Tangui Le Bahers
email: tangui.le_bahers@ens-lyon.fr

Kazuhiro Takanabe
email: takanabe@chemsys.t.u-tokyo.ac.jp

Combined theoretical and experimental characterizations of semiconductors for photoelectrocatalytic applications

Keywords: photocatalysis, absorption coefficient, exciton binding energy, effective mass, DFT

Photoelectrocatalysis studies have emerged tremendous knowledge and experience, which has led to discovery of a number of active materials. Due to complexity of the processes involved, however, development of suitable semiconductor and catalyst for the photoelectrocatalysis still requires a try-and-error approach as different parameters are coherently varied to determine overall performance. This contribution provides insight into the key optoelectronic properties of the semiconductors, such as absorption coefficient, dielectric constant, effective masses, exciton binding energy, and band positions. The combined assessment from theoretical and experimental points of view is reviewed, which may help thorough understanding of the semiconductors and the way to reliably characterize them, and in turn to develop improved photoelectrocatalytic materials.

Introduction

The scientific community has devoted much effort to developing efficient photovoltaic devices that can convert sunlight (i.e., photons) into electricity (i.e., electrons and holes in an external circuit). This important topic of research has led scientists to the development of a large variety of photovoltaic materials [START_REF] Peng | Organohalide Lead Perovskites for Photovoltaic Applications[END_REF][START_REF] Goetzberger | Photovoltaic materials, history, status and outlook[END_REF][START_REF] Ameri | Highly efficient organic tandem solar cells: a follow up review[END_REF][START_REF] Stuckelberger | Review: Progress in solar cells from hydrogenated amorphous silicon[END_REF][START_REF] Shah | Photovoltaic Technology : The Case for Thin-Film Solar Cells[END_REF][START_REF] Green | Solar cell efficiency tables (version 51)[END_REF][START_REF] Grätzel | Recent advances in sensitized mesoscopic solar cells[END_REF] whose photoconversion efficiencies are compared on the frequently updated NREL chart. [START_REF]Photovotlaic Efficiency Chart[END_REF] In parallel to this research activity, the researchers developed the idea that photogenerated electrons and holes can be effectively and directly consumed to perform electrochemical redox reactions. [START_REF] Nozik | Photoelectrochemistry: Applications to solar energy conversion[END_REF][START_REF] Yoneyama | Electrochemical aspects of light-induced heterogeneous reactions on semiconductors[END_REF] Such photoelectrocatalytic reactions can produce valuable molecules with the ability to generate energy on demand or valuable molecules for the chemical industry. [START_REF] Schneider | Understanding TiO2 photocatalysis: Mechanisms and materials[END_REF][START_REF] Lang | Heterogeneous visible light photocatalysis for selective organic transformations[END_REF][START_REF] Tu | Photocatalytic conversion of CO2 into renewable hydrocarbon fuels: State-of-the-art accomplishment, challenges, and prospects[END_REF][START_REF] Yang | Roles of cocatalysts in photocatalysis and photoelectrocatalysis[END_REF][START_REF] Jiang | Photoelectrochemical devices for solar water splitting-materials and challenges[END_REF][START_REF] Kudo | Heterogeneous photocatalyst materials for water splitting[END_REF][START_REF] Braslavsky | Glossary of terms used in photochemistry, 3rd edition (IUPAC Recommendations[END_REF][START_REF] Braslavsky | Glossary of terms used in photocatalysis and radiation catalysis (IUPAC Recommendations 2011)[END_REF][START_REF] Kisch | Semiconductor photocatalysis -Mechanistic and synthetic aspects[END_REF] The two main investigated reactions are CO2 reduction [START_REF] Habisreutinger | Photocatalytic Reduction of CO2 on TiO2 and Other Semiconductors[END_REF][START_REF] Morris | Molecular approaches to the photocatalytic reduction of carbon dioxide for solar fuels[END_REF][START_REF] Chang | CO2 photo-reduction: insights into CO2 activation and reaction on surfaces of photocatalysts[END_REF] and water splitting. [START_REF] Chen | Semiconductor-based photocatalytic hydrogen generation[END_REF][START_REF] Hisatomi | Recent advances in semiconductors for photocatalytic and photoelectrochemical water splitting[END_REF][START_REF] Bard | Artificial Photosynthesis: Solar Splitting of Water to Hydrogen and Oxygen Water Splitting[END_REF][START_REF] Chen | Particulate photocatalysts for overall water splitting[END_REF][START_REF] Tachibana | Artificial photosynthesis for solar watersplitting[END_REF] Recently, new types of photoelectrocatalytic reactions are being investigated, such as biomass valorization, among which glycerol oxidation seems most promising. [START_REF] Colmenares | Heterogeneous photocatalytic nanomaterials: Prospects and challenges in selective transformations of biomass-derived compounds[END_REF][START_REF] Zhang | Nanochemistry-derived Bi2WO6 nanostructures: Towards production of sustainable chemicals and fuels induced by visible light[END_REF][START_REF] Li | A sustainable approach for lignin valorization by heterogeneous photocatalysis[END_REF][START_REF] Puga | Photocatalytic production of hydrogen from biomass-derived feedstocks[END_REF] Since water splitting is one of the most investigated reactions in photoelectrocatalysis, this manuscript is oriented toward this reaction as an excellent benchmark of the energetically uphill reaction. [START_REF] Osterloh | Photocatalysis versus Photosynthesis: A Sensitivity Analysis of Devices for Solar Energy Conversion and Chemical Transformations[END_REF] The main concepts presented herein can be transferred to other photoelectrocatalytic reactions. Water splitting is the combination of two half-electrochemical reactions called oxygen evolution reaction (OER, equation 1) and hydrogen evolution reaction (HER, equation 2). To achieve solar-driven water splitting, multiple device configurations have been proposed. [START_REF] Nielander | A taxonomy for solar fuels generators[END_REF][START_REF] Lewis | Research opportunities to advance solar energy utilization[END_REF] The first solar-derived water splitting technology is simply accomplished by connecting a solar panel to an electrolyzer via an external circuit to split water and produce H2. Another technology consists of solar panels directly immersed in water to perform OER and HER with appropriate surface modifications to make photoelectrodes, [START_REF] Gerischer | The impact of semiconductors on the concepts of electrochemistry[END_REF][START_REF] Ding | Photoelectrocatalytic Water Splitting: Significance of Cocatalysts, Electrolyte, and Interfaces[END_REF] which might not require wiring. [START_REF] Walter | Solar water splitting cells[END_REF] Also, photon absorbers can be dispersed in water as a powder form with the high surface area, so-called photocatalyst, which has a dual function to absorb the sunlight and to perform the two half-reactions. [START_REF] Gerischer | A mechanism of electron hole pair separation in illuminated semiconductor particles[END_REF][START_REF] Brus | Electronic wave functions in semiconductor clusters: Experiment and theory[END_REF][START_REF] Nosaka | Separation Mechanism of a Photoinduced Electron-Hole Pair in Metal-Loaded Semiconductor Powders[END_REF][START_REF] Abe | Recent progress on photocatalytic and photoelectrochemical water splitting under visible light irradiation[END_REF][START_REF] Osterloh | Inorganic nanostructures for photoelectrochemical and photocatalytic water splitting[END_REF][START_REF] Goto | A Particulate Photocatalyst Water-Splitting Panel for Large-Scale Solar Hydrogen Generation[END_REF]

Multiple articles have addressed the economic viability of these water splitting technologies. [START_REF] Pinaud | Technical and economic feasibility of centralized facilities for solar hydrogen production via photocatalysis and photoelectrochemistry[END_REF][START_REF] Shaner | A comparative technoeconomic analysis of renewable hydrogen production using solar energy[END_REF][START_REF] Seitz | Modeling practical performance limits of photoelectrochemical water splitting based on the current state of materials research[END_REF][START_REF] Sathre | Life-cycle net energy assessment of large-scale hydrogen production via photoelectrochemical water splitting[END_REF] Several architectures were proposed to estimate their capital cost, assuming several working conditions. As of now, the highest efficiency for producing H2 from sunlight is to connect two separate devices of a photovoltaic cell and an electrolyzer. [START_REF] Ager | Experimental demonstrations of spontaneous, solar-driven photoelectrochemical water splitting[END_REF] It is difficult to rigorously compare these studies because the assumptions made about the properties and architectures of the materials are typically different. However, it seems likely that a direct water splitting technology, using photoelectrochemical and photocatalytic systems, would be more profitable than a solar panel connected to an electrolyzer. [START_REF] Pinaud | Technical and economic feasibility of centralized facilities for solar hydrogen production via photocatalysis and photoelectrochemistry[END_REF] Unfortunately, despite the potential economic advantage, efficient photoelectrocatalytic materials have not yet met the requirements for commercialization.

Therefore, for this technology to be considered viable for H2 production, it is necessary to develop efficient water splitting photoelectrocatalytic materials that are applicable to an integrated photoelectrochemical system or a photocatalyst powder system.

Understanding the fundamental phenomena involved in the photocatalytic process should help to improve this efficiency. As we will see in the next section, however, this is a difficult task because of the complex elementary steps involved in the photocatalytic process. [START_REF] Takanabe | Photocatalytic Water Splitting: Quantitative Approaches toward Photocatalyst by Design[END_REF] This review focuses on characterization of photocatalyst powder materials, but we note that fundamental aspects for optoelectronic properties are very much in common for photoelectrochemical reactions.

Focus on the photocatalysts

Heterogeneous photocatalysts developed for overall water splitting using a single semiconductor typically have the architecture presented in Figure 1. [START_REF] Hisatomi | Recent advances in semiconductors for photocatalytic and photoelectrochemical water splitting[END_REF][START_REF] Fabian | Particle suspension reactors and materials for solar-driven water splitting[END_REF] The photocatalyst consists of semiconductor particles decorated by cocatalyst nanoparticles to facilitate OER or HER. [START_REF] Takanabe | Solar Water Splitting Using Semiconductor Photocatalyst Powders[END_REF] The fundamental principles of photoelectrochemical OER or HER are generally the same under open circuit conditions. Figure 1 focuses on the respective half-reactions. The photocatalytic reaction can be divided into several sequential processes (Figure 1). The efficiency of these steps is mainly governed by the properties of the semiconductor, the cocatalyst or the interface between the two. Successful photoelectrocatalysis proceeds via the following six steps.

Step 1 corresponds to the light absorption by the semiconductor creating an electron (e -) and hole (h + ) pair, called exciton. [START_REF] Liang | Excitons[END_REF] The light absorption is mainly governed by the bandgap (Eg) in a semiconductor that can be seen as the difference in energy between the top Step 1

Step 2

Step 3

Step 4 Step 4

Step 5 Diffusion Convection ... Diffusion Convection ...

Step 6

Step 6

HER + Ion drift

Step [START_REF] Shah | Photovoltaic Technology : The Case for Thin-Film Solar Cells[END_REF] of the valence band and the bottom of the conduction band. In short, the bandgap must be low enough to absorb the largest number of photons contained in sunlight and large enough to produce electrons and holes having sufficient energies to perform the two half-reactions. For water splitting reaction, the thermodynamic potential difference is 1.23 eV. In reality, it has been shown that to take into account the unavoidable energy loss due to all the electron transfers and entropic loss, [START_REF] Osterloh | Maximum Theoretical Efficiency Limit of Photovoltaic Devices: Effect of Band Structure on Excited State Entropy[END_REF][START_REF] Omelchenko | Excitonic effects in emerging photovoltaic materials: A case study in Cu2O[END_REF] the bandgap of the semiconductor is required to be above 1.8 eV or, more ideally, between 2.0 and 2.5 eV (to maximize the attainable efficiencies with some variations). [START_REF] Pinaud | Technical and economic feasibility of centralized facilities for solar hydrogen production via photocatalysis and photoelectrochemistry[END_REF][START_REF] Sivula | Semiconducting materials for photoelectrochemical energy conversion[END_REF][START_REF] Hu | An analysis of the optimal band gaps of light absorbers in integrated tandem photoelectrochemical water-splitting systems[END_REF][START_REF] Haussener | Modeling, simulation, and design criteria for photoelectrochemical water-splitting systems[END_REF] Figure 2(a)-(c) presents the variation of the solar to hydrogen (STH) efficiency as a function of the bandgap of the semiconductor, along with the classical Shockley-Queisser limit of a photovoltaic device based on a p-n junction. [START_REF] Pinaud | Technical and economic feasibility of centralized facilities for solar hydrogen production via photocatalysis and photoelectrochemistry[END_REF][START_REF] Shockley | Detailed Balance Limit of Efficiency of p-n Junction Solar Cells[END_REF] This simulation includes some models of the energetic losses, including overpotentials for the halfreactions. [START_REF] Pinaud | Technical and economic feasibility of centralized facilities for solar hydrogen production via photocatalysis and photoelectrochemistry[END_REF] It is interesting to see the large difference between the theoretical maximum photoconversion efficiencies between a standard p-n junction photovoltaic cell (around 35 %; power efficiency) and the maximum STH efficiency (around 12%; energy efficiency based on bandgap energy and hydrogen product energy) using a single photocatalyst. Because a single semiconductor based photocatalyst has a limited attainable efficiency, multiple semiconductor configurations are considered, such as Z-scheme structures or so-called type-II heterostructure. [START_REF] Zhou | All-solid-state Z-scheme photocatalytic systems[END_REF][START_REF] Li | Z-Scheme Photocatalytic Systems for Promoting Photocatalytic Performance: Recent Progress and Future Challenges[END_REF][START_REF] Maeda | Z-scheme water splitting using two different semiconductor photocatalysts[END_REF][START_REF] Maeda | Efficient nonsacraficial water splitting through two-step photoexcitation by visible light using a modified oxynitride as a hydrogen evolution photocataylst[END_REF] One of the driving forces motivating the development of such architecture is the possibility to achieve, in principle, much higher attainable efficiencies (Figure 2(b)-(d)). [START_REF] Pinaud | Technical and economic feasibility of centralized facilities for solar hydrogen production via photocatalysis and photoelectrochemistry[END_REF][START_REF] Hu | An analysis of the optimal band gaps of light absorbers in integrated tandem photoelectrochemical water-splitting systems[END_REF] The maximum attainable STH energy efficiency using twosemiconductor-based materials is ~25% when a tandem configuration is used together with highly efficient electrocatalysts. Step 2: The exciton dissociation. After light absorption, the electron and the hole are bound, creating a particle called an exciton. The electron and the hole must be separated to perform the redox reactions. The exciton ionization energy is called exciton binding energy, noted Eb. To have free charge carriers, this energy should be lower than the thermal energy (~25 meV at room temperature). [START_REF] Bahers | Semiconductors used in photovoltaic and photocatalytic devices: Assessing fundamental properties from DFT[END_REF] If the exciton is not dissociated, it will quickly recombine, losing the energy of the photon. The exciton binding energy is mainly governed by the dielectric constant of the material, noted εr. The dielectric constant represents the ability of the material to screen charges. The larger the dielectric constant, the easier it will be to dissociate the exciton. In fact, materials frequently used in photovoltaics (e.g., Si, CdTe, GaAs) have a dielectric constant higher than 10, and their Eb is always lower than 25 meV. [START_REF] Madelung | Semiconductors: Data Handbook, Third edit[END_REF] This value of 10 for εr can be seen as a target to develop new semiconductors for water splitting.

