be aware that ignoring the effect of spin contamination can lead to misinterpretations and to relevant errors.153,154 In spite of the inherent simplifications, LDA (and LSDA) predictions on molecular geometries and vibrational frequencies are surprisingly good. However, bonding energies are much less accurate and require going beyond this level of theory. This is also the case when dealing with more difficult systems such as biradicals or more delicate properties such as magnetic coupling in binuclear complexes or ionic solids.153,155 In 2000, Perdew and Schmidt introduced the term “Jacob’s ladder” 156 to refer to the hierarchy of density functional methods of increasing complexity and, hopefully, accuracy, where each rung on the ladder gets closer to the exact universal functional. Following this ladder, the DF methods that go beyond the LDA can be grossly classified into three groups: GGAs, meta-GGAs, and hybrids. GGAs are functionals based on the generalized gradient approach,157−159 where the explicit calculation of the EXC[ρ] contributions involves not only the density (ρ), but also its gradient (∇ρ), as suggested earlier by Becke for the exchange term160 and by Lee−Yang−Parr for the correlation potential161 (the BLYP functional). Note the LYP correlation potential in BLYP is indeed based on the work of Colle and Salvetti on the correlation factor.162 Meta-GGA functionals also add the Kohn−Sham orbital kinetic energy terms.163 Hybrids, finally, include part of the nonlocal Fock exchange, following the original idea of Becke in 1993.164 More recently, very accurate density functionals have been proposed, such as the Minnesota meta-hybrid functionals developed by Zhao and Truhlar165,166 with very new and promising developments167,168 and the range-separated methods of Scuseria et al.169,170 The most popular hybrid method is B3LYP, where the number indicates that three parameters are fit to the experimental data. Fitting to experiment introduces a certain degree of empiricism, but, on the other hand, has led to the most accurate methods. This is particularly the case for M11-L developed by Peverati and Truhlar.167,168 This functional is of a local type, which makes it attractive for large-scale calculations in solids. Indeed, M11-L has the best average accuracy for a set of 338 chemical data and performs extraordinarily well for a subset of 328 data outside the reference database. The authors of the so-called SOGGA11-X167 claim that this functional has a better overall performance for a broad chemical database than any previously available global hybrid generalized gradient functional. In addition, SOGGA11-X includes second-order corrections in the density gradient. Aside from the great advances in the development of new exchange-correlation functionals mentioned above, it is clear that the question of which functional provides the best chemical accuracy is still under discussion, and solid-state physics and molecular chemistry tend to have different answers. In the

Hartree−Fock methods, and the orbitals minimizing eq 16, while preserving orthonormality, are those satisfying a oneelectron eigenvalue problem such as ̂ ϕ (r ) = ε ϕ (r ) hKS i i i

(17)

Here, in addition to the kinetic energy of the noninteracting electrons and external (nucleus−electron) and classical Coulomb operators, similar to the terms in the Fock operator of the Hartree−Fock method, the ĥKS one-electron operator contains the exchange and correlation effective potentials (vx[ρ] and vc[ρ]) as well. In this formulation, vx[ρ](r) and vc[ρ](r) are local one-electron operators acting on r coordinates only. The orbitals and one-electron energies in eq 17 are referred to as Kohn−Sham orbitals and Kohn−Sham orbital energies, which, although not equivalent to the Hartree−Fock orbitals, are used for interpretation purposes in a similar way.80 The Kohn−Sham orbitals allow a ground-state trial density, ρ0, to be obtained in a straightforward way as ρ0 (r ) =

∑ |ϕi(r)|2 i

(18)

Mathematically, the exchange and correlation potentials are the functional derivative of the corresponding energy contributions in eq 16.80 Once the effective potentials vx[ρ] and vc[ρ] are known, solving the Kohn−Sham equations is similar to solving the Hartree−Fock equation, with the important difference that here one could find the exact solution if VXC[ρ] were the exact one. Notice that there is no guarantee that the final electron density arises from a proper wave function of the corresponding Hilbert space through eq 15. This is the well-known representability problem, which, however, does not affect the practical use of DFT and therefore will not be further discussed here. The interested reader is addressed to more specialized literature.80 Several approaches to VXC[ρ] have been proposed in the past several years with increasing accuracy and predictive power.73,140,144 However, in the primitive version of DFT, the correlation functional was ignored and the exchange part approximated following Slater’s ρ1/3 proposal; this method was known as Xα. In 1980, Vosko, Wilk, and Nusair145 succeeded in solving the electron correlation for a homogeneous electron gas and establishing the corresponding correlation potential; the resulting method also included the exchange part and is nowadays known as the local density approximation (LDA). It has been successful in the description of metals, bulk, and surfaces, although it has experienced more difficulties in the description of molecules and ionic systems. LDA, for instance, incorrectly predicts NiO to be a metal,146,147 whereas experimentally this is a well-characterized antiferromagnetic insulator.148 Here, it is worth recalling that the Kohn−Sham equations were initially proposed for a system with a closedshell electronic structure and hence were suitable to study singlet electronic states only. The study of open-shell systems can be carried out using a spin-unrestricted formalism, similar to that proposed by Pople and Nesbet for the Hartree−Fock wave function, thus leading to the so-called unrestricted Hartree−Fock (UHF) method,149 where different spatial orbitals are used for α and β spin orbitals. In the case of LDA, the resulting formalism is known as the local spin density approximation (LSDA) or simply local spin density (LSD)150,151 and is the basis for almost all applications of DFT to open-shell-containing systems. However, spin-polar4463


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concerns the calculation of excited states, a description of the methods used to approach them is also necessary. 2.3.1. Configuration Interaction and Related Methods. The configuration expansion in eq 3 is the basis of the family of methods traditionally used to approach excited states. The basic principle is that since the roots of the secular equation are obtained from the linear variational method, they correspond to upper bounds of the corresponding electronic states. The different truncated CI expansions, especially the one including only single excitations (CIS), together with the more accurate MRCI or CASPT2 methods provide a direct way to approach excited states, as previously discussed. However, one must notice that the quantity of interest is not the total energy of different electronic states but their difference. Specific techniques have been introduced which are aimed at directly obtaining the relevant energy difference, such as the differencededicated configuration interaction (DDCI) method of Caballol et al.185 This method is based on the fact that there are contributions which are expected to cancel in energy differences and hence are directly excluded from the CI expansion, with a substantial reduction of the computational cost. The closely related spectroscopy-oriented configuration interaction method introduced later by Neese114 makes use of the ideas behind DDCI. Finally, one should be aware that, at present, this type of method can only be applied to finite systems, implying that periodic symmetry cannot be exploited. 2.3.2. Time-Dependent Density Functional Theory. Excitations between states of different symmetries, either spin or space, have long been approximately calculated as the energy difference of the two corresponding ground states. This approach is referred to as ΔSCF, as it follows the ideas used in the earlier days of quantum chemistry for HF calculations. Nevertheless, while the ΔSCF strategy is well justified in the case of HF calculations, the situation is less clear with DFT, since it neglects any symmetry dependence of the exchangecorrelation functional.80 This becomes very problematic when one aims to obtain the relative energy of different atomic multiplets, especially when these cannot be represented by a single Slater determinant. In such a case, a possibility is to use the sum rule of Ziegler, Rauk, and Baerends,186similarly to the approach of Bagus and Bennet187 in their pioneering work concerning the energies of atomic multiplets in the framework of SCF-Xα. Another possibility is to rely on broken symmetry approaches.153−155 The most accurate way to approach excitations in the framework of DFT is its time-dependent implementation, usually called TD-DFT, based on the Runge−Gross theorems,188 which may be seen as a consequence of the timedependent Schrödinger equation. TD-DFT thus extends the standard formulation of DFT described above to timedependent phenomena. Moreover, it allows one to continue using the reference noninteracting system and to obtain an equivalent set of one-electron Kohn−Sham equations which now are time dependent. This set of equations is usually referred to as Runge−Gross equations or time-dependent KS equations (TDKS):

wave-function-based methods, it is possible to check the accuracy of a given level of theory by systematic improvement of the basis set and the level of treatment of electron correlation. Unfortunately, this is not completely feasible in the framework of DFT, although there are prescriptions for the design and selection of density functional approximations,171 and the work of Truhlar et al.165−168 shows a way to systematically improve EXC[ρ] even if this sometimes requires adjusting some of the parameters included in the functional definition. Nevertheless, the accuracy of a given choice for EXC[ρ] has to be established, whenever possible, by comparison to experimental data or accurate wave-function-based calculations. Otherwise, one can study the dependence of a test-case property by varying the adopted functional. This approach, however, assesses the sensitivity of the property under study to the functional, but does not clarify which functional is more accurate. Fortunately, for many systems the choice of a DFT approximate method is not critical. In particular, energy differences do not show a dramatic dependence on the adopted functional. However, in other cases, such as the description of transition-metal−oxide interfaces, the choice of the functional has been shown to be crucial. This is also the case for the adsorption of CO on MgO(100), which constituted a challenge for wave-function-based methods as well as for current density functional approximations because of its weak interaction character, which renders the experimental determination difficult too. Using a method of local increments and highlevel explicitly correlated wave functions, Staemmler172 has recently suggested that the adsorption energy of CO on MgO(100) is 2.86 kcal/mol, which is also in agreement with the latest experimental estimate.173−175 Valero et al.176 have reported adsorption energy values calculated with the M06-2X and M06-HF functionals165,166 which are ∼6 kcal/mol, in qualitative agreement with experiment, although from a quantitative point of view the result is not so satisfactory since it overestimates the best experimental value by 50− 65%.173−175 Interestingly, the PBE functional also gives reasonable results for the binding energy (1.9−2.8 kcal/mol) and is comparable in accuracy to that predicted by M06-HF, whereas the results predicted by LDA, BLYP, and B3LYP are less satisfactory. The interaction of CO and NO with NiO or Ni-doped MgO provides yet another example where DFT methods need to be used with special care.177 Finally, it is important to note that one of the main advantages of DFT methods, compared to wave function methods, is the potential N3 scaling with respect to the dimension of the basis set (N) when using an appropriate density fitting, whereas for Hartree−Fock one has an N4 dependence which grows even more for explicitly correlated wave functions. Moreover, DFT methods have been proposed which exhibit a linear dependence (order N) on the number of electrons or the dimension of the basis set,178−181 although progress has also been achieved in linear scaling for wavefunction-based methods.182,183 Nevertheless, order N methods continue to be mostly developed in the context of DFT as recently reviewed by Bowler and Miyazaki.184 2.3. Describing Excited States

i

The methods described so far are applicable to the electronic ground state and, in some cases, to excited states as well, with an enormous number of applications published over the past 30 years. However, since the main focus of the present review

∂ ̂ (t ) ϕ (r , t ) ϕ(r , t ) = hKS i ∂t i

(19)

where the Kohn−Sham one-particle Hamiltonian and orbitals depend on the time, and so does the density 4464


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∑ |ϕi(r , t )|2 i

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requirement when using a plane wave basis set, to date this formalism is only implemented for finite systems.

(20)

2.4. Excited States from Band Structure and Quasiparticle Excitation Calculations

.In particular, one should stress that the external potential is now time dependent and that the exchange and correlation potentials are now a functional of both the time-dependent density and the initial state of the system, which normally is the ground state. Rigorously speaking, vx[ρ](r) and vc[ρ](r) should now be written as vx[ρ](r,t) and vc[ρ](r,t) to highlight the dependency of both the functional and density on time. The time-dependent exchange-correlation potential, vxc[ρ](r,t), is a much more complex mathematical object than the universal ground-state vxc[ρ](r), since it includes all the quantum mechanics of all electrons subjected to any possible time-dependent perturbation. Therefore, all practical applications of TD-DFT rely on an adiabatic approximation in time, stating that v xc[ρ(r , t )] = vxcgs[ρ]ρ = ρ(r , t )

Excited states in solids, for instance the band gap in oxides and semiconductors, have long been approached from energy differences between one-electron energy levels in the reciprocal space. This information can be extracted from the usual band structure obtained by either HF or DFT methods. Because of the neglect of instantaneous electron−electron correlation, the HF-calculated band gaps of oxides are usually overestimated. On the other hand, the calculation of the classical Coulomb interaction in the Kohn−Sham implementation of DFT inevitably includes the so-called self-interaction error, i.e., the interaction of each electron with its own electron density, which results in a severe underestimation of band gaps in oxides. A qualitatively good description of the band gap is provided by hybrid functionals, as earlier demonstrated by Muscat et al.204 for periodic calculations using GTO basis sets and, later, by the work of Scuseria et al.205 on UO2 using periodic boundary conditions and a GTO basis set206,207 and the work of Kresse et al.208,209 in periodic calculations using plane waves. The application of hybrid functionals to the study of the electronic structure of oxides and other strongly correlated solids is, hence, rather recent, and even if hybrid DFT approaches provide a notable improvement over the description arising from GGA, they still suffer from a certain degree of empiricism, due to the amount of nonlocal Fock exchange that needs to be included in the exchange potential. Related to this issue is the finding by Izmaylov and Scuseria199 revealing that when employing semilocal functionals (such as LDA, GGA, or meta-GGA), the lowest excitations in TD-DFT are equivalent to the KS minimal band gap values. On the contrary, a dependence on the k vectors of the reciprocal space is introduced through the nonlocal Fock exchange terms in hybrid functionals, which makes the lowest excitation a collective excitation of independent Kohn−Sham transitions.199 The introduction of a part of the Fock exchange is not the only strategy to improve the description of the band gap in oxides and related material where LDA and GGA fail even qualitatively. A commonly used approach is to add two parameters to the LDA or GGA exchange-correlation functional with the idea to mimic an effective onsite Coulomb (U) term, as in the Hubbard model, and an exchange (J) term between electrons with the same orbital angular momentum.210,211 A simpler implementation consists of using a single effective parameter defined as Ueff = U − J.212 The resulting method is often referred to as DFT+U, or, more specifically, as LDA+U or GGA+U. One advantage of LDA+U or GGA+U over the hybrid functionals described above is that the additional computational cost is minimal, although it has the problem that the U parameter is usually effective only on one band (3d, 4d, or 5d in the case of transition-metal oxides), whereas the introduction of a part of the Fock exchange modifies the whole electron density of the system. The value of U may be determined empirically or estimated from first-principles approaches.213−215 Among the various possible approaches going beyond the DFT band structure calculations to predict the band gap of oxides, a growing interest is nowadays dedicated to the socalled GW quasiparticle approach. The quasiparticle approach permits simulation of the ejection or the absorption of an

(21)

which allows one to replace the time-dependent exchangecorrelation potential, vxc[ρ](r,t), with the ground-state exchange-correlation potential, vgs xc[ρ]ρ=ρ(r,t), acting on the density corresponding to time t. This permits use of all the approximations implemented in standard static DFT codes for ground-state calculations in the TD-DFT formalism. Nevertheless, the resulting technical aspects are not trivial, and the reader is referred to the specialized recent literature.189−191 Here we add that, in general, applications of TD-DFT are restricted to a linearized form of eqs 19 and 20 assuming a small external perturbation and attempt to obtain the first-order density response in the frequency domain. To extract electronic excitations, the response of the system to an external electric field is calculated as in standard perturbation theory. Several practical routes have been adopted for extracting excitation energies from TD-DFT response theory. An important contribution is the one due to Casida,192,193 who converted the optical response problem into an eigenvalue problem and showed that TD-DFT largely resembles a configuration interaction limited to single excitations (CIS), especially in the so-called Tamm−Dancoff approximation194 and using the Kohn−Sham orbitals to define both the reference system and the excited states.195,196 Nevertheless, TD-DFT appears to be more robust than CIS based on an HF reference197 and has the advantage of its computational simplicity, compared to other quantum-chemical methods such as general CI or MRCI approaches. In any case one should not forget that the performance of TD-DFT calculations is affected by the well-known shortcomings of the state-of-theart density functionals. Almost all applications of TD-DFT have been carried out for molecular (nonperiodic) systems, and their use to approach excited states in oxide surfaces is therefore resctricted to embedded cluster models (see section 3.1). The only exceptions so far are the theoretical development of Hirata et al.198 to extend CIS, TDHF, and TD-DFT to onedimensional systems, the application to a one-dimensional chain of hydrogen molecules,199 and the recent implementation of TD-DFT by Bernasconi et al.200 to periodic systems in the CRYSTAL code.201 Here, it is important to point out that it is possible to carry out TD-DFT calculations using a plane wave basis set as, for instance, in the TurboTDDFT code202 based on the formalism developed by Walker et al.203 However, in spite of using periodic boundary conditions, which is a necessary 4465


