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

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 single-electron approximation [49,50]. These two approximations are the key to solving multiparticle systems.

2.2 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.

2.3 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).

2.4 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).

2.5 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].

3.1 Crystalline structure of TiO₂

Three main crystal types of TiO₂ have been identified in nature, namely, brookite, rutile, and anatase [70]. Both rutile and anatase TiO₂ belong to the tetragonal system, while brookite TiO₂ belongs to the orthorhombic system. The structure of brookite TiO₂ is unstable, and its preparation is relatively difficult [71]; therefore, it is usually used for basic research. Meanwhile, rutile and anatase TiO₂ play a vital role in the field of photocatalysis applications.

Some similarities have been observed between the anatase and rutile crystal structures, namely, TiO₆ 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 TiO₂ octahedron. The TiO₆ octahedron formed by rutile TiO₂ 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 TiO₂, the TiO₆ octahedron formed by anatase TiO₂ 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 TiO₂ [74]. The mass density of anatase TiO₂ is 3.894 g/cm², which is slightly smaller than the value of 4.250 g/cm² for rutile.

According to recent studies, the photocatalytic property of rutile TiO₂ is slightly lower than anatase TiO₂ [75,76,77]. In fact, the energy gap (3.0 eV) of rutile phase TiO₂ 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 TiO₂ is wider. However, the rutile phase TiO₂ 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 TiO₂ than the rutile phase.

3.2 Photocatalytic mechanism of TiO₂

As shown in Figure 1, based on the solid energy band theory, the energy band structure of semiconductor TiO₂ 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 TiO₂, 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 TiO₂, and react with the water molecules adsorbed on the surface of TiO₂ in the corresponding hydrogen evolution reaction (HER) or oxygen evolution reaction (OER) before finally decomposing the water molecules into H₂ and O₂ [83].

Figure 1

Schematic of the TiO₂ photocatalytic water-splitting mechanism, reprinted with permission from ref. [77], Copyright (2008) American Chemical Society.

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 TiO₂ 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 valence-band maximum (VBM) of TiO₂ [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 TiO₂ 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 H₂O molecules on the TiO₂ 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.

4.1 Non-metal doping

In 2001, Asahi et al. [96] prepared the TiO_2−x N _x photocatalyst with nonmetallic element N-doped TiO₂ in a study conducted in Japan, resulting in a new field of visible light-mediated catalysis using nonmetallic-doped TiO₂. In the nitrogen-doped anatase TiO₂, the N 2p and O 2p hybrid orbital energies are very close, which reduces the band gap of N-TiO₂ and broadens the absorption range to visible light. The doping of TiO₂ 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 TiO₂. Concretely, nonmetallic doping usually uses carbon, nitrogen and other atoms that contain more empty p orbitals than oxygen to replace the oxygen atoms in TiO₂, and thus the VB of TiO₂ 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 TiO₂ 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 TiO₂ 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 TiO₂ [101].

Figure 2

(a) Electron transition energy for the configurations of pure anatase and anatase with various P doping levels, reprinted with permission from ref. [98], Copyright (2011) American Chemical Society. Band structures of (b) TiO₂, and (c) Ti_0.98Mo_0.2O₂, reprinted with permission from ref. [103], Copyright (2013) Elsevier. XPS spectra of (d) Pt 4f and (e) N 1s levels in Pt-, N-, and Pt, N-TiO₂, reprinted with permission from ref. [108], Copyright (2014) Elsevier. (f) The optical absorption spectra of undoped, N-, Co-doped, Co-1N-, Co-2N, and Co-3N co-doped TiO₂, reprinted with permission from ref. [40], Copyright (2017) Elsevier.

4.2 Metal doping

Metal ions are effective electron acceptors. When doped with metal ions in TiO₂ crystals, the trapping of electrons in TiO₂ by the metal ions prolongs the recombination time of photogenerated electrons and holes in TiO₂ to improve the photocatalytic efficiency of TiO₂. 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 TiO₂ will decrease [104]. Lian et al. [105] synthesized Mo-doped TiO₂ nanoparticles using the sol–gel method and observed that Mo⁶⁺ ions occupy the lattice position of anatase phase TiO₂ and reduce the band gap from 3.19 to 3.05 eV. The doping of Mo also increases the light absorption of TiO₂ in the ultraviolet and visible range, and the increase in ultraviolet light absorption is basically related to the charge transfer process of O²⁻ (2p) and Ti⁴⁺ (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 TiO₂. The doping of V³⁺, Mn³⁺, Cr³⁺, and Fe³⁺ ions into the anatase phase TiO₂ shortens the band gap of the doped system, but V³⁺ and Fe³⁺ ions basically have no effect on the band gap size of the rutile phase TiO₂. In addition, the doped metal ions extend the range of light absorbed to the visible light region, ensuring that TiO₂ 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.

4.3 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 TiO₂ 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 TiO₂ by 1 eV [108].