Step 3: Charge carrier diffusion. Once the exciton is dissociated, electrons and holes diffuse to the surface, according to semiconductors physics. [START_REF] Bisquert | Theory of the Impedance of Electron Diffusion and Recombination in a Thin Layer[END_REF] This diffusion is characterized by the diffusion coefficient, D, of the charges related to the charge carrier mobilities, μ, by the Einstein relationship (equation ( 3)). The mobility is linked to the effective masses, m * , and the collision time, τ (equation ( 4)).

B k T D e µ = (3)
* e m τ µ = (4)
There is no fundamental lower limit for the value of the charge carrier mobilities or effective masses. But empirically, by analyzing all semiconductors efficient for photovoltaic applications, it appears that they all have a mobility higher than 10 cm 2 s -1 V -1 and an effective mass lower than 0.5 electron mass at least in one crystallographic direction. Based on this empirical observation, all semiconductors having an effective mass lower than 0.5 electron mass (9.1 × 10 -31 kg, noted m0 hereafter) for holes and electrons can be considered as having good charge carrier mobilities.

Step 4: Charge transfer. In practice, the semiconductors developed to efficiently generated charge carriers are not necessarily adapted to catalyze the two-half reactions of water splitting. In other words, the activation energy or the overpotential of the electrocatalytic reactions are generally large on the surface of the semiconductor. This is the reason why the semiconductor surface is often decorated by cocatalysts having better catalytic activity toward the electrochemical reactions. The electron or hole transfer must be efficient to have a maximum of charges collected by the cocatalysts. This step depends on the quality of the interface between the cocatalyst and the semiconductor.

Step 5: The electrochemical redox reactions. For water splitting, the electrocatalytic reactions are HER and OER, both of which require multiple electrons (see equations ( 1) and

(2)). The potentials used to split water in photocatalysis come from the charge dynamics inside the semiconductor. In other words, photocatalytic water splitting can be seen as electrocatalytic water splitting, in which the applied potential is not chosen by an external electric circuit as in conventional electrolyzer but depends on the efficiency of the semiconductor to generate a charge flow to the surface as a consequence of steps 1-4 (see above). The efficiency of this step is mainly governed by the overpotential (noted η) associated with the cocatalyst. The overpotential corresponds to the extra energy (or the extra potential) needed to perform one half-reaction beyond the thermodynamic limit. We chose photoelectrochemical performance as an example to show how electrocatalytic performance might influence the case of a photoanode for oxygen evolution reaction, as presented in Figure 3. [START_REF] Coridan | Methods for comparing the performance of energy-conversion systems for use in solar fuels and solar electricity generation[END_REF] Figure 3. Conceptual schemes representing attainable current as a function of potential in a two-photoelectrode configuration. The same cathodic and anodic currents achieved by the respective photocathode and photoanode will be the total current generated by the entire device. Two types of electrocatalysts (cocatalysts) are arbitrarily chosen to represent their excellent and poor performances. Black dotted lines represent potentially-achievable photoelectrochemical performance from only charge separation, and red and green dotted lines are actual performance obtainable with excellent and poor electrocatalysis.

Step 6: Fluid mechanics and diffusion of ions and molecules. This step presents phenomena happening mainly inside the electrolyte, where the reactions take place. [START_REF] Qureshi | Contribution of electrolyte in nanoscale electrolysis of pure and buffered water by particulate photocatalysis[END_REF] This is an important aspect in the working principle of a water-splitting device, and further improvements of the photocatalytic efficiencies could be obtained by a smart design of the electrolyte. The optimization of the electrolyte at near-neutral pH was reviewed elsewhere. [START_REF] Shinagawa | Towards versatile and sustainable hydrogen production via electrocatalytic water splitting: Electrolyte engineering[END_REF] Development of new semiconductors and photocatalysts (i.e., semiconductor and cocatalyst) fulfilling this list of requirements has been a major source of work for the photoelectrochemistry community, starting from oxides [START_REF] Hara | Cu2O as a photocatalyst for overall water splitting under visible light irradiation[END_REF][START_REF] Osterloh | Inorganic Materials as Catalysts for Photoelectrochemical Splitting of Water[END_REF][START_REF] Wang | Scalable water splitting on particulate photocatalyst sheets with a solar-to-hydrogen energy conversion efficiency exceeding 1%[END_REF][START_REF] Li | Photoelectrochemical Water Splitting with Mesoporous Hematite Prepared by a Solution-Based Colloidal Approach[END_REF][START_REF] Brillet | Highly efficient water splitting by a dual-absorber tandem cell[END_REF] but quickly progressing to more sophisticated materials, such as sulfides [START_REF] Osterloh | Inorganic Materials as Catalysts for Photoelectrochemical Splitting of Water[END_REF][START_REF] Hu | Mass production and high photocatalytic activity of ZnS nanoporous nanoparticles[END_REF][START_REF] Bao | Self-Templated Synthesis of Nanoporous CdS Nanostructures for Highly Efficient Photocatalytic Hydrogen Production under Visible Light Self-Templated Synthesis of Nanoporous CdS Nanostructures for Highly Efficient Photocatalytic Hydrogen Production under[END_REF][START_REF] Kandiel | Electronic structure and photocatalytic activity of wurtzite Cu-Ga-S nanocrystals and their Zn substitution[END_REF], oxysulfides [START_REF] Lardhi | Ab initio assessment of Bi1-xRExCuOS (RE=La, Gd, Y, Lu) solid solutions as a semiconductor for photochemical water splitting[END_REF][START_REF] Baqais | Bismuth Silver Oxysulfide for Photoconversion Applications: Structural and Optoelectronic Properties[END_REF][START_REF] Meng | Synthesis, Crystal Structure, and Photoelectric Properties of a New Layered Bismuth Oxysulfide[END_REF][START_REF] Ishikawa | Oxysulfide Sm2Ti2S2O5 as a stable photocatalyst for water oxidation and reduction under visible light irradiation (λ ≤ 650 nm)[END_REF], nitrides [START_REF] Nurlaela | Temperature Dependence of Electrocatalytic and Photocatalytic Oxygen Evolution Reaction Rates Using NiFe Oxide[END_REF][START_REF] Hisatomi | Recent advances in semiconductors for photocatalytic and photoelectrochemical water splitting[END_REF][START_REF] Ong | Graphitic Carbon Nitride (g-C3N4)-Based Photocatalysts for Artificial Photosynthesis and Environmental Remediation: Are We a Step Closer to Achieving Sustainability?[END_REF][START_REF] Bhunia | Dendritic Tip-on Polytriazine-Based Carbon Nitride Photocatalyst with High Hydrogen Evolution Activity[END_REF], and oxynitrides. [START_REF] Maeda | Efficient nonsacraficial water splitting through two-step photoexcitation by visible light using a modified oxynitride as a hydrogen evolution photocataylst[END_REF][START_REF] Maeda | Overall Water Splitting on (Ga1-xZnx)(N1-xOx) Solid Solution Photocatalyst: Relationship between Physical Properties and Photocatalytic Activity[END_REF][START_REF] Fuertes | Metal oxynitrides as emerging materials with photocatalytic and electronic properties[END_REF][START_REF] Ziani | Photophysical Properties of SrTaO2N Thin Films and Influence of Anion Ordering: A Joint Theoretical and Experimental Investigation[END_REF] These new systems are frequently the subject of reviews, highlighting the wealth of new semiconductors designed for photoelectrocatalysis. [START_REF] Kudo | Heterogeneous photocatalyst materials for water splitting[END_REF][START_REF] Sivula | Semiconducting materials for photoelectrochemical energy conversion[END_REF][START_REF] Hisatomi | Recent advances in semiconductors for photocatalytic and photoelectrochemical water splitting[END_REF][START_REF] Li | Engineering heterogeneous semiconductors for solar water splitting[END_REF] In parallel, the tools developed by quantum chemists are ever more frequently used to investigate photocatalyst properties, [START_REF] Bhatt | Recent theoretical progress in the development of photoanode materials for solar water splitting photoelectrochemical cells[END_REF][START_REF] Liao | New concepts and modeling strategies to design and evaluate photo-electro-catalysts based on transition metal oxides[END_REF][START_REF] Pham | Modelling heterogeneous interfaces for solar water splitting[END_REF][START_REF] Blood | Factors influencing local membrane curvature induction by N-BAR domains as revealed by molecular dynamics simulations[END_REF][START_REF] Garcia-Esparza | Full in silico DFT characterization of lanthanum and yttrium based oxynitride semiconductors for solar fuels[END_REF] mainly based on the Density Functional Theory (DFT). [START_REF] Cohen | Challenges for density functional theory[END_REF] However, rationalizing the differences in activity of these semiconductors is difficult, mainly because of the lack of facile experimental characterization of their optoelectronic properties.

The current review article is dedicated to the characterization of semiconductors, which can be coherently achieved both experimentally and computationally. To make this contribution unique, we focus on the characterizations of optoelectronic properties that can be directly related to the electronic structures of semiconductor solids. We will present the different techniques available to measure and compute the following key properties, all directly involved from Step 1 to Step 4 (Figure 1).

I. Bandgaps and absorption coefficient.

The bandgap, Eg, is an essential property of semiconductors, and many publications presenting new semiconductors predominantly focus on reporting Eg to characterize their electronic structures. Eg is an important property since the primary attainable efficiency of a photocatalytic device from photon absorption is directly related to this value (Figure 2).

However, the definition of Eg is ambiguous and controversial, sometimes leading to confusion between the optical gap, electronic gap, and HOMO-LUMO gap. A recent article dedicated to the comparison between the different types of bandgaps has been discussed by Breda (Figure 4). [START_REF] Bredas | Mind the gap![END_REF] Based on this work, we can define the fundamental gap corresponding to the difference in energy between the first ionization potential (IP) and the first electron affinity (EA).

( )

g fund EA IP E = - (5)
We can also define the optical gap, corresponding to the energy of transition between the ground state (noted S0) and the first excited state (noted S1). This fundamental gap is larger than the optical gap, measured by absorption spectroscopy or photoluminescence. The energy difference between these two gaps originates from the formation of an exciton for optical measurements. The exciton formation energy is noted as Eb.

( )

g b opt EA IP E E = -- (6)
Eb is generally relatively small for standard covalent semiconductors (few dozens of meV) when compared to the fundamental bandgap (few eV). Thus, in general, Eg(fund) and Eg(opt) are very close. ( )

g CB VB mono E ε ε = - (7)
These energies are obtained by self-consistently solving the monoelectronic Kohn-Sham equations. Removing or adding an electron to a system, with the associate energies IP and EA, change all the electronic structure of the system because the electrons interact with each other leading to orbital energies variation after the electron addition/removal. Thus, to compute IP and EA, the full electronic structure relaxation induced the change in the number of electrons must be taken into account, that is time consuming to compute or even impossible for periodic system (this aspect is discussed later in the manuscript). In 1934, Koopman demonstrated that under the assumption that if the orbital relaxation is not important under electron addition/removal or in other words if the orbitals does not change in energy and in shape, the orbital energies correspond to IP and EA (IP ≈ -εHOMO and EA ≈ εLUMO). [START_REF] Koopmans | Uber die Zuordnung von Wellenfunktionen und Eigenwerten zu den Einzelnen Electronen eines Atoms[END_REF] This is called the Koopman's theorem and is the most frequent assumption used to estimate IP and EA of a system. This theorem was demonstrated in the framework of Hartree-Fock theory. [START_REF] Koopmans | Uber die Zuordnung von Wellenfunktionen und Eigenwerten zu den Einzelnen Electronen eines Atoms[END_REF] An attempt to adapt this theorem to DFT was performed by Janak in 1978. [START_REF] Janak | Proof that / i i E n ε ∂ ∂ = in density-functional theory[END_REF] The application area of Janak's theorem is smaller, because this theorem works only for the HOMO orbital (i.e., IP ≈ -εHOMO). Nevertheless, the community has taken the habit to apply the Koopman's theorem to DFT allowing to estimate the EA from the LUMO energy or in some scientific fields to use core orbitals to estimate core level energies.

( ) ( )

g CB VB g fund EA IP mono E E ε ε = -≈ - = (8)
This assumption is very rough for molecules but works relatively well (for the precision we are looking for) for solids, which will be discussed later. [START_REF] Perdew | Understanding band gaps of solids in generalized Kohn-Sham theory[END_REF] Because the exciton binding energy usually requires only a small correction to go from the fundamental bandgap to the optical bandgap, the monoelectronic bandgap is usually directly compared to the optical bandgap.

