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. 2020 Jun 16;5(25):15344–15352. doi: 10.1021/acsomega.0c01438

Machine Learning Band Gaps of Doped-TiO2 Photocatalysts from Structural and Morphological Parameters

Yun Zhang 1,*, Xiaojie Xu 1,*
PMCID: PMC7331044  PMID: 32637808

Abstract

graphic file with name ao0c01438_0005.jpg

Titanium dioxide (TiO2) photocatalysts in the form of thin films are of great interest due to their tunable optical band gaps, Eg’s, which are promising candidates for applications of visible-light photocatalytic activities. Previous studies have shown that processing conditions, dopant types and concentrations, and different combinations of the two have great impacts on structural, microscopic, and optical properties of TiO2 thin films. The lattice parameters and surface area are strongly correlated with Eg values, which are conventionally simulated and studied through first-principle models, but these models require significant computational resources, particularly in complex situations involving codoping and various surface areas. In this study, we develop the Gaussian process regression model for predictions of anatase TiO2 photocatalysts’ energy band gaps based on the lattice parameters and surface area. We explore 60 doped-TiO2 anatase photocatalysts with Eg’s between 2.280 and 3.250 eV. Our model demonstrates a high correlation coefficient of 99.99% between predicted Eg’s and their experimental values and high prediction accuracy as reflected through the prediction root-mean-square error and mean absolute error being 0.0012 and 0.0010% of the average experimental Eg, respectively. This modeling method is simple and straightforward and does not require a lot of parameters, which are advantages for applications and computations.

1. Introduction

Titanium dioxide, TiO2, shows great promises in several environmental applications due to its distinct properties over other materials, such as the nontoxicity, low cost, ease of preparation, water insolubility, superior acid resistance, and superhydrophilicity.1 Examples of application areas include the air purification, water treatment, renewable energy processes, solar cells, and conversion of CO2 to hydrocarbons.26 Among TiO2 polymorphs, anatase TiO2 is preferred over brookite and rutile because it has a higher surface energy of its {001} facets and better photocatalytic activities and is more stable than the other two forms. However, anatase TiO2 has a relatively wide band gap (∼3.20 eV), which only allows the material to absorb UV light. As UV light only accounts for merely 5% of solar photons, the large band gap of TiO2 limits the quantum yield in light-to-energy conversion.710

One effective way to modify the band gap of anatase TiO2 is chemical doping with foreign elements. Different elements, metals and nonmetals, affect the band gap in different ways. Metal ions, such as Zr, Cr, and W, are reported to inhibit the anatase-to-rutile phase transformation.11,12 Transition metals, such as Cu, and rare-earth metals, such as La, lead to the lattice deformation and the formation of oxygen vacancies, resulting in an impurity state in the TiO2 band gap, which improves the absorption of visible light by narrowing the band gap.13,14 Nonmetal doping, such as the nitrogen incorporation into the TiO2 lattice or on its surface, has been reported to benefit the improvement of photoefficiency under UV/visible light.1 Both single doping and codoping methods have been applied to the TiO2 photocatalyst fabrication by incorporating various elements into the crystal structure.10,1324 The addition of foreign elements results in lattice distortions and changes in the Eg due to electronegativities, ionic radius differences, and introductions of impurity states.25 In addition to chemical doping, various preparation methods of TiO2 photocatalysts can influence the band gap narrowing differently. Typical fabrication methods include the coprecipitation, sol–gel process, spray pyrolysis, hydrothermal process, low-temperature solvothermal method, and plasma treatment.13,18,20,21 Processing parameters, including but not limited to the precursor materials, substrate temperature, deposition rate, and annealing temperature, affect the crystal structure and microstructure significantly. As a result, lattice parameters and the surface area are changed upon different combinations of synthesis steps.1,2529 Previous research has demonstrated that the photocatalytic activity of TiO2 strongly depends on its phase structure, crystallinity, and morphology.30,31 Among various phases of TiO2, anatase is reported to have a better photocatalytic activity than the other two polymorphs.32 A good crystallinity is required to achieve the formation of an optimal amount of electron traps, which affects the photocatalytic efficiency. Lattice deformation caused by nonequilibrium crystal growth and chemical doping affects the electronic structure by modifying orbital hybridization and introducing additionally available electrons for conduction.33,34 Both the crystallinity and lattice deformation can be characterized by lattice parameters. Furthermore, other crystal defects, such as residual strain, impurities, dislocation densities, and defect energy, have significant influences on band gap structures and are correlated with surface morphology, which can be characterized by the surface area.35,36 High surface areas also promote quantum confinement effects in the semiconductor space charge and surface reaction, which greatly increase the photocatalytic efficiency.37 For example, N-doped TiO2 obtained by the reduction-nitridation method via the nonthermal plasma treatment is more favorable than the simple nitridation treatment, as the former promotes Ns doping and narrows the band gap more efficiently.38 Further, the recombination of photogenerated electron-hole pairs limits the photocatalytic activity. Some research has been carried out to reduce the recombination rate of the photoelectron–hole pairs and increase the interfacial charge-transfer efficiency. The surface microstructure, mainly characterized by the surface area, shows additional influences on photocatalyst quality and optical performance. The surface area is correlated with the residual strain, dislocation density, crystallinity, defect energy, impurities, and other structural defects and is shown to contribute to the band gap of TiO2 structures.35,36 Hence, with various synthesis methods and dopant selection, combination possibilities of TiO2 with the tunable Eg are enormous. It is, therefore, of great importance to investigate correlations among the tunability of the Eg, lattice parameters, and the surface area. Qualitative analysis on the effect of dopant types and levels on the Eg of TiO2 photocatalysts has been conducted through experiments.10,1324 Quantitative analysis through thermodynamics models and first-principle models has been utilized to aid the understanding of the optical performance of these materials and facilitate the tuning of doped-TiO2Eg.39,40 However, these models require a significant amount of data inputs, such as variables for equations of state and orbital configurations, which can only be obtained by extensive measurements. The requirement of computational power also increases significantly when it comes to the codoping situation.

