Theoretical calculation of a TiO2-based photocatalyst in the field of water splitting: A review

Currently, energy and environmental problems are becoming more serious. The use of solar energy to split water and produce clean, renewable hydrogen as an energy source is a feasible and effective approach to solve these problems. As the most promising semiconductor material for photocatalytic water splitting, TiO2-based nanomaterials have received increasing attention from researchers in academia and industry in recent years. This review describes the research progress in the theoretical calculations of TiO2-based photocatalysts in water splitting. First, it briefly introduces some commonly used theoretical calculation methods, the crystal structure of TiO2 and its photocatalytic mechanism, and the principle of doping and heterojunction modification to improve the photocatalytic performance of TiO2. Subsequently, the adsorption state of water molecules with different coverages on the surface of TiO2, the ratelimiting steps of the splitting of water molecules on the surface of TiO2, and the transfer process of photogenerated current carriers at the interface between water molecules and TiO2 are analyzed. In addition, a brief review of research into the theoretical calculations of TiO2-based commercial photocatalysts in the field of water splitting is also provided. Finally, the calculation of TiO2-based photocatalytic water-splitting simulations is summarized, and possible future research and development directions are discussed.


Introduction
The energy crisis is one of the two major problems facing the world today, and the development and utilization of new energy sources represented by solar energy and hydrogen energy are imminent. Since Fujishima and Honda [1] reported the production of hydrogen from the photocatalytic water splitting of TiO 2 in 1972, with high catalytic activity, good stability, nontoxicity to the human body, and low cost [2], it has become one of the most extensively studied and applied semiconductor materials in the field of photocatalysis [3].
Although the electron-hole pair generated by photoexcitation of TiO 2 possesses a strong redox ability as a result of the wide band gap (E g = 3.0-3.2 eV), it does not effectively absorb visible light [4]. According to numerous experiments and theoretical calculations (Liu T. et al., December 2019, China United Test & Evaluation Qingdao CN110568122-A), the doping or heterojunction modification of TiO 2 not only effectively broadens its light absorption range but also substantially reduces the probability of photogenerated electron-hole pair recombination [5][6][7][8][9][10][11][12][13][14]. However, due to the complexity of the photocatalytic water-splitting reaction system, considerable controversy still exists regarding its reaction mechanism, reaction process, and kinetic behavior. Importantly, studies of the interaction between the surface of TiO 2 and water molecules, the energy barrier and rate-limiting steps of photocatalytic reactions, and the process of charge transfer are important methods to understand the mechanism of photocatalytic reactions. At the same time, this research also provides an important theoretical basis for improving the efficiency of hydrogen production from water splitting and improving the design of TiO 2 -based photocatalysts [15,16].
With the rapid development of modern computer technology, molecular simulation has become a very important method and tool in many fields. Specifically, the methods of molecular simulation are generally divided into the following categories: quantum mechanics methods [17], molecular mechanics methods [18], classical molecular dynamics [19], and molecular Monte Carlo methods [20]. The most widely used density functional theory (DFT) approach [21][22][23][24][25] was developed based on quantum mechanics methods (Guo S. et al., March 2019, Univ North Minzu CN109545286-A). The molecular mechanics method mainly relies on a molecular force field to calculate the various properties of the molecule, and its advantage is that it quickly obtains the various properties of the molecule. It is often used to analyze drugs, clusters, and biochemical macromolecules [26][27][28]. The molecular Monte Carlo method is based on the random movement of the particles in the system. It is the earliest nonquantum calculation method used for large systems, but this method is not suitable for the movement principles in physics. Compared with the molecular Monte Carlo method, the movement of system particles has the correct physical basis, and the calculation accuracy is high in classical molecular dynamics simulations. Meanwhile, the dynamic and thermodynamic statistics of the system are obtained, which are widely applicable to the discussion of various systems and characteristics. Currently, the theoretical calculation methods used to study photocatalytic water splitting of TiO 2 mainly include DFT based on quantum mechanics, classical molecular dynamics, and first-principles molecular dynamics combining DFT and classical molecular dynamics [29][30][31].
In addition to theoretical calculations of the structure of TiO 2 itself and the doped or heterojunction system [32][33][34][35][36][37][38], scholars have also conducted numerous theoretical calculations and experiments to determine the interaction between the TiO 2 surface and water molecules, the migration mechanism of photoproduced electron-hole pairs, recombination of electron-hole pairs and other steps in the process to obtain an understanding of the mechanism and essential laws of photocatalytic water splitting [39][40][41][42][43][44][45][46].
This review describes the research progress in theoretical calculations on the application of TiO 2 -based photocatalysts in water splitting. First, it briefly introduces some commonly used theoretical calculation methods, the crystal structure of TiO 2 and its photocatalytic mechanism, and the principle of doping and heterojunction modification to improve the photocatalytic performance of TiO 2 . Subsequently, the results of simulations and calculations of the adsorption and distribution structure of water molecules with different levels of coverage on the surface of TiO 2 are summarized. Then, the progress in simulations and calculations related to the rate-limiting steps of the water molecule splitting reaction on the surface of TiO 2 and the progress in the simulations and calculations of the photogenerated current carrier transfer process occurring at the interface of water molecules and TiO 2 are reviewed. In addition, a brief review of studies performing theoretical calculations of TiO 2 -based commercial photocatalysts in the field of water splitting is also provided. Finally, the simulation and calculation of TiO 2 -based photocatalytic water splitting are summarized, and possible future research and development directions are discussed.
2 Theoretical method 2.1 The first-principles calculation The first-principles method is based on the principle of interactions between the nucleus and electrons and its law of motion, combined with the basic principles of quantum mechanics through some approximate processing steps to solve Schrodinger's equation. The generalized first principles include ab initio calculations based on density functional and ab initio calculations based on the Hartree-Fock self-consistent calculation. The former uses electron density as the basic variable, and by solving the Kohn-Sham equation, the ground state electron density of the system is procured iteratively and self-consistently to solve the ground state properties of the system. The latter obtains the wave function of the system by self-consistently solving the Hartree-Fock equation. Two basic approximations are included in the first-principles calculation method for solving the ground state properties, namely, the Bonn-Oppenheimer approximation [47,48] and singleelectron approximation [49,50]. These two approximations are the key to solving multiparticle systems.

DFT
DFT originated from the electron density functional of kinetic energy based on the uniform electron gas model proposed by Thomas and Fermi in 1927, also known as the Thomas-Fermi model [51,52]. Subsequently, the Hohenberg-Kohn theorem and Kohn-Sham theorem were proposed based on this model, and the DFT was truly developed.
DFT is a method to reduce a 3N dimensional problem to a density problem of a three-dimensional particle, which substantially reduces the number of calculations in practical applications. This reduction makes DFT completely different from the classical electronic structure theory based on complex multielectron wave functions. Currently, this theory is already one of the most commonly used methods in the fields of condensed matter physics and computational chemistry, and its appearance has facilitated the development of materials physical chemistry and other fields.