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where f m and f n are the occupations of the mth and nth states, respectively, while ε indicates the corresponding eigenvalues. The self-energy operator, M, can be thus written as a series expansion of “GW” products, which explains the name normally attributed to this method, according to the following equation, where, for simplicity, (1,2) stands for ([x,t],[x′,t]) and so on:

electron caused by an incident photon, i.e., the transition from an N-electron state to an N ± 1 state. This in principle allows for a direct comparison of the calculated results with photoemission or inverse-photoemission experiments. The first formulation of the GW method dates back to the work from Hedin in 1965.216 In this seminal paper, a set of selfconsistent equations are derived, providing a series expansion of Green’s one-particle equation (referred to as G) in terms of the screened electrostatic potential (referred to as W). As a starting point to explain this approach, it is worth mentioning that Green’s function is a mathematical tool to solve any nonlinear differential equation. We can thus reformulate HF eq 13 in terms of Green’s functions. For each eigenvalue, ε, the one-particle Green function, G(x,x′,ε), satisfies [ε − h(x) − V (x)]G(x , x′, ε) −

M(1, 2) = iℏG(1, 2) W (1+ , 2) − ℏ2

G(3, 4) G(4, 2) W (1, 4) W (3, 2) d(3) d(4)

The symbol 1 in eq 25 stands for [x,t+δt]. The whole formalism is rather complex, and a more detailed description can be found in the review paper of Huang and Carter.73 The mathematical complexity of this formalism is accompanied by an equal complexity in the computational implementation. Nevertheless, the ever-growing computational facilities have made GW calculations indeed attractive in solidstate computational simulations. In particular, we mention the recent implementation by Shishkin and Kresse in the code VASP.218 These authors suggested a first approximation to adapt this method to the projector-augmented wave (PAW) description of the core electrons. It consists of splitting the problem of the calculation of the self-interaction energy into two parts: inside the pseudopotential spheres (i.e., close to the nucleus positions), the GW Hamiltonian is approximated as the Hartree−Fock Hamiltonian, while on the plane waves grid the self-energy operator is calculated on a set of discrete grid points, ω, and has the form

(22)

In eq 22, x represents the coordinates of one generic electron (space and spin), h(x) is a one-electron Hamiltonian operator containing the kinetic energy and the external potential (nucleus−electron), V(x) is just the classical Coulomb potential, and the right-hand term is the usual Dirac δ function. M in eq 22 represents the self-energy operator, describing the correlation effect of the many-particle system. A series expansion of M gives the Hartree−Fock exchange potential, MHF, as the first term. If MHF is used as the self-energy operator in eq 22, a self-consistent solution of this equation will provide Green’s function built up as one-particle functions within the Hartree−Fock approximation. In this sense, the HF methods can be regarded as a particular case of Hedin’s formulation. Therefore, the eigenvalues, ε, will provide a manifold of occupied and virtual HF states. By considering further terms in the expansion of M, it is possible to describe a multielectronic system going beyond the mean-potential approach of Hartree− Fock. The most obvious option would be to expand M as a series of Coulomb electron−electron interactions. However, the convergence of such an expansion of Coulomb potential, V, is rather slow. A good solution is to consider, instead of V, an expansion in terms of the screened dielectric potential, W. The screened potential between two generic electrons, W([x,t], [x′,t]), gives the potential at time t in position x generated by an electron at x′, taking into account the reciprocal screening of the electrons. It consists of a Coulomb term multiplied by an inverse dielectric tensor, η−1: W (x , x′) =

4πe 2 −1 η (x , x′) |x − x′|

M(x , x′, ω) =

∑ (fm m,n

− fn )

∫ eiωδG(x , x′, ω + ω′) W (26)

where δ is a positive infinitesimal. The screened dielectric potential is constructed as described in eqs 23 and 24. For each special k-point, the quasiparticle energies, εQP nk , can be iteratively calculated as εnkQP = Re[⟨ψnk|h + V + M(εnkQP , ψnk)|ψnk⟩]

(27)

Since eq 27 already depends on εQP nk and ψnk, it is necessary to provide an initial guess for the eigenvalues and eigenvectors of the system. These are normally taken from a prerun performed at the DFT level. The simplest way to obtain the quasiparticle energies is to proceed perturbatively, performing a single iteration on eq 27 starting from the εnk and ψnk Kohn−Sham eigenvalues and eigenvectors. This approach is normally referred to as G0W0. A second possibility, namely, GW0, consists of iterating on eq 27 until convergence on the quasiparticle eigenvalues, εQP nk , is achieved, updating at every cycle Green’s function in M(εQP nk ), but keeping the screened dielectric potential obtained at the G0W0 level. The GW approach consists of updating both Green’s function and the dielectric potential at every iteration. It is worth noting that, in all three aforementioned schemes, the iterative process concerns the eigenvalues, εQP nk , but not the eigenvectors, |ψnk⟩, which are the ones calculated at the chosen DFT level. It is also possible to reach a fully self-consistent solution of eq 27 by updating both eigenvalues and eigenvectors (i.e., orbitals), and this leads to the so-called self-consistent GW methods (sc-GW0 and sc-GW). While in principle sc-GW is theoretically more correct, it implies a

(23)

⟨ψm|e iGx|ψn⟩⟨ψm|e iGx ′|ψn⟩ εm − εn

i + 4π

(x , x′, ω′) d(ω′)

It is worth mentioning that, for highly polarizable systems in particular, W ≪ V, ensuring though a more rapid convergence of the series expansion of M. The dielectric tensor, in the formulation derived from the random-phase approximation,217 depends on the quantity χq(x,x′) called the one-particle polarizability, which is itself a functional of the eigenvalues and eigenvectors of eq 22: χq (x , x′) =

(25)

+ ... +

∫ M(x′, x″, ε) d(x″)

= δ(x , x′)

∫ G(1, 3)

(24) 4466


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Scheme 1

theless a very important improvement which opens the door to a more rigorous study of excited states in extended solids in general and oxide surfaces in particular. As remarked above, it should be noted that the quasiparticle approach describes non-neutral excitations related to photoemission phenomena. Single neutral excitations, such as the ones involved in optical spectroscopy and electron energy loss spectroscopy (EELS), can be regarded as an excitation involving two (quasi)particles, a (quasi)electron, and a (quasi)hole. The excitonic effects can be described in terms of a linear combination of particle−hole pair states, ∑phcph|ph⟩, where |ph⟩ is a Hartree-like product containing a particle in state p and a hole in state h and cph are the amplitudes of the particle−hole mixed states. The amplitudes and the excitation energies, E, can be obtained from the Bethe−Salpeter equation (BSE), which can be written as an eigenvalue equation according to

dramatic augment in terms of computer requirements compared to GW and sometimes leads to a crude overestimation of the band gap compared to the experiment. This is often attributed to the spurious energy transfer from quasiparticle peaks to satellites and the consequent reduction of the screening. According to the discussion that appeared in the literature, this might be due to neglect of the vertex corrections, i.e., truncation of the series expansion of the M operator at the first term after the HF zero term. In a recent paper,219 Shishkin and Kresse assessed the performance of the G0W0, GW0, and GW approximations by calculating the band gap of a series of insulators and semiconductors, spanning from ionic crystals such as MgO and CaO to covalent semiconductors such as GaAs to weakly bound crystals such as Ne and Ar. The results indicate that G0W0 already remarkably improves the poor performance of GGA in predicting the band gap of these materials, but still systematically underestimates the band gaps of each considered solid compared to the experiment. The performances of GW0 and GW are in much nicer agreement with the experiment. Carter and co-workers220 have obtained interesting results in calculating the band gap and edges of the valence and conduction bands of several metal oxides by correcting the DFT+U band gap amplitudes simply with G0W0 calculations. This surely allows for a modification of the whole density, whereas the inclusion of the U term only corrects for the band where the correction is applied. Beyond the calculation of the band gap of stoichiometric bulk materials, two independent studies have recently been published where the GW technique is used to study electronic excitations of oxygen vacancy (F and F+) centers in MgO,221 and in MgO, CaO, and ZnO,222 and results will be described in a forthcoming section. Here, we mention that the results thus obtained highlight the good performance of GW, in particular in the case of these point defects, extending the applicability of this approach beyond bulk oxides and related systems. The GW methods thus appear to be a rather general tool, applicable to a rather wide range of, stoichiometric or defective, materials. The results represent a clear improvement of the poor GGA performance in predicting electronic excitation energies. The main drawbacks, at the state of the art, are due to the high computational cost, which makes these calculations unfeasible for supercells larger than 50−60 atoms, and to the lack of a general, correct, self-consistent GW implementation. The partially self-consistent schemes, where only the eigenvalues are updated, depend indeed on the GGALDA wave function provided as input and represent never-

[(εpQP − εhQP) − E]cph +

∑ K ph,p ′ h′cp ′ h′ = 0 p′h′

εQP p

(28)

εQh

where and are the quasiparticle eigenvalues of the particle and the hole, respectively. The sum in eq 28 runs over the kernel Kph,p′h′, which can be split into two terms: exc dir K ph , p ′ h ′ = K ph , p ′ h ′ + K ph , p ′ h ′

(29)

Kexc ph,p′h′ is the particle−hole exchange operator, which has the usual form exc K ph ,p′h′ =

∫ dx dx′ ψp*(x) ψh(x) |x −1 x′| ψp′(x) ψh*′(x′) (30)

Kdir ph,p′h′

is the screened Coulomb operator, where a term depending on the inverse screened dielectric tensor is applied, similarly to the case for GW (see eq 23): dir K ph ,p′h′ =

∫ dx dx′ ψp*(x) ψh(x′) η−1(x , x′) |x −1 x′| ψp′(x) ψh*′(x′)

(31)

. The solution of the BSE involves, thus, a DFT calculation to obtain a set of Kohn−Sham eigenvalues and eigenvectors, followed by a GW calculation to obtain the quasiparticle energies and the dielectric screening, prior to eqs 28−31 being solved. Interestingly, Onida et al. have shown189 that the BSE is 4467


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for several quantum methods with modern computer facilities. However, very demanding methods, for instance coupled cluster, would require smaller models. As a general rule, the local quantum region contains the atoms directly involved in the property under study and the first shell of atoms around those centers. However, note that there are many cases (e.g., geometry optimizations) for which such a local region is not sufficient and a larger number of atoms need to be included. In any case, one should always check that the property of interest is converged with respect to the size of the quantum region. This offers an internal check on the consistency of the model. 3.1.2. Adding Static Short-Range Interactions. Continuing with the example of MgO(100), once the quantum region is defined, it is necessary to set up, around the quantum region, an embedding scheme which must account for three different interactions: short-range Coulomb forces, long-range Coulomb effects, and long-range polarization. In the first place, it is necessary to consider the short-range electrostatic repulsion between the atoms in the quantum region and those surrounding them. The quantum mechanical nature of this interaction does not permit one to rely on a simple embedding based on classical models, and more elaborate schemes have to be invoked. Several approaches have been developed over the years, but we will only mention here the total ion potential (TIP) embedding230,231 and the more rigorous ab initio model potential (AIMP) scheme232,233 recently improved to better describe the Pauli repulsion term234 and implemented as embedding fragment ab initio model potentials in CASSCF/ CASPT2 calculations of doped solids.235 Both embedding schemes absorb the interaction of the quantum cluster electrons with the surroundings into an effective one-electron Hamiltonian acting on the electrons of the atoms in the local region only. In other words, the static short-range repulsion of the cluster with its surroundings can be accounted for by simply adding specific one-electron matrix elements to the standard energy expression of the total cluster energy, and this is common to both wave function and density functional methods. Hence, both TIP and AIMP methods provide a computational scheme that accounts for the short-range repulsion at a low computational cost. A very pragmatic way to include these matrix elements is to approximate the charge distribution of the ions external to the cluster with effective core potentials but without adding electrons to the corresponding centers. This approach is equivalent to the total ion potentials. It was first applied by Winter, Pitzer, and Temple in their study of a Cu+ impurity in NaF,230 and since then, it has been used as a standard tool to embed cluster models for MgO surfaces and similar systems,236−238 as well as to study magnetic coupling interactions in superconducting cuprates20,21,239 and other ionic solids.155 This approach is, however, restricted to cations only, since sufficiently large core pseudopotentials cannot be constructed for anions. In other words, the charge distribution of the O2− anion does not have any resemblance to the charge distribution represented by any effective core potential for oxygen. The AIMP approach provides a more complete and more rigorous embedding method, although the resulting potentials are material specific and need to be constructed for each crystal separately, which hinders somehow a more intensive use of this approach. 3.1.3. Adding Long-Range Electrostatic Interactions. In the case of the MgO(100) surface, this interaction arises from the electric field generated by all ions in the crystal. Ignoring this contribution may lead to severe misunderstand-

intimately related to the linear-response formulation of TDDFT.192

3. MODELING OXIDE SURFACES The models usually employed to represent extended oxide surfaces do not largely differ from those used to represent other materials, although in oxides the presence of cations and anions interacting through ionic or covalent bonds introduces some peculiarities. In fact, for dominantly ionic oxide surfaces, it is necessary to account for the long-range electrostatic effects. This is not a trivial issue for extended covalent oxide surfaces because of the presence of highly polarizable oxygen atoms in the crystal. The case of surfaces of oxide nanoparticles represents yet a third category requiring a different approach. In the case of extended oxide surfaces, one may choose to represent the oxide surface by choosing either a properly embedded cluster model or a suitable supercell periodic approach. The latter is rather straightforward, although some caveats are necessary, whereas the former requires more intervention from the modeler. If one is interested in ground-state properties, the supercell approach provides a reliable representation of the oxide surface. On the other hand, the embedded cluster model approach is more appropriate if the focus is on excited states and related properties simply because this type of model allows one to make use of configuration interaction wave functions or TDDFT in a straightforward way; this is summarized in Scheme 1, which also shows the compatibility between electronic structure methods and surface models. Finally, notice that the accurate prediction of excited states of three-dimensional periodic systems through TD-DFT has become a real possibility only very recently.200 3.1. Embedded Cluster Model Approach

3.1.1. Defining the Quantum Region. Let us begin with a simple case such as MgO(100), which is the paradigm of simple ionic surfaces and is used in a large number of applications, mostly regarding chemisorption of probe molecules,223−225 which indeed serves as a benchmark for new theoretical methods176 or as an initial simple model for supported metal clusters.226−229 The design of an embedded cluster model for this surface begins by defining the so-called quantum region and the external embedding zone. The quantum region contains the atoms, whose electrons will be explicitly treated with the most accurate feasible quantum mechanical method, whereas the outer region should provide an appropriate representation of the rest of the crystal. The quantum region necessarily includes the atoms that play a fundamental role in the property under study. Thus, it should be large enough to accommodate any possible adsorbate or to contain the atoms involved in the electronic transitions of interest. Clearly, the larger the quantum region, the better the representation of the surface, but the larger the computational cost as well. Hence, the choice of the local quantum region is crucial and will always be a balance between precision and feasibility. Too many atoms in the quantum region will lead to unnecessarily large calculations, while a too conservative choice will result in an unrealistic description of the physics under study. In the case of MgO(100), a reasonable model may be Mg100O100, consisting of a stacking of four layers, two Mg12O13 layers and two Mg13O12 layers with 25 atoms per layer. Depending on the stacking, the surface layer is O centered or Mg centered. This local region will have a total of 200 atoms, a reasonable number 4468