In this work, the Gaussian process regression (GPR) model is developed to elucidate the statistical relationship among the lattice parameters, a (Å) and c (Å), surface area, and energy band gap for doped-TiO2 anatase photocatalysts. Among the three descriptors, lattice parameters are structural parameters as direct representatives of the phase structure and crystallinity, and the surface area is the morphological parameter. Empirical studies show that crystal defects introduced by doping, such as foreign ions at substitutional or interstitial lattice positions, can shift the band gap toward the visible-light region. Depending on ionic radii, electronegativities, and valence, however, specific types and extents of crystal defects are difficult to estimate. Experimentally, crystal defects require significant amounts of analytical work to characterize, which eventually may be used as inputs into further theoretical work. First-principle simulations calculate the probability of each type of defects and their effects on the band gap tuning based on known atomic interactions, but these methods are known to be susceptible of underestimations of Eg values, particularly when the TiO2 lattices are doped with transition metals.41,42 Besides, effects of the morphological parameter, the surface area, on Eg values are difficult to incorporate into first-principle simulations. Although a high surface area is generally preferred, it is hard to quantify the required surface area while also considering lattice deformation in a practical application. Our GPR model, however, avoids depending on quantum mechanics theories for calculations, which may be susceptible to over- or underestimations due to unknown atomic interactions. In this method, the known experimental lattice parameters are used as macroscopic descriptors to find the relationship with experimentally measured Eg values. The model generalizes well in the presence of a few descriptive features, where intelligent algorithms are able to learn and recognize the patterns. This modeling approach demonstrates a high degree of accuracy and stability, contributing to efficient and low-cost estimations of the energy band gap of anatase TiO2 and understandings of which are based on the lattice parameters and surface area. As one of the computational intelligence techniques, the GPR model has already been utilized in other materials systems to predict significant physical parameters in different fields of applications.4345 This model can serve as a guideline for searching for anatase TiO2 with tunable Eg when a specific range of band gaps is required for a practical application. It can also be used as part of machine learning to aid understandings of the effects of crystal structures and morphology on the optical performance of TiO2 photocatalysts.

The remaining of this work is organized as follows. Section 2 describes the data. Section 3 presents and discusses results, and Section 4 concludes. Section 5 contains details of the GPR model.

2. Description of Data Set

The experimental data used, shown in Table 1 (columns 1–5), are obtained from the literature.10,1324 The data set covers a wide range of anatase TiO2 that are prepared through different synthesis methods and doped with various elements. A total of 60 TiO2 photocatalysts with the energy band gap, Eg, ranging from 2.280 to 3.250 eV are explored. The lattice parameters, a (Å) and c (Å), and measured surface area are used as descriptors. Eg values are calculated using the Tauc relationship13,22 after acquiring the transmittance data by the UV–vis spectrometer in each reference in Table 1. Data visualization in Figure 1 reveals nonlinear relationships, which are modeled through the GPR.