Exchange-correlation functional
DFT holds that all quantities are theoretically accurate, and only the exchange-correlation functional uses approximate methods. The usual approximation methods include local density approximation (LDA) and generalized gradient approximation (GGA) (Liu B. et al., August 2018, Univ Shenyang Technology CN108387636-A).

LDA
LDA is a simple and effective approximation in which electrons have a uniform electron density behavior in a uniform field, and their exchange-correlation energy is accurately calculated using the quantum Monte Carlo algorithm. When the external potential field does not change drastically, the uniform electron gas density function is used instead of the nonuniform electron gas density function to calculate the exchange-correlation energy in the local area [53,54].
The LDA method was widely used in the early DFT calculations [55], but it still has certain errors when calculating the solutions to some problems. For example, the lattice constant is usually too small during the calculation leading to a larger adsorption energy and bulk modulus of the system. The calculated diffusion barrier is often too low. The calculated band gap of a semiconductor usually produces a deviation greater than 50%, and sometimes an effective band gap value is not able to be obtained.

GGA
In practical applications, the performance of most semiconductor materials will improve after the introduction of some intrinsic defects or doping of some impure atoms. Due to the introduction of defects and impure atoms, the electron density in these semiconductor materials will undergo abrupt changes; therefore, the LDA method based on the uniform electron gas model is no longer suitable.
The GGA is developed based on LDA, and it not only uses the electron density as a variable of the exchangecorrelation functional but also considers the gradient of density (▽ρ), thus strengthening its nonlocal description of charge density [56,57] and improving the calculation accuracy. Commonly used schemes include PBE, PW91, and RPBE.

Hybrid functional and Hubbard + U
When calculating values for transition metal oxides, LDA and GGA tend to underestimate their band gap size, and the value obtained is generally 1.0-2.0 eV less than the experimental value [58]. This discrepancy is attributed to the calculation of the properties of the ground state of the system by the common LDA and GGA methods, which are unable to accurately describe the local and delocalized characteristics of electrons. The potential function based on the LDA and GGA methods must be modified to more accurately describe the electronic properties of materials. Currently, researchers often use hybrid functional (HF) or Hubbard + U methods to solve these problems [59,60].
The so-called HF adds other functionals to modify the traditional LDA or GGA functionals. Compared with the two types of functionals described above, HF provides a more accurate description of the ion potential and lattice constants of the system. Common HFs include B3LYP (for molecular systems), PBE0 (25% HF and 75% DFT functionals; for solid systems), and HSE (for solid systems). However, compared with traditional functionals, the use of HF often requires more computing resources, which has become an important factor limiting its application.
In addition to using HF calculations, the Hubbard + U method is a more commonly used calculation method. The U in LDA + U or GGA + U refers to the strong correlation repulsion energy of electrons with opposite spins in the Hubbard model. Since both the LDA and GGA methods use a single-electron approximation to address valence electrons, they usually ignore the strong correlation between these two electrons when they calculate a strong correlation system containing d and f state electrons. Generally, a U value must be added in the calculation process to modify this method, which is generally only applicable to the strong correlation system containing d and f state electrons. In the TiO 2 system, the U value generally ranges from 2.5 to 4.5 eV, and different U values will also affect the lattice constant of the system, particularly the electronic peculiarities, and thus a unified U value is required in the calculation [58,61,62].

Transition state theory
The transition state theory developed in the 1830s states that the chemical reaction from reactant to product undergoes a high-energy transition state structure. During the reaction, partial chemical bonds of the reactant molecules break to form a transition state and then recombine to form a product. The energy of the transition state is higher than the reactants and products, the energy increases from the reactant to the transition state, and the energy decreases from the transition state to the product. The transition state is the saddle point on the potential energy surface of the reaction and is the highest point of energy connecting the reactant and the product. The transition state has high activity and an unstable structure, and its existence in the reaction is short. The transition state structure is difficult to obtain experimentally; fortunately, the transition state structure can be effectively simulated through theoretical calculations [63]. The synchronous transition method proposed by Halgren et al. [64] automatically generates a continuous path search and predicts the transition state structure between known reactants and products, including linear synchronous transition (LST) and secondary synchronous transition (QST).

Molecular dynamics simulation
Molecular dynamics provide a definite microscopic description of a physical system, either a few-body system or a multibody system. The basic principle of this method is to establish a particle system, simulate the microscopic phenomenon under study, and determine the interaction between the particles using quantum mechanics [65]. For many particle systems conforming to the laws of classical Newtonian mechanics, the motion law and trajectory of the particle in the phase space are obtained from the numerical solution of the particle kinematics equation set, and then the corresponding macroscopic physical properties of the system are obtained according to the principles of statistical physics [66]. The motion equation of the particle is a set of ordinary differential equation sets, and the basic principle of the solution is to use the finite difference method to integrate the equation along the time axis with a certain time step. Although the accuracy of the molecular dynamics simulation is not as good as the first-principles simulation, its advantages, such as a convenient calculation process, small number of calculations, and high computable atomic system complexity, greatly exceed the latter and thus have broad development and application prospects [67].
In the process of a molecular dynamics simulation, the basic steps include determining the research object, establishing the potential energy model, establishing the corresponding molecular dynamics equation, initializing the configuration of the established microstructure, establishing the periodic boundary conditions of the computational model, and truncating the initial potential energy of the calculation model. The model of the target material system is simulated, calculated, and analyzed [68].
In the adhibition of molecular dynamics simulation methods, simulations are mainly conducted for the two types of system states. One is the simulation of the equilibrium state, and the other is the simulation of the nonequilibrium state. The main difference between the two systems is whether the amount of interest and the boundary conditions of the equilibrium system are related to time. A molecular dynamics simulation is always performed under a certain ensemble to accurately reflect the actual physical process [69].
3 Crystalline structure of TiO 2 and its photocatalytic mechanism 3.1 Crystalline structure of TiO 2 Three main crystal types of TiO 2 have been identified in nature, namely, brookite, rutile, and anatase [70]. Both rutile and anatase TiO 2 belong to the tetragonal system, while brookite TiO 2 belongs to the orthorhombic system. The structure of brookite TiO 2 is unstable, and its preparation is relatively difficult [71]; therefore, it is usually used for basic research. Meanwhile, rutile and anatase TiO 2 play a vital role in the field of photocatalysis applications. Some similarities have been observed between the anatase and rutile crystal structures, namely, TiO 6 octahedrons constitute the most basic structure. However, at the same time, some differences also exist between the two crystal structures, including the degree of distortion and connection mode of the TiO 2 octahedron. The TiO 6 octahedron formed by rutile TiO 2 is closely connected along the space c axis in the form of an edge sharing chain, and the octahedrons have a slight orthogonal distortion [72]. Compared with rutile TiO 2 , the TiO 6 octahedron formed by anatase TiO 2 displays a more substantial orthorhombic distortion [73], and the symmetry is also poor. Regarding the connection mode, each octahedron in the rutile phase is coupled with 10 adjacent octahedrons, while in the anatase phase, each octahedron is coupled with 8 adjacent octahedrons. These differences in the lattice structure result in different mass densities and electronic band structures of the two forms of TiO 2 [74]. The mass density of anatase TiO 2 is 3.894 g/cm 2 , which is slightly smaller than the value of 4.250 g/cm 2 for rutile.
According to recent studies, the photocatalytic property of rutile TiO 2 is slightly lower than anatase TiO 2 [75][76][77]. In fact, the energy gap (3.0 eV) of rutile phase TiO 2 is narrower than the anatase phase (3.2 eV), and the energy required for electrons to be excited to transition is relatively small; therefore, the light response range of rutile phase TiO 2 is wider. However, the rutile phase TiO 2 tends to have a larger particle size and form an agglomerated structure, leading to easy recombination of the electron-hole pairs [78]. Conversely, the anatase phase crystal lattice contains more defects and dislocations, which increase the separation of electron-hole pairs generated in this lattice, resulting in a stronger photocatalytic property of' anatase phase TiO 2 than the rutile phase.