Benchmarks of DFT functionals to simulate Eg(mono), thus by using the top of the valence band and the bottom of conduction band energies, have been the subject of numerous works in the literature. [START_REF] Shishkin | Accurate Quasiparticle Spectra from Self-Consistent GW Calculations with Vertex Corrections[END_REF][START_REF] Paier | Screened hybrid density functionals applied to solids[END_REF][START_REF] Tran | Accurate band gaps of semiconductors and insulators with a semilocal exchange-correlation potential[END_REF][START_REF] Hai | Accurate band gaps for semiconductors from density functional theory[END_REF][START_REF] Alkauskas | Defect levels through hybrid density functionals: Insights and applications[END_REF][START_REF] Jain | Reliability of hybrid functionals in predicting band gaps[END_REF][START_REF] Labat | Modeling ZnO phases using a periodic approach: from bulk to surface and beyond[END_REF][START_REF] Labat | Density functional theory analysis of the structural and electronic properties of TiO2 rutile and anatase polytypes: performances of different exchange-correlation functionals[END_REF][START_REF] Labat | Structural and electronic properties of selected rutile and anatase TiO2 surfaces: An ab initio investigation[END_REF][START_REF] Gaspari | Semiconducting and optical properties of selected binary compounds by linear response DFT+U and hybrid functional methods[END_REF] As a matter of fact, some of us presented a benchmark focused on semiconductors used in photoelectrocatalysis and photovoltaic, using functionals frequently used in solid-state chemistry to computed electronic structures: PBE [START_REF] Perdew | Rationale for mixing exact exchange with density functional approximations[END_REF] (GGA), B3LYP (global hybrid, 20% of exact exchange), PBE0 [START_REF] Adamo | Toward reliable density functional methods without adjustable parameters: The PBE0 model[END_REF] (global hybrid, 25% of exact exchange) and HSE06 [START_REF] Heyd | Hybrid functionals based on a screened Coulomb potential[END_REF][START_REF] Heyd | Erratum: "Hybrid functionals based on a screened Coulomb potential[END_REF] (range-separated hybrid). The results of this are presented in If we focus on the bandgap calculation, GGA and global hybrids give bandgaps with a relatively large error. Actually, GGA strongly underestimates bandgaps while global hybrids overestimate it. [START_REF] Jain | Reliability of hybrid functionals in predicting band gaps[END_REF] The range-separated hybrid HSE06 gives bandgaps with 10% of error. It is interesting to focus on this functional and attempt to understand its reliability. This is a range-separated hybrid, meaning that the exact exchange (or Hartree-Fock (HF) exchange) depends on the interelectronic distance in the evaluation of the exchange integrals, unlike for global hybrids, such as PBE0 and B3LYP functionals, whose exact exchange is constant (Figure 6). Contrary to the well-known range-separated hybrids, such as cam-B3LYP, [START_REF] Yanai | A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)[END_REF] designed to increase the HF exchange when increasing the interelectronic distance, the HSE06 functional decreases this fraction (Figure 6). The reason for this behavior is practical rather than physical. Decreasing the HF exchange with the interelectronic distance allows reducing the number of exchange integrals to compute on long-range that save a large amount of computational time in solid-state calculation. [START_REF] Heyd | Hybrid functionals based on a screened Coulomb potential[END_REF] The parameters governing the variation of HF exchange were fitted to reproduce the bandgap of several semiconductors. [START_REF] Heyd | Hybrid functionals based on a screened Coulomb potential[END_REF] HSE06 works well because it is designed specifically for that purpose. Hybrid functionals are frequently considered to be prohibitive in term of computational time, but their implementation in localized basis set (such as Gaussian basis set) codes offers very accurate performances [START_REF] Dovesi | Ab Initio Quantum Simulation in Solid State Chemistry[END_REF], and they can be easily applied to very large systems (up to several hundreds of atoms). [START_REF] Labat | First principles modeling of eosin-loaded ZnO films: a step toward the understanding of dye-sensitized solar cell performances[END_REF][START_REF] Dovesi | Quantum-mechanical condensed matter simulations with CRYSTAL[END_REF] It is interesting to note the high reliability of DFT for bandgap simulation based on band (i.e., orbital) energies. In the molecular case, it is well-documented that to take the HOMO-LUMO gap to simulate the first optical transition is a very bad approximation, with errors of several eV. [START_REF] Blase | The Bethe-Salpeter equation in chemistry: relations with TD-DFT, applications and challenges[END_REF] One reason for this is the large exciton binding energy for molecules because of its very localized nature, unlike the case of the solids. Nevertheless, the HOMO-LUMO gap also fails to reproduce the fundamental gap for molecules. This discrepancy comes from the large orbital relaxation induced by electron addition/removal in molecules, making Koopmans's theorem not applicable. The review proposed by Blase et al.

on that topic is particularly interesting. [START_REF] Blase | The Bethe-Salpeter equation in chemistry: relations with TD-DFT, applications and challenges[END_REF] For solids, this orbital relaxation is very moderate because of the periodic nature of the system. Adding or removing one electron creates an extra charge (electron or hole) delocalized on several unit cells, thus with a diluted influence on the other electrons of these unit cells leading to a moderate orbital relaxation and making the Koopmans's theorem more applicable. Interested readers are invited to refer to the work published on that topic by Perdew et al. [START_REF] Perdew | Understanding band gaps of solids in generalized Kohn-Sham theory[END_REF] In addition to the fact that the exciton binding energy is generally a weak correction, the monoelectronic bandgap in solid-state systems is generally close to the fundamental and optical bandgaps.

The development of new functionals for solid-state simulation and, more precisely for semiconductor properties calculations, is still a topic of research. One approach proposed is to use global hybrid functionals, but changing the HF fraction for each semiconductor. A first attempt is to assume that a good HF fraction is equal to 1/ε∞. [START_REF] Alkauskas | Defect levels through hybrid density functionals: Insights and applications[END_REF][START_REF] Gaspari | Semiconducting and optical properties of selected binary compounds by linear response DFT+U and hybrid functional methods[END_REF][START_REF] Conesa | Band structures and nitrogen doping effects in zinc titanate photocatalysts[END_REF][START_REF] Gerosa | Electronic structure and phase stability of oxide semiconductors: Performance of dielectric-dependent hybrid functional DFT, benchmarked against GW band structure calculations and experiments[END_REF][START_REF] Menéndez-Proupin | Self-consistent relativistic band structure of the CH3NH3PbI3 perovskite[END_REF][START_REF] Skone | Self-consistent hybrid functional for condensed systems[END_REF][START_REF] Fritsch | Self-Consistent Hybrid Functional Calculations: Implications for Structural, Electronic, and Optical Properties of Oxide Semiconductors[END_REF] A less empirical approach is to use the Green-Screened Coulomb interaction formalism (better known as GW) and to extract an HF exchange ratio using a screened exchange operator, as proposed by Galli and coworkers, which can also be adapted to build range-separated hybrid functionals. [START_REF] Brawand | Generalization of dielectric-Dependent hybrid functionals to finite systems[END_REF][START_REF] Skone | Nonempirical range-separated hybrid functionals for solids and molecules[END_REF] The use of tuned hybrid functionals, based on dielectric constant, works relatively well, leading to errors on the bandgaps below 10%, but has the main drawback of requiring a change of functional when going from one semiconductor to another. New functionals are sometimes presented with superior performance than HSE06. [START_REF] Skone | Nonempirical range-separated hybrid functionals for solids and molecules[END_REF][START_REF] Verma | HLE17: An Improved Local Exchange-Correlation Functional for Computing Semiconductor Band Gaps and Molecular Excitation Energies[END_REF] Meta-GGA functionals, such as the one developed by Becke and Johnson (mBJ) [START_REF] Tran | Accurate band gaps of semiconductors and insulators with a semilocal exchange-correlation potential[END_REF][START_REF] Becke | A simple effective potential for exchange[END_REF] and Verna and Truhlar (HLE17) [START_REF] Verma | HLE17: An Improved Local Exchange-Correlation Functional for Computing Semiconductor Band Gaps and Molecular Excitation Energies[END_REF], can also lead to errors around or even below 10%. These functionals are pure functionals, meaning that there is no exact exchange calculation. The consequence is faster calculations compared to hybrid functionals, giving an advantage compared to HSE06 for similar reliability. Unfortunately, the SCF convergence of meta-GGA for some systems and surfaces is sometimes difficult. [START_REF] Gräfenstein | Avoiding singularity problems associated with meta-GGA (generalized gradient approximation) exchange and correlation functionals containing the kinetic energy density[END_REF][START_REF] Yang | More realistic band gaps from meta-generalized gradient approximations : Only in a generalized Kohn-Sham scheme[END_REF][START_REF]The VASP Manual[END_REF] The good performance assessed of density functionals presented above is valid only for semiconductors based on filled (d 10 ) or empty (d 0 ) d-shells elements. In general, all the density functionals fail to reproduce the bandgap and electronic properties of semiconductors containing partially filled d-orbitals. [START_REF] Liao | Testing variations of the GW approximation on strongly correlated transition metal oxides: Hematite (α-Fe2O3) as a benchmark[END_REF][START_REF] Filippetti | Self-interaction-corrected pseudopotential scheme for magnetic and strongly-correlated systems[END_REF][START_REF] Nohara | Electronic structure of perovskite-type transition metal oxides LaMO3 (M=Ti-Cu) by U+GW approximation[END_REF][START_REF] Singh | Putting DFT to the test: A first-principles study of electronic, magnetic, and optical properties of Co3O4[END_REF][START_REF] Rödl | Quasiparticle excitations in the photoemission spectrum of CuO from first principles: A GW study[END_REF][START_REF] Jiang | Revisiting the GW approach to d-and f -electron oxides[END_REF][START_REF] Petit | How Should Iron and Titanium Be Combined in Oxides to Improve Photoelectrochemical Properties?[END_REF] This is particularly true for the third-row transition metals. The reason is the strong electronic correlations between electrons of d-shells not properly described by DFT. Two approaches can be used to overcome this important difficulty. The first consists of forcing (artificially) the correlation in the DFT Hamiltonian by adding a phenomenological term, called the Hubbard term. [START_REF] Dudarev | Electron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U study[END_REF][START_REF] Hafner | Ab-Initio Simulations of Materials Using VASP : Density-Functional Theory and Beyond[END_REF][START_REF] Liechtenstein | Density-functional theory and strong interactions: Orbital ordering in Mott-Hubbard insulators[END_REF][START_REF] Kulik | Density functional theory in transition-metal chemistry: A self-consistent hubbard U approach[END_REF] Once added, the total energy of the system is given by Formula 9, where U is called the Hubbard parameter, J the screened Stoner-like exchange parameter, and ρ σ is the density matrix. DFT+U DFT + 2

U J E E Tr σ σ σ σ ρ ρ ρ -   = -   ∑ (9)
This term promotes the localization of the electrons on d-orbitals inducing a larger correlation. Nevertheless, the strength of this localization is parameterized, corresponding to the U value in Equation 9. This parameter can be adapted, such as reproducing property of the system (bandgap, ionization potential, and atomization energy). As a matter of illustration, Figure 7 presents the influence of the U parameter on the computed bandgap of TiO2 (anatase and rutile) and compares these values to experimental and HSE06-computed values. [START_REF] Arroyo-De Dompablo | DFT+U calculations of crystal lattice, electronic structure, and phase stability under pressure of TiO2 polymorphs[END_REF] The strength of this approach is the speed of calculation, which is close to that of GGA functionals. The weakness is the parameterized character of the approach, making the results parameter dependent. Nevertheless, this approach, due to its easiness of application, is the most used for strongly correlated systems. [START_REF] Liao | Testing variations of the GW approximation on strongly correlated transition metal oxides: Hematite (α-Fe2O3) as a benchmark[END_REF][START_REF] Nohara | Electronic structure of perovskite-type transition metal oxides LaMO3 (M=Ti-Cu) by U+GW approximation[END_REF][START_REF] Singh | Putting DFT to the test: A first-principles study of electronic, magnetic, and optical properties of Co3O4[END_REF][START_REF] Dudarev | Electron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U study[END_REF][START_REF] Hafner | Ab-Initio Simulations of Materials Using VASP : Density-Functional Theory and Beyond[END_REF][START_REF] Liechtenstein | Density-functional theory and strong interactions: Orbital ordering in Mott-Hubbard insulators[END_REF][START_REF] Arroyo-De Dompablo | DFT+U calculations of crystal lattice, electronic structure, and phase stability under pressure of TiO2 polymorphs[END_REF][START_REF] Hamdad | New theoretical investigation on the electronic structure and magnetic interaction for both cubic SrFeO3 and CaFeO3 oxides: Comparison between GGA and GGA + U approaches[END_REF][START_REF] Noura | GGA+U-DFT+U modeling structural, electronic and magnetic properties investigation on the ferromagnetic and anti-ferromagnetic BaFeO3 characteristics: Insights from First-principle calculation[END_REF] The most elegant way to improve the description of correlation by DFT is to go beyond the approximations done by DFT in a rigorous manner. Several approaches have been developed in that direction, but the GW approach is mostly used in solid-state chemistry. [START_REF] Shishkin | Accurate Quasiparticle Spectra from Self-Consistent GW Calculations with Vertex Corrections[END_REF][START_REF] Blase | The Bethe-Salpeter equation in chemistry: relations with TD-DFT, applications and challenges[END_REF][START_REF] Hafner | Ab-Initio Simulations of Materials Using VASP : Density-Functional Theory and Beyond[END_REF][START_REF] Onida | Electronic excitations : density-functional versus many-body Green ' s-function approaches[END_REF][START_REF] Scherpelz | Implementation and Validation of Fully Relativistic GW Calculations: Spin-Orbit Coupling in Molecules, Nanocrystals, and Solids[END_REF][START_REF] Govoni | Large Scale GW Calculations[END_REF][START_REF] Hybertsen | First-principles theory of quasiparticles: Calculation of band gaps in semiconductors and insulators[END_REF][START_REF] Ping | Electronic excitations in light absorbers for photoelectrochemical energy conversion: First principles calculations based on many body perturbation theory[END_REF] GW is sometimes presented as a "post-DFT" approach.