Table 1. Experimental Data and Predictionsa.

sample a (Å) c (Å) surface area (m2/g) Eg (eV) prediction reference
pure TiO2 3.7650 9.4860 43.00 3.122 3.12198 (21)
1.0% Cu–3.5% In–TiO2 3.7760 9.4620 85.00 3.087 3.08695 (21)
5 mol % Cu-doped TiO2 (700°C) 3.7700 6.0190 110.70 2.430 2.43004 (15)
5 mol % Cu-doped TiO2 (600°C) 3.7960 9.4310 150.90 2.510 2.51004 (15)
5 mol % Cu, 15 mol % Zr co-doped TiO2 (700 °C) 3.8340 11.4420 127.20 2.280 2.28005 (15)
5 mol % Cu, 15 mol % Zr co-doped TiO2 (600 °C) 3.8080 10.0000 156.60 2.320 2.32005 (15)
undoped-TiO2 3.7900 9.6300 146.07 2.950 2.94999 (20)
S-doped TiO2 (K2S2O8/Ti = 0.25) 3.7800 9.5900 183.45 2.850 2.84999 (20)
S-doped TiO2 (K2S2O8/Ti = 0.5) 3.7900 9.5800 216.22 2.830 2.83001 (20)
pure TiO2 3.7740 9.4480 44.80 3.230 3.22994 (10)
S0.05/TiO2 3.7750 9.5480 48.60 2.950 2.95000 (10)
S0.05V0.001/TiO2 3.7770 9.4610 64.40 2.810 2.81002 (10)
S0.05Fe0.001/TiO2 3.7820 9.6670 66.60 2.880 2.88001 (10)
S0.05Zn0.001/TiO2 3.7930 9.4140 61.80 3.230 3.22997 (10)
undoped-TiO2 3.7848 9.4826 216.00 3.110 3.10996 (19)
V-doped TiO2 3.7882 9.4949 203.00 2.910 2.91000 (19)
N-doped TiO2 3.7917 9.4868 181.00 2.920 2.91999 (19)
V,N co-doped TiO2 3.7996 9.4976 172.00 2.760 2.76000 (19)
pure TiO2 3.7860 9.5260 80.37 3.180 3.17999 (13)
N–TiO2 3.7850 9.4710 96.49 2.900 2.90003 (13)
La–TiO2 3.7800 9.5180 89.42 3.020 3.02000 (13)
N/La–TiO2 3.7860 9.4780 116.25 2.840 2.84001 (13)
undoped-TiO2 3.7760 9.4860 51.30 2.910 2.91001 (18)
Ce-doped TiO2, “C0.03T1 3.7740 9.4540 63.52 2.720 2.72002 (18)
Ce-doped TiO2, “C0.01T1 3.7720 9.4420 67.43 2.670 2.67002 (18)
Ce-doped TiO2, “C0.05T1 3.7710 9.4500 60.14 2.700 2.70002 (18)
Ce,Si co-doped TiO2, “C0.01T1S0.05 3.7700 9.4570 102.41 2.650 2.65006 (18)
Ce,Si co-doped TiO2, “C0.01T1S0.5 3.7600 9.4620 164.48 2.510 2.51005 (18)
Ce,Si co-doped TiO2, “C0.01T1S1 3.7600 9.4810 168.54 2.710 2.70997 (18)
Ce-doped TiO2, “C0.005T1 3.7580 9.4420 53.57 2.820 2.82001 (18)
undoped-TiO2 (475 °C) 3.8220 10.6100 71.40 3.210 3.20997 (14)
Ce(2%)Co(4%)–TiO2 (600 °C) 3.8060 11.1800 43.60 3.200 3.19997 (14)
Ce(2%)Co(4%)–TiO2 (700 °C) 3.8130 10.0000 39.80 3.180 3.17997 (14)
undoped-TiO2 3.7760 9.3410 85.93 3.220 3.21996 (24)
1% Nb-doped TiO2 3.7860 9.3470 150.61 3.250 3.24996 (24)
pure TiO2 3.7760 9.4860 67.70 2.980 2.97994 (17)
Ag-doped TiO2(400 °C) 3.7822 9.5023 106.37 2.450 2.45006 (17)
Ag-doped TiO2 (500 °C) 3.7770 9.5010 78.40 2.510 2.51009 (17)
Ag-doped TiO2 (600 °C) 3.7760 9.4860 29.33 2.540 2.54004 (17)
Ag-doped TiO2 (700 °C) 3.7822 9.5023 1.93 2.590 2.59002 (17)
undoped-TiO2 3.7945 9.5079 37.90 3.180 3.17997 (16)
1 wt % Mn-doped TiO2 3.7945 9.4860 52.59 3.100 3.10000 (16)
5 wt % Mn-doped TiO2 3.7956 9.4993 80.77 2.700 2.70002 (16)
10 wt % Mn-doped TiO2 3.7922 9.4546 95.19 2.700 2.70003 (16)
20 wt % Mn-doped TiO2 (TMA400) 3.7762 9.4681 212.71 2.450 2.45003 (16)
20 wt % Mn-doped TiO2 (TMB400) 3.7808 9.4720 203.00 2.500 2.50004 (16)
20 wt % Mn-doped TiO2 (TMN400) 3.8014 9.4105 205.36 2.550 2.55003 (16)
pure TiO2 3.7835 9.4907 86.11 3.180 3.18000 (22)
0.05 wt % W-doped TiO2 3.7858 9.4862 88.14 3.190 3.18997 (22)
0.1 wt % W-doped TiO2 3.7858 9.4817 86.08 3.120 3.12004 (22)
0.5 wt % W-doped TiO2 3.7813 9.4773 91.71 3.180 3.17996 (22)
1 wt % W-doped TiO2 3.7835 9.4817 88.18 3.190 3.18998 (22)
pure TiO2 3.7850 9.5021 59.00 3.220 3.21998 (23)
0.1 mol % Sn-doped TiO2 3.7864 9.4958 66.00 3.200 3.19999 (23)
0.5 mol % Sn-doped TiO2 3.7863 9.4927 75.00 3.180 3.18000 (23)
1 mol % Sn-doped TiO2 3.7866 9.4915 87.00 3.210 3.20997 (23)
3 mol % Sn-doped TiO2 3.7878 9.4884 106.00 3.190 3.18996 (23)
2.0% In–TiO2 3.7630 9.4740 62.00 3.186 3.18597 (21)
3.5% In–TiO2 3.7680 9.4620 98.00 3.195 3.19494 (21)
5.0% In–TiO2 3.7920 9.4360 123.00 3.217 3.21696 (21)
minimum 3.7580 6.0190 1.93 2.280 2.28005  
mean 3.7843 9.5220 101.17 2.910 2.90962  
median 3.7842 9.4860 86.56 2.935 2.93499  
maximum 3.8340 11.4420 216.22 3.250 3.24996  
standard deviation 0.0141 0.5906 53.67 0.285 0.28542  
correlation coefficient with Eg 0.76% 10.59% –34.18%   99.99%  
a