Photocatalytic mechanism of TiO 2
As shown in Figure 1, based on the solid energy band theory, the energy band structure of semiconductor TiO 2 is very different from that of metals or insulators, as it contains a valence band (VB), a conduction band (CB), and a region between the CB and the VB called the band gap. The formula used to calculate the band gap (E g ) is E g = E VB − E CB , where E CB corresponds to the bottom potential of the CB, and E VB corresponds to the top potential of the VB [79,80]. When the energy of the incident photon is equal to or greater than the E g of TiO 2 , the electrons in the VB will shift to the CB, and the holes will be generated in the VB, thus forming a photogenerated electron (e − )-hole (h + ) pair [81,82]. Subsequently, under the action of an electric field, the formed electrons and holes will be separated, gradually transferred to the active sites on the surface of TiO 2 , and react with the water molecules adsorbed on the surface of TiO 2 in the corresponding hydrogen evolution reaction (HER) or oxygen evolution reaction (OER) before finally decomposing the water molecules into H 2 and O 2 [83].
The high overpotential in the OER will cause a substantial loss of energy, limiting the efficiency of photocatalytic reactions and practical applications. Therefore, the mechanism of OER must be elucidated to improve the photocatalytic efficiency, and it is a research hotspot in TiO 2 photocatalytic reactions. As shown in a previous study [84], the OER is a four-step reaction process, and every electron transfer reaction is accompanied by the loss of protons. The first step of the reaction is the reaction of photogenerated holes with water molecules or the reaction of holes with OH-and Ti-OH groups [85,86]. However, photoelectric emission spectroscopy data show that the O 2p energy level of water species is below the valenceband maximum (VBM) of TiO 2 [87], revealing the irrationality of the traditional four-step reaction mechanism from a thermodynamic perspective. Subsequently, through further experimental technical measurement and characterization, two new mechanisms, the redox photooxidation (RP) mechanism [88,89] and nucleophilic (NA) reaction mechanism [90,91], have been proposed in succession. In both of these mechanisms, the O atoms on the TiO 2 lattice are the active sites for trapping holes, and not the O atoms on the water molecules. However, significant differences have also been identified between these mechanism, mainly including the initial hole capture site, the role of water molecules in the reaction, and the source of oxygen. To date, the mechanisms of RP and NA have been extensively discussed and supported by some theoretical studies. However, most theoretical studies still adhere to the earlier mechanism, namely, the first step in the OER is to capture holes in the H 2 O molecules on the TiO 2 surface. Researchers propose that the theory that the hole is initially localized in the protonated O br in the RP mechanism is not accurate. Theoretically, this site is not the most advantageous hole capture site in the solution medium. Overall, a consensus on the active site of hole capture has not been achieved, and further research and discussion are needed.

Doping modification of TiO 2
Doping refers to the method of replacing some atoms in TiO 2 with atoms of other elements, changing its band gap by changing its VB or CB edge.

Non-metal doping
In 2001, Asahi et al. [96] prepared the TiO 2−x N x photocatalyst with nonmetallic element N-doped TiO 2 in a study conducted in Japan, resulting in a new field of visible light-mediated catalysis using nonmetallic-doped TiO 2 . In the nitrogen-doped anatase TiO 2 , the N 2p and O 2p hybrid orbital energies are very close, which reduces the band gap of N-TiO 2 and broadens the absorption range to visible light. The doping of TiO 2 with nonmetal elements is the main method to improve its photocatalytic performance, because the configuration of extranuclear electrons of non-metal elements is similar to the structure of oxygen atoms, with little effect on the original structure of TiO 2 . Concretely, nonmetallic doping usually uses carbon, nitrogen and other atoms that contain more empty p orbitals than oxygen to replace the oxygen atoms in TiO 2 , and thus the VB of TiO 2 is shifted upwards to reduce its band gap [97,98]. According to the first principles of DFT, Tian and Liu [99] calculated that S-doped anatase TiO 2 produces an S 3p hybrid energy level in the forbidden band and the VB shifts upward; thus, the band gap decreases, resulting in a redshift of the absorption spectrum. As the dopant concentration increases, the band gap is reduced and the redshift of the absorption spectrum is increased. As shown in Figure 2(a), Peng et al. [100] analyzed the effect of the P doping concentration on the absorption spectrum of anatase TiO 2 through theoretical research and found that the absorption spectrum exhibited a redshift as the doping concentration increased under low-concentration doping, but the absorption spectrum exhibited a blueshift as the doping concentration increased under high-concentration doping. In addition, the doping of non-metallic elements may also form isolated impurity energy levels above the VB, which will also broaden the light absorption range of TiO 2 [101].