Presenting the full equations of the GW theory goes beyond the topic of this review, and interested readers are invited to look at specific publications. [START_REF] Blase | The Bethe-Salpeter equation in chemistry: relations with TD-DFT, applications and challenges[END_REF][START_REF] Aryasetiawan | Reports on Progress in Physics Related content[END_REF] The GW method involves two non-local operators called Green's function (G) and the screened Coulomb repulsion (W). Briefly, the GW approach uses orbitals previously obtained by a DFT calculation and performs either a single point calculation (called G0W0) or a self-consistent calculation (called scGW). There are several types of "flavors" of GW calculations depending on the level of self-consistency. Finally, the GW calculation gives new energy levels exactly corresponding to the ones measured experimentally by photoelectron spectroscopy. [START_REF] Blase | The Bethe-Salpeter equation in chemistry: relations with TD-DFT, applications and challenges[END_REF] In other words, the GW bandgap should correspond to Eg(fund) (equation 5). In practice, GW calculations are notably more computational resource demanding than DFT. GW provides very accurate bandgaps for d 0 and d 10 materials, with the same accuracy as for good DFT functionals (see Figure 8). [START_REF] Shishkin | Accurate Quasiparticle Spectra from Self-Consistent GW Calculations with Vertex Corrections[END_REF][START_REF] Gerosa | Electronic structure and phase stability of oxide semiconductors: Performance of dielectric-dependent hybrid functional DFT, benchmarked against GW band structure calculations and experiments[END_REF][START_REF] Mosconi | Electronic and optical properties of MAPbX3 perovskites (X = I, Br, Cl): a unified DFT and GW theoretical analysis[END_REF][START_REF] Umari | Relativistic GW calculations on CH3NH3PbI3 and CH3NH3SnI3 Perovskites for Solar Cell Applications[END_REF] For partially occupied d-orbitals basedsemiconductors, GW leads to much more accurate bandgaps than DFT. [132][135] Interestingly, GW methods are increasingly available in quantum chemistry codes. While these are an improvement compared to DFT calculations, the large computational resources needed to perform such calculations prevents their use in large systems. From the knowledge of monoelectronic energies and associated orbitals, whether they come from DFT or GW calculations, the imaginary part of the frequency dependent dielectric constant can be computed based on band-to-band transitions using the formula ( 10)-( 12). [START_REF] Gajdoš | Linear optical properties in the projector-augmented wave methodology[END_REF] From this formula, it appears that only the transitions at the same k-point (vertical transition in the band structure) are taken into account. In other words, indirect transitions are not considered by this approach. Finally, the real part of the dielectric constant is computed from the imaginary part, using the Kramers-Kronig transformation (formula [START_REF] Lang | Heterogeneous visible light photocatalysis for selective organic transformations[END_REF]) and the absorption coefficient is computed from the real and imaginary part of the dielectric constant (formula ( 13)-( 14)). This approach is now frequently used to evaluate the absorption spectra of semiconductors. [START_REF] Ziani | Photophysical Properties of SrTaO2N Thin Films and Influence of Anion Ordering: A Joint Theoretical and Experimental Investigation[END_REF][START_REF] Gajdoš | Linear optical properties in the projector-augmented wave methodology[END_REF][START_REF] Zhong | Electronic structures and optical properties of Cu1-xAgxInTe2(x = 0, 0.25, 0.5, 0.75 and 1) chalcopyrite compounds[END_REF][START_REF] Walsh | Band Edge Electronic Structure of BiVO4: Elucidating the Role of the Bi s and V d Orbitals[END_REF][START_REF] Du | Hybrid graphene/titania nanocomposite: Interface charge transfer, hole doping, and sensitization for visible light response[END_REF][START_REF] Saparov | Thin-Film Preparation and Characterization of Cs3Sb2I9: A Lead-Free Layered Perovskite Semiconductor[END_REF][START_REF] Landmann | The electronic structure and optical response of rutile, anatase and brookite TiO2[END_REF][START_REF] Borges | DFT study of the electronic, vibrational, and optical properties of SnO2[END_REF][START_REF] Bahers | Assessing the Use of BiCuOS for Photovoltaic Application: From DFT to Macroscopic Simulation[END_REF][START_REF] Nurlaela | Combined experimental and theoretical assessments of the lattice dynamics and optoelectronics of TaON and Ta3N5[END_REF] This works fairly well if (i) the methodology used to compute the electronic structure gives a reasonable bandgap and (ii) the monoelectronic approach is a good approximation of the electronic structure.

( ) ( ) ( ) ( ) ( )

1 2 i ε ω ε ω ε ω = + ( 10
) ( ) ( ) ( ) 2 2 2 0 , , 4 1 lim 2 k ck qe vk vk ck qe q v c k ck vk e w u u u u q α β αβ π ε ω δ ω + + → = Ω -- ∑ ò ò (11) ( ) ( ) ( ) ( ) 2 1 2 2 0 ' ' 2 1 ' ' d αβ αβ ε ω ω ε ω ω π ω ω ∞ = + - ∫ (12) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 1 2 1 2 αβ αβ αβ αβ ε ω ε ω ε ω κ ω + - = (13)
( ) ( )

4 αβ αβ π α ω κ ω λ = ( 14
)
The community of computational solid-state chemistry is also working on the implementation of TD-DFT that would give access to a much more accurate simulation of the frequency-dependent dielectric constant, and ultimately a more accurate absorption coefficient (Figure 9). Copyright (2018) The American Physical Society.

While bandgap energy is directly associated with the maximum attainable efficiency of the photoelectrocatalysis, the most beneficial property involved in photon absorption is the absorption coefficient as a function of wavelength. UV-visible spectroscopy is a direct method used to determine this property. [START_REF] Chen | Photoelectrochemical Water Splitting[END_REF] Based on the Beer-Lambert law, the light intensity I(x) as a function of the point x from the surface to the bulk, which has a dimensionality of power (or photon numbers) per area (e.g., W m -2 or photons m -2 ), is defined as ( ) ( )

0 exp I x I x α = - ( 15
)
where α is the absorption coefficient, which has a dimensionality of inverse length (e.g., cm -1 ). This exponential decay of light intensity is no longer true if non-linear phenomena happen (multi-photon absorption). At the typical light intensities used to perform the measurement, however, these non-linear phenomena are extremely uncommon and can be ignored in the analysis. Because of this proportionality coefficient, it is convenient to describe the absorption measurement with dimensionless values, such as absorptance, a, and transmittance, T, which are the relative amount of the light being absorbed by, and passing through the sample, respectively, with a thickness, l. In the ideal case where the incident light contributes solely to absorption with the material, the absorptance can be described as;

1 1 l in out in I I a T e I α - - = = -= - (16)
The reciprocal of absorption coefficient gives absorption depth (i.e., the thickness of the semiconductor requiring sufficient number of photons at a given wavelength). In reality, however, the incident photons might be lost by additional processes, such as specularly reflected (Rs), forward-scattered (S), and back-scattered (Rd), [START_REF] Chen | Photoelectrochemical Water Splitting[END_REF] and thus;

a + T + Rs + S + Rd = 1 (17)
The frequently used dimensionless parameter, the absorbance, A, should not be confused with these values, which is defined as:

1 10 A α - = - (18)
For a given semiconductor, it is useful to obtain the absorption coefficient as a function of incident wavelength. A thin-film configuration of the bulk semiconductor that minimizes the contribution of scattering is The measurement becomes more accurate if a thin film with absolute thickness is defined. Swanepoel proposed a useful method to determine film thickness using interference fringes of the transmittance spectrum [START_REF] Swanepoel | Determination of the thickness and optical constants of amorphous silicon[END_REF]. In a transmittance mode, [START_REF] Muth | Absorption coefficient, energy gap, exciton binding energy, and recombination lifetime of GaN obtained from transmission measurements[END_REF] the complete light absorption should be avoided to extract the information of α. It is, therefore, necessary to fabricate reliable and sufficiently-thin films that reflect the characteristics of the solid (i.e., the same displacement of atoms as the bulk solid). In practice, it is conducted to measure the absorption coefficient focusing on two measured parameters, which are transmittance, T, and reflectance, R, [START_REF] Wemple | Optical Transmission Through Multilayered Structures[END_REF] by minimizing other contributions such as scattering. When the reflectance is small, the absorption coefficient is reported using the following relationship [START_REF] Lodenquai | Determination of absorption coefficients of thin films[END_REF].

( )

2 1 ln 1 T d R α = - - (19)
It is important to give extra attention to the type of substrates used in the absorption measurement because it is known to drastically affect the obtained properties. [START_REF] Nama Manjunatha | Investigation of optical properties of nickel oxide thin films deposited on different substrates[END_REF] The common approach to measure absorption characteristics of powder semiconductor materials is diffuse reflectance measurement. Use of an integrating sphere to collect all the scattered light is effective at improving the accuracy of the measurement. The Kubelka-Munk model is based on the measurement assuming scattering and absorption, the relationship of which is given by [START_REF] Džimbeg-Malčić | Kubelka-Munk theory in describing optical properties of paper[END_REF][START_REF] Dahm | Interpreting Diffuse Reflectance and Transmittance: A Theoretical Introduction to Absorption Spectroscopy of Scattering Materials[END_REF];

( ) ( ) 2 1 2 R f R R s α - = = (20)
The Kubelka-Munk function is theoretically proportional to the absorption coefficient, as can be seen in Equation [START_REF] Habisreutinger | Photocatalytic Reduction of CO2 on TiO2 and Other Semiconductors[END_REF]. The bandgap is thus obtained using this relationship, even if the scattering coefficient s is unknown. However, Murphy discussed uncertainty to determine bandgap using the Kubelka-Munk function, especially for doped semiconductors [START_REF] Murphy | Band-gap determination from diffuse reflectance measurements of semiconductor films, and application to photoelectrochemical water-splitting[END_REF]. The weak part of the absorption spectrum is frequently overestimated. According to Murphy, the absorption coefficient due to doped electronic state is so weak that the contribution to photoelectrocatalysis using this transition is expected to be minimal [START_REF] Murphy | Band-gap determination from diffuse reflectance measurements of semiconductor films, and application to photoelectrochemical water-splitting[END_REF]. Also, it is important to analyze the data with zero absorption background beyond the bandgaps as this would provide important information about the dopants and defects [START_REF] Qureshi | Insights on measuring and reporting heterogeneous photocatalysis: Efficiency definitions and setup examples[END_REF].

Using the obtained absorption coefficient, the bandgap is commonly determined using the Tauc plots. The Tauc relationship [START_REF] Wood | Weak Absorption Tails in Amorphous Semicondcutors[END_REF], which is supported by Davis and Mott [START_REF] Schubert | Light-Emitting Diodes[END_REF],

provides the following relationship when the reasonable absorption is achieved, typically α > 10 4 cm -1 ;

( )

1 n g h h E α ν ν ∝ - (21)
where n can take values of 3, 2, 3/2, or 1/2, corresponding to indirect (forbidden), indirect (allowed), direct (forbidden), and direct (allowed) transitions, respectively [START_REF] Chen | Photoelectrochemical Water Splitting[END_REF][START_REF] Murphy | Band-gap determination from diffuse reflectance measurements of semiconductor films, and application to photoelectrochemical water-splitting[END_REF][START_REF] Munir | Effect of carrier concentration on the optical band gap of TiO2 nanoparticles[END_REF] The nature of the bandgap (direct, indirect…) is generally obtained from DFT calculations of band structures. Tauc plots, i.e., (αhν) n as a function of hν (n is the same as above) give Eg from the intersection of a line tangent to the slope in the linear region of the absorption onset with the baseline (Figure 10). [START_REF] Chen | Photoelectrochemical Water Splitting[END_REF][START_REF] Murphy | Band-gap determination from diffuse reflectance measurements of semiconductor films, and application to photoelectrochemical water-splitting[END_REF][START_REF] Munir | Effect of carrier concentration on the optical band gap of TiO2 nanoparticles[END_REF] Bandgaps of many semiconductors are compiled in ref.

[64] Photoluminescence spectroscopy is another experimental method to the determine bandgap [START_REF] Sanguinetti | Characterization of Semiconductor Heterostructures and Nanostructures[END_REF]. In theory, the onset of the luminescence signal should correspond to the bandgap described previously. The intrinsic band structures are reflected by the measurement at low temperatures, avoiding thermal contribution. The bandgap determination by this technique, however, cannot be applied to nonradiative transitions and the samples with indirect bandgap semiconductors. It is also noted that many dopant states appear in the signals and, thus, often lead to ambiguity to determine bandgaps.

II. Dielectric constant.

When an electric field ( E ur ) is applied on a dielectric material, an electric displacement field ( D ur ) appears within it. Both electric fields are linked by the dielectric constant εr.