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ings.240 On the other hand, it can be easily incorporated into the material model by calculating the Madelung field of the ions not included in the quantum region and adding this potential to the cluster Hamiltonian. To this purpose, the exact Madelung field is first calculated by an Ewald summation241 with formal point charges at the lattice positions. The contribution of the cluster’s ions is subtracted from this total potential. The use of formal charges is consistent with the assignment of an integer number of electrons to the quantum region. One may also suggest using fractional chargese.g., Mulliken chargesto calculate the Madelung field. However, we must warn that this may introduce some severe conceptual problems. For example, the violation of the overall charge neutrality and the number of electrons to be assigned to the cluster is no longer unique. In practice, the Madelug field of the ions not included in the quantum region is easily incorporated by adding an array of point charges at the ion sites of the lattice. Essentially there are three methods to include the effects of the resulting Madelung field. The first one is the Evjen method,242 which consists of setting the value of the point charges at the lattice positions in the cluster edge and far from the local region according to the restriction of overall charge neutrality. This restriction often results in a fast convergence of the Madelung potential with the number of point charges included in the outer region. The overall charge neutrality is achieved by setting formal charges at all lattice positions except for the charges in the outermost shell. For a cubic lattice, half of the formal charge is used for the faces, a quarter for the edges, and one-eighth for the vertices. The method is valid for MgO, and Figure 1 presents a

more complex oxide surfaces such as CeO2(111). Precisely because of the complex interplay between ionicity and covalent interactions in this oxide, Müller and Hermansson found that the results are rather sensitive to the details of the embedding procedure.247 Their paper also contains a nice report on the historical developments of embedding methods. Finally, we mention the recently proposed general electrostatic embedding method of Sushko and Abarenkov.248 This new development provides a systematic way to construct an accurate electrostatic embedding potential for bulk, surfaces, and nanostructures of crystalline materials. The main idea behind this approach is the modification of the original lattice to eliminate its electric multipole moment up to any given order and, thus, to calculate the electrostatic potential starting from a cell of short range. 3.1.4. Adding Long-Range Polarization. The adsorption of atoms, molecules, or metal clusters on the surface of an oxide can cause a long-range response of the crystal to changes in the electronic structure of the quantum region induced by the adsorbate. In the prototypal MgO(100) surface this effect is small, at least for ground-state-related properties but may become important for excited states. For other more polarizable oxides, such as CaO or ZnO, this type of physical effect can play an important role, and several attempts to include it in a rigorous way have been published. A real ab initio scheme to include this effect was earlier suggested by Barandiarán and Seijo,249 although the method has not been efficiently implemented, probably because it also implies an exceedingly large computational cost. Most of the embedding schemes designed to introduce the polarization of the crystal surface in response to the presence of a charge, created either by an adsorbate or by charge transfer electronic excitations, are based on inclusion of the polarization contribution to the total energy (Epol) as estimated by the classical formula due to Born:250 2 ⎛ 1⎞ q Epol = −⎜1 − ⎟ ⎝ ε ⎠ 2R

(32)

where ε is the dielectric constant of the material, q is the absolute value of the charge, and R is the radius of the spherical cavity in which the charge is distributed. Since a certain degree of ambiguity remains in the definition of R, this correction is only qualitative. An elegant and efficient way to introduce these effects is provided by the so-called shell model introduced long ago by Dick and Overhauser,251 later popularized by Catlow.252 This approach has been implemented in a variety of codes based on the use of interatomic potentials,253−255 interfaced with quantum chemistry codes as in the case of the GUESS code,256,257 which allows treatment of the cluster electronic structure and the lattice polarization in a self-consistent way,258−260 or as a direct option to simulate a polarizable environment as implemented in the PARAGAUSS code developed by Rösch et al.261−268 The shell model is nowadays a mature technology which has been applied to several problems, although initially mainly related to point defects in solids and, more recently, in nanostructures.269−277 An increasing number of applications to oxide surfaces have also been reported in recent years.261−268,278−280 Nevertheless, most of the applications to oxide surfaces involve ground-state chemistry, and hence, a substantial amount of work remains to be done to further apply this accurate embedding methodology to problems involving excited states of oxide surfaces, although

Figure 1. Schematic representation of a cluster model for bulk MgO (or NiO) consisting of a MgO6 quantum region (left panel) surrounded by several AIMPs representing Mg2+ cations and O2− anions (middle panel), which in turn is surrounded by an array of point charges (right panel). For simplicity not all necessary AIMPs are shown.

schematic view of an embedded cluster model for bulk MgO, which can be readily modified to represent MgO(100). However, this simple Evjen approach is not applicable to any type of crystal and does not give accurate representations of the Madelung field except for the simplest lattices such as facecentered cubic (fcc). A more general scheme consists of fitting a small set of point charges at lattice positions within a certain radius around the quantum region to the exact value of the potential at a large number of points in a grid around the cluster local region. Finally, we mention a third approach in which all charges within a hemisphere of radius r around the center of the cluster are taken.243 The radius is, however, taken such that the overall charge of all centers within the sphere is zero to ensure a good convergence of the resulting potential. This method has been successfully applied to represent the αAl 2 O 3 (0001) surface, which does not have a simple structure.244−246 This approach has also been used to simulate 4469


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either force fields or less accurate quantum chemical methods. The main problem consists of defining the interface between both regions, which is made precisely by saturating the dangling bonds with hydrogen atoms. In this sense the IMOMM and ONIOM methods permit one to handle large models and to include all important effects of the crystal. This strategy has been successfully applied to the study of metal−support interactions in silica303 and in an increasing number of studies concerning the reactivity of zeolites.304−317 A general QM/MM embedding scheme for applications in catalysis has also been reported318 which has been shown to be massively parallelizable and can hence be applied to heterogeneous catalysis systems of industrial interest.319

we must acknowledge considerable progress in this field as well.256,281,282 An even more rigorous treatment to account for the longrange polarization has been developed by Pisani and coworkers.283−285 This is the perturbed cluster method based on the EMBED computer program.286 The method is efficient; however, the number of applications has been so far rather limited. The method relies on the knowledge of the oneelectron Green function for the unperturbed host crystal, which is obtained by means of the periodic program CRYSTAL.201 A cluster C containing the adsorbate is defined with respect to the rest of the host H. The molecular solution for the cluster C in the field of H is corrected self-consistently by exploiting the information contained in the Green function to allow a proper coupling of the local wave function to that of the outer region. Most of the applications of this methodology involve groundstate properties of point defects287,288 or adsorbates on a variety of surfaces.289,290 Nevertheless, studies related to excited states have also been reported. In particular, we mention the study of charge transfer between adsorbed species and paramagnetic oxygen vacancies of the MgO(100) surface291 and the model of the structure and spectroscopic properties of oxygen divacancy in yttrium-stabilized zirconia.292 3.1.5. Case of Covalent Oxides. The techniques described in the previous section are valid for very ionic oxides such as MgO, which constitute our main example. However, there are many interesting oxides such as SiO2, TiO2, ZnO, or CeO2 where the covalent interactions also play an important role. Designing an embedded cluster model for these materials is not a simple task, and validation by comparison to experiment or to periodic calculations is mandatory, as illustrated by Müller and Hermansson in the case of CeO2(111).247 Nevertheless, using cluster models is almost compulsory when the interest lies in the description of excited states and related properties. In some cases the bonding between cations and anions is highly directional as in SiO2, and cutting a cluster from the bulk necessarily results in the presence of dangling bonds. In these cases, a simple embedding consists of saturating the cluster dangling bonds by H atoms.293−295 Once the quantum region has been defined by orienting the bulk along the desired direction and cutting a portion of the surface, the geometry of the cluster is usually optimized, especially for cluster models representing point defects, but with the embedding hydrogen atoms fixed in space to provide a simple, though efficient, representation of mechanical embedding with important applications in the description of excited states of point defects in silica.296−299 The saturation of the cluster broken bonds is an important aspect of the embedding, but not the only one. In fact, in this way the crystalline Madelung field is almost completely neglected, except for the electrostatic potential built in the quantum mechanical region. While the Madelung field is certainly of minor importance in more covalent materials such as silica, it is crucial in the description of surfaces of solid materials with a more ionic character such as ZnO or TiO2 and also to represent zeolite surfaces. A good compromise between accuracy and computational cost is provided by the different types of quantum mechanics/molecular mechanics (QM/MM) methods, as reviewed recently by Lin and Truhlar,300 and in particular by the integrated molecular orbital molecular mechanics (IMOMM)301 or ONIOM302 method. All these approaches have the particularity of designing a small quantum region treated accurately and a surrounding region treated with

3.2. Periodic Slab Model Approach

The periodic slab model permits a realistic description of surfaces, in general, and of oxide surfaces, in particular. The idea behind the slab model is very simple; it consists of defining a periodic unit cell exhibiting the crystal surface of interest and including in the unit cell a finite number of atomic layers in the direction perpendicular to the surface, while periodic symmetry is used to replicate the unit cell in the directions parallel to the surface. The area of the unit cell of the slab can be made as large as necessary to represent a given coverage of adsorbates or a point defect with its relaxed surroundings. The number of atomic layers in the slab can be controlled by simply computing the properties of interest as a function of the slab thickness until convergence is achieved. In general, the atomic position of the atoms in the bottom layers is kept fixed at the bulk value, whereas those of the uppermost layers are fully relaxed. Choosing a large enough supercell allows one to model relaxation and reconstructions in a realistic and accurate way. A suitable supercell model of CeO2(111) is presented in Figure 2.

Figure 2. Top view of the CeO2 bulk (left), alignment of the bulk perpendicular to the [111] direction and cut of a (111) slab of thickness ϕ (middle), and 2 × 2 supercell for CeO2(111) featuring nine atomic layers (right).

Moreover, the availability of efficient and parallel computer codes such as VASP,320−322 CASTEP,323 ABINIT,324,325 and Quantum Espresso,326 to mention a few, makes the slab model the choice for most applications in the theoretical study of surface-related phenomena. Note also that TurboTDDFT202 is connected to Quantum Espresso, but as mentioned earlier, TDDFT calculations with this periodic code are so far restricted to molecular systems. The aforementioned codes use plane waves as the basis set, and since these functions are intrinsically periodic, it is necessary to replicate the slab model in the third direction as well as leave a sufficiently large vacuum region between the interleaved slabs. Similar to the slab thickness, the vacuum width needs to be controlled in the model since the computational cost increases with the unit cell size. The use of a repeated slab is not necessary when adopting local basis sets such as the Gaussian-type orbitals (GTOs), as in most quantum chemical codes. This is precisely the case for the CRYSTAL 4470


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code,201 which allows one to carry out calculations in 0, 1, 2, or 3 periodic dimensions with no need for spurious replications, and in the latest version, it also allows TD-DFT calculations for solids.200 From the discussion above, it appears that the periodic slab model is really appealing, and no doubt this has become the default choice for theoretical studies of surfaces, with hundreds if not thousands of applications in the past few years. The slab model, however, is so far restricted to methods applicable to the ground state only or at most to a spin-polarized solution with a prefixed or variable number of unpaired spins. Because of its periodic nature, it is not possible, at this moment, to make use of configuration interaction wave functions or time dependent density functional theory, although recent theoretical developments200 can change this panorama. For this reason, most of the applications described in the forthcoming sections are based on the use of embedded cluster models.

morphological variables such as the particle size and shape have on photocatalytic activity.

4. EXCITED STATES IN CLEAN OXIDE SURFACES The theoretical study of the electronic structure of clean oxide surfaces is, to a large extent, based on previous studies considering the corresponding bulk phases. For bulk materials, a considerable amount of work has been devoted to the study of core-level states which are relevant to X-ray photoelectron spectroscopy (XPS). The pioneering study of the multiplet structure of the Mn2+ in MnO and MnF2 by Bagus et al.339 is probably one of the first examples of configuration interaction calculations, although the model here is very crude: a single Mn2+ cation. Nevertheless, the careful analysis of the different electronic states has provided a qualitative picture of the experimental spectra, including the multiplet splitting and the relative intensities. A more elaborated cluster model for wustite (FexO with x = 0.90−0.95), which accounted for crystal field splitting due to the anions directly coordinated to the Fe2+ or Fe3+ cations, depending on the final state, and the Madelung field simulated by an array of point charges, as described in the previous section, was used in a subsequent study of the core levels of this system.340 A similar model, studied with a more sophisticated theoretical approach which made use of fully relativistic four-component spinor wave functions, correctly accounts for most of the features observed in the MnO 2p and 3p level XPS spectra.341 A different point of view has been used by Canepa et al.342 in their recent theoretical study of the X-ray absorption near-edge structure (XANES) spectrum of hematite (α-Fe2O3). The authors compared the calculated band gap and the value arising from the measured spectrum to assess the quality of different DFT methods. In particular, they found that the main features of the electronic structure are well described by the so-called F40LYP hybrid functional or by DFT+U with U > 6 eV. Finally, Fronzoni et al.343,344 explored the applicability of TD-DFT to describe core excitation spectra (XAS) in transition-metal oxides such as TiO2 and V2O5. Regarding valence-type electronic excitations in transitionmetal oxides, the d−d transitions are experimentally well characterized and provide the first examples for ab initio cluster model calculations in this type of material.345,346 This class of excited states, which constitutes a broad field of study, arises from the crystal field splitting of the d-manifold in going from an atom (spherical symmetry) to structures where the symmetry of the metal site is due to the octahedral or tetrahedral arrangement of the nearest anions. Generally speaking, one would tend to argue that the valence electronic structure of transition-metal oxides has little relationship with that of the corresponding surfaces. However, the interest in photocatalysis has brought these materials to the main focus as they constitute potential photocatalysts and photoelectrodes. Thus, Toroker et al. used DFT+U and GW periodic calculations to investigate the band edge position of MnO, FeO, α-Fe2O3, Cu2O, and NiO.220 Moreover, the optical excitations of hematite have been studied by means of CASPT2 calculations using cluster models,347 also assumed as a benchmark for different approximate GW calculations.348 The study of point defects in bulk oxides provides another type of reference system to test methods and models prior to applying them to the case of oxide surfaces. A well-known type of point defect in oxides which we have already mentioned is the one provided by oxygen vacancies.57,58 The removal of an oxygen atom in an ionic, nonreducible oxide leaves one or two

3.3. Oxide Nanoparticle Surfaces

The surface of oxide nanoparticles exhibits an increasing interest, because the real photocatalytic systems do not involve extended ideal surfaces but nanoparticles. Moreover, oxide nanostructures are of importance, because it is possible to control the chemistry and physical properties through control of the shape and size.327 For instance, recent advances have shown that it is possible to stabilize highly active anatase {001} facets in anatase TiO2 nanocrystals,41 which add further interest to this already hot field. In nonconventional titania powders with morphology-controlled nanocrystals, it is expected that the presence of only one crystal phase will suffice to achieve highquality materials, because both oxidation and reduction reactions are expected to happen in different facets of the same crystalline particle.328 Therefore, modeling oxide nanostructures is becoming a crucial task. However, this is not straightforward, since one needs to consider different terminations and large enough nanoparticles.329 Large oxide nanoparticles up to 10 nm in size and comprising about 16 000 ions can be adequately modeled by means of molecular dynamics simulations carried out with empiric interatomic potentials.330 However, the usual strategy to model oxide nanoparticles for electronic structure studies is to cut models from the bulk and then to reoptimize the structure. This has been successfully carried out for ceria where the optimum structures have also been investigated by means of global optimization techniques331−335 and for anatase stoichiometric nanoparticles of increasing size and up to (TiO2)38. In the latter study the optical spectra of different particles have been obtained from TD-DFT calculations.336 Interestingly, the lowest excitation carrying intensity is not the one corresponding to the HOMO−LUMO, which hence casts doubts on the use of the density of states of extended models of either surfaces or bulk polymorphs to gain an understanding of the optical spectra of these nanoparticles. Obviously, smaller nanoparticles have their own particularities, including the minimum energy atomic structure, which has to be determined from global optimization techniques as illustrated in the case of (TiO2)n clusters with n = 1−7, 10, and 13.337 From the brief discussion above, it is clear that the study of the surface of nanostructured oxides and their electronic spectra is an emerging field with a great technological future, although so far the number of theoretical studies is rather limited. Therefore, we will not consider this part further, and we refer to the recent review by Kubacka et al.338 on the influence that 4471


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electrons well localized and trapped in the cavity left by the missing anion. These electrons are stabilized by the Madelung field, are usually referred to as color centers, and are denoted as F, F+, or F2+, depending on whether two, one, or zero electrons are trapped. The localization is so well-defined that F centers have the topological features of pseudoatoms,349 as illustrated also by the electron localization function plots in Figure 3.