Metal doping
Metal ions are effective electron acceptors. When doped with metal ions in TiO 2 crystals, the trapping of electrons in TiO 2 by the metal ions prolongs the recombination time of photogenerated electrons and holes in TiO 2 to improve the photocatalytic efficiency of TiO 2 . In metal doping, transition metals such as Ni, Cu, Cr, Co, and Fe are usually used to replace titanium atoms [102,103]. If the CB of the doped metal is lower than titanium, the band gap of the doped TiO 2 will decrease [104]. Lian et al. [105] synthesized Mo-doped TiO 2 nanoparticles using the sol-gel method and observed that Mo 6+ ions occupy the lattice position of anatase phase TiO 2 and reduce the band gap from 3.19 to 3.05 eV. The doping of Mo also increases the light absorption of TiO 2 in the ultraviolet and visible range, and the increase in ultraviolet light absorption is basically related to the charge transfer process of O 2− (2p) and Ti 4+ (3d). The increase in visible light absorption results from electron transfer from the O 2P state to the Mo 4D state and the transition of the Mo 4D and Ti 3D states in the forbidden band (Figure 2(b) and (c)). Karvinen et al. [106] used the first-principles Hartree-Fock calculation method to establish a supercell system with atomic clusters and studied the electronic properties of a series of metal cations after doping into TiO 2 . The doping of V 3+ , Mn 3+ , Cr 3+ , and Fe 3+ ions into the anatase phase TiO 2 shortens the band gap of the doped system, but V 3+ and Fe 3+ ions basically have no effect on the band gap size of the rutile phase TiO 2 . In addition, the doped metal ions extend the range of light absorbed to the visible light region, ensuring that TiO 2 display a strong response to visible light. However, most of the doped metal ions are heavy metal ions, which will cause certain damage to the environment and limit its development and application.

Nonmetal-metal co-doping
Regardless of whether metal doping or nonmetal doping are used, a certain defect state likely occurs. Transition metal doping will produce an occupied interstitial state in the semiconductor forbidden band, and the charge of this interstitial state will be localized in doped metal atoms and may become exciton recombination centers and cause a decrease in photocatalytic ability. Experiments have also confirmed this hypothesis. Nonmetallic doping of nitrogen and carbon is also believed to produce empty p orbitals in the band gap. This p orbital is also considered the recombination center of excitons. Notably, the co-doping of metals and nonmetals solves this problem. Using nitrogen-chromium co-doped TiO 2 as an example, the basic idea of this doping method is that chromium will produce occupied d orbitals and nitrogen will produce empty p orbitals. During co-doping, d electrons will enter the empty p orbitals, resulting in a full orbit of the VB and completely empty orbit of the CB, thereby avoiding the generation of exciton recombination centers [107]. At the same time, co-doping has been shown to shorten the band gap of TiO 2 by 1 eV [108].
Kim et al. [109] synthesized Pt/N co-doped TiO 2 using a wet chemical method. The specific surface area of Pt/N co-doped TiO 2 (178 m 2 /g) is larger than undoped TiO 2 (110 m 2 /g). A time-resolved diffuse reflectance spectroscopy (TDR) study of Pt/N-TiO 2 , revealed a synergistic effect of co-doping. In addition, Pt ions only exist in the form of Pt 4+ in Pt/N-TiO 2 . As shown in Figure 2(d) and (e), the results of photoelectron spectroscopy (XPS) indicate an intervalence charge transfer transition. First-principles calculations also show that the electronic interaction of Pt and N in TiO 2 accelerates the mobility of current carriers, thereby improving the photocatalytic activity. Sun et al. [110] used a sol-gel method to prepare TiO 2 co-doped with Ce and N and found that Ce doping effectively inhibits the transformation of TiO 2 from the anatase phase to the rutile phase. Moreover, He et al. [40] studied the properties of Co-N co-doped TiO 2 through first-principles calculations. Due to the reduction of the band gap and ILs between VB and CB, the light absorption limitation and the absorption scope of the Co-2N co-doped system increased significantly (Figure 2(f)). For the precious metal/TiO 2 prepared using the sol-gel method, the precious metal elements relatively easily enter the TiO 2 lattice to replace Ti ions. The appearance of this heterojunction extends the absorption band edge of TiO 2 to the visible region and even near infrared region.
Generally, the doping of a small amount of precious metal produces impurity energy levels above the VB or below the CB of TiO 2 . These impurity energy levels have been used as shallow acceptor levels or shallow donor levels and then become effective traps for photogenerated electrons. As shown in Figure 3(a), Yang et al. [111] used theoretical calculations based on DFT to study the photodeposition of Au or Ru precious metal clusters with excellent electronic properties as cocatalysts on the anatase phase TiO 2 (101) surface and the mechanism of enhanced photocatalytic activity. As a result, the synthesized Ru/TiO 2 and Au/TiO 2 both showed excellent HER activity (Figure 3(b)). This significant increase in HER activity is due to the interfacial interaction of the catalyst, namely, the strong chemical bonds at the interface serve as electron traps for providing photoinduced electrons (Figure 3(c)). Due to the presence of Ru-4d and Au-5d electronic impurity states, the formation of new degenerate energy levels causes the band gap of the catalyst to narrow. Moreover, the synthesized Au/TiO 2 exhibits a faster HER rate than Ru/TiO 2 , which is attributed to the effect of surface plasmon resonance (SPR). As shown in Figure 3(d), a synergistic effect of plasma-induced "hot" electrons is observed that enhances the final built-in electric field harvest, promotes the migration and separation of photogenerated current carriers, and effectively increases the precipitation rate of hydrogen in the overall photocatalytic water-splitting reaction.
On the other hand, as a promising photocatalytic material for overall water splitting, TiO 2 effectively improves its low efficiency for HER by loading appropriate metal cocatalysts. Zhou et al. [12] used first-principles simulation calculations to deeply explore the working mechanism of a TiO 2 photocatalyst loaded with the non-noble metal Ni in the HER. Compared with the (001) crystal plane of TiO 2 , Ni n clusters more easily aggregate on the (101) crystal plane of TiO 2 , and the structural difference between the two crystal planes determines this characteristic. The Ni n clusters adsorbed on the surface of TiO 2 facilitate the separation of photogenerated current carriers and introduce the interstitial state into the forbidden band of TiO 2 , which increases the Fermi level to a higher energy region, effectively improving the efficiency of the HER. Subsequently, electrochemical calculations were used to further study the process of hydrogen production by TiO 2 . The active site of HER in the Ni n /TiO 2 composite material is the O 2c atom that is bound to or close to the Ni n cluster. Compared with clean surfaces, the loading of Ni n clusters significantly reduces the Gibbs free energy of the HER. The authors reveal the growth behavior of the Ni cocatalyst on the surface of anatase TiO 2 and provide a reasonable explanation for the experimental improvement in the HER efficiency of TiO 2 after the introduction of Ni cocatalyst.