0 r D E ε ε = ur ur (22)
This dielectric constant represents the ability of dielectric material to screen the external electric field by the apparition of a polarization. This polarization comes from the reorganization of the electron density or the motion of the ions constituting the material. The dielectric constant induced by the electron density is named ε∞, and the contribution of the dielectric constant involving the ionic motions is noted εvib. Finally, we have:

r vib ε ε ε ∞ = + (23)
A study on dielectric materials itself is an important field. The dielectric constants of various materials are compiled elsewhere. [START_REF] Madelung | Semiconductors: Data Handbook, Third edit[END_REF][START_REF] Vurgaftman | Band parameters for III-V compound semiconductors and their alloys[END_REF][START_REF] Young | Compilation of the Static Dielectric Constant of Inorganic Solids[END_REF] There are many ways to experimentally measure the dielectric constant of a semiconductor, and one of the common ways is to conduct capacitance measurements with varied frequency after making a disc device with two metal contacts at each side. For example, dielectric properties of TiO2 powder was measured by making pellets of the sample. [START_REF] Wypych | Dielectric properties and characterisation of titanium dioxide obtained by different chemistry methods[END_REF] For the powder semiconductor, the measurement may give overestimation due to extrinsic microstructural features. [START_REF] Kim | Epitaxial thin-film deposition and dielectric properties of the perovskite oxynitride BaTaO2N[END_REF] Another common technique to measure dielectric properties for powder sample is ellipsometry (Figure 11). [START_REF] Madelung | Semiconductors: Data Handbook, Third edit[END_REF][START_REF] Morbec | Optoelectronic properties of Ta3N5: A joint theoretical and experimental study[END_REF] Copyright (2018) The American Physical Society.

Computationally, the electronic contribution to the static dielectric tensor can be evaluated through a coupled-perturbed Hartree-Fock/Kohn-Sham (CPHF) scheme adapted to periodic systems [START_REF] Skone | Nonempirical range-separated hybrid functionals for solids and molecules[END_REF][START_REF] Ferrari | Ab Initio Calculation of the Ultraviolet-Visible (UV-vis) Absorption Spectrum, Electron-Loss Function, and Reflectivity of Solids[END_REF][START_REF] Ferrero | Coupled perturbed Hartree-Fock for periodic systems: the role of symmetry and related computational aspects[END_REF][START_REF] Ferrero | The Calculation of Static Polarizabilities of 1-3D Periodic Compounds . The Implementation in the CRYSTAL Code[END_REF][START_REF] Maschio | Ab initio analytical infrared intensities for periodic systems through a coupled perturbed Hartree-Fock/Kohn-Sham method[END_REF] or by applying an electric field on the system during the SCF procedure [START_REF] Nunes | Berry-phase treatment of the homogeneous electric field perturbation in insulators[END_REF]. While more time consuming, determining the total dielectric constant by performing SCF calculations with an electric field considers local field effects because of the self-consistent nature of the calculation. Nevertheless, CPHF and electric field approaches usually lead to similar results, but more sensitive on the choice of the functional (for DFT calculation) than the way to compute the electronic contribution to the dielectric constant. The reliability of DFT to compute the electronic contribution to the dielectric constant has been assessed in several publications. [START_REF] Bahers | Semiconductors used in photovoltaic and photocatalytic devices: Assessing fundamental properties from DFT[END_REF][START_REF] Paier | Dielectric properties and excitons for extended systems from hybrid functionals[END_REF][START_REF] Kootstra | Application of time-dependent densityfunctional theory to the dielectric function of various nonmetallic crystals[END_REF][START_REF] Petousis | Benchmarking density functional perturbation theory to enable high-throughput screening of materials for dielectric constant and refractive index[END_REF][START_REF] Petousis | High-throughput screening of inorganic compounds for the discovery of novel dielectric and optical materials[END_REF] Results are typically close to experimental data, and the exchange-correlation functional typically has a weaker effect on the dielectric constant than the for the bandgap. In fact, the standard deviation of the DFT for a series of the semiconductor is given in Figure 5. The standard deviation is below 20% for all functionals. It is important to note that the ordering of functionals for the bandgap simulation is the same for ε∞. Consequently, range-separated functionals are the best for simulating ε∞. However, as mentioned above, the choice of the function has, in general, not a large impact on the final dielectric constant. Thus, global hybrid or even GGA can reasonably be used to compute that property.

The vibrational contribution is computed from the phonon spectrum, following Formula [START_REF] Hisatomi | Recent advances in semiconductors for photocatalytic and photoelectrochemical water splitting[END_REF]. νp is the phonon frequency of mode p, V is the unit cell volume, and Zp is the mass-weighted mode effective Born vector. The intensity Ip of IR absorbance for a given mode p is proportional to |Zp| 2 . [START_REF] Maschio | Ab initio analytical infrared intensities for periodic systems through a coupled perturbed Hartree-Fock/Kohn-Sham method[END_REF] From this formula, it appears that the softer the vibrational modes, the more they contribute to the dielectric constant.

2 p 2 p p 4 r vib Z V v π ε ε ε ε ∞ ∞ = + = + ∑ (24)
The agreement between theory and experiment for the vibrational dielectric constant is also very good, whatever the functional chosen (Figure 5). But this conclusion is no longer available for very large dielectric constants (>100). In these cases, the very soft vibrational modes leading to the large dielectric constant are non-harmonic, thus badly reproduced by DFT in the harmonic approximation. For these difficult systems, DFT results are qualitative informing only on the presence of these soft modes.

III. Charge transport properties

The electron and hole transport is an important phenomenon, not only in photoelectrocatalysis but also for several other applications, including but not limited to photovoltaic and thermoelectric. In some aspect of semiconductor physics, the charge carrier drift due to an electric field of a space charge zone plays a fundamental role (such in a p-n junction), but as pointed out by Grätzel et al. in nanoparticulate materials with a high density of impurities, the space charge zone plays only a marginal role and charge carrier transport are mainly governed by thermal diffusion. [START_REF] Kavan | Electrochemical and photoelectrochemical investigation of single-crystal anatase[END_REF][START_REF] Wang | Characteristics of high efficiency dye-sensitized solar cells[END_REF][START_REF] Kavan | Preparation of TiO2(anatase) films on electrodes by anodic oxidative hydrolysis of TiCl3[END_REF][START_REF]Highly Efficient Semicondcuting TiO2 Photoelectrodes Prepared by Aerosol Pyrolysis[END_REF] To quantify the transport at a first guess, the measurement of the conductivity is considered. This property depends on the charge carrier density n (generally in cm -3 ), the charge of the charge carrier density q (in C) and the mobility of the charge carrier density μ (in cm 2 V -1 s -1 ) as presented in equation [START_REF] Bard | Artificial Photosynthesis: Solar Splitting of Water to Hydrogen and Oxygen Water Splitting[END_REF]. In intrinsic (i.e. without any doping) semiconductors in the dark, the conductivity increases with increasing the temperature because of conduction band population with thermal energy (increase of n in equation ( 25)). But under doping or light exposure, the charge carrier densities (electrons in the conduction band and holes in the valence band) increase from another reason than temperature. This can lead to a different behavior of the conductivity has a function of the temperature. The value of the conductivity is dependent on temperature and purity and synthesis conditions. For that reason, the mobility μ is considered to be a better intrinsic property of the material to quantify the transport of charge carriers.

nq σ µ = (25)
Furthermore, the conductivity and diffusion coefficient are diretly related (see Equation ( 4)).

Two main theories are used to interpret the transport properties in optoelectronic materials.

The first theory is called the band-like mechanism and is related to the curvature of the bandstructure leading to the quantity called effective mass. This theory works well for delocalized charge carriers. Thus, this theory is adapted for highly covalent d 0 or d 10 semiconductors. The band-like model is sometimes called the rigid band-model. This means that, by applying this model, we assume that the bands or, more generally, the electronic structure is almost unaffected by the electron and hole generation. This is a good approximation if the charge carriers are delocalized on several unit cells, thus having a diluted influence on the ions. For systems having the more localized character of their orbitals, the charge carriers can be centered only on few ions, leading to a large perturbation around these atoms. The main consequence is generally a geometrical relaxation around the atoms, stabilizing the system and increasing the localization of the electron and the hole. The term self-trapping is generally used to name this phenomenon. The particle made of the charge carrier and the geometrical relaxation is called a small polaron. [START_REF] Shluger | Small polarons in real cyrstals: concepts and probelms[END_REF] The polaronic conductivity is based on a hopping mechanism, well described by the Marcus theory. [START_REF] Difley | Electronic properties of disordered organic semiconductors via QM/MM simulations[END_REF][START_REF] Barbara | Contemporary issues in electron transfer research[END_REF][START_REF] Ostroverkhova | Organic Optoelectronic Materials: Mechanisms and Applications[END_REF] In this theory, the polaron jumps from one site to another, leading to mobility given by Equation ( 26). [START_REF] Morbec | Charge transport properties of bulk Ta3N5 from first principles[END_REF]

] B 2 B Ea k T eR n e k T µ ν - = ( 26
)
where R is the distance between the two hopping sites and n is density of hopping sites. This mobility is thermally activated, which increases with increasing the temperature, following an Arrhenius behavior, in contrast to the mobility governed by a band-like mechanism, which decreases with the temperature. [START_REF] Shuai | Evaluation of charge mobility in organic materials: From localized to delocalized descriptions at a first-principles level[END_REF][START_REF] Lee | Thin Films of Highly Planar Semiconductor Polymers Exhibiting Band-like Transport at Room Temperature[END_REF] To determine if the transport mechanism is a bandlike mechanism or a polaronic mechanism, the measurement of the mobility as a function of the temperature is generally performed. An increase of mobility as a function of the temperature indicates a thermally activated mechanism, and thus a hopping mechanism for the polaron. A decrease of the mobility with the temperature is more in favor of a band-like mechanism of charge transport, affected by phonons generated by increasing the temperature. This is illustrated in the work by Lee et al. on a donor-acceptor type thiophene-thiazole copolymer thin film, where the nature of the transport mechanism can change from band-like to polaronic depending on the drain field applied on the film (Figure 12). [63] Band-like transport. In the band-like transport theory, the charge carrier mobility is given by the formula (4), including the collision time τ and the effective masses m * . The concept of effective masses comes from solid-state physics and is related to the curvature of the top of the valence band (for holes) and bottom of the conduction band (for electrons) in the reciprocal space along one direction of the band structure. [START_REF] Brus | Electronic wave functions in semiconductor clusters: Experiment and theory[END_REF][START_REF] Kittel | Introduction to Solid State Physics[END_REF][START_REF] Hautier | How does chemistry influence electron effective mass in oxides? A high-throughput computational analysis[END_REF][START_REF] Janssen | Precise effective masses from density functional perturbation theory[END_REF] The collision time, τ, is extremely experimental dependent and involves all the scattering events (electron-electron, phonon-electron, and electron-impurities) and cannot be easily theoretically evaluated. The effective mass can be obtained by fitting these band extrema by the formula:

( ) ( ) ( ) ( ) 2 2 2 2 0 0 0 * * E k E k k k k k m m ⊥ ⊥ = ± - ± - h h ( 27
)
The ± is + for the conduction band and -for the valence band. k0 is the vector of the reciprocal space where the band is at its extremum. The symbols and ⊥ refer to the longitudinal and perpendicular axes that depend on the structure of the crystal. The effective masses are obtained by fitting the bottom of the conduction band (for the electron effective mass) and the top of the valence band (for the hole effective mass) with a parabolic function.

To calculate the effective mass in that way, the band structure must be computed in several directions. For highly symmetrical crystals, one generally chooses the most symmetric directions. For low symmetry systems, the choice of the crystallographic directions is more difficult and generally arbitrary. Ideally, the effective masses should be computed in a large number of directions to obtain a contour plot allowing to attribute the parallel and perpendicular directions of equation ( 26), as illustrated in Figure 13 from the work of Feng et al. [START_REF] Feng | Crystal Structures, Optical Properties, and Effective Mass Tensors of CH3NH3PbX3 (X = I and Br) Phases Predicted from HSE06[END_REF] But, in general, researchers have focused on the simplest directions. The effective mass calculation from the band-structure is the most frequent approach used by the quantum chemists to characterize the charge carrier mobilities. [START_REF] Bahers | Semiconductors used in photovoltaic and photocatalytic devices: Assessing fundamental properties from DFT[END_REF][START_REF] Baqais | Bismuth Silver Oxysulfide for Photoconversion Applications: Structural and Optoelectronic Properties[END_REF][START_REF] Ishikawa | Oxysulfide Sm2Ti2S2O5 as a stable photocatalyst for water oxidation and reduction under visible light irradiation (λ ≤ 650 nm)[END_REF][START_REF] Walsh | Band Edge Electronic Structure of BiVO4: Elucidating the Role of the Bi s and V d Orbitals[END_REF][START_REF] Borges | DFT study of the electronic, vibrational, and optical properties of SnO2[END_REF][START_REF] Nurlaela | Combined experimental and theoretical assessments of the lattice dynamics and optoelectronics of TaON and Ta3N5[END_REF][START_REF] Harb | Critical difference between optoelectronic properties of α-and β-SnWO4 semiconductors: A DFT/HSE06 and experimental investigation[END_REF][START_REF] Kim | Accurate band structures and effective masses for InP, InAs, and InSb using hybrid functionals[END_REF][START_REF] Harb | Predicting suitable optoelectronic properties of monoclinic VON semiconductor crystals for photovoltaics using accurate first-principles computations[END_REF][START_REF] Gamon | Substituting Copper with Silver in the BiMOCh Layered Compounds (M = Cu or Ag; Ch = S, Se, or Te): Crystal, Electronic Structure, and Optoelectronic Properties[END_REF][START_REF] He | Firstprinciples study of the structural, electronic, and optical properties of Ga2O3 in its monoclinic and hexagonal phases[END_REF] This property can also be used to analyze the excited state entropy of the semiconductor, considered as a source of efficiency decrease, as presented by Osterloh. [START_REF] Osterloh | Maximum Theoretical Efficiency Limit of Photovoltaic Devices: Effect of Band Structure on Excited State Entropy[END_REF] Assessing the reliability of DFT for that specific property is difficult since we lack experimental data to compare the results. Nevertheless, some classes of semiconductors were deeply characterized leading to the effective mass dataset. Kim et al.

compared the PBE (GGA), mBJ (meta-GGA), and HSE06 (range-separated hybrid) functionals and found that the non-local nature of the exchange part of HSE06 was necessary to accurately reproduce the effective masses. [START_REF] Kim | Towards efficient band structure and effective mass calculations for III-V direct band-gap semiconductors[END_REF] GW band structures can also be used to evaluate effective masses, typically those close to the masses obtained with hybrid functionals. [START_REF] Mosconi | Electronic and optical properties of MAPbX3 perovskites (X = I, Br, Cl): a unified DFT and GW theoretical analysis[END_REF][START_REF] Umari | Relativistic GW calculations on CH3NH3PbI3 and CH3NH3SnI3 Perovskites for Solar Cell Applications[END_REF][START_REF] Svane | Quasiparticle self-consistent GW theory of III-V nitride semiconductors: Bands, gap bowing, and effective masses[END_REF][START_REF] Filip | GW Band Structures and Carrier Effective Masses of CH3NH3PbI3and Hypothetical Perovskites of the Type APbI3: A = NH4, PH4, AsH4, and SbH4[END_REF] Beyond the quantitative evaluation of the effective mass, it is important to recall that this property is deeply related to the covalency of the chemical bonds. [START_REF] Hautier | How does chemistry influence electron effective mass in oxides? A high-throughput computational analysis[END_REF][START_REF] Canadell | Orbital Approach to the Electronic Structure of Solids[END_REF][START_REF] Ha | Structural design principles for low hole effective mass s-orbital-based p-Type oxides[END_REF] It is sometimes difficult to find the "good" crystallographic direction for the effective mass calculation, or sometimes, there are several bands extrema close in energy. In these specific cases, the effective masses can be computed using Boltzmann transport theory [START_REF] Madsen | A code for calculating band-structure dependent quantities[END_REF] with the equations ( 28)- [START_REF] Li | A sustainable approach for lignin valorization by heterogeneous photocatalysis[END_REF], where σ is the conductivity of the material and τ is the collision time.