Figure 4. Schematic representation of the energy levels involved in the electronic transitions of bulk F and M centers. The energies reported refer to the computed excitations for the bulk centers at the CASPT2 level. Contour plots of the relevant molecular orbitals are shown.

The electronic properties of impurities in reducible oxides provides another clear example of the need to go beyond standard LDA and GGA approaches. Let us consider the case of the aluminum-doped rutile TiO2, which has been studied by Islam et al.364 using a supercell periodic approach and different functionals. For the oxygen vacancy related defect, these authors find that the best agreement between the calculated excitation energy and optical spectroscopy is obtained with the hybrid PW1PW exchange-correlation potential. These authors also considered the substitution of a single Ti by Al, which introduces an unpaired electron localized on one of the oxygen atoms closest to Al and an unoccupied peak which reduces the band gap to 1.6 eV. The consequences on color-doped TiO2 samples are also discussed. Nevertheless, one must realize that this corresponds to a one-electron-type description and, i.e., the electron−hole interaction is neglected, differently from Bethe− Salpeter or TD-DFT formalism.189 Saal et al.365 considered the effect of ZnO doping with Mn using a supercell periodic model and a hybrid functional and focused on the effect of doping on the band gap. The most important conclusion of this study is the strong dependence of the optical gap on interatomic distances, which nicely reproduces the experimental trends. In some cases simple model systems can offer some clues on the nature of the excited states in these systems. We mention for instance the LDA+U and GGA+U study of Jejidi et al.366 aimed at modeling localized photoinduced electrons in rutile. These authors suggested taking photoexcitation into account by imposing two unpaired electrons per cell of the same spin. In this way, by varying the size of the cell, one can control the number of excitations per surface area.

Figure 3. Localization domains of the F vacancy in the bulk MgO. The limiting isosurfaces correspond to a value of 0.8 for the electron localization functions (ELFs). The Mg, O, and vacancy basins are represented in magenta, orange, and blue colors, respectively.

These centers also exhibit well-defined and unambiguously assigned spectroscopic features,350 thus providing an excellent reference for theory. Excitations of F centers in MgO,351,352 CaO,353 α-Al2O3,354 and ZnO355 have been studied with the embedded cluster model approach using configuration interaction (MgO, ZnO) or CASPT2 and/or TD-DFT (CaO, α-Al2O3), with generally good agreement between theory an experiment, as reviewed by Zhukovskii et al.356 Very recently, two studies have been published where supercell periodic models for these point defects are designed and excitations are estimated from the GW method.221,222 Note also that although F centers can hardly diffuse through the bulk or from the bulk to the surface,357 it is possible to find F center aggregates. The so-called M center contains two near-neighbor oxygen vacancies, again with well-localized electrons, and the interaction between the energy levels of these point defects results in new states (Figure 4) and, consequently, in a peculiar electronic structure with a recognizable spectral fingerprint.358 The cases illustrated above have a common feature, as they represent excited states with a strong local character, and hence, the corresponding system is well described by a properly embedded cluster model. This is indeed the case for excitations from core levels, since the ionized electron has a strong local character. Similarly, F centers exhibit strong localization because of the trapped nature of the electrons, as illustrated by electron density maps359 or a topological analysis of the electron density.349 A different situation is found when considering the lowest optical excitation in an oxide such as MgO, which is responsible for the optical gap of this material. In this case, embedded cluster models may be useful just to indicate the delocalized nature of the excited state,360−362 which hinders an accurate prediction as recently shown in the systematic study of Kanan et al.363 In these cases a proper prediction of the band gap requires the use of hybrid functionals204−209 or more sophisticated techniques such as the GW method discussed above.218,219

4.1. Stoichiometric Surfaces

The aforementioned cases for excited states of bulk oxides have prompted similar studies on oxide surfaces. For instance, Pascual and Pettersson367 used an embedded cluster model to study O 1s core excitons of the MgO(100), CaO(100), and SrO(100) surfaces. The corresponding excited states involve the excitation of a core electron, in this case the O 1s to empty states, a process which is observed in near-edge X-ray absorption fine structure (NEXAFS) spectroscopy. These authors show that the calculated NEXAFS spectra for different cluster models are in reasonable agreement with the experiment and, in particular, that NEXAFS spectra for the different surface cluster models exhibit a well-defined discrete peak approx4472


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functions, Satitkovitchai et al.383 studied optical secondharmonic generation below the gap of NiO(001) and identified transitions which are not allowed in the bulk, thus indicating the existence of other specific spectroscopic features of this oxide surface. The electronic structure of NiO(001) and the corresponding d−d excitations have also been studied with periodic Hartree−Fock calculations.384 These authors also compared their results with the previous multireference CEPA381 and CASSCF/CASPT2382 calculations for embedded clusters and showed that because of their local character, the energies of these excitations are determined mainly by the coordination number of the excited atom rather than the nature of the local environment. The same authors investigated the electronic structure of an unsupported NiO(001) monolayer, which, in spite of being a barely theoretical model, provides useful information for the study of thin films grown on metals. These authors also considered ferromagnetic, ferrimagnetic, antiferromagnetic, and fully frustrated spin alignments as a function of the lattice constant.385 In a more recent work on NiO(100), Gomez-Abal et al.386 combined optical control theory with ab initio quantum chemistry and electronic crystal field theory to explore the laser-induced femtosecond spin dynamics and predict for the first time the possibility of alloptical spin switching within 100 fs. Finally, the optical and magnetic states in bulk CoO and CoO(001) have been studied by means of the embedded cluster model approach using a class of configuration interaction wave function including relativistic effects. The presence of the surface is found to reduce both the optical excitations and the magnetic coupling between Co cations.387 The case of low-coordinated sites at stoichiometric surfaces deserves some additional comments, since model calculations show a dramatic dependence of the calculated optical absorption and luminescence energies on the coordination of O atoms at the MgO surface and also in MgO nanoclusters.258 The strong dependence of the excitation and luminescence energies on ion coordination shown by Shluger et al.258 has been attributed to a combination of several interrelated factors which include the reduction of the Madelung potential at lowcoordinated sites with a concomitant large relaxation with respect to ideal geometry and to strong electron-density redistribution. However, it is argued that the reduction of the Madelung potential alone cannot quantitatively explain the experimental data without considering the degree of localization of the excited state and its location.

imately 1.5 eV below the ionization potential, which is tentatively assigned to a transition from the 1s state to the empty polarized 3pz state. Ochs et al.368 and Puchin et al.369 have focused on the ionization of valence electrons of MgO and α-Al2O3 surfaces of single crystal samples, respectively. In the case of MgO, thin films were also considered and the difference between a single crystal and thin films was analyzed. Experimentally, the ionization of valence electrons is studied by metastable impact electron spectroscopy (MIES) and ultraviolet photoemission spectroscopy (UPS). These authors use slab models and band-structure-type calculations to interpret MIES and UPS experiments through the analysis of the density of states. An interesting conclusion from this study368 is that the MgO(100) crystal surface exhibits a lower concentration of defects than the surfaces of thin oxidized films supported on silicon. Cluster models and DFT-based calculations were also used in a series of studies aimed at elucidating the different types of oxygen sites on vanadium oxide surfaces. The stoichiometrydependent oxidation states of the cation are also considered.370−375 Even more complex systems, such as VxOy films supported on silica,376 also mimicking realistic reaction conditions,377 have been studied. Note that the, in principle, simple V2O5(010) surface contains terminal (vanadyl) O atoms located directly above V atoms, O atoms bridging two vanadyl groups sticking out of the surface, O atoms bridging two vanadyl groups pointing into the bulk, and O atoms connecting three vanadyl groups of different surface orientations.370 Each one of these distinct atoms has its own spectroscopic XPS signature, and a contribution from theory is very useful to properly assign them. Very often the combined use of theory and experiment370,375 allowed gaining considerable microscopic insight into the atomic and electronic structures of these materials, which are relevant to various catalytic processes. This successful strategy has also been used to study other oxide surfaces such as MoO3(010) with the aim to assign the observed spectral features to different types of O sites,378 as well as relate the spectroscopic fingerprints to their particular reactivity. Characteristic features in the NEXAFS spectra that arise from specific vacancy sites present at oxygen-deficient molybdenum oxide surfaces are also identified and interpreted.379 Beyond the model of molybdena surfaces, a detailed picture of the structure of molybdenum oxide supported on silica has also been proposed, which is useful as a model for supported molybdena catalysts.380 The studies of excited states regarding stoichiometric oxide surfaces also include some examples involving valence electrons. The case of NiO(100) is a paradigm which has been investigated by EELS and ab initio cluster model calculations.381 Interestingly, Freitag et al. find that, in addition to the previously commented bulk excitations of NiO corresponding to d−d transitions, two new states at energies of 0.57 and 1.62 eV are found, which are attributed to d−d transitions of the Ni surface ions. The d−d transitions on the NiO(100) surface have also been studied by Geleijns et al.382 using CASSCF/CASPT2 calculations on embedded cluster models. These authors analyzed the 15 lowest states related to the Ni2+ cation in the d8 local configuration, and they attributed the signal at 0.6 eV to d−d transitions. In addition, they confirmed the assignment to surface d−d transitions of the peak observed at 2.1 eV. They investigated the charge transfer excitations and predicted the lowest one to occur at 2.0 eV. Using a similar embedded cluster model, but mainly CIS wave

4.2. Defective Surfaces

The presence of oxygen vacancies in oxides is not limited to the bulk. Indeed, these point defects have also been detected at the surface; in nonreducible oxides, the presence of the vacancy also leads to F centers, which in this case are denoted as Fs for neutral O vacancies with two electrons in the cavity or Fs+ and Fs2+ when the cavity contains one or zero electrons, respectively. Moreover, these point defects can be located at terrace sites or can occupy low-coordinated sites. The presence of defective sites characterized by a low coordination of the metal cation depends on the specific oxide surface and on its preparation. In fact, optical spectra of MgO samples containing surface F centers were reported quite long ago for polycrystalline samples388 and later on for single crystal surfaces.389 The introduction of oxide thin films4−7 has also permitted optical spectra of ultrathin MgO films grown on Mo(100) to be measured390 and their presence to be evidenced by means of 4473


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Figure 5. Schematic representation of embedded cluster models for oxygen vacancies (F centers) at the terrace, step, and corner sites of MgO surfaces. Atoms are displayed as colored spheres, and TIP is displayed as gray spheres.

spectroscopy. The embedded cluster calculations of Fink,397 using a sophisticated wave-function-based method, show that while F+ centers in bulk ZnO exhibit a characteristic blue absorption at 3.19 eV, in agreement with experiment, the corresponding surface excitations are more than 0.5 eV higher in energy, thus indicating that it is not possible to observe them experimentally, because the excitation energy is well above the optical band gap of ZnO. Note also that, at variance with the case of MgO discussed above as well as α-Al2O3(0001),354 where the presence of the surface lowers the transition energy relative to the bulk F center, here the presence of the surface increases it, which is interpreted as an effect of the enhancement of the Pauli repulsion originated by the relaxation of the nearest Zn centers. Impurities at the surface of oxides also exhibit particular spectroscopic features. The case of the Cr3+ defect in MgO, a classical well-known system,350 has been considered by Kantorovich et al.398 using an embedded cluster approach and CASSCF and CI calculations. These authors considered various electronic states of the impurity at Mg sites in the bulk and at the MgO(001) surface, simulating the effect of a scanning force microscopy (SFM) tip on the Cr spectroscopic properties. Indeed, these appear to be well reproduced by the simplest possible embedded cluster model where the quantum region contains the Cr3+ ion only. In fact, for the bulk impurity the first two excited states (4P, 2G) are found to be separated from the ground state 4F by 1.91 and 2.01 eV, which is close enough to the experimental values of 1.71 and 1.82 eV, respectively. To obtain a more quantitative estimate, a rather large quantum region is needed, and the authors did not report converged values. This is also the case for the surface impurity, for which a value of 1.85 eV is predicted for the 4A → 4E transition using the CASSCF method and a quantum region containing 19 ions. Here, the surface lowers the symmetry, making this transition, even though weak, electric-dipole allowed. Electron density plots show that the 4A and 4E states involved in these d−d transitions are strongly localized around the Cr site, thus justifying the use of an embedded cluster model and of some approximations in the estimate of the lattice relaxation. We close this subsection by mentioning rather recent studies focused on the spectroscopic features of oxygen vacancies in the bulk and at the surface of SrTiO3, a perovskite-type oxide representative of a broad class of materials with potential technological applications. Alexandrov et al.399 have considered F and F+ centers in the bulk and on the SrO- and TiO2terminated SrTiO3(001) surfaces. These authors used periodic supercells and hybrid functionals to focus on the density of states and band structure of the different possible F and F+ centers. Their calculations show that the charged oxygen vacancy, compared to the neutral one, exhibits a stronger local

scanning tunneling microscopy and electron paramagnetic resonance.391,392 Interestingly, each type of surface exhibits typical features related to the presence of oxygen vacancies. Thus, optical measurements on fine powder samples of MgO using diffuse reflectance techniques showed a feature at 2.05 eV which was attributed to five-coordinated Fs+ (F5c+) centers.388 Henrich et al.389 observed a peak at 2.3 eV on MgO single crystals using EELS, a feature which was tentatively assigned to surface F centers, although the assignment is not univocal. In fact, Underhill and Gallon393 assigned the loss to a surface V− center (a cation vacancy with a hole trapped on an adjacent oxygen ion). More recently, Wu, Truong, and Goodman390 studied the defects generated in an ultrathin film of MgO grown on Mo(100) with high-resolution EELS. They observed three features at 1.15, 3.58, and 5.33 eV. The 5.33 eV band was assigned to bulk F centers, the band at 1.15 eV to surface F centers, and the band at 3.58 eV to F aggregates.390 A proper assignment of each of these optical transitions requires additional information. Embedded cluster models representing low-coordination sites (Figure 5), in connection with sophisticated DDCI calculations, have enabled the proposition that transitions at ∼5 eV correspond to bulk F centers (F6c), those at ∼3 eV correspond to terraces (F5c), and those at ∼2.2−2.5 and ∼2.1−2.2 eV arise from step (F4c) and corner (F3c) sites, respectively. Clearly, the peaks at 1.15 and 3.58 eV should have another origin. Pfnür et al.394 used EELS to analyze MgO thin films after electron bombardment and tentatively assigned the transitions at around 1 eV to surface M centers (oxygen divacancies). A series of model calculations on embedded clusters using both CASPT2 and TD-DFT have shown that bulk M and M+ centers give rise to intense absorptions at about 4.4 and 4.0 eV, respectively, as schematically shown in Figure 4 with another less intense transition at 1.3 eV also due to the M+ center.358 The same authors found that on the surface M centers the transitions occur at 1.6 eV (Ms+) and 2 eV (Ms). In particular, it has been suggested that the band at 1.6 eV could be the origin of a band at 1−1.3 eV observed in electron-irradiated MgO films and tentatively attributed to M centers.390 Thus, comparison between experiment and calculations on suitable models allowed a rather complete picture of the spectroscopic observations related to surface oxygen vacancies in MgO to be obtained. The polar ZnO(0001̅) surface provides another important example related to the optical excitations arising from electrons trapped at oxygen vacancies. This surface, in addition, is important in catalysis, because of its direct relationship to the Cu/ZnO catalysts used for methanol synthesis from syngas; indeed, some studies argue that oxygen vacancies at the polar surface constitute the active site.395,396 Here, a key question is whether it is possible to observe F+ centers by optical 4474