Type II heterostructure
In the type II heterojunction, the positions of CB and VB of semiconductor A are all higher than semiconductor B. Based on the difference in the chemical potential between semiconductors A and B, the CB and VB at the interface of the type II heterojunction will be bent, and then a built-in electric field will be formed, enabling the spatial separation of photoexcited electrons and holes (Li H. et al., Univ Qingdao CN107096548-A). Considering the energy band structure of TiO 2 , researchers usually choose narrow band gap semiconductors to combine with TiO 2 to form a type II heterojunction, such as CdS/TiO 2 [112], Cu 2 O/ TiO 2 [113], CeO 2 /TiO 2 [114], Bi 2 S 3 /TiO 2 [115], MoS 2 /TiO 2 [116], and other heterojunctions, which are all effective methods to increase the light response of TiO 2 to the visible region. In recent years, ternary narrow band gap semiconductors that match the solar spectrum are often selected to increase the absorbance coefficient of TiO 2 to a greater extent. For example, Pillai et al. [37] successfully synthesized a new type of heterojunction AgBiS 2 -TiO 2 composite photocatalyst using a solvothermal method. Then, the electronic and optical capabilities of AgBiS 2 and TiO 2 were studied using DFT (Figure 3(e) and (f)). XRD results revealed tetragonal crystal forms of both AgBiS 2 and TiO 2 . XPS and TEM results illustrate the formation of a heterostructure. As shown in Figure 3(g), due to the coupling between AgBiS 2 and TiO 2 , the UV-DRS spectra of all composite photocatalysts show increased absorption of visible light. As the dopant concentration increases, the band gap of the composite photocatalyst also decreases. Upon exposure to visible light, the photocatalytic hydrogen production by the AgBiS 2 /TiO 2 heterojunction is significantly increased, and the output is 1,000 times higher than pure TiO 2 (Figure 3(h)). As confirmed by the PL measurement, the reinforced photocatalytic efficiency is ascribed to the reduced recombination rate of photogenerated excitons. In addition, the author also performed free radical scavenging experiments and a theoretical analysis to define a reasonable photocatalytic mechanism.

Z-scheme heterostructure
The Z-scheme heterojunction is usually composed of a photocatalyst that produces H 2 , a photocatalyst that produces O 2 , and an electron mediator [117,118]. The main advantage of the Z-scheme heterojunction is that it combines the strongest reduction and oxidation capabilities of the two photocatalysts. Therefore, the Z-scheme heterojunction has been extensively used in water-splitting [119]. Yi et al. [120] reported the visible light-responsive Z-scheme heterojunction photocatalyst CdS/Au/TiO 1.96 C 0.04 . As shown in Figure 3(i), upon exposure to visible light, the photoinduced electrons on the surface of TiO 1.96 C 0.04 are transferred to CdS through Au and recombined with the photoinduced holes on the CdS surface such that the CdS surface retains the photogenerated electrons with higher reducing power; at the same time, the TiO 1.96 C 0.04 surface retains the photogenerated holes with higher oxidizing power. Using this approach, the CdS photocatalytic hydrogen production and TiO 1.96 C 0.04 photocatalytic oxygen production are effectively improved at the same time (Figure 3(j) and (k)). Yang et al. [121] constructed an 0D/1D g-C 3 N 4 /TiO 2 Z-scheme heterostructure containing an interfacial oxygen vacancy layer. The g-C 3 N 4 /TiO 2 photocatalyst shows the best photocurrent density at 1.23 V upon irradiation with light in the visible region (relative to the reversible hydrogen electrode), with a value as high as 0.72 mA/cm 2 , which is 8 times higher than the sample without the interfacial oxygen vacancy layer. EIS also shows that the g-C 3 N 4 /TiO 2 heterostructure photoanode exhibits the lowest charge transfer resistance among all prepared photoanodes. This improved photoelectrochemical performance is attributed to the Z-scheme heterostructure generated by establishing an interfacial oxygen vacancy layer between TiO 2 and g-C 3 N 4 , and this interfacial layer promote the separation and transport of charge carriers. Subsequently, the author further confirmed the formation of this Z-scheme heterostructure through hydroxyl fluorescence capture and characterization and spin polarization DFT calculations.
6 Adsorption of water molecules on the surface of TiO 2 and the formation of hydrogen bonds 6.1 Adsorption of a single water molecule on the surface of TiO 2 In the photocatalytic water-splitting reaction, the adsorption state of water molecules on the surface of TiO 2 will directly affect its catalytic activity. Scholars in the field of theoretical calculations have conducted numerous studies on the interfacial adsorption configuration of water molecules and TiO 2 , but controversies still exist regarding the adsorption morphology and structure of water molecules. The most controversial question is whether water molecules self-dissociate on the surface of TiO 2 [122,123]. This problem is important because the dissociated water molecules will provide TiO 2 an OH group adsorbed on the titanium atom on its surface, and the OH group on the surface of TiO 2 plays an important role in the photocatalytic reaction.
In recent years, the results of these theoretical calculations of the adsorption energy of the single water molecule in the molecular state and dissociated state on the surface of TiO 2 depend strongly on the selection of parameters, such as functionals, the number of layers of the TiO 2 surface model and crystal planes. For example, most of the results obtained from the RPBE functional calculations revealed a lower dissociative adsorption energy of water molecules, while the results of the PW91 and PBE functional calculations produced a lower molecular adsorption energy of water molecules. The number of layers of the TiO 2 surface model will also exert a substantial effect on the results of the theoretical calculations. As shown in Figure  4(a), Kowalski et al. [124] observed a strong odd-even oscillation behavior of the adsorption energy as the number of layers of the TiO 2 surface model changed. A similar oscillation behavior exists even on a clean TiO 2 surface, and the difference in the surface hybridization of Ti-O bonds between odd and even layers is the main explanation for this result [125]. In addition, as shown in Figure  4(b), Zhao [126] used DFT and reported that a single water molecule adsorbed on rutile phase TiO 2 (100) is mainly in the molecular state, while dissociation state adsorption mainly occurred on the surface of (001) and (110), which was also confirmed by a series of subsequent studies [127,128]. Although the dissociation adsorption on the (110) surface is more stable, the reaction energy of the molecular adsorption structure and the dissociation adsorption structure are very similar, with a difference of only 0.089 eV.
In addition to calculating the surface adsorption energy of TiO 2 , thermodynamic and kinetic studies also improve our understanding of the state of the single water molecule on the surface of TiO 2 . Luo et al. [129] used a 3 × 2 primitive cell with 1/6 of the water molecular layer coverage and GGA functional calculation to obtain a water molecular splitting potential barrier of 0.20 eV, while the water molecular splitting potential barrier obtained using the GGA + U method and the same model is only 0.08 eV (Figure 4(c)). These low splitting potential barriers allow water molecules to dissociate on the TiO 2 surface with low coverage.

Adsorption of a single layer of water molecules on the surface of TiO 2
Similarly, in the field of theoretical calculations, researchers have also disputed whether the adsorption of the single layer of water molecules occurs in a molecular state or a dissociated state. Notably, when considering whether the single layer of water is dissociated on the surface of TiO 2 in a 1 × 1 surface primitive cell model, the molecular state of water molecules is unable to be described well. The choice of a surface primitive cell model exceeding 2 × 1 can avoid this problem well. Analogously, the results of these theoretical calculations of the adsorption energies of the single layer of water molecules in the molecular state and dissociation state on the TiO 2 surface also depend on the selection of parameters, such as functionals and the number of layers of the TiO 2 surface model. As shown in Figure 4(d), Harris and Quong [130] used the PW91 functional and the surface primitive surface cell model (2 × 2) of TiO 2 with three layers for the calculation and found that the simultaneous presence of mixed adsorption state of the molecular state and the dissociated state is the most stable configuration. However, as the number of TiO 2 surface model layers continues to increase, the most stable state of the monolayer water molecules on the TiO 2 surface changes from a mixed adsorption state to a molecular adsorption state. As shown in Figure 4(e), Kumar et al. [131,132] used the functional with van der Waals correction to obtain the result that the single layer of water molecules does not dissociate when the number of TiO 2 surface model layers is sufficient. In addition, the authors also considered the effect of the U value on the calculated result. When the U value is sufficiently large, the monolayer of water molecules will exist in the form of the mixed adsorption state. Notably, the selection of the appropriate empirical U value to accurately describe the properties of the TiO 2 surface is itself a very complicated problem. Some research groups have also used HFs [133,134] to study the monolayer of water molecules on the surface of TiO 2 and to solve the problem of an inaccurate description of the electronic molecules of semiconductors by GGA; all of these researchers have obtained a single conclusion that the monolayer of water molecules does not dissociate from the surface of TiO 2 .

Multilayer structure of water molecules on the surface of TiO 2
In the actual photocatalytic water-splitting system, the surface of TiO 2 usually contacts water molecules with higher coverage. At this time, the water molecules easily form a highly ordered hierarchical structure on the surface of TiO 2 , and this structure will follow the increase in the surface distance and gradually weaken. Since the time scale of the classical molecular dynamics simulation that can be calculated is longer [135] and the number of TiO 2 surface model layers is also increased, this review mainly introduces some reports on classical molecular dynamics simulation that strongly depend on the force field. The force fields currently used to simulate the system of TiO 2 and water molecules mainly include the Matsui Akaogi force field suitable for TiO 2 , which is applicable to both the surface and bulk phase of TiO 2 [136]. In the water molecule model, TIP3P and SPC/E are the most widely used force fields [137][138][139]. The Buckingham potential and Lennard-Jones potential are usually used to describe the interaction between water molecules and the TiO 2 surface [140][141][142].
As shown in Figure 4(f), Mamontov et al. [143] performed classical molecular dynamics simulations and found that water molecules were stratified on the surface of rutile phase TiO 2 (110), forming a three-layer structure. Among these layers, the oxygen atoms in the first layer of water molecules exert a strong effect on the Ti atoms on the surface of TiO 2 ; the hydrogen atoms in the second layer of water molecules exert a strong effect on the oxygen atoms on the surface of TiO 2 . Both the first and second layers of water molecules have formed a highly ordered structure and are stably adsorbed on the (110) crystal face of TiO 2 , while the third layer of water molecules is rarely affected by the surface, consistent with the results of the molecular dynamics simulation reported by Kavathekar et al. [136]. Furthermore, the authors also studied the distributed structure of water molecules on other crystal faces of rutile phase TiO 2 and anatase phase TiO 2 . Water molecules will also form a double-layer ordered structure on the other crystal faces (Figure 4(g) and (h)).
On the other hand, the development of the reaction force field provides a new theoretical approach for studying the interfacial characteristics of TiO 2 and water molecules [144,145]. As shown in Figure 4(i), Huang et al. [146] used the reaction force field (ReaxFF) to analyze the structure of water molecules on the surface of TiO 2 and their dissociation and adsorption. The dissociation adsorption of water molecules occurred near the rutile phase TiO 2 (110), (011), anatase phase TiO 2 (001) and TiO 2 -B(100) crystal faces, while only molecular adsorption was observed at the crystal surface of TiO 2 -B(001). As shown in Figure 4(j), water molecules form a three-layered, ordered structure on the rutile TiO 2 (011) crystal face. Futera et al. [147] studied the static and dynamic structural characteristics of the anatase TiO 2 (101) crystal face, rutile TiO 2 (110) crystal face, and the water molecule interface using ReaxFF. The results of the theoretical calculation revealed a clear, layered, and well-organized structure of water molecules in the interface area within the range of 6.5 Å from the TiO 2 surface. The molecules of the first hydration layer adsorbed on the unsaturated surface titanium atoms will spontaneously dissociate, but controversy exists regarding whether H + will completely cover O 2c /O b and OH − will partially cover Ti 5c (Figure 4(k)). Moreover, the expected change in the inherent electric field in the interface was large, and the drop in the electrostatic potential was also large. The interfacial water is severely restricted, and its self-diffusion constant is two orders of magnitude lower than the 2.28 × 10 −9 m 2 /s constant reported for the bulk water (Figure 4(l)). The pivoted motion of adsorbed water molecules is also considerably hindered. The author also calculated that the hydrogen bond lifetime on the interface is also shorter than in the bulk water region. Finally, the Fourier transform infrared (FT-IR) spectra procured from the collective water dipole changes in the interface area revealed a stronger effect of water molecules on the stretching vibration of the anatase (101) crystal plane than on the rutile (110) crystal plane (Figure 4(m)). However, the detection of the presence of the liquid water bond stretching vibration is often limited by the loss of accuracy of the applied reaction potential.

Hydrogen bonds formed by water molecules on the surface of TiO 2
At the interface between TiO 2 and water molecules, the possible hydrogen bonding sites are mainly observed between two water molecules and the H atom in the water molecule and between dissociated H + , dissociated OH − and O atoms on the surface of TiO 2 . The strength of the hydrogen bond formed will also exert a certain effect on the structure of water molecules formed on the surface of TiO 2 [148,149]. Kumar et al. [150] used first-principles molecular dynamics simulations to calculate the various hydrogen bonds formed during the interaction of water molecules with the surface of TiO 2 . Table 1 presents the structural information of the four types of hydrogen bonds with the strongest acting force. Similarly, English [151] performed FT-IR spectroscopy and found that in the wavelength range of 835 to 920 cm −1 , water molecules formed stable hydrogen bonds with the surface of rutile phase TiO 2 , while the oxygen vacancy on the surface of TiO 2 formed weak hydrogen bonds with water molecules and bridged oxygen atoms on the surface. As shown in Figure 4(n), the results of the calculations performed by Mattioli et al. [152] using DFT and first-principles molecular dynamics simulations reveal that the hydrogen bond formed by water molecules on the surface of the anatase phase TiO 2 (101) is stronger than the bond formed on the surface of the rutile phase, and the role of hydrogen bonding in the permutation of atoms on the surface of TiO 2 and the vibration characteristics of intermediate products in the water-splitting process is very important. Additionally, the selection of crystal planes will also exert a certain effect on the stability of hydrogen bonds. This point was also confirmed in a subsequent study by English et al. [153]. The authors simulated the molecular dynamics of hydrogen bonds formed by water molecules and rutile TiO 2 (110), (101), (100) and anatase TiO 2 (101) crystal faces at room temperature. The hydrogen bonds on the anatase phase TiO 2 (101) crystal planes were the most stable, followed by the rutile phase TiO 2 (110) crystal planes. Water molecules adsorbed on these crystal faces potentially increase the number of hydrogen bonds and their stability, particularly on the surface of anatase phase TiO 2 , which may be beneficial to accelerate the activity of TiO 2 for water splitting.
7 Rate-limiting step of the watersplitting reaction on the surface of TiO 2 The photocatalytic water-splitting reaction on the surface of TiO 2 is composed of several different elementary reactions [154][155][156]. In previous studies, the energy barrier to be overcome during the reaction was one of the main factors affecting the reaction [157]. As the OER is the most important process [158], many researchers have conducted many theoretical reports on the OER mechanism in the last 10 years [159,160]. As shown in Figure 5(a), Hu et al. [158] used a first-principles method to study the photocatalytic OER at the water/TiO 2 (110) interface. Theoretical calculations show that all factors affecting the OER rate will vary with the concentration of photogenerated holes ( + C h ) reaching the surface of TiO 2 . Under the experimental conditions, the intrinsic catalytic activity of TiO 2 (110) is not the major factor that alters the increase in the OER rate, as all steps involving photogenerated holes are very slow due to the low + C h reaching the surface of TiO 2 ( Figure 5(b)-(d)). Therefore, the key to improving the efficiency of OER is to increase + C h reaching the surface of TiO 2 as much as possible toward its estimation threshold ( + C h ≈ 10 −4 ). Lang et al. [161] thoroughly studied the mechanism of OER on the TiO 2 (001) crystal plane covered by water. The authors discovered a novel OER pathway on the (001) crystal plane, namely, the straight transfer of protons from molecular water and surface hydroxyl groups, as well as the O-O coupling process. In this pathway, the O-O coupling step exhibits the largest thermodynamic energy and the highest energy barrier, indicating that this process is the rate-limiting step in the entire approach (Figure 5(e)). The authors' discovery provides novel insights into the strong dependence of the water adsorption mode of the anatase phase TiO 2 (001) crystal plane on the coverage rate, and they attempted to explain the strong oxidation activity on the surface of TiO 2 (001) in a representative photocatalytic water-splitting environment. Liu et al. [16] systematically and comparatively studied the mechanism and energetics of water oxidation on the surface of rutile phase TiO 2 (110), anatase phase TiO 2 (101), and brookite phase TiO 2 (210) models using DFT. The results of the theoretical calculation show that the easiest mechanism for water oxidation on these three different crystalline phases is the formation of H 2 O 2 double electrons/protons through surface peroxide O* intermediates.
As shown in Figure 5  not subtle to the local surface structure. In addition, the authors also concluded that the overpotential of the reaction is mainly affected by the position of the top of the VB. In the reaction process, the main rate-limiting step is the formation of hydroxyl radicals (OH*) from the water molecules and the vacancy of the TiO 2 surface. This point was also confirmed in the study by Valdes et al. [163]. As shown in Figure 5(i), Chen et al. [164] performed a detailed kinetic analysis of the rate-limiting steps in the water-splitting reaction on the TiO 2 surface based on the theoretical calculations using DFT. The reaction is accompanied by the continuous electron transfer process of proton coupling, and proton transfer precedes electron transfer. As shown in Figure 5(j), Siahrostami and Vojvodic [165] analyzed the effect of water coverage on the rutile TiO 2 (110) crystal surface on the reaction energy barrier. In the four elementary reactions, the coverage of water had little effect on the formation of (O*) and (HO*), but it significantly reduced the reaction energy barrier for the formation of (HOO*) ( Figure 5(k)).
In summary, the reaction kinetics of water splitting on the TiO 2 surface are mainly determined by the ratelimiting steps of the elementary reaction in the process, and the difficulty of the elementary reaction is determined by the free energy of the reaction. The water coverage on the surface of TiO 2 will also exert a certain effect on the size of the reaction energy barrier.
8 Charge transfer process occurring on the surface of TiO 2 In the photocatalytic reaction process, the effective utilization of the photogenerated electron-hole pair of TiO 2 is an important factor affecting its photochemical performance, and it is the pivotal to improving the efficiency of TiO 2 photocatalytic water-splitting reaction to produce hydrogen. When TiO 2 is excited by incident light, the photogenerated electrons will transition and be injected into the CB of TiO 2 . Then, these electrons either migrate to the surface of TiO 2 through diffusion and are effectively used, are captured by the defect state of the TiO 2 surface, or recombine with the positive charge of the electrolyte or the sensitizer. The aforementioned processes all occur simultaneously. According to experimental [166,167] and theoretical studies [168,169], photogenerated electrons are trapped by defects on the TiO 2 surface. As shown in Figure 6(a), Luo et al. [170] used DFT to show that photogenerated holes on the rutile TiO 2 (110) crystal plane are more likely to be adsorbed by the bridge oxygen atoms and have the characteristics of p orbitals. When interacting with water molecules on the surface of TiO 2 , the holes will hybridize with the highest occupied orbital of water molecules (Figure 6(b)). Furthermore, based on the simultaneous removal of a proton and electron from the crystal face of rutile TiO 2 (110) and the maintenance of the neutrality of the unit cell, Cheng et al. Minot et al. [172] established a model of the photoexcited state at the interface between water molecules and TiO 2 . DFT calculations found that the photogenerated electrons are located on the five-coordinated Ti atom on the TiO 2 surface, while the holes are located between two oxygen atoms adjacent to the Ti atom. Moreover, the concentration of electron-hole pairs on the surface of TiO 2 is related to the size of the unit cell, as a smaller size results in a higher concentration. As shown in Figure 6(d), Nakamura et al. [173] conducted a DFT calculation to study the photoexcited solvated electrons on the interface of water molecules and TiO 2 and found that the distribution of solvated electrons on the interface is mainly affected by the hydrogen bonds on the interface. This electron is mainly distributed around the dangling hydrogen bond, and a type of "wet electron" is formed. Prezhdo et al. [174] also used the DFT method to study "wet electrons." Figure 6(e) shows the process by which "wet electrons" are generated by photogenerated electrons at the interface of water molecules and TiO 2 to the surface state to the bulk phase. Due to the strong coupling between the surface of TiO 2 and water molecules, the electron transfer process is extremely fast, but the adiabatic mechanism and the nonadiabatic mechanism contribute equally to electron transfer. Compared with the vibration of the hydroxyl group, the motion of water molecules exerts a greater effect on the electron transfer dynamics. This "wet electron" represents the lowest energy path for electron transfer at the interface of the metal oxide and aqueous solution. It may play important role in photocatalysis and the photoelectrochemical energy conversion process of TiO 2 and other oxides [175].
Zhao et al. [176] studied the interaction between graphene and anatase phase TiO 2 (110) crystal planes with or without oxygen vacancies using first-principles calculations. The tight but nondestructive contact at the interface promotes the transfer of photoexcited electrons between graphene and TiO 2 . Since the work function is smaller than a TiO 2 substrate without oxygen vacancies, graphene usually depletes electrons. However, the introduction of surface oxygen vacancies significantly reduces the work function of TiO 2 to a smaller value than graphene, which causes electrons to reverse their transfer and accumulate on the graphene sheet ( Figure 6(f)). In particular, as shown in Figure 6(g), the obvious redshift of the optical absorption edge and the markedly reinforced absorption intensity in the visible light region of graphene/anatase phase TiO 2 with or without oxygen vacancies illustrates the escalation mechanism of the photocatalytic efficiency of TiO 2 .
In summary, DFT is mainly used for the simulation and calculation of the charge transfer process on the TiO 2 surface. By establishing the photoinduced excited state model and the photogenerated hole model on the interface of water molecules and TiO 2 , relevant information, such as the electron trapping sites, the hole trapping sites, and the concentration of electron-hole pairs, can be obtained, and the lowest energy path theory using "wet electrons" for electron transfer at the interface of water molecules and TiO 2 may help researchers analyze the process of photocatalysis water-splitting energy conversion. In addition, the loading of some cocatalysts on the surface of TiO 2 , such as graphene, also helps promote the transfer of photogenerated charges on its surface. 9 Application of theoretical calculations to TiO 2 -based commercial photocatalysts for hydrogen production through a water-splitting reaction As the most promising photocatalyst for hydrogen production mediated by water splitting, TiO 2 has already  (Figure 7(b)). The model estimates a series of model parameters and accurately predicts the hydrogen release rate of a given mass of Pt/TiO 2 granules and an extended bed height. Subsequently, based on the model, the hydrogen evolution rate and apparent quantum efficiency of the system are rapidly dominated by the mass transfer rate in the separator. If the gas-liquid separator is optimized or the fluidization flow is increased to ensure that the mass transfer rate is far greater than the reverse reaction rate of H 2 and O 2 catalyzed by Pt, the performance of the system will be maximized (Figure 7(c) and (d)). The author also confirmed this inference through experimental verification of the model, as a small improvement in the separation device increased the hydrogen evolution rate of the system by 350%. As a formidable tool for designing and optimizing photocatalytic fluidized bed systems, this model not only maintains the property of existing TiO 2 -based photocatalysts but also minimize the energy required to produce hydrogen.
As shown in Figure 7(e), Hu et al. [178] used a simple mechanical lapping method to attach organics with electron supply capabilities to the surface of commercial P25.
The results of the theoretical calculation show that the electron-donating organic matter attached to the surface of commercial P25 produces more electrons. These electrons expand the range of light absorbed by commercial P25 via impeding the attraction of holes to photoinduced electrons and improving the separation efficiency of its photogenerated current carriers. The property of the commercial P25 photocatalytic water-splitting reaction for hydrogen production has also been significantly improved, with a continuous increase in the electron-donating capacity of functional groups in organics, from small to large: PeEG < P-MEA < P-EDA. Among these functional groups, P-EDA has the highest yield of H 2 that is 43 times higher than the marketed P25 (Figure 7(f)). This study may result in the development of a new field for the commercialization of TiO 2 designed to enhance the performance of photocatalytic water-splitting reactions to produce hydrogen by attaching various organic substances to the surface of TiO 2 -based photocatalysts.
10 Conclusions and future development directions 10.

Conclusions
As the most promising semiconductor material for photocatalytic water-splitting reactions, TiO 2 -based nanomaterials have received extensive attention from researchers in academia and industry in recent years. This review describes the research progress in theoretical calculations related to TiO 2 -based photocatalytic water splitting, focusing on the analysis of the adsorption state of water molecules with different levels of coverage on the TiO 2 surface, the rate-limiting steps of the decomposition of water molecules on the TiO 2 surface, and the transfer process of photoexcited carriers at the interface between water molecules and TiO 2 . Finally, a brief introduction of the theoretical calculations of TiO 2 -based commercial photocatalysts in water splitting is provided.

Existing problems
The theoretical calculation of TiO 2 -based photocatalytic water splitting is currently limited to only pure TiO 2 or a relatively simple system, such as elemental doping or metal loading of TiO 2 . Some studies involving more complex photocatalytic systems, such as heterojunction systems of TiO 2 and other semiconductors, have mostly focused on the experimental aspect, and relatively few applications in photocatalytic water-splitting reactions for hydrogen production have been reported; thus, studies of theoretical calculations are even less frequently reported.

Future development directions
The microstructure and related properties of the heterojunction interface formed by TiO 2 and other semiconductors require further study to solve these problems. For a photocatalytic system, the system is in a nonequilibrium state throughout the process from the absorption of photons to the excitation of electrons and holes to the various behaviors of electrons and holes after excitation. A very accurate description of the electronic structure and electronic vibration dynamics is needed for theoretical calculations to simulate the nonequilibrium state process occurring at the molecular scale and nanoscale at the atomic level. In other words, the appropriate crosslinking function and density functionals must be studied to more accurately describe long-range electron transport, multielectron excited states, bond breakage, and other situations that create the nonequilibrium state of the system. The development of time-domain (TD) DFT and nonadiabatic molecular dynamics theory [179][180][181][182] describes the time scale for the photoexcited electron transport, energy transport, thermal relaxation and charge recombination processes that occur on the photocatalytic interface system. Simulation calculations provide the conditions [183][184][185]. At the same time, this method also provides unique and universal theoretical support for studies of different nonequilibrium processes that occur in photocatalytic systems. However, the theory is only able to provide theoretical calculations for smaller systems at present, and the simulation time scale is only a few hundred picoseconds. Therefore, the further development of this theory for application to more complex photocatalytic systems will be the focus of future research. On the other hand, the in-depth understanding of the interface properties and the charge behavior at the interface of the heterojunction formed by TiO 2 and other semiconductors through simulations and calculations will provide a solid theoretical basis for the design of the architecture, and TiO 2 -based photocatalytic water splitting represents a new method to improve the efficiency of hydrogen production.