This corresponds to an effective mass averaged on the density of state of the conduction band and the valence band parameterized by the scattering time τ. Using that approach, the calculation of the effective mass is a post-treatment of a full band structure. Several codes now offer the possibility to perform such analysis. [116,217-

dk e v i k v i k f E i k T E αβ α β σ τ µ π ∂ = - ∂ ∑ ∫ (28) ( ) ( ) , 1 , E i k v i k k α α ∂ = ∂ h (29) * 2 m e σ τ = 219] ( ) ( ) ( ) ( ) 3 2 3 , , , , , 4 i
Experimentally, the determination of effective mass is not an easy task. This is mainly based on the measurement of the cyclotron frequency (Equation 31). The sample is placed in a static magnetic field, B. From a classical point of view, free charge carriers start to have a helicoidal motion with an angular frequency equal to the cyclotron frequency noted ωc. From a quantum point of view, their energy levels are quantified and separated by a constant energy ħωc. [START_REF] Dresselhaus | Cyclotron resonance of electrons and holes in silicon and germanium crystals[END_REF] Transitions between these states are possible under radio-frequency radiations, leading to the measurement of the cyclotron frequency and the determination of the effective mass. [START_REF] Dresselhaus | Cyclotron resonance of electrons and holes in silicon and germanium crystals[END_REF][START_REF] Galkowski | Determination of the exciton binding energy and effective masses for methylammonium and formamidinium lead tri-halide perovskite semiconductors[END_REF][START_REF] Miyata | Direct measurement of the exciton binding energy and effective masses for charge carriers in organic-inorganic tri-halide perovskites[END_REF][START_REF] Merino | Cyclotron effective masses in layered metals[END_REF][START_REF] Rzepniewski | Cyclotron resonance in the layered perovskite superconductor Sr2RuO4[END_REF][START_REF] Schwab | On the Cyclotron Resonance of Carriers in Cu2O[END_REF] c *

eB m ω = (31)
To have a measurable cyclotron resonance, the mean free path of charge carriers must be large. Considering a charge carrier scattering time τ, it is assumed that the product ωcτ must be larger than 1 to lead to well-defined cyclotron resonance. In other words, the crystal must be as pure as possible and cool down below 100 K. Thus, this type of experiment is rather difficult for semiconductors synthesized as a powder with a large number of impurities; making the experimental determination of effective mass difficult to find for weakly investigated semiconductors.

Another approach has been proposed in 1966 by Zhitinskaya et al. to determine the effective mass of the charge carriers at the Fermi level from the experimental measurement of the Hall (RH), Seebeck (α), conductivity (σ) and Nernst (Q) coefficients using the following formula (where q, ħ, kB and T are the charge of the particle, the reduced Planck constant, the Boltzmann constant and the temperature respectively):[226,227]

2 2 3 * 2 H B H 3 q Q m R q k T R α π σ     = -             h (32)
The demonstration of the equation presented above is based on the Boltzmann transport theory, thus the effective mass obtained in that way is a density averaged effective mass. This method to estimate the effective mass is sometimes called "the method of four coefficients" because it involves the knowledge of the four coefficients presented above. Because the measurement of the Hall, Seebeck and conductivity coefficients is a kind of routine work in the field of thin solid films, the determination of effective mass using the equation ( 32) has been mainly employed by scientists in that field. [START_REF] Lee | Effective density-of-states distribution of polycrystalline silicon thin-film transistors under hot-carrier degradation[END_REF][START_REF] Young | Growth and characterization of radio frequency magnetron sputter-deposited zinc thin films Growth and characterization of radio frequency magnetron sputterdeposited zinc stannate , Zn2SnO4 , thin films[END_REF][START_REF] Tang | performance in CoSb3 skutterudites[END_REF] This experimental way to determine the effective mass can be applied to materials being less pure than the one used by the cyclotron resonance experiment, but the result is an averaged effective mass while the cyclotron resonance gives access to hole and electron effectives mass in several crystallographic directions.

Polaron. In terms of computational chemistry, the investigation of polaron conductivity is not easy in periodic systems. A charged system must be created to address the problem of treating such systems in periodic boundary conditions. [START_REF] Schultz | Charged Local Defects in Extended Systems[END_REF][START_REF] Jarvis | Supercell technique for total-energy calculations of finite charged and polar systems[END_REF][START_REF] Bailey | Defect physics of CuGaS2[END_REF] Then the charge must be localized, that is usually done first by distorting the geometry around the central atom and by adding a +U parameter on this atom. Next, the energy of the system is computed along several directions for the charge transfer (i.e., the hopping) to find the highest energy and define the activation energy. Because of all these steps, the simulation of hopping is difficult.

Furthermore, this type of simulation is methodology dependent because of the charge compensation, because of the +U parameter and the initial distortion artificially imposed to localize the charge. Nevertheless, some examples are proposed in the literature for Ta3N5 [START_REF] Morbec | Charge transport properties of bulk Ta3N5 from first principles[END_REF] (Figure 15), TiO2 [START_REF] Deskins | Electron transport via polaron hopping in bulk TiO2: A density functional theory characterization[END_REF], Fe2O3 [START_REF] Iordanova | Charge transport in metal oxides: A theoretical study of hematite α-Fe2O3[END_REF] or BiVO4 [START_REF] Kweon | Electron small polarons and their transport in bismuth vanadate: A first principles study[END_REF]. The main information extracted from this type of simulation is the influence of the crystallographic direction on the polaron diffusion and the activation energy for the thermal activation of the diffusion (Equation 25). Copyright (2018) The American Physical Society.

Some works solve some of the drawbacks of using periodic boundary conditions by extracting a cluster from the crystal structure and performing post-Hartree-Fock calculations (such as CASSCF). [START_REF] Liao | Electron transport in pure and doped hematite[END_REF] The advantages of this approach are to avoid the problem of the electric charge in periodic boundary conditions and to use a more accurate method than DFT to investigate correlated materials. The drawback of the cluster approach is the removal of the Madelung potential of the infinite crystal that is known to have an important effect on chargetransfer states.

The activation energy of polaron hopping can be obtained experimentally by performing an Arrhenius analysis of charge carrier mobilities or conductivity. [START_REF] Lu | A review for polaron dependent charge transport in organic semiconductor[END_REF][START_REF] Zhao | Polaron localization at interfaces in high-mobility microcrystalline conjugated polymers[END_REF][START_REF] Sampaio | Activation Energies for Electron Transfer from TiO2 to Oxidized Dyes: A Surface Coverage Dependence Correlated with Lateral Hole Hopping[END_REF][START_REF] Rettie | Combined charge carrier transport and photoelectrochemical characterization of BiVO4 single crystals: Intrinsic behavior of a complex metal oxide[END_REF] The main difficulty is to attribute the activation energy to a specific mechanism since several mechanisms can be active for the conduction.

IV. Exciton binding energy.

The exciton is a quasi-particle developed in solid-state physics to describe the bound state between an electron and a hole. The formation of the exciton upon light absorption leads to a decrease of the fundamental gap energy, by an amount called exciton binding energy,

Eb, [START_REF] Bastard | Exciton binding energy in quantum wells[END_REF] explaining why the fundamental bandgap is larger than the optical one (Figure 4, equation 6). Excitons are generally classified in two categories depending on their delocalization: (i) The Mott-Wannier exciton delocalized on several unit cells and having a radius as high as several hundred of Å and generally having a relative low exciton binding energy (few meV). This is the exciton found in silicon. (ii) The Frenkel exciton, more localized (a radius of few Å) and generally having a large exciton binding energy (up to several eV). Other types of excitons can sometimes be defined in specific materials. The exciton plays an important role in photoelectrocatalysis since it must be dissociated to obtain free charge carriers that will be able to transport toward the surface of the semiconductor. To be dissociated efficiently at room temperature, the exciton must have a binding energy of the order of magnitude of the room thermal energy (kBT ~ 25 meV) or lower. If Eb is notably larger, like in polymer semiconductors, the exciton cannot dissociate in the bulk material and must diffuse toward an interface to dissociate at this interface. This type of architecture is called bulk-heterojunction has been developed by the organic photovoltaic community. [START_REF] Clarke | Charge Photogeneration in Organic Solar Cells[END_REF] The most classical way to determine the exciton binding energy is by absorption or photoluminescence spectroscopy. Some other less usual approaches can be used to determine exciton binding energies, such as STM spectroscopy [START_REF] Alvarado | Direct determination of the exciton binding energy of conjugated polymers using a scanning tunneling microscope[END_REF] or UV-Vis spectroscopy under strong magnetic field (up to 65 T) [START_REF] Galkowski | Determination of the exciton binding energy and effective masses for methylammonium and formamidinium lead tri-halide perovskite semiconductors[END_REF][START_REF] Miyata | Direct measurement of the exciton binding energy and effective masses for charge carriers in organic-inorganic tri-halide perovskites[END_REF]. In general, it is rather difficult to experimentally access exciton binding energies because of the good purity of the sample required and the very accurate spectroscopic measurements needed.

Due to the excited state nature of excitons, standard DFT, developed for ground state systems, is unable to treat them. Consequently, only theories developed to investigate excited states can perform a reliable analysis of exciton properties. These theories have been briefly

presented in the discussion on bandgap simulations and are GW/BSE and TD-DFT methods.

The BSE procedure (from Bethe-Salpeter Equations) is a post-treatment of the GW quasi-particle calculation to get access to the exciton binding energy. The computation time of the BSE procedure is similar to the TD-DFT one. [START_REF] Jacquemin | Benchmarking the Bethe-Salpeter Formalism on a Standard Organic Molecular Set[END_REF] The efficiency of TD-DFT and GW/BSE are not frequently compared in solid-state physics since TD-DFT is marginally used in that field. Their comparison is more documented for molecular systems. [START_REF] Jacquemin | Benchmarking the Bethe-Salpeter Formalism on a Standard Organic Molecular Set[END_REF][START_REF] Li | Correlated electron-hole mechanism for molecular doping in organic semiconductors[END_REF][START_REF] Escudero | Modeling the Photochrome-TiO2 Interface with Bethe-Salpeter and Time-Dependent Density Functional Theory Methods[END_REF][START_REF] Jacquemin | 0-0 energies using hybrid schemes: Benchmarks of TD-DFT, CIS(D), ADC(2), CC2, and BSE/GW formalisms for 80 real-life compounds[END_REF] The recent review of Blase et al. shows that hybrid functional generally gives optical gaps close to the ones of GW/BSE. [START_REF] Blase | The Bethe-Salpeter equation in chemistry: relations with TD-DFT, applications and challenges[END_REF] In term of computational time, these approaches are now quite similar, and because the large improvement performed in the recent years in the speed of GW/BSE calculations, we can expect that in the near future, such calculations will be performed in a daily manner.

If the exciton belongs to the Mott-Wannier category (i.e., delocalized on several unit cells), it can be treated as a hydrogen atom (the hole being the proton) offering an analytical formula for the binding energy. 37 In that model, the absolute value of the 1S state energy of the exciton is its binding energy, Eb, and it can be computed with the formula:

b H 2 r E E µ ε = - (33)
EH is the energy of the 1s orbital of hydrogen (-13.6 eV), and μ is the reduced mass of the exciton.

* * e h 1 1 1
m m µ = + (34)
Consequently, if the dielectric constant and the effective masses of the semiconductor were already computed, the exciton binding energy can be easily estimated through Formula 33.

The strength of this simple model is to highlight the dominant effect of the dielectric constant, representing the ability of the material to screen charges, confirmed by more sophisticated calculations. [START_REF] Dvorak | Origin of the variation of exciton binding energy in semiconductors[END_REF] The Mott-Wannier model has been used to compute the exciton binding energy used to build Figure 5. As expected, hybrid functionals (global or range-separated) lead the best agreement with the experiment, but with a standard deviation of around 30%.

This error is relatively large, but considering the simplicity of Formula (33), it is an excellent ratio precision/time. Of course, this model fails to predict Frenkel exciton binding energy.