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4.0−5.5 eV. TD-DFT calculations with the B3LYP functional, carried out on suitable embedded cluster models, interpreted these features in terms of the excitations of nanocrystal interfaces. This has important consequences, since it implies that photons with an energy as low as 4.6 eV can excite electrons not only at MgO surfaces, as previously believed, but also at interfaces inside the powder. The combination of theory and experiment in this study is fundamental and shows that it is possible to identify regions of the spectra that are associated with particular types of interfaces. More importantly, it is likely to provide new means for functionalization of nanostructures and chemical activation, which in turn may improve the performance and reliability for many nanopowder applications, such as photocatalysts and solar cells. Embedded cluster models and TD-DFT with the hybrid B3LYP functional were also employed to assign the photoluminescence spectra of dehydroxylated409 and, later on, hydroxylated410 MgO powders. The reason to rely on photoluminescence experiments is that this is one of the few techniques enabling characterization of oxide ions as a function of their coordination. However, the appearance of broad bands in the spectra of MgO powders enormously complicates the assignment. This is not surprising because a very large number of sites with different coordinations are present at the surface. The combined experimental and theoretical study of Chizallet et al.409 shows that the excitation energy depends not only on the coordination number of the surface ions but also on the local environment around them. Moreover, the calculations provide compelling evidence that a single defect can lead to several bands, and a single band can contain contributions of several defects. This study thus provides another excellent illustration of the need to combine theory and experiment to understand the physics and chemistry of these complex systems. All cases discussed above concern MgO nanostructures; this is because of the simplicity of preparation of MgO powders405,406 and because their composition and their resulting chemistry are rather simple. MgO, moreover, due to its ionic character, represents a particularly suitable case for embedded cluster calculations, allowing combination of sophisticated theoretical approaches on small but reasonable model structures, thus providing an excellent material to join insights from theory and experiment. However, MgO is not the only material where this type of study has been carried out. In fact, a considerable amount of work has been devoted to ZnO because of its relationship to catalysis and photocatalysis. De Angelis and Armelao411 have recently reported a systematic work where one-, two-, and three-dimensionanal ZnO nanostructures are examined and their lowest optical transitions explored by means of TD-DFT using B3LYP and PBE0 hybrid functionals. Moreover, these authors explicitly considered the effect of water as a solvent using a polarizable continuum model (PCM).412 These authors considered oxygen vacancies on a realistic model of a particle, including relaxation for the excited states, and calculated a transition at 2.45 eV for the neutral oxygen vacancy in agreement with the experiment. From the calculated data for charged vacancies the authors concluded that the neutral ones are likely to be the source of green luminescence in ZnO nanostructures. The large influence of defects on the optical spectra of ZnO nanoparticles has also been shown in a previous work by Ischenko et al.,413 although these authors did not find any relationship between oxygen vacancy content and green luminescence. Nevertheless, the embedded cluster calculations using MCCEPA-type wave

lattice relaxation, a deeper one-electron energy level, and a considerably stronger propensity for surface segregation. Nevertheless, a common feature of both types of vacancies is that when present at the surface, they exhibit more shallow energy levels than their bulk counterparts, particularly on TiO2 termination, although the effect is less pronounced for F+ centers. An extension of this study to other perovskites such as PbTiO3 and PbZrO3 and comparison to SrTiO3 have been reported by Zhukovskii et al.400 using the same theoretical approach and materials model. 4.3. Nanostructures

Oxides are materials of interest also in the growing fields of nanoscience and nanotechnology, as evidenced by the large number of publications devoted to this topic and recently reviewed by Bromley et al.329 In synthetic processes, there is increasing control over the shape and chemical properties of oxide nanoparticles,41,401−404 which also triggered more fundamental studies on the excited states of oxide nanostructures. Diwald et al.405 have studied the selective excitation of surface oxygen atoms on highly dispersed MgO produced by UV irradiation and identified paramagnetic species by electron paramagnetic resonance (EPR). From the measured spectral dependence of formation of paramagnetic O−, O2−, and O3−, confirmed by a variety of theoretical models aimed at predicting the EPR data, it is suggested that optical excitation of the studied samples at about 5.4 eV creates holes (O− species) at step edges, which are localized at ideal highly symmetric corners. The authors also concluded that O− species are directly produced by optical excitation at about 4.6 eV and that excitations at 4.3−5.6 eV at low O2 pressure lead to O2−. This study provided direct evidence of site-selectively created active centers of known geometry and electronic structure, which become available on polycrystalline MgO and may have implications in catalysis. In a subsequent paper by the same authors, the dynamics of the photoinduced electron and holes was studied.406 It is suggested that the observed formation of a considerable amount of hole and electron centers is due to a mechanism implying the sequential absorption of two photons, which is supported by DFT calculations. To obtain a deeper insight into optical properties arising from the interface between MgO nanocrystallites, McKenna et al.275 used embedded cluster models and TD-DFT. They found that photon energies lower than 5 eV reveal a number of features inside the MgO powder, which is a rather surprising and useful result. They also provide evidence that interfaces between MgO nanocrystallites in a powder are unlikely candidates for trapping localized holes, differently from the case of the surface. On the contrary, this is certainly possible for delocalized holes, thus illustrating the different chemistry of MgO nanoparticles compared to extended surfaces. Further works on the same type of systems have focused on the elementary photoinduced processes at ideal and defect-containing surfaces of these nanoparticles.407 From TD-DFT calculations carried out for cluster models representing the nanoparticles, a theoretical model is derived, predicting that sub-band gap photons can induce photoexcitation and ionization of low-coordinated sites and formation of surface defects via atomic desorption. The influence of the density of the optical properties of MgO nanoparticles has also been considered in a recent study combining experiment and theoretical modeling.408 The UV diffuse reflectance spectra of MgO nanopowders after compaction show new absorption features in the range of 4475


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and whether the adsorbed adatom maintains the magnetic ground state of the gas-phase atoms or if, on the contrary, the presence of the oxide surface induces some changes. Isolated transition-metal atoms usually exhibit a high-spin ground state according to Hund’s rule. The interaction with the oxide surface can modify this trend because, in principle, exchange terms favor the high-spin state for a given electronic configuration and the chemical bonding tends to form an electron pair with a concomitant reduction in the magnetic moment of the adsorbed atom. The interaction with the surface can also modify the electronic configuration as discussed in detail below. In the case of a reducible oxide, one must also consider possible charge transfer between the adatom and the oxide. Hence, even determining the electronic ground state of transition-metal atoms at surfaces requires exploring various electronic states, often of different multiplicities. With the exception of Cr, the electronic ground state of firstrow transition-metal atoms corresponds to a 3dn4s2 electronic configuration coupled high spin (HS), whereas upon adsorption on the regular sites (O sites) of the MgO(100) surface the electron populations on the 4s orbital decrease. Depopulating the 4s atomic orbital allows bonding and antibonding states to be formed and reduces the Pauli repulsion between the adatom and the surface, thus contributing to enhancement of the interaction. Nevertheless, the number of unpaired electrons is maintained as in the isolated atom, even if, depending on the spin coupling, the resulting electronic configuration for the supersystem can be HS or low spin (LS) as shown in a systematic study using periodic and embedded cluster models and different functionals (PW91 and B3LYP) by Markovits et al.426 These authors found that, at very low coverage, the electronic ground state of the transition metal was always HS, as in the isolated atoms, although, interestingly, the energy required to modify the spin of first-row transition metals (Ti, V,Cr, Fe, Co, and Ni) supported on the MgO(100) surface is lower than the corresponding value of the isolated atoms. In some cases such as V, Ni, and Co, a change in the multiplicity of the ground state with respect to the free atom is observed. Ni is a special case, because the PW91 functional describes incorrectly its atomic electronic ground state.427,428 To investigate whether a stronger interaction can completely quench the magnetic moment of the adsorbed atom, Florez et al. studied the same systems using embedded cluster models and the hybrid B3LYP functional, but considering low-coordinated sites and adsorption at F centers as well.429 The interaction above lowcoordinated anion sites and Fs centers was found to be stronger than on the regular terrace O sites. Hence, in spite of the larger interaction at low-coordinated and Fs centers of the MgO surface, first-row transition-metal atoms tend to preserve the number of unpaired electrons observed in the free atom. However, the larger interaction at these sites induces a reduction of the energy required to switch from high spin to low spin, although, in general, the decrease is not enough to even partially quench the spin of the adsorbed atom. This is important, since it demonstrates that the spin state of adsorbed metal atoms on oxide supports has to be explicitly considered. This statement is reinforced by other authors who studied the same problem using either embedded clusters430 or periodic models431 and by the study of the interaction of Co, Rh, and Ir atoms with the Al-terminated α-Al2O3(0001) surface at 1/3 monolayer (ML) coverage carried out within a supercell periodic approach using a standard GGA functional.432 This is

functions predicted an absorption for the oxygen vacancy of 3.19 ± 0.3 eV, which is in agreement with the experimental value in the 2.95−3.03 eV energy range. The difference between these two studies lies probably in the structure used as a model for the ZnO nanoparticles. In fact, the former study used a realistic nanoparticle and included solvent effects, whereas the latter used the bulk structure. Doping in ZnO nanocrystals has also been the subject of theoretical studies, often combined with experiment. For instance, Locmelis et al.414 considered the effect of substituting O with S recurring to periodic supercell calculations with a hybrid functional, and they attributed the reduction of the band gap to the structural changes induced by the substitution.

5. EXCITED STATES RELATED TO ADSORBATES AT OXIDE SURFACES The adsorption of chemical species at surfaces, in general, and oxide surfaces, in particular, modifies the electronic structure of both the surface and adsorbate, generating new electronic states which result from the interaction. In the case of weak interactions (physisorption), the adsorbed molecule maintains its atomic structure, and consequently, the electronic structure is only slightly modified. In other cases the geometry of the adsorbed molecule (or metal particle) is heavily modified by the interaction with the surface with a concomitant relevant change in the electronic structure. In this section we will review several cases which are relevant to oxide surfaces starting with the adsorption of metal atoms and metallic clusters which are of importance to understand the structure and reactivity of supported catalysts. Next we will focus on the existing studies concerning the excited states of probe molecules at oxide surfaces and photodesorption. Sections 6 and 7 are devoted to theorerical studies related to photocatalytic systems based on metal oxides and models for dye-sensitized solar cells, respectively. 5.1. Metal Atoms and Metal Clusters

An intensive study of the interaction between metal atoms, or metal clusters, and oxide surfaces has been carried out in the past few years, mainly because of its possible applications in supported catalysis,415−418 but also in other technological fields.419 The earlier theoretical studies concentrated mainly on the appropriate modeling of the system and on the nature and strength of the interaction between metal atoms and the oxide surface; MgO(100) has often been chosen to represent a typical nonreducible oxide.226−228,420−424 Here, it is worth mentioning that the interaction energy between transition metals and MgO(100) is not a simple issue and that in the case of using embedded cluster models one must correct for the basis set superposition error.226−228 In a systematic study, Yudanov et al.226 considered the adsorption of transition-metal atoms on MgO(100) at low coverage and found that the transition-metal atoms can be classified into two main groups, depending on whether they tend to form relatively strong bonds with the surface oxygen anions of MgO (Ni, Pd, Pt, and W) or have a weak interaction with the MgO surface (Cr, Mo, Cu, Ag, and Au). Later on, Cinquini et al.425 provided a discussion of the main bonding mechanism and how the morphology, presence of defects, doping and functionalization, redox properties, and substrate thickness affect the properties of supported metal atoms and nanoparticles. Regarding the interaction of transition-metal atoms on oxides, a key issue is the nature of the electronic ground state 4476


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The vertical transitions of Cu atoms, dimers, and tetramers deposited on various sites of the MgO(100) surface, including low-coordinated sites and Fs centers, have been investigated by means of CASSCF/CASPT2 and TD-DFT calculations carried out on embedded cluster models.447 This is one of the few studies dealing directly with excited states of supported particles and reveals that, upon deposition on the surface, a substantial modification of the energy levels of the supported cluster is induced by the Pauli repulsion with the substrate. This causes shifts in the optical transitions relative to those of the free unsupported clusters, whereas the changes in cluster geometry induced by the substrate have a much smaller effect on the optical absorption bands. Interestingly, the results for isolated Cu atoms deposited on the MgO(100) allow assignment of the bands in the EELS spectra obtained for Cu on MgO thin films.448,449 The case of the Fs centers is special because here the presence of filled impurity levels in the band gap of MgO results in a strong mixing with the empty levels of the Cu atoms and clusters with consequent deep changes in the optical properties of the color centers. Also using a cluster model approach and TD-DFT, Walter and Hakkinen studied the optical spectra of Au clusters up to eight atoms supported on the terrace and Fs sites of MgO(100) and considered explicitly optical polarization.450 These authors found that polarization-resolved optical spectra contain information about the bonding and structure of the supported cluster and also other species such as CO or O2 adsorbed on the metal cluster. Later on, Bonacic-Koutecky et al.451 applied a similar methodology, embedded cluster models and TD-DFT with the PBE functional, to analyze the structural properties and optical spectra of Ag clusters with two, four, six, and eight atoms on the stoichiometric MgO(100) surface and on the Fs center in the same surface. This study shows that the absorption and emission patterns are due to the specific interaction between the excitations within the cluster and the support site, which, in addition, is strongly dependent on the cluster size and structure. Results from the studies commented above show that absorption and emission properties of small Cu and Ag clusters supported on MgO can be tuned by the cluster size, cluster structure, and support site. Nevertheless, when deposited at the regular sites of this surface, the optical properties of the supported clusters are not heavily perturbed. Similar conclusions were reached by Bosko et al.452 in their study of Cu, Ag, and Au atoms and dimers supported in the same sites of the same surface and using a similar embedded cluster model and TD-DFT approach, although with the BP86 exchange-correlation functional and a different computational code. The optical properties of small noble-metal clusters at the Fs site of the MgO(100) were also studied by Burgel et al.453 These authors focused on comparing absorption and emission properties of supported silver and gold clusters and found that the leading absorption features are similar for the supported Ag and Au clusters. These authors also carried out molecular dynamics (MD) simulations in the excited electronic states to unravel the relaxation mechanism and to propose that dimers and tetramers are good candidates for emissive centers whereas larger supported Ag8 and Au8 are unlikely to fluoresce. To interpret the optical spectra of metal clusters at oxide surfaces, Huix-Rotllant et al.454 proposed a new strategy based on the combined use of natural transition orbital and fragment molecular orbital analyses. These authors applied this methodology to the case of coinage metal dimers supported also on the Fs center of MgO, represented by an embedded cluster model.