Eg (eV)” and “prediction” represent the experimental and GPR predicted band gaps, respectively, which are visualized in Figure 3.

Figure 1.

Figure 1

Data visualization.

3. Results and Discussion

3.1. Computational Methodology

MATLAB is utilized for computations and simulations in this work. The relationship between model performance and training data sizes is investigated in Figure 2, which shows the benefit of training the GPR using all observations. The stability of the GPR approach is confirmed by bootstrap analysis in Section 3.3.

Figure 2.

Figure 2

Model performance and training data sizes. When the training data set size is between 30 and 57, 2000 random subsamples are drawn without replacements from the whole sample for model training. When the training data set size is 58, 59, or 60, 60C58, 60C59, or 60C60 subsamples are drawn without replacements from the whole sample based on exhaustive sampling for model training. Each trained model based on a certain subsample is used to score the whole sample and obtain the associated model performance. The GPR here uses the exponential kernel and constant basis function, with standardized predictors. Given a model performance measure, box plots show the median, 25th percentile, and 75th percentile. The whiskers extend to the most extreme values (i.e., ±2.7 standard deviation coverage) not considered as outliers, and the outliers are plotted using the “+” symbol.

3.2. Prediction Accuracy

Metal ions, such as Sn4+, Zr4+, and Cu2+, are incorporated into the anatase structure by the substitution of Ti4+ due to similar ionic radii, while Ag+ is favorably stabilized at an interstitial site. Nonmetal ions, such as N and S, are incorporated into the lattice and coexisted at both substitutional and interstitial sites. Changes in lattice parameters depend on ionic radii, electronegativities, valence, and incorporation mechanisms. On one hand, these crystal defects allow additional electronic levels to be created in the band structure, which effectively narrow the band