GW/BSE simulation of exciton binding energies has been compared to Mott-Wannier ones for C3N4 based-semiconductors (Figure 17). [START_REF] Steinmann | Challenges in Calculating the Bandgap of Triazine-Based Carbon Nitride Structures[END_REF] The very large exciton binding energies were in agreement with the known Frenkel exciton character of this semiconductor. The error between the scGW/BSE exciton binding energy and the Mott-Wannier energy can reach 100%. But interestingly, while quantitatively wrong, the Mott-Wannier model succeed to predict the very large values of the exciton binding energy of this system, confirming that this approach could be used at least qualitatively. ln

N E E k T n   = -     (35) eff,V F,h V B h ln N E E k T n   = +     (36)
Osterloh proposes a thermodynamic interpretation of the previous formula where EC and EV are an enthalpy contribution to the chemical potentials and the logarithm term represents the entropic contribution. [START_REF] Osterloh | Maximum Theoretical Efficiency Limit of Photovoltaic Devices: Effect of Band Structure on Excited State Entropy[END_REF] The entropic contribution can be assimilated to the different ways to distribute particles in the effective states of the CBM and VBM. Those who are interested in the relationship between the different quantities (quasi-Fermi energies, chemical potentials, band positions, etc.) and the origin of this entropy contribution are invited to read the very clear publication of F. Osterloh. [START_REF] Osterloh | Maximum Theoretical Efficiency Limit of Photovoltaic Devices: Effect of Band Structure on Excited State Entropy[END_REF] In general, Neff is in the order of 10 19 cm -3 . [START_REF] Lee | Effective density-of-states distribution of polycrystalline silicon thin-film transistors under hot-carrier degradation[END_REF] The typical charge carrier density ranges from 10 17 -10 20 cm -3 . [START_REF] Garcia-Esparza | A simplified theoretical guideline for overall water splitting using photocatalyst particles[END_REF] As a consequence, the quasi-Fermi levels are very close to the position of the CBM and VBM and generally researchers in the field of photoelectrocatalysis assimilate the CBM and VBM to the redox potentials of the semiconductor. A better estimation of the redox potential involves the knowledge of Neff and ne/h during the working of the device. This is possible computationally to calculate Neff from the band structure obtained by DFT [START_REF] Bahers | Assessing the Use of BiCuOS for Photovoltaic Application: From DFT to Macroscopic Simulation[END_REF] and to obtain ne/h by simulating the electron and flux inside the semiconductor [START_REF] Garcia-Esparza | A simplified theoretical guideline for overall water splitting using photocatalyst particles[END_REF] or by using semiconductors semiclassical equations [START_REF] Cendula | Calculation of the Energy Band Diagram of a Photoelectrochemical Water Splitting Cell[END_REF]. Experimentally, quasi-Fermi energies can be obtained by impedance spectroscopy as explained by Bisquert et al. [START_REF] Bisquert | Theory of the Impedance of Electron Diffusion and Recombination in a Thin Layer[END_REF] It is important to mention that quasi-Fermi energies, thus redox potentials, can change from the bulk to the surface due to the change in charge carrier concentrations because of their consumption for the HER and OER and because of charge transfer with the electrolyte. [START_REF] Bisquert | Energy diagram of semiconductor/electrolyte junctions[END_REF] This change is quantifiable only under a good simulation of ne/h or experimentally by an accurate fitting of the impedance spectra. For water splitting reaction, the potentials must be adapted to perform the two half-reactions, the HER and OER (equations 1 and 2). [START_REF] Van De Walle | Universal alignment of hydrogen levels in semiconductors , insulators and solutions[END_REF][START_REF] Maeda | New non-oxide photocatalysts designed for overall water splitting under visible light[END_REF] As we can see in Figure 18, [START_REF] Sivula | Semiconducting materials for photoelectrochemical energy conversion[END_REF] some semiconductors, like WO3, have suitable bandgap (~2.8

eV) [START_REF] Wang | Electronic and structural properties of WO3: A systematic hybrid DFT study[END_REF], but their band positions (the conduction band in the case of WO3) are not adapted to water splitting; this is why these properties are necessary to fully characterize a semiconductor. The measurement of binding energies of occupied valence electrons is possible by conducting photoemission spectroscopies. This can then be translated to estimate the valence band maxima. One of the useful techniques is the photoelectron spectroscopy in the air (PESA) [START_REF] Uda | Open Counter for Low Energy Electron Detection[END_REF]; the measurement is easily conducted in the open air with relatively simple sample preparation. Accordingly, extra attention is necessary to check whether the surface is at an oxidized state, which might change the observed Fermi level. A similar method that requires ultrahigh vacuum (UHV) is ultraviolet photoelectron spectroscopy (UPS) [START_REF] Uda | Open Counter for Low Energy Electron Detection[END_REF][START_REF] Chun | Conduction and valence band positions of Ta2O5, TaON, and Ta3N5 by UPS and electrochemical methods[END_REF].

The reference point can be the vacuum level, which is rationalized to obtain absolute potential, e.g., standard hydrogen electrode (SHE) potential is at -4.44 eV. [START_REF] Trasatti | The Absolute Electrode Potential : an Explanatory Note[END_REF] In the form of photoelectrode, the flatband potential can be estimated from impedance spectroscopy using Mott-Schottky analysis. [START_REF] Schottky | Zur Halbleitertheorie der Sperrschicht-und Spitzengleichrichter[END_REF][START_REF] Schottky | Halbleitertheorie der Sperrschicht[END_REF][START_REF] Mott | Note on the Contact Between a Metal and an Insulator or Semi-Conductor[END_REF][START_REF] Turner | Energetics of the Semiconductor-Electrolyte Interface[END_REF][START_REF] Albery | Interpretation and use of Mott-Schottky plots at the semiconductor/ electrolyte interface[END_REF] The practical methodology can be found in the references. [START_REF] Chen | Photoelectrochemical Water Splitting[END_REF][START_REF] Van De Krol | Photoelectro-chemical Hydrogen Production[END_REF][START_REF] Zhang | Band Bending in Semiconductor Chemical and Physical Consequences at Surfaces and Interfaces[END_REF][START_REF] Gelderman | Flat-Band Potential of a Semiconductor: Using the Mott-Schottky Equation[END_REF] In short, the space-charge capacity of the semiconductor, C, is given by the following equation;

app FB 2 2 0 r D 1 1 kT E E C e A N e ε ε   = - -     ( 37
)
where A is the surface area of the electrode, and Eapp is the applied potential. By varying the applied potential, the Mott-Schottky plot, 1/C 2 vs. the applied potential, is obtained and then the line is extrapolated when 1/C 2 becomes zero to derive the flatband potential. The important aspect of the Mott-Schottky analysis is the fact that the measurement can be conducted with the sample immersed in liquid, and thus the conditions should reflect the actual situation of the relevant photoelectrocatalysis. The importance of the surface state, potentially determining ions and pH, was emphasized to significantly contribute to the determination of flatband potential. [START_REF] Cardon | On the determination of the flat-band potential of a semiconductor in contact with a metal or an electrolyte from the Mott-Schottky plot[END_REF][START_REF] Singh | Surface charge and specific ion adsorption effects in photoelectrochemical devices[END_REF][START_REF] Matsumoto | Dependence of the band bending of the oxide semiconductors on pH[END_REF][START_REF] Rothenberger | Spectroscopy of conduction band electrons in transparent metal oxide semiconductor films: Optical determination of the flatband potential of colloidal titanium dioxide films[END_REF][START_REF] Kavan | Electrochemical and Photoelectrochemical Investigation of Single-Crystal Anatase[END_REF] Therefore, it is challenging to accurately interpret the Mott-Schottky analysis of the high surface area electrodes, typical for powdersemiconductor based electrodes. Similarly, there are some interesting discussions about the correlation of flatband potential with atomic electronegativities of oxide. [START_REF] Kavan | Electrochemical and Photoelectrochemical Investigation of Single-Crystal Anatase[END_REF] In computational chemistry, one difficulty for defining the energy levels of semiconductors is to obtain the ionization potential and electron affinity with respect to a reference electrode. A first assumption frequently made is the Koopman's theorem assuming that the crystalline orbitals energy can be considered as the ionization potential and the electron affinity, as we have discussed in the bandgap section. The natural reference system used in quantum chemistry is the vacuum level, being at +4.44 V vs. SHE. [START_REF] Bredas | Mind the gap![END_REF][START_REF] Cahen | Electron energetics at surfaces and interfaces: Concepts and experiments[END_REF][START_REF] Kahn | Fermi level, work function and vacuum level[END_REF] In a quantum chemical calculation, a part of the system must be a vacuum to have orbital energies obtained with respect to this reference. In other words, the band positions are not obtained for a bulk material but for a surface. Each computed band position depends on the surface chosen to be exposed to the vacuum, and the results are surface dependent, as illustrated by the work of Stevanovic et al. [START_REF] Stevanović | Assessing capability of semiconductors to split water using ionization potentials and electron affinities only[END_REF] (Figure 19). In general, quantum chemists choose non-polar surfaces because polar surfaces are prone to reconstruction and defects, and from a technical point of view, this leads to a non-stoichiometry or strong dipole moment, ultimately affecting the electronic structure, as can be seen in the case of SrTiO3 in Figure 19. [START_REF] Stevanović | Assessing capability of semiconductors to split water using ionization potentials and electron affinities only[END_REF][START_REF] Hinuma | Band alignment of semiconductors from densityfunctional theory and many-body perturbation theory[END_REF][START_REF] Noguera | Polar oxide surfaces Claudine[END_REF][START_REF] Martirez | Synergistic oxygen evolving activity of a TiO2rich reconstructed SrTiO3(001) surface[END_REF][START_REF] Erdman | Surface structures of SrTiO3(001): A TiO2-rich reconstruction with a c(4 x 2) unit cell[END_REF][START_REF] Heifets | Ab initio modeling of surface structure for SrTiO3 perovskite crystals[END_REF][START_REF] Li | Zinc oxide as a model transparent conducting oxide: A theoretical and experimental study of the impact of hydroxylation, vacancies, interstitials, and extrinsic doping on the electronic properties of the Polar ZnO (0002) surface[END_REF][START_REF] Lauritsen | Stabilization principles for polar surfaces of ZnO[END_REF][START_REF] Tasker | The stability of ionic crystal surfaces[END_REF] As for the bandgap calculation, the only very accurate computational protocol can lead to reliable band positions, such as range-separated hybrid calculation or GW calculations. [START_REF] Ping | Electronic excitations in light absorbers for photoelectrochemical energy conversion: First principles calculations based on many body perturbation theory[END_REF][START_REF] Bahers | Assessing the Use of BiCuOS for Photovoltaic Application: From DFT to Macroscopic Simulation[END_REF][START_REF] Stevanović | Assessing capability of semiconductors to split water using ionization potentials and electron affinities only[END_REF][START_REF] Hinuma | Band alignment of semiconductors from densityfunctional theory and many-body perturbation theory[END_REF][START_REF] Harb | Tuning the properties of visible-light-responsive tantalum (oxy)nitride photocatalysts by non-stoichiometric compositions: a first-principles viewpoint[END_REF][START_REF] Toroker | First principles scheme to evaluate band edge positions in potential transition metal oxide photocatalysts and photoelectrodes[END_REF] Even with the most accurate computational chemistry approach, to obtain a band position close to the experiment is not guaranteed. Indeed, the slab directly cut in the bulk material is frequently far to be a good model of the real surface. Surfaces can experience important reconstruction, especially for the polar ones, such as in the case of SrTiO3, or change of composition compared to the bulk like for Ta3N5. [START_REF] Erdman | Surface structures of SrTiO3(001): A TiO2-rich reconstruction with a c(4 x 2) unit cell[END_REF][START_REF] Heifets | Ab initio modeling of surface structure for SrTiO3 perovskite crystals[END_REF][START_REF] Nurlaela | Critical role of the semiconductor-electrolyte interface in photocatalytic performance for water-splitting reactions using Ta3N5 particles[END_REF] Furthermore, put into the electrolyte of water splitting device, the nature of the surface can change. Thus, determining the band edge positions can be a difficult task. The description of a realistic interface by computational chemistry has been the subject of several developments, summarized by the review of Phan et al. [START_REF] Pham | Modelling heterogeneous interfaces for solar water splitting[END_REF] Modeling highlighted the influence of water adsorbed on the semiconductor surface on the electronic properties of the surface, including band positions. [START_REF] Ping | Solvation effects on the band edge positions of photocatalysts from first principles[END_REF] The most important achievements in that field are clearly the description of the semiconductor/water interface by quantum chemical molecular dynamics. [START_REF] Car | Unified Approach for Molecular Dynamics and Density-Functional Theory[END_REF] This type of calculation has been applied to several types of semiconductors/electrolyte interfaces, such as the TiO2/water interface simulation [START_REF] Cheng | The electric double layer at a rutile TiO2 water interface modelled using density functional theory based molecular dynamics simulation[END_REF][START_REF] Cheng | Aligning electronic energy levels at the TiO2/H2O interface[END_REF][START_REF] Sun | Titania-water interactions: a review of theoretical studies[END_REF], Fe2O3/water interface simulation [START_REF] English | Photo-active and dynamical properties of hematite (Fe2O3)-water interfaces: An experimental and theoretical study[END_REF], ZnO/water interface simulation [START_REF] Kharche | First-principles approach to calculating energy level alignment at aqueous semiconductor interfaces[END_REF], and Al2O3/water interface simulation [START_REF] Huang | /Water Interface: Structural Properties and Infrared Spectra from First-Principles Molecular Dynamics Simulations[END_REF]. While ab initio molecular dynamics is probably the best tool to model the surface of the semiconductor placed in water, the very demanding computational resources and time prevent the use of this approach for high-throughput simulation of semiconductors. Another difficulty to manage during the simulation of the semiconductor/electrolyte interface is the electrolyte composition, frequently containing a large concentration of ions. Several approaches can be used to model such systems, including explicit simulation of the ions [START_REF] Opalka | The ionization potential of aqueous hydroxide computed using many-body perturbation theory[END_REF][START_REF] Wan | Electronic structure of aqueous sulfuric acid from first-principles simulations with hybrid functionals[END_REF][START_REF] Gaiduk | Structural and electronic properties of aqueous NaCl solutions from ab initio molecular dynamics simulations with hybrid density functionals[END_REF] or implicit simulation generally based on the Poisson-Boltzmann equations [START_REF] Borukhov | Steric effects in electrolytes: A modified poisson-boltzmann equation[END_REF][START_REF] Jinnouchi | Electronic structure calculations of liquid-solid interfaces: Combination of density functional theory and modified Poisson-Boltzmann theory[END_REF][START_REF] Gunceler | The importance of nonlinear fluid response in joint density-functional theory studies of battery systems[END_REF][START_REF] Sundararaman | Evaluating continuum solvation models for the electrode-electrolyte interface: Challenges and strategies for improvement[END_REF][START_REF] Ringe | Transferable ionic parameters for first-principles Poisson-Boltzmann solvation calculations: Neutral solutes in aqueous monovalent salt solutions[END_REF] of the electrolyte. The simulation of band positions will probably gain in reliability in the future due to the large work devoted to the simulation of surface/electrolyte interfaces. [START_REF] Bisquert | Energy diagram of semiconductor/electrolyte junctions[END_REF] The band positions of semiconductors are also affected by the interaction of the surface with the co-catalyst deposited on the surface. This influence can be modeled by directly simulating the interface between the two materials, for example, the WO3/IrO2 interface by Ping et al. [START_REF] Ping | Energetics and solvation effects at the photoanode/catalyst interface: Ohmic contact versus Schottky barrier[END_REF] and TiO2/IrO2 interface by Pastore et al. [START_REF] Pastore | First-principles modeling of a dye-sensitized TiO2/IrO2 photoanode for water oxidation[END_REF] In photocatalytic water splitting, the decoration with active electrocatalysts is essential to achieve high efficiency, so this catalyst decoration must be taken into consideration. Since this local electronic structure generated at the catalyst-semiconductor interface is effectively utilized to separate charges, this should be a critical investigation for the future direction of the research in photoelectrocatalysis. Also, it is important to determine whether the interface is constructed with inclusion of an electrolyte (semiconductor-electrolyte interface) or effectively isolated with an electrolyte by a catalyst [START_REF] Pham | Modelling heterogeneous interfaces for solar water splitting[END_REF][START_REF] Nakato | Structures and functions of thin metal layers on semiconductor electrodes[END_REF][START_REF] Mills | Theory and simulations of electrocatalyst-coated semiconductor electrodes for solar water splitting[END_REF][START_REF] Sze | Physics Of Semiconductor Devices[END_REF] or protecting layer. [START_REF] Hu | Amorphous TiO2 coatings stabilize Si, GaAs, and GaP photoanodes for efficient water oxidation[END_REF][START_REF] Labrador | Enhanced Performance of Si MIS Photocathodes Containing Oxide-Coated Nanoparticle Electrocatalysts[END_REF] Concurrent investigations on both experimental measurements and theoretical description of such complicated interface would be suggested to be the key for improved photocatalytic reactions.