as well a nonreducible oxide surface, but with a more complex crystal structure compared to MgO(100). However, adsorption on O surface sites, as in the case of MgO(100), turned out to be preferable for all the considered metals. Co and Ir maintain the same number of unpaired electrons as in the free atom, whereas a spin quenching is observed for Rh.432 Spin polarization was also found to be important for Ni on the Alterminated α-Al2O3(0001) surface.433 Nevertheless, one must be aware that the presence of impurities at the surface may change this trend, enhancing the adsorption energy and the tendency toward the quenching of the spin of the adatom.434 The case of reducible oxides such as TiO2 or CeO2 introduces some complexity, because, in this case, the adsorbed adatom or metal cluster can be oxidized by electron transfer to the substrate. This in turn is difficult to describe within the current DFT approaches, as discussed by Pacchioni for the bulk materials.435 The case of Au on the O site of the perfect, stoichiometric, CeO2(111) surface is a clear example of these difficulties, since, to a large extent, the charge state of the adatom depends on the details of the adopted methods and models436−440 and, on the other hand, experiments do not yet provide a definitive answer. This is because the electronic states corresponding to neutral and oxidized Au on this surface are very close and have the same symmetry and spin state. The only difference lies in the localization center where the unpaired electron of Au is located: Au for the neutral state or a substratereduced Ce3+ cation in the case of the ionic, charge transfer state. This is not the case for Cu and Ag, where charge transfer is always predicted. Recently, using periodic models and the DFT+U approach, Nolan has also shown that, on the CeO2(110) surface, which is not the most stable but is nevertheless detected in ceria nanoparticles, Au, Ag, Cu, Al, Ga, In, La, Ce, V, Cr, and Fe become all oxidized.441 Similar results were also reported by Cui et al.442 using a very similar approach, although these authors concluded that at this surface Cu and Au are oxidized to Cu2+ and Au2+, which is somewhat surprising and requires further analysis. The situation seems to be more clear for the interaction of Cu, Ag, and Au on the rutile TiO2(110) surface. Using embedded cluster models with the hybrid B3LYP functional and periodic models with the PW91 functional, Giordano et al.443 found that Cu and Ag become clearly oxidized, whereas in the case of Au the charge transfer is much smaller, which is in agreement with subsequent studies including theory and experiment.444 The charge state of Cu, Ag, and Au on TiO2(110) and α-Al2O3(0001) has been considered by Marquez et al.445 using a new and interesting strategy consisting of a combined use of Bader charges and simulated infrared spectra for CO interacting with these adatoms. The examples above show that even the ground state of metal atoms at oxide surfaces is not straightforward and various electronic states have to be considered. The situation is even more complex when considering metal clusters. In the case of magnetic metals such as Ni or Co, even the number of unpaired electrons cannot be anticipated and a systematic search is required. Giordano et al.446 used an embedded cluster model approach to study Ni4 and Co4 supported on the regular MgO(100) surface. In spite of the relatively strong bond between the metal clusters and the MgO(100) surface (2.0 eV for Co4 and 2.4 eV for Ni4), the perturbation induced on the cluster electronic structure is small and only a partial magnetic quenching is observed, which is restricted to the metal atoms in direct contact with the surface. 4477


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Merging TD-DFT calculations with this method of analysis allowed the authors to present a straightforward and transparent quantitative characterization of the main spectral features. The studies described so far constitute perhaps the advent of a new field related to the excited states of supported metal clusters and their corresponding chemistry. At this early stage, however, most of the works involve MgO surfaces. The only exception to this substrate is surely in the experimental and theoretical work of Del Vitto et al.455 on the optical properties of mass-selected Au atoms and dimers supported on amorphous SiO2, which allowed identification of possible anchoring centers for the supported Au species. Finally, we must mention that metal atoms and clusters adsorbed at the surface of oxide films supported on metals exhibit a rich and different chemistry, in particular due to the possibility of electron tunneling through the film.456 Oxide thin films constitute a specific field8,425 which is reviewed in another paper of this special issue457 and, hence, will not be further discussed here.

The work of Klinkmann et al.467 on autoionization spectroscopy of CO on metal oxides constitutes one of the first and few examples where quantum chemical theoretical models are used to interpret experiments regarding the excited states of a molecule adsorbed on an oxide surface. These authors reported a considerable amount of experimental information regarding the C 1s to 2π* excitation on various oxides. The experimental assignment was then confirmed by CI-type calculations carried out for CO in the gas phase and for CO adsorbed on an embedded cluster model for NiO(100), where a single Ni2+ cation is considered and the rest are substituted by Mg2+ cations. In spite of the simplicity of the model, the very local character of the involved excited states allowed the authors to show that the relative energetic positions of the electronic states involved in the process can be represented reasonably well by their CI calculations, although the quite crude model did not describe the difference between the autoionization spectra of the gas phase and the adsorbed CO.

5.2. Probe Molecules

Photodesorption of atoms or small molecules from surfaces is one of the most fundamental processes in surface photochemistry. In principle, the process is simple: adsorbed species are removed from the surface by light. However, understanding the physics behind photodesorption implies studying the nuclear motion on the potential energy surface of electronically excited states. This is quite a complicated issue, even for the case of single adatoms. In the work of Hess et al.468 a previously proposed exciton mechanism of desorption, developed for alkali-metal halides, is extended to the adsorption of adatoms on a simple oxide surface such as MgO. These authors discussed the mechanisms of laser-induced photodesorption of alkali-metal halides from MgO using frequency-selected laser pulses. The MgO surface is modeled by an embedded quantum cluster, the ground-state energy is computed with B3LYP, and the same functional is used to explore excited states through TD-DFT. Laser-induced desorption leads to two types of desorbed species, depending on their kinetic energy distribution. Thermally desorbed atoms correspond to kinetic energy (KE) distribution centered below 0.05 eV, whereas hyperthermal atoms have a KE distribution centered between 0.1 and 0.7 eV and are produced by electronic excitations of the surface. The model calculations of Hess et al.468 suggest that hyperthermal desorption of oxygen atoms may take place from MgO corner sites as a result of sequential absorption of photons. The authors also showed that the observed control is due to preferential excitation of surface excitons. In spite of the apparent simplicity of the system, the suggested model is the first attempt to treat the mechanisms of photoinduced desorption of hyperthermal atoms at oxide surfaces. It is rewarding to see that the developed model is consistent with the available desorption and spectroscopic data from MgO films and powder samples, respectively. In a recent joint experiment and theoretical study this methodology has been applied to the hyperthermal atomic oxygen desorption from nanostructured CaO samples.282 In a rather long and complete series of papers, Klüner et al. have studied quite in detail the photodesorption of CO and NO from NiO(100),469−473 CO from Cr2O3(001),474−478 including the analysis of the angular momentum of desorbed molecules479 and lateral velocity distribution,480 and, more recently, CO on TiO2(110).481 These works are all based on

5.3. Photodesorption

Small molecules such as CO, CO2, SO2, and NO are commonly used as probes in surface science and model catalyst studies, often with the purpose of identifying adsorption sites and probing the electron density and acidity/basicity of surface sites or investigating the presence of radical centers.3−7 In these cases, the adopted experimental techniques are mostly based on infrared spectroscopy458−461 and much less is known about the electronic states of adsorbed molecules, especially at oxide surfaces. Most of the published theoretical works do not help in clarifying this issue either. Nevertheless, we mention the collection of papers in the special issue of Chemical Reviews devoted to photophysics and photochemistry of surfaces, which allows one to have a general and rather updated view of this field.462 The limited number of theoretical studies regarding the excited states of small molecules at oxide surfaces comes, at least in part, from the generally low adsorption energy, which makes an accurate theoretical modeling difficult. We already mentioned the case of CO on MgO, where even the most recent and state of the art theoretical methods have not reached a full satisfactory description of the adsorption energy for the ground state.172,176 A similar situation occurs for NO on NiO(001) or on Ni-doped MgO, due to the multiconfigurational character of the electronic ground state, which complicates the application of DFT-based techniques.71,177,463 There is a study on MgO(100)−CO focusing on the excited states, using a quasiparticle (GW) approach, and taking into account the hole−electron interaction through the Bethe− Salpeter approximation.464 The BSE calculations indicated that the exciton spectrum of adsorbed CO is dominated by charge transfer excitons which couple strongly to the molecular excitations of CO. However, the GW calculations are based on either the LDA or GGA exchange-correlation functionals, which, as mentioned above, have problems describing the ground-state adsorption energy. More experimental information would be needed to clarify whether the problems in the description of the adsorption energy affect the calculated spectrum of the adsorbed molecule. Nevertheless, this study is important, as it allowed further developments, such as considering the dynamics of the electronic states of clean MgO(100) and of the CO-covered surface.465,466 4478


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6. EXCITED STATES AT OXIDE SURFACES RELATED TO PHOTOCATALYSIS As discussed earlier, the basics of photocatalysis consist of the absorption of photons of energy higher than or equal to that corresponding to the band gap of the photocatalyst with the consequent generation of photoelectrons and photoholes. The former are able to reduce adsorbed species and the latter to oxidize them. This field has also been the subject of several reviews,34−38 and hence, we will focus on the description of electronic excitations in photocatalytic systems by means of theoretical calculations. We must warn, however, that this field is in very rapid expansion and a complete revision is therefore difficult. Photoreactions taking place at the surface of semiconducting and insulating oxides have also been reviewed by Osgood,494 who focused essentially on the most fundamental aspects and, in particular, on the several details governing the reaction dynamics. Titanium dioxide is no doubt the most widely used material in heterogeneous photocatalysis, although an important limitation of TiO2-based photocatalysis is its low efficiency. Nevertheless, several strategies have been proposed to enhance the photocatalytic activity, such as doping39 or photoinduced superhydrophilicity,495 which implies using the energy from light to change the catalyst surface itself, among others.496 The first issue for a theoretical description of photocatalytic systems concerns the role of oxygen vacancies and other relevant point defects. Motivated by the experimental results indicating a low defect concentration on anatase surfaces,497,498 DFT calculations were carried out by Cheng and Selloni499 to investigate the formation energies and diffusion pathways of oxygen vacancies and Ti interstitials on anatase TiO2(101). Unlike on the widely studied rutile (110) surface, it was found that on anatase (101) both defects are energetically more stable at bulk and subsurface sites than on the surface and the computed energy barriers for the defects to diffuse from the surface to the bulk are rather low, while the opposite is true for the barriers to diffuse from the bulk to the surface. Later on, DFT+U calculations of surface and subsurface defect formation energies at the reduced anatase and rutile surfaces500 and STM experiments501 confirmed that the (101) surfaces of anatase single crystals display no evidence of surface oxygen vacancies, differently from the case of the rutile TiO2(110), where these are quite common. In any case, the presence of oxygen vacancies, dopants, and other point defects has a direct impact on the electronic structure of TiO2 and, hence, on the photocatalytic properties of the resulting material. In the case of n-defects, acting as electron donors, new electronic states arise, commonly attributed to the presence of “Ti3+ centers”. Whether these centers exist has been a matter of debate for many years, mainly because of the already commented failure of standard LDA and GGA functionals to properly describe the electronic structure of oxides. Consequently, many of the experimentally observed fingerprints of the formation of Ti3+ centers in reduced titania cannot be reproduced by DFT simulations. The case of N-doping provides a clear example of the difficulties in properly describing the electronic structure of the resulting system. An earlier study using the GGA-type PBE functional concluded that, due to different crystal structures and densities, N-doping has opposite effects on the photoactivity of anatase and rutile TiO2, leading to a red shift and a

modeling the oxide surface by an embedded cluster model and obtaining suitable approximations for the ground state and relevant excited states, initially, from CASSCF wave functions and, later, including dynamical correlation effects through the second-order CASPT2 approach. More and more complex dynamical studies have been carried out on the obtained potential energy surfaces, including an increasing number of degrees of freedom. In fact, the detailed study of photodesorption also requires performance of high-dimensional quantum dynamic of molecules on surfaces, which leads to the development of new methods exploiting the architecture of massive parallel computers.482 The dynamical aspects are not straightforward, and new methodologies are required to derive a microscopic model for the excitation and relaxation processes in photochemistry at surfaces. On the basis of ab initio calculations and the so-called surrogate Hamiltonian method, Koch et al.483 were able to compute desorption probabilities and velocities for NO on NiO(001), falling in the experimentally observed range, and to obtain excited-state lifetimes. The surrogate Hamiltonian method was also used to study electronic quenching in the femtosecond laser-induced photodesorption of NO from NiO(001),484 which is caused by the interaction of the excited adsorbate−substrate complex with electron hole pair (O 2p → Ni 3d) states in the surface. This methodology, together with the massively parallel implementation of multidimensional quantum dynamics,482 has also been used to interpret laser-induced desorption of NO from NiO(001)485 and the effect of the time scale, showing that the common assumption of time-scale separation of excitation and relaxation does not hold if ultrashort femtosecond laser pulses are used,486 and to take thermal effects into account.487 To allow for a more complete simulation of the multidimensional dynamics, a mixed quantum−classical scheme has been implemented for the simulation of the laser-induced desorption from surfaces and applied to the laser-induced desorption of NO from NiO(001), which illustrates the significant role of the multidimensionality in the desorption process.488,489 In line with this result a further quantum mechanical treatment was carried out using a 4D potential energy surface for the electronic ground state and a representative excited state, using an embedded cluster model and CASPT2 and CI wave functions.490 The analysis of the wave packet dynamics demonstrates that essentially the lateral coordinate, which was neglected in previous studies,471 is responsible for the experimentally observed bimodality. The case of CO on NiO(001) has also been revisited recently by Mehdaoui and Klüner.491 These authors used quantum dynamics on various potential energy surfaces constructed from CASSCF/CASPT2 calculations using a properly embedded cluster model. The analysis of the results confirms previous results492 by the same authors indicating that a 5σ → 2π* (a3Π) like transition in the CO molecule, which is at variance with a substrate-mediated process, is in excellent agreement with the laser pulse energy used in the experiment. Here, the formation of a covalent bond between the adsorbate and substrate in the electronically excited state is proposed as the driving force for photodesorption, which is markedly different from other cases such as NO on Ni(001) or CO on Cr2O3(0001), where electrostatic forces are the dominant interactions in the electronically excited state.491 For a more detailed description of the theoretical study of photodesorption of diatomic molecules from oxide surfaces, we refer to the recent review by Klüner.493 4479


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blue shift, respectively, of the absorption band edge, but without evidence of Ti3+ species.502 This is in line with the experiments of Chambers et al.503,504 on N-doped rutile, in which they observed substantial red shifting with N-doping. This has been recently analyzed by Govind et al.505 using linear response (LR), real-time (RT) TDDFT and high-level active space equation-of-motion coupled cluster (EOMCC) calculations to study the excited states of the pure and N-doped rutile. The three methods consistently indicate a lowering of the band gap. Moreover, both RT-TDDFT and EOMCC calculations reveal that, compared to the N-doped system, the excitations in pure TiO2 are more delocalized. Regarding the surface electronic states, a GGA study of the N-doped rutile TiO2(101) surface, using the PW91 functional, has also been reported where adsorbed and implanted atomic N is considered. It is also found that it is not likely that N-doping will improve the photocatalytic behavior of this polymorph.506,507 Two related studies focus on the C-doping of both anatase and rutile, one of them using LDA508 and the other one applying the PBE density functional:509 therefore, one must warn on the conclusions regarding the details of the electronic structure. The study by Di Valentin et al.509 takes into account the temperature and oxygen pressure effects through ab initio thermodynamics510,511 and finds that C-doping leads to the formation of numerous different species: in an oxygen-poor environment, C−O substitution and formation of oxygen vacancies are favorable, while, under oxidizing conditions, C tends to substitute for Ti ions in the lattice or to remain in interstitial positions. The interplay of these different defects induces occupied gap states, significantly reducing the electronic transition energy required for photocatalysis. In spite of the controversial results arising from DFT simulations, in a subsequent study, EPR experiments unambiguously show that N-doping induces the presence of paramagnetic centers, which is also predicted by high-level ab initio calculations;512−514 a similar situation is also encountered for F-doped anatase.515 In fact, more sophisticated methods, which go beyond standard LDA or GGA, such as DFT+U and hybrid functional calculations, provide strongly localized solutions (Figure 6) where an excess electron is centered on a single Ti3+ ion, as described in detail in the feature paper by Di Valentin et al.516 However, these states are also very close in energy or even degenerate with partly or highly delocalized solutions, where the extra charge is distributed over several Ti ions. The location in the band gap of the defect states corresponding to these different solutions has implications on the conductivity mechanism in reduced or n-type doped titania. In a recent paper, N-doping of anatase has been studied also with hybrid functionals, including a thermodynamic study to investigate the effect of the oxygen chemical potential and of various chemical doping agents.517 These authors also considered the optical properties by the calculation of the simulated UV and visible absorption spectra using the imaginary part of the frequency-dependent dielectric function. This allowed the authors to unambiguously show that the TiO2−3xN2x diamagnetic system exhibits an enhanced optical absorption under visible light irradiation, with a concomitant band gap narrowing of 0.6 eV induced by delocalized impurity states located at the top of the valence band of TiO2. This study is a remarkable achievement of comparison between theoretical and experimental data. More recently, Ceotto et al.518 focused on the exact location of nitrogen in the TiO2 nanoparticles. The

Figure 6. Spin density (B3LYP) plot for N-doped anatase showing the presence of a single Ti3+ cation. Reprinted from ref 516. Copyright 2009 American Chemical Society. Courtesy of Cristiana Di Valentin and Gianfranco Pacchioni.

comparison between calculated interatomic distances and X-ray absorption experiments shows that N substitutes for oxygen at low levels of doping, whereas oxygen vacancy creation is observed at higher dopant concentrations. In some cases codoping has also been considered since this may have a synergic beneficial effect. This is the case for N−F codoping of anatase, which gives rise to paramagnetic centers, since more impurities are introduced into the lattice. This increases the optical absorption in the visible and is also related to a higher photocatalytic activity toward degradation of methylene blue.519 Hybrid functionals have also been applied to the study of B-doping of anatase. The comparison of calculated and measured hyperfine coupling constants from electron paramagnetic resonance experiments and core-level binding energies suggests that B defects exist both substitutional for oxygen and interstitial.520 These authors also showed that electronic characteristics of interstitial boron are rather independent of the site where the atom is incorporated. More recently, the structure and energetics of photogenerated electrons and holes in the bulk and at the (101) surface of anatase have been investigated using hybrid density functionals,521 thus extending previous work where the N-doping at the (101) surface of anatase was studied by means of the standard PBE functional only.522 This interesting new work521 shows that excitons formed upon UV irradiation become selftrapped, which is consistent with the observation of temperature-dependent Urbach tails in the absorption spectrum and a large Stokes shift in the photoluminescence band of anatase. The electronic structure of B−N-codoped anatase has also been considered, again in a recent combined experimental and theoretical study where hybrid functionals are employed.523 An interesting result has been reported concerning N- and Aucodoping of TiO2 rutile surfaces, where from both theory and experiment it is found that Au preadsorption significantly increases the reachable amount of N implanted in the oxide.524 4480


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calculations carried out using the LDA functional, and hence, the localized nature of these states and their position in the band gap remain to be confirmed by more accurate calculations using hybrid functionals or quasiparticle methods. In a subsequent work, some of these authors extended this study to the case of single-crystalline anatase TiO2(101) nanobelts which exhibit an enhanced photocatalytic activity.541 Finally, in a recent paper Li and Liu presented a comprehensive systematic study of particle size and shape in the photocatalytic activity of anatase nanoparticles in aqueous surroundings.542 These authors studied models of more than 10 differently shaped TiO2 anatase nanoparticles with sizes ranging from 1.2 to 2.7 nm and investigated their atomic and electronic structure carried out with the PBE functional and integrated with a modified Poisson−Boltzmann solver for modeling the solid− liquid interface. In this way a quantitative correlation between the particle morphology and the photocatalytic activity is obtained which is subsequently used to rationalize the experimental observations. In particular, the authors showed that the band gap shift converges rapidly with particle size and that the equilibrium shape of the nanoparticle is quantitatively correlated with the size provided that the effects of the edge and corners are taken into account. However, no specific treatment of the excited states was carried out, and only estimations of the HOMO−LUMO gap are given. The effect of B- and N-doping on TiO2 nanostructures has also been considered in a study comparing the electronic properties of small TiO2 clusters with that of one-dimensional nanorods and nanotubes, two-dimensional layers, and bulk phases using different approximations within GGA density functional theory and GW calculations. The general conclusion is that quantum confinement and doping contribute to reduction of the electronic band gap.543 From the preceding discussion and from the aforementioned specialized literature, it is clear that, as far as photocatalysis is concerned, TiO2 is definitely the most considered oxide. However, other systems have also been described as potential photocatalysts, even under visible sun light irradiation, although many of them often require the presence of a cocatalyst and exhibit more complex structures.544 For instance, it has been shown that appropriate doping of indium−tantalum oxide with nickel yields a series of photocatalysts of In1−xNixTaO4 general formula with x ranging from 0 to 0.2, which induces direct splitting of water into stoichiometric amounts of oxygen and hydrogen under visible light irradiation with a quantum yield of about 0.66%.545 The complicated crystal structure and nonstoichiometry hinder the application of theoretical approaches to these interesting systems, although some relevant papers have already been published. This is the case for PbMoO4, which shows activities for both H2 and O2 evolution, although in the presence of sacrificial reagents and under UV irradiation. Replacing Cr6+ with Mo6+ leads to activity under visible light, which according to DFT calculations is due to the transition from the valence band consisting of Pb 6s and O 2p levels to the electron acceptor level composed of Cr 3d empty orbitals.546 Nevertheless, these studies focus on the electronic structure of the material as emerging from the band structure of the bulk material, without analyzing in detail the effect of the different surfaces and often using LDA- or GGA-type approaches. Jensen et al.,547 on the contrary, studied the structural and electronic properties of the (Ga1−xZnx)(N1−xOx) solid solution as a function of the relative composition using the DFT+U approach. Downward bowing of the band gap over

This study has been further extended to anatase; the analysis of the electronic structure shows a charge transfer among implanted N, adsorbed Au, and oxygen vacancies. These authors also found that the creation of vacancies on the anatase surface modified with both implanted N and supported Au atoms produces migration of substitutional N impurities from bulk to surface sites.525 The mechanism for photocleavage of several organic molecules such as chlorobenzene, p-chlorophenol, or 4chlorophenolon on anatase TiO2 surfaces of different types has been investigated in a series of papers,526−531 where the surface is represented by cluster models terminated by H atoms and the energy is calculated by means of a semiempirical (MSINDO) approach which mimics Hartree−Fock and CIS wave functions. These works are based on an earlier study concerning the photocatalytic generation and reaction of hydroxyl radicals at the surface of anatase nanoparticles.532 These authors focused on the role of three types of oxygen species, namely, atomic (O), singlet (1O2), and superoxide radical anion (O2−), in the aromatic ring-opening step. The computational simplicity of the MSINDO method and of the adopted cluster model allowed these authors to investigate the energy profile for the bond cleavage in the ground (S0) and first excited singlet (S1) electronic states. The final conclusions of these authors remained somehow speculative, but there is no doubt that these works might inspire further investigations adopting a more sophisticated ab initio method. The papers reviewed so far involve the study of bulk or surface mainly of the anatase polymorph of TiO2. Nevertheless, one must realize that the photocatalytic systems involve the contact between the TiO2 (or any other relevant photocatalytic material) and an aqueous solution. The interaction of water with different TiO2 surfaces, stoichiometric or not, has been the subject of several relevant theoretical studies,533,534 mainly based on DFT approaches but also including semiempirical methods,535,536 and has been recently reviewed.537 The origin of various point defects (such as oxygen vacancies) at the TiO2−water interface has been widely studied, giving raise to some controversies.538 The case of the stoichiometric surfaces is not necessarily more simple; the rutile TiO2(011) surface, for instance, has been recently found to prefer, in an aqueous environment, a simple bulk-terminated structure rather than the 2 × 1 reconstruction observed under UHV conditions.539 The question of how this structural change affects the electronic structure of the material and its photocatalytic performance requires, however, further intensive work. Other important aspects determining the photocatalytic activity of TiO2 nanoparticles are the effect of nanostructuring the TiO2 substrate as well as the particle size. These questions have only been addressed rather recently. For instance, in a combined experimental and theoretical paper, Wang et al.540 attempted to understand the underlying photocatalysis mechanism of the nitrogen-doped titania nanobelts. The authors found that N-doping leads to an add-on shoulder on the edge of the valence band which is attributed to the localized N 2p levels above the valence band maximum and to the 3d states of Ti3+. These authors also concluded that the visible light photocatalytic activity originates from the N 2p levels near the valence band, whereas it is suggested that oxygen vacancies and the associated Ti3+ species act as the recombination centers for the photoinduced electrons and holes. Nevertheless, one must consider, once again, that the electronic structure is analyzed from the density of states plots obtained from DFT 4481


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electronic structure calculations, usually within TD-DFT, with a semiclassical description of the electron-vibrational dynamics. Often, model Hamiltonians are used to describe the interfacial electron transfer dynamics, which contain different well-ground physical contributions. In a more recent application, Duncan et al. studied the alizarin−TiO2 interface separately and in the presence of an electrolyte, which provides a model for the interface in a Grätzel-type solar cell and for studying the electron dynamics in systems with both molecular and bulk components.573 This study focused on the dynamics that occur after the initial electron transfer, including the relaxation of the injected electron inside the conduction band of TiO2, the recombination of the injected electron with the chromophore, the electron loss from the semiconductor to an electrolyte mediator, and regeneration of the neutral chromophore by electron transfer from the mediator. In subsequent work, Duncan and Prezhdo provided an elegant explanation of the experimentally observed insensitivity to temperature of the interfacial electron transfer processes in DSSCs by showing that the electron transfer process is promoted by atomic vibrational motions and, hence, the electron transfer rates do depend on temperature, concluding that the dependence is due to other experimental factors.574 For a more detailed description of the electron transfer process, we refer to the excellent reviews by Prezhdo et al.575−577 Nevertheless, in some cases, important information is obtained by resolving the electronic spectra of the dye, either isolated or in solution, without taking into account the oxide surface and also neglecting the dynamics of the electron injection. This was the approach used in the earlier theoretical works on this topic, as well as in more recent studies, and in many cases it has been shown that very useful information can be extracted from these calculations.578−591 In a recent study by Casanova et al.592 the ground and first excited singlet states of several molecular dyes of a considerable size, such as 3-(5-(4(diphenylamino)styryl)thiophene-2-yl)-2-cyanoacrylic acid, have been studied by means of TD-DFT and wave-functionbased methods. Clearly the size of the dyes justifies a simplified approach where the role of the oxide surface is neglected. A comparison to experimental data allows the assessment of several theoretical methods in terms of compromise between accuracy and computational cost and rationalizes the role of different structural features and chemical substitutions of the dye. There have been attempts to study the interaction of sensitizers with periodic models of TiO2593−597 or ZnO598,599 surfaces, although in this case the description of the electronic structure and excited states is rather crude and relies on the information extracted from band structure analysis and density of states or spin density plots. As an alternative approach, the effect of the oxide surface has been explicitly considered in a study of the sensitization of TiO2 nanoparticles by the [Fe(CN)6)]4− anion, where a stoichiometric anatase (TiO2)38 cluster (Figure 7) of nanometric dimensions exposing (101) surfaces was used as a model.600 The authors adopted TD-DFT with the hybrid B3LYP functional to compare the energy levels of the isolated [Fe(CN)6)]4− and TiO2 components with those corresponding to the dye−surface complex in reciprocal interaction. Optimized structures have been obtained from DFT calculations using a plane wave basis set with the PBE functional and considering explicitly the presence of the solvent, which was indeed found to be crucial. From this study, it is concluded that a direct charge injection process from

the entire composition was reported, and the minimum band gap was estimated to be about 2.29 eV for the intermediate concentration equal to x = 0.525. The work of Wu et al.548 on the Cd1−xZnxS solid solutions also analyzed the relationship between photocatalytic activity and electronic structure by means of the short-range screened HSE06 hybrid functional. Also in this case, however, the analysis is based on the electronic structure of the bulk material only. A similar strategy has also been used by many authors; here we mention the study of the photoelectrocatalytic activity of BiVO4 and the effect of doping with Mo or W,549 the study of native defects and transition-metal doping in CdS photocatalysts,550 and the studies of the electronic structure of RuO2-dispersed ZnGa2O4 and RuO2-loaded PbWO4 photocatalysts551,552 as representative examples which also indicate that a considerable amount of work remains to be done from the modeling theoretical point of view.

7. EXCITED STATES AT OXIDE SURFACES RELATED TO DYE-SENSITIZED SOLAR CELLS As already mentioned above, dye-sensitized solar cells (DSSCs) constitute peculiar chemical systems, implying an oxide surface and an adsorbed dye. The electrodynamics in DSSCs involve excited states and charge transfer processes between the adsorbed dye and the oxide surface. The basic physics of DSSCs and the state of the art has been reviewed by Grätzel,51,553 and consequently, we will focus on the most relevant theoretical studies only. Here, we point out that a sufficiently accurate description of the excited states involved in DSSCs is required to further understand the microscopic mechanism of the time evolution of the charge transfer process between the dye and the oxide surface electron injection process. This is a quite complicated process with significant advances due to the pioneering and continued work of several groups.554−571 Batista et al.563,564 investigated the dynamics of electron injection from the relatively high excited states of catechol to TiO2 employing a strategy consisting of a combination of ab initio and semiempirical approaches. These authors used the semiempirical Hückel Hamiltonian for electron relaxation dynamics combined with an ab initio description of atomic configurations. This approach allowed a quite large system to be simulated and anisotropic charge diffusion through the TiO2 crystal to be followed and has been recently applied to propose an inverse design methodology aimed at assisting the synthesis and optimization of molecular sensitizers for DSSCs.572 However, the first real-time atomistic simulations of the interfacial electron transfer were carried out slightly earlier by Prezhdo et al., who studied the electron injection from isonicotinic acid into TiO2 using ab initio nonadiabatic molecular dynamics.554,555 The approach of Prezhdo et al. considered treatment of coupled electron−nuclear dynamics, which allowed them to describe both adiabatic and nonadiabatic electron transfer mechanisms. Note that in nonadiabatic molecular dynamics atomic motions can induce transitions between electronic states, which is not possible in the adiabatic approximation. Moreover, the coupling of ab initio simulations and electron−nuclear dynamics allowed the electron injection mechanism that varied at low and high temperatures to be established and also the nuclear motions promoting the electron transfer to be identified. Therefore, the main common idea behind this series of papers is to combine a quantum mechanical description of the interface through 4482


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fundamental requisite to obtain high open-circuit potentials in DSSC devices,604 a conclusion which may be helpful to further improve this important technology. Precisely, joint experimental and theoretical work resulted in the fabrication of a new DSSC with an extra-high voltage exceeding 1 V when using a less expensive organic sensitizer and resulting in an overall efficiency of 3.17%.605 Transient photovoltage decay measurements and DFT calculations show that this high voltage arises from a shift of the TiO2 conduction band edge rather than from slow recombination. Nevertheless, in spite of the considerable number of studies already commented600−605 and others following the same strategy,606−613 it is necessary to continue on the line of fundamental studies to be able to evaluate separately the effect of the many components of a real DSSC. For instance, a key issue of the electron injection mechanism is the localization of the HOMO well inside the dye and preferably with the electron density condensed in the region far from the anchorage to the TiO2 surface and a virtual orbital (LUMO + n) whose electron density is located in the anchoring region and preferably with an overlap with the dye in the substrate region. The adsorbed structures (obtained from Car−Parrinello MD) of the N719 (bis(tetrabutylammonium) cis-bis(isothiocyanato)bis(2,2′-bipyridyl-4,4′-dicarboxylato)ruthenium(II)) sensitizerfor which record cells delivered photocurrent densities of 17.7 mA cm−2 and open-circuit potentials of 846 mV reaching an efficiency of 11.2%on a (TiO2)82 cluster with various counterions are shown in Figure 8. Figure 9 shows the HOMO and LUMO + 11 (the first virtual orbital with dye character) density plots clearly illustrating that the condition above is fulfilled.612 In a recent work, a new panchromatic transdithiocyanatoruthenium(II) sensitizer has been synthesized, characterized, tested in DSSC devices, and theoretically described by means of TD-DFT calculations with the B3LYP functional.614 The calculated absorption spectrum of the complex in solution and adsorbed onto an extended (TiO2)82 surface slab was then compared with experiment. Later,615 adsorption geometries and electronic properties of dyesensitized TiO2 interfaces were studied by periodic DFT calculations for organic dyes and solvent (water or acetonitrile) molecules coadsorbed on the (101) surface of anatase TiO2. The predictions from the calculations are consistent with a shift of the conduction band toward higher energies measured in acetonitrile compared with water and highlight the relevant role of solvation in determining the dye−semiconductor electronic coupling. The modeling of DSSCs has gone one step further with one of the latest papers by Pastore and De Angelis,616 in which a computational framework model to interpret the Stark shifts experimentally observed by photoinduced absorption spectroscopy is presented. The results of the calculations show that the presence of oxidized dye molecules induces major spectral changes on the adjacent neutral dyes, which, along with the simulated effect of injected charge into TiO2, provide Stark shifts nicely reproducing the experimental observations. Dynamical effects on the electronic structure of the adsorbed solvated dye have also been considered in a first-principles molecular dynamics and TD-DFT study of the squaraine dye attached to a TiO2 surface model surrounded by solvent molecules that are treated at the same level of theory as the dye molecule.617 It is found that the dynamical effects induced by thermal fluctuations have a strong effect on the optical properties and that satisfactory agreement with experiments is achieved only when those thermal effects are accounted for

Figure 7. Optimized geometry of the stoichiometric anatase Ti38O76 cluster used in ref 600. Reprinted from ref 600. Copyright 2004 American Chemical Society.

an occupied dye molecular state to a nanoparticle excited state localized on a few Ti atoms takes place in this system. This finding is in agreement with experimental evidence. The same type of approach, namely, PBE optimization within a plane wave approach on the same cluster model of the surface, followed by a TD-DFT calculation with the hybrid B3LYP functional for the adsorbed dye in solution using a continuum model for the solvent, was used in a subsequent study aimed at investigating the role of sensitized protonation.601 This study confirmed an injection mechanism for Ru(II) dyes on TiO2 mediated by the dye excited states and pointed out a remarkable effect of dye protonation on the electronic properties of TiO2 nanoparticles sensitized by the cis(NCS)2−Ru(II)−bis(2,2′-bipyridine-4,4′-dicarboxylate) dye. The authors concluded that two different electron injection mechanisms may be present in DSSCs employing dyes carrying a different number of protons. In a further step toward the understanding of more realistic systems, a combined experimental and theoretical study focused on the origin of the different open-circuit potentials observed in dye-sensitized solar cells using Ru(II)−polypyridyl sensitizers.602 Photovoltaic measures for different sensitizers and DFT calculations allow understanding the electronic structure of dye-sensitized TiO2 nanoparticles. It is concluded that sensitizers which have two equivalent bipyridine ligands adsorb onto TiO2 via a single bipyridine, leading to a TiO2 conduction band downshift and overall reduction of the cell open-circuit potential. In a subsequent work,603 always within the aforementioned approach, the excited-state oxidation potential for several isolated dyes and TiO2 nanoparticles in solution was determined as the difference between the ground-state oxidation potential and the lowest vertical excitation energy. The values obtained for the transition energy between the two lowest singlet states reproduce almost exactly the experimental values, indicating that this type of approach is valuable to understand these complex and important systems. In fact, a recent work along these lines concluded that the existence of three anchoring sites for the sensitizer at the surface appear as a 4483


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Figure 9. HOMO and LUMO + 11 (first unoccupied orbital with dye character) of the N719 sensitizer adsorbed onto a (TiO2)82 surface as obtained from B3LYP/3-21G*/C-PCM single-point calculations on Car−Parrinello-optimized geometries from ref 612. Reprinted from ref 612. Copyright 2011 American Chemical Society. Courtesy of Filippo De Angelis. Figure 8. Adsorbed structures of N719 on the (TiO2)82 cluster with varying counterions as obtained from Car−Parrinello calculations from ref 612. Reprinted from ref 612. Copyright 2011 American Chemical Society. Courtesy of Filippo De Angelis.

38 and used in subsequent studies of catechol621 and coumarin622 derivatives on TiO2. In the case of coumarin, the authors suggested that the position and width of the first band in the electronic absorption spectra, the absorption threshold, and the LUMO energy with respect to the conduction band edge are the key parameters controlling the efficiency of coumarin derivatives as sensitizers in DSSCs. In a related study,623 the electronic structure and the optical response of cathecol and alizarin adsorbed to (TiO2)9 models, which according to previous work620,621 reproduces the results obtained by means of a larger (TiO2)38 cluster, has been analyzed and compared to the electronic structure of the free molecules using both LR-TD-DFT and RT-TD-DFT, revealing characteristic aspects of each sensitizer having different electron injection mechanisms. DSSC devices based on materials other than TiO2, such as ZnO, have attracted the attention of theoretical groups and deserve some additional comments. For instance, Labat et al.624

explicitly, which opens the way to more realistic DSSC models but at the cost of very demanding calculations. The works reviewed above are all based on the standard linear response implementation of TD-DFT (LR-TD-DFT) described in section 2.3.2. An alternative approach is to make use of real-time TD-DFT (RT-TD-DFT),618,619 which allows the whole spectrum to be obtained from a single calculation but has the disadvantage that transitions contributing to the absorption spectra cannot be resolved. On the other hand, the real-time version can be used to find the injection time of an electron. A combination of the two approaches permits the most complete picture available to be obtained, as shown by Sanchez de Armas et al.620 in the study of the absorption spectrum of alizarin adsorbed on (TiO2)n clusters with n up to 4484


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have considered the case of the eosin dye on ZnO(1010̅ ) using a periodic hybrid DFT approach. From the electronic structure analysis, mainly density of states and spin density plots, corresponding to the state-optimized structure, it is found that both the HOMO and the LUMO of the resulting system are very well localized on the dye and, consequently, a direct HOMO → LUMO excitation cannot lead to electron injection into the semiconductor. Therefore, a two-electron mechanism is proposed to explain the low efficiency of this type of solar cell. In a following study,625 these authors used a strategy in which the spectral features of the isolated dyes of interest are first computed and compared to the calculated edges of the valence and conduction bands of the ZnO substrate, with the effect of the electrolyte being considered explicitly by inclusion of solvent molecules following previous studies from the same group.626 This approach, recently reviewed,627 allows the surface to be described using a periodic model, opposite the studies described above, where the TiO2 substrate is simulated using a sufficiently large cluster model.600−605 Conversely, this approach implies the use of a crude method (density of state plots) to estimate the absorption spectrum of the system containing the dye and the substrate, whereas in the case of the TiO2 studies commented above the absorption spectrum is calculated using TD-DFT. The recent developments concerning the implementation of TD-DFT for periodic systems200 will for sure contribute to avoidance of the dilemma in the choice of the surface model.

and engineering of more efficient DSSCs. However, reaching this horizon, even if already in sight, will still require a continued effort in the forthcoming years. Photocatalysis, on the other hand, requires further development from the theoretical point of view. In fact, most of the applications reviewed in this work focus on the modifications of the electronic structure of the oxide materials of interest so that they will efficiently absorb photons in the visible part of the solar spectrum, and little or nothing is known about the reactivity on the excited states. This constitutes no doubt one of the more complicated challenges in the field, since it requires not only the possibility of describing the excited states of interest but of exploring the corresponding energy surfaces and coupling the reaction dynamics on these excited surfaces to the deactivation dynamics of the excited states. This will require development of new methods and techniques; the road is still to be built, but the destination is clear and known.

AUTHOR INFORMATION Corresponding Author

*Phone: +34934021229. Fax: +34934021231. E-mail: francesc. [email protected].

8. CONCLUDING REMARKS AND PERSPECTIVES At first sight, excited states at oxide surfaces could, perhaps, appear to be a rather exotic field, mainly related to basic science and to the will to understand these systems at the deepest possible level. However, the discussion in the two previous sections made it clear that excited states at oxide surfaces have become a field of enormous interest because of the direct relationship to photocatalysis and DSSCs, the core of two relevant technologies of paramount importance for a sustainable society. The input from technology has been a driving force for the basic research, for both theory and experiments. Theoretical methods together with appropriate surface models are nowadays capable of treating very large oxide systems with an increasing predictive power, especially for the ground-state properties and chemical reactivity. Nevertheless, one must warn that in some cases the situation is less clear and even the nature of the ground state constitutes a challenge for the present methods. This is especially the situation encountered when several open shells are involved and the electronic states arising from the different possible spin couplings are near degenerate. In any case, the progress in CI methods specially designed to reproduce excitation energies, such as DDCI, together with the remarkable advances in DFT and TD-DFT methods, which are nowadays applicable to cluster or periodic surface models, has paved the way for the study of more realistic systems. The spectroscopic characterization of point defects at oxide surfaces and the theoretical study of the low-lying states of probe molecules at oxide surfaces can nowadays be studied in an almost routine way. The modeling of DSSCs discussed in the previous section where the oxide surface, the adsorbed dye, and the effect of the solvent are all taken into account simultaneously to predict the absorption spectrum of the corresponding system constitute no doubt a landmark toward the theoretically assisted prediction

Notes

The authors declare no competing financial interest. Biographies

Carmen Sousa graduated with a degree in chemistry from the University of Barcelona in 1989. She received her Ph.D. in Physical Chemistry at the same university in 1994 under the supervision of Prof. Francesc Illas. She moved to the University of Groningen for two years as a postdoctoral fellow with Prof. W. C. Nieuwpoort and Prof. R. Broer. In 1997 she joined the Physical Chemistry Department of the University of Barcelona, where she has been Associate Professor since 2003. Her research concerns the study of the electronic structure and optical properties of molecules, clusters, and solids with strong electronic correlation effects. 4485


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LIST OF ACRONYMS

AIMP B(3)LYP BCS BSE CAS-MP2

ab initio model potential Becke−Lee−Yang−Parr Bardeen, Cooper, and Schrieffer Bethe−Salpeter equation complete active space second-order Möeller− Plesset CAS-PT2 complete active space second-order perturbation theory CASSCF complete active space self-consistent field CC coupled cluster CEPA coupled electron pair approximation CGTO contracted Gaussian-type orbital CI configuration interaction CIPSI configuration interaction by perturbation of a multiconfiguration wave function selected iteratively CIS configuration interaction, single excitations DCI double-excited configuration interaction DDCI difference-dedicated configuration interaction DFT density functional theory DSSC dye-sensitized solar cell EELS electron energy loss spectroscopy EPR electron paramagnetic resonance FCI full configuration interaction GGA generalized gradient approach GTO Gaussian-type orbital H space Hilbert space HF Hartree−Fock HF-SCF-LCAO Hartree−Fock self-consistent field linear combination of atomic orbitals HOMO highest occupied molecular orbital HS high spin IMOMM integrated molecular orbital molecular mechanics KE kinetic energy KS Kohn−Sham LCGTO-DF linear combination of Gaussian-type orbitals density functional LDA local density approximation LR-TD-DFT linear response time-dependent density functional theory LS low spin LSD(A) local spin density (approximation) LUMO lowest unoccupied molecular orbital LYP Lee−Yang−Parr MCCEPA multiconfiguration coupled electron pair approximation MCSCF multiconfiguration self-consistent field MD molecular dynamics MIES metastable impact electron spectroscopy MO molecular orbital MOCI molecular orbital configuration interaction MO-LCAO molecular orbital linear combination of atomic orbitals MPn nth-order Mö e ller−Plesset perturbation theory MR(SD)CI multireference (single- and double-excited) configuration interaction MRCI multireference configuration interaction NEXAFS near-edge X-ray absorption fine structure

Sergio Tosoni was born in Torino, Italy, in 1979. He obtained his Ph.D. in Chemistry from the University of Torino in 2007 under the supervision of Prof. Piero Ugliengo. After three years as a postdoctoral fellow with Prof. Joachim Sauer at the Humboldt University in Berlin, he is currently working in the group of Prof. Francesc Illas at the University of Barcelona. His research interests are in the area of modeling of surface interactions, reactivity, and photocatalysis.

Francesc Illas was born in Barcelona, Spain, in 1954. He studied chemistry at the University of Barcelona (UB) and obtained his Ph.D. on the application of quantum chemistry to surface problems. He has been a visiting scientist at the IBM Almaden Research Center and Los Alamos National Laboratory and an invited professor at the Pierre and Marie Curie University (Paris, France) and University of Calabria (Italy). He is currently Full Professor at UB, where he also serves as director of the Institute of Theoretical and Computational Chemistry. His research focuses on theoretical and computational studies in materials science, surface science, and heterogeneous catalysis.

ACKNOWLEDGMENTS We acknowledge F. De Angelis, P. S. Bagus, T. Bredow, E. A. Carter, K. Fink, N. M. Harrison, K. Hermann, T. Klüner, E. A. Kotomin, C. Di Valentin, L. G. M. Pettersson, G. Pacchioni, J. F. Sanz, P. Sautet, A. Shluger, and P. V. Sushko for sending relevant references and/or useful comments which made this review possible. This research has been supported by the Spanish MICINN through Program INNPACTO Project CASCADA IPT-120000-2010-19 and Research Grant FIS2008-02238. We are also grateful to the Generalitat de Catalunya for partial support through Grants 2009SGR1041 and XRQTC. Finally, F.I. acknowledges additional support through a 2009 ICREA Academia award for excellence in research. 4486


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our own N-layered integrated molecular orbital and molecular mechanics projector-augmented wave polarizable continuum model plane wave quantum mechanics/molecular mechanics real-time time-dependent density functional theory self-consistent field single- and double-excited configuration interaction single-, double-, triple-, and quadruple-excited configuration interaction scanning force microscopy spectroscopy-oriented configuration interaction Slater-type orbital time-dependent density functional theory time-dependent Hartree−Fock time-dependent Kohn−Sham total ion potential unrestricted Hartree−Fock ultrahigh vacuum ultraviolet photoemission spectroscopy ultraviolet valence bond X-ray absorption near-edge structure X-ray absorption near-edge spectroscopy X-ray photoelectron spectroscopy

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