Kim et al. [109] synthesized Pt/N co-doped TiO₂ using a wet chemical method. The specific surface area of Pt/N co-doped TiO₂ (178 m²/g) is larger than undoped TiO₂ (110 m²/g). A time-resolved diffuse reflectance spectroscopy (TDR) study of Pt/N-TiO₂, revealed a synergistic effect of co-doping. In addition, Pt ions only exist in the form of Pt⁴⁺ in Pt/N-TiO₂. 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 TiO₂ accelerates the mobility of current carriers, thereby improving the photocatalytic activity. Sun et al. [110] used a sol–gel method to prepare TiO₂ co-doped with Ce and N and found that Ce doping effectively inhibits the transformation of TiO₂ from the anatase phase to the rutile phase. Moreover, He et al. [40] studied the properties of Co–N co-doped TiO₂ 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)).

5.1 Metal/TiO₂

For the precious metal/TiO₂ prepared using the sol–gel method, the precious metal elements relatively easily enter the TiO₂ lattice to replace Ti ions. The appearance of this heterojunction extends the absorption band edge of TiO₂ 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 TiO₂. 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 TiO₂(101) surface and the mechanism of enhanced photocatalytic activity. As a result, the synthesized Ru/TiO₂ and Au/TiO₂ 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/TiO₂ exhibits a faster HER rate than Ru/TiO₂, 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.

Figure 3

(a) Density of states and difference in the charge density of the configuration model of anatase TiO₂, Ru/TiO₂, and Au/TiO₂, respectively. (b) Photocatalytic H₂ evolution reactions mediated by various photocatalysts. (c) Schematic of the mechanism of electron-traps for photocatalytic overall water splitting. (d) Schematic of the mechanism of intensification of SPR, reprinted with permission from ref. [111], Copyright (2020) Royal Society of Chemistry. Calculated electronic structure and PDOS of (e) TiO₂ and (f) AgBiS₂, and (g) DRS spectra of TiO₂ and AgBiS₂–TiO₂ composites. (h) The H₂ production rate of the solution at varying weight percentages of AgBiS₂–TiO₂ catalysts, reprinted with permission from ref. [37], Copyright (2019) Elsevier. (i) CdS/Au/TiO_1.96C_0.04 was designed to mimic the natural photosynthesis system. (j) UV-vis absorbance spectrum of the CdS/Au/TiO_1.96C_0.04 powder and the effect of the light wavelength on H₂ evolution via water splitting on Pt/CdS/Au/TiO_1.96C_0.04. (k) H₂ production via photocatalytic water-splitting, reprinted with permission from ref. [120], Copyright (2011) American Chemical Society.

On the other hand, as a promising photocatalytic material for overall water splitting, TiO₂ 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 TiO₂ photocatalyst loaded with the non-noble metal Ni in the HER. Compared with the (001) crystal plane of TiO₂, Ni _n clusters more easily aggregate on the (101) crystal plane of TiO₂, and the structural difference between the two crystal planes determines this characteristic. The Ni _n clusters adsorbed on the surface of TiO₂ facilitate the separation of photogenerated current carriers and introduce the interstitial state into the forbidden band of TiO₂, 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 TiO₂. The active site of HER in the Ni _n /TiO₂ 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 TiO₂ and provide a reasonable explanation for the experimental improvement in the HER efficiency of TiO₂ after the introduction of Ni cocatalyst.

5.2 Semiconductor/TiO₂

6.1 Adsorption of a single water molecule on the surface of TiO₂

In the photocatalytic water-splitting reaction, the adsorption state of water molecules on the surface of TiO₂ 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 TiO₂, 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 TiO₂ [122,123]. This problem is important because the dissociated water molecules will provide TiO₂ an OH group adsorbed on the titanium atom on its surface, and the OH group on the surface of TiO₂ 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 TiO₂ depend strongly on the selection of parameters, such as functionals, the number of layers of the TiO₂ 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 TiO₂ 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 TiO₂ surface model changed. A similar oscillation behavior exists even on a clean TiO₂ 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 TiO₂(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.

Figure 4

(a) Convergence of the hydrogen adsorption energy E _ad ^H with slab thickness at full monolayer coverage, reprinted with permission from ref. [124], Copyright (2009) American Physical Society. (b) The comparison of decomposition reaction pathways of water on the low-index rutile TiO₂ surfaces, reprinted with permission from ref. [126], Copyright (2014) American Chemical Society. (c) Adsorption and dissociation of H₂O molecule at ground state calculated with GGA and GGA + U methods, reprinted with permission from ref. [129], Copyright (2013) American Chemical Society. (d) Plan view of the three 1 ML adsorption modes: molecular, dissociative, and “mixed”, reprinted with permission from ref. [130], Copyright (2004) American Physical Society. (e) Difference in the adsorption energy between associated and partially dissociated configurations using various dispersion and DFT + U corrections with varying slab thicknesses, reprinted with permission from ref. [131], Copyright (2013) American Chemical Society. (f) MD snapshots of rutile hydrated surfaces, reprinted with permission from ref. [143], Copyright (2007) American Chemical Society. (g) Representative configurations of various rutile-water interfaces: (110), (100), (101), and (001), (h) Representative configurations of various anatase–water interfaces: (101) and (001), reprinted with permission from ref. [136], Copyright (2011) Taylor & Francis. (i) Partial final configurations of the five studied TiO₂/H₂O systems. (j) Cartoon illustration of the hydrogen bond network on reactive TiO₂ surfaces, reprinted with permission from ref. [146], Copyright (2014) American Chemical Society. (k) Dissociative adsorption of water on (a) anatase (101) and (b) rutile (110) surfaces, (l) on the logarithmic scale, where the transition from ballistic to the random-walk regime is clearly visible, and (m) IR spectrum of water calculated as a power spectrum of the system-collective-dipole autocorrelation function, reprinted with permission from ref. [147], Copyright (2017) American Chemical Society. (n) The structure and the difference density map of a pair of water molecules adsorbed on the (101) anatase surface are presented in parts, reprinted with permission from ref. [152], Copyright (2008) American Chemical Society.

In addition to calculating the surface adsorption energy of TiO₂, thermodynamic and kinetic studies also improve our understanding of the state of the single water molecule on the surface of TiO₂. 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 TiO₂ surface with low coverage.

6.2 Adsorption of a single layer of water molecules on the surface of TiO₂

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 TiO₂ 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 TiO₂ surface also depend on the selection of parameters, such as functionals and the number of layers of the TiO₂ 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 TiO₂ 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 TiO₂ surface model layers continues to increase, the most stable state of the monolayer water molecules on the TiO₂ 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 TiO₂ 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 TiO₂ 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 TiO₂ 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 TiO₂.

6.3 Multilayer structure of water molecules on the surface of TiO₂

In the actual photocatalytic water-splitting system, the surface of TiO₂ usually contacts water molecules with higher coverage. At this time, the water molecules easily form a highly ordered hierarchical structure on the surface of TiO₂, 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 TiO₂ 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 TiO₂ and water molecules mainly include the Matsui Akaogi force field suitable for TiO₂, which is applicable to both the surface and bulk phase of TiO₂ [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 TiO₂ 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 TiO₂(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 TiO₂; the hydrogen atoms in the second layer of water molecules exert a strong effect on the oxygen atoms on the surface of TiO₂. 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 TiO₂, 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 TiO₂ and anatase phase TiO₂. 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 TiO₂ 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 TiO₂ and their dissociation and adsorption. The dissociation adsorption of water molecules occurred near the rutile phase TiO₂(110), (011), anatase phase TiO₂(001) and TiO₂-B(100) crystal faces, while only molecular adsorption was observed at the crystal surface of TiO₂-B(001). As shown in Figure 4(j), water molecules form a three-layered, ordered structure on the rutile TiO₂(011) crystal face. Futera et al. [147] studied the static and dynamic structural characteristics of the anatase TiO₂(101) crystal face, rutile TiO₂(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 TiO₂ 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 m²/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.

6.4 Hydrogen bonds formed by water molecules on the surface of TiO₂

At the interface between TiO₂ 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 TiO₂. The strength of the hydrogen bond formed will also exert a certain effect on the structure of water molecules formed on the surface of TiO₂ [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 TiO₂. 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 TiO₂, while the oxygen vacancy on the surface of TiO₂ 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 TiO₂(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 TiO₂ 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 TiO₂(110), (101), (100) and anatase TiO₂(101) crystal faces at room temperature. The hydrogen bonds on the anatase phase TiO₂(101) crystal planes were the most stable, followed by the rutile phase TiO₂(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 TiO₂, which may be beneficial to accelerate the activity of TiO₂ for water splitting.

10.1 Conclusions

As the most promising semiconductor material for photocatalytic water-splitting reactions, TiO₂-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 TiO₂-based photocatalytic water splitting, focusing on the analysis of the adsorption state of water molecules with different levels of coverage on the TiO₂ surface, the rate-limiting steps of the decomposition of water molecules on the TiO₂ surface, and the transfer process of photoexcited carriers at the interface between water molecules and TiO₂. Finally, a brief introduction of the theoretical calculations of TiO₂-based commercial photocatalysts in water splitting is provided.

10.2 Existing problems

The theoretical calculation of TiO₂-based photocatalytic water splitting is currently limited to only pure TiO₂ or a relatively simple system, such as elemental doping or metal loading of TiO₂. Some studies involving more complex photocatalytic systems, such as heterojunction systems of TiO₂ 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.

10.3 Future development directions

The microstructure and related properties of the heterojunction interface formed by TiO₂ 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 TiO₂ and other semiconductors through simulations and calculations will provide a solid theoretical basis for the design of the architecture, and TiO₂-based photocatalytic water splitting represents a new method to improve the efficiency of hydrogen production.