Conclusions and perspectives.

New semiconductors are now frequently proposed in the literature for photocatalytic applications. A deep characterization of new semiconductors is important to assess their potentialities for the desired application. Unfortunately, these characterizations are sometimes lacking or performed using unreliable techniques. Collective efforts should be dedicated to establishing very reliable approaches to characterize semiconductors, both experimentally and theoretically. For some properties, the quantum chemical approaches have reached accuracy very close to the experiment with very reasonable computational resources. Those calculations should be performed for all newly synthesized semiconductors, such as to obtain a kind of identity card of the new material.

Computational chemistry can go beyond the characterization of a single material.

Because the tools of quantum chemists have achieved a degree of precision close to the experiment, projects of high throughput prediction of materials properties are appearing. [START_REF] Singh | Computational screening of 2D materials for photocatalysis[END_REF][START_REF] Jain | Performance of genetic algorithms in search for water splitting perovskites[END_REF][START_REF] Greeley | Computational high-throughput screening of electrocatalytic materials for hydrogen evolution[END_REF][START_REF] Emery | High-Throughput Computational Screening of Perovskites for Thermochemical Water Splitting Applications[END_REF][START_REF] Choudhary | High-throughput Identification and Characterization of Two-dimensional Materials using Density functional theory[END_REF][START_REF] Castelli | New cubic perovskites for one-and two-photon water splitting using the computational materials repository[END_REF] The objective is to predict new materials with the desired properties before starting a long and costly experimental synthesis.

Figure 1 .

 1 Figure 1. Structure of a photocatalyst with the semiconductor in orange and the cocatalysts in grey. Valence and conduction bands of the semiconductor are in red and blue, respectively.

Figure 2 .

 2 Figure 2. (a) Single semiconductor photocatalyst architecture used for the STH efficiency simulation of (c). (b) Tandem cell architecture used to perform the simulation of the STH efficiency of (d). (c) Comparison between the Schockley-Queisser limit of a photovoltaic cell and the Solar-to-Hydrogen (STH) efficiency of photocatalysts as a function of the bandgap of the semiconductor. (d) STH efficiency of two-junction photocatalysts as a function of the bandgaps of the two semiconductors.

Figure

  Figure reprinted with permission from "B.A. Pinaud et al. Energy Environ. Sci. 2013, 6, 1983" with the permission of the royal society of chemistry.

Figure 4 .

 4 Figure 4. An energetic diagram presenting the different bandgap and their relationships. EB in this figure corresponds to Eb in the text. Figure reprinted with permission from "J.-L. Breda Mater. Horiz. 2014, 1, 17-19" with the permission of the royal society of chemistry.[95]

Figure 5 .

 5 Figure5.[START_REF] Bahers | Semiconductors used in photovoltaic and photocatalytic devices: Assessing fundamental properties from DFT[END_REF]

Figure 5 .

 5 Figure 5. Mean absolute error (MAE) of the four investigated functionals on the selected properties. MAE is obtained by performing the arithmetic average of the absolute error (compared to experiment). Adapted with permission from "Le Bahers, T.; Rérat, M.; Sautet, P. J. Phys. Chem. C 2014, 118, 5997". Copyright (2018) American Chemical Society.[START_REF] Bahers | Semiconductors used in photovoltaic and photocatalytic devices: Assessing fundamental properties from DFT[END_REF]

Figure 6 .

 6 Figure 6. The fraction of exact exchange as a function of the interelectronic distance for several functionals.

Figure 7 .

 7 Figure 7. Illustration of the Hubbard Hamiltonian of the bandgap of TiO2 (anatase and rutile).Figure reprinted with permission from "M.E. Arroyo-de Dompablo et al. J. Chem. Phys. 2011, 135, 054503". Copyright (2018) The American Physical Society.

Figure 8 .

 8 Figure 8. Comparison between DFT (PBE GGA Functional) and GW bandgaps for selected semiconductors. Figure reprinted with permission from [M. Shishkin et al. Phys. Rev. Lett. 2007, 99, 246403]. Copyright (2018) The American Physical Society.[95]

  [START_REF] Paier | Dielectric properties and excitons for extended systems from hybrid functionals[END_REF][START_REF] Ferrari | Ab Initio Calculation of the Ultraviolet-Visible (UV-vis) Absorption Spectrum, Electron-Loss Function, and Reflectivity of Solids[END_REF][START_REF] Refaely-Abramson | Solid-state optical absorption from optimally tuned time-dependent range-separated hybrid density functional theory[END_REF]

Figure 9 .

 9 Figure 9. Example of a simulation of the imaginary part of the dielectric constant by TD-DFT for Si, GaAs, SiC, C, and LiF from top to bottom respectively. Figure reprinted with permission from [J. Paier et al. Phys. Rev.B 2008, 78, 121201].

Figure 10 .

 10 Figure 10. Absorption spectra of Cu2O (a) along with the related absorption coefficient (b) and several ways to determine the bandgap from the Tauc plot, assuming direct-indirect/allowed-forbidden transitions (c)-(f).Figure reprinted with permission from [Z. Chen et al. Photoelectrochemical Water Splitting, 2013, Springer-Verlag New-York]. Copyright (2018) Springer-Verlag New-York.

Figure 11 .

 11 Figure 11. Experimental (obtained by ellipsometry) and computed (GW/BSE) frequency dependent dielectric constant (ε1 real part and ε2 imaginary part) of Ta3N5. Figure reprinted with permission from [J. Morbec et al. Phys. Rev. B 2014, 90, 1]

Figure 12 .

 12 Figure 12. Temperature-dependent mobility as a function of the drain field applied on polymer based on the thiophenethiazole groups. Adapted with permission from "Lee, J. et al. J. Am. Chem. Soc. 2015, 137, 7990". Copyright (2018)American Chemical Society.[START_REF] Bahers | Semiconductors used in photovoltaic and photocatalytic devices: Assessing fundamental properties from DFT[END_REF]

Figure 13 .

 13 Figure 13. The contour plots of the effective masses of holes and electrons close to the Γ point for CH3NH3PbX3 (X = I and Br). (a) me* of CH3NH3PbI3; (b) mh* of CH3NH3PbI3; (c) me* of CH3NH3PbBr3; (d) mh* of CH3NH3PbBr3. Figure reprinted with permission from [J. Feng and B. Xiao, J. Phys. Chem. Lett. 2014, 5, 1278-1282]. Copyright (2018) The American Chemical Society.

  Favoring the covalency of the chemical bounds reduces the effective mass and consequently increases the charge carrier mobilities. The covalency can be tuned by changing the chemical nature of the elements or by changing the local geometries, as illustrated by Ha et al. for the hole effective mass dominated by the Sn-O interaction in tin oxides (Figure 14).

Figure 14 .

 14 Figure 14. Hole effective mass as a function of the Sn-O-Sn bond in several tin oxides. Figure reprinted with permission from [V.-A. Ha et al., J. Mater. Chem. C. 2017, 5, 5772-5779]. Copyright (2018) The Royal Society of Chemistry.

Figure 15 .

 15 Figure 15. Simulation of the potential energy surface for the electron polaron hopping in two different directions (green and red curves) in Ta3N5. Figure reprinted with permission from [J. M. Morbec and G. Galli Phys. Rev. B 2016, 93, 035201].

[ 244 ]Figure 16 .

 24416 photoluminescence spectroscopy.[START_REF] Chernikov | Exciton binding energy and nonhydrogenic Rydberg series in monolayer WS2[END_REF] If the exciton is a Mott-Wannier one (i.e., delocalized on several unit-cells), researchers frequently compare it to a hydrogen atom, namely a positive (the hole) and a negative (the electron) charges interacting through the Coulomb potential, because the Hamiltonian of this exciton is the same as the one of the hydrogen atom. The hydrogen-like exciton has the same wavefunctions and energies as the hydrogen atom, adapted to the dielectric constant of the material and the hole mass (lighter than a proton). Due to the hydrogen-like nature of the exciton, it is sometimes possible to observe, in the absorption spectra of the semiconductor, energetic levels inside the fundamental bandgap associated with the hydrogen-like levels of the exciton. As a matter of illustration, we can

Figure 17 .

 17 Figure 17. Electronic band gaps for (a) gt-C6N9H3, (b) corrugated gt-C3N4 are reported. The Wannier-Mott analysis using the CRYSTAL14 code is depicted in an alternative color scheme. The optical band gaps Eopt,nv g are depicted semi-transparently. The Evb-corrected band gaps are provided in italics. The WM and scGW-BSE exciton binding energies Eb are provided to the right of the bandgaps, with v and nv denoting vibrationally and not vibrationally corrected, respectively. The 2D unit cells for the studied structures are given with lattice vectors a and b. Figure reprinted with permission from [S. T. A. G. Melissen et al. J. Mater. Chem. A 2017, 5, 5115-5122.] and The Royal Society of Chemistry.[START_REF] Bredas | Mind the gap![END_REF]

Figure 18 .

 18 Figure 18. Band positions with respect to SHE reference of several semiconductors used for photoelectrocatalysis. Figure adapted with permission from [K. Sivula and R. vand de Krol, Nat. Rev. 2016, 1, 15010]. Copyright (2018) Springer.

Figure 19 .

 19 Figure 19. Calculated band edge energies (vs. vacuum) from GW single point calculations on top of PBE optimized surfaces. Experimental energies (obtained by photoemission) are given as bold lines. Figure reprinted with permission from [V. Stevanović et al. Phys. Chem. Chem. Phys. 2014, 16, 3706-3714]. Copyright (2018) The Royal Society of Chemistry.

  I. Bandgaps and absorption coefficient. ................................................................................... 12 II. Dielectric constant. .............................................................................................................. 25 III. Charge transport properties ................................................................................................ 29 IV. Exciton binding energy. ..................................................................................................... 37 V. Band positions. .................................................................................................................... 42

A complete semiconductor characterization is also necessary for multi-scale simulations. This type of modeling aims at simulating the full process of photoelectrocatalysis from light absorption to chemicals production [START_REF] Haussener | Modeling, simulation, and design criteria for photoelectrochemical water-splitting systems[END_REF][START_REF] Cendula | Calculation of the Energy Band Diagram of a Photoelectrochemical Water Splitting Cell[END_REF][START_REF] Gaudy | Determination and optimization of material parameters of particle-based LaTiO2N photoelectrodes[END_REF][START_REF] Haussener | Simulations of the irradiation and temperature dependence of the efficiency of tandem photoelectrochemical watersplitting systems[END_REF][START_REF] Gaudy | Utilizing modeling, experiments, and statistics for the analysis of water-splitting photoelectrodes[END_REF] and involves several fields of physics and chemistry, as well as with several scales in space and time. Among the phenomena simulated by multi-scale modeling, charge carrier generation and transport in the semiconductor, leading to the flux of electrons and holes arriving at the surface of the semiconductor, are governed by semi-classical semiconductor equations involving several properties discussed in the manuscript (dielectric constants and mobilities). [START_REF] Garcia-Esparza | A simplified theoretical guideline for overall water splitting using photocatalyst particles[END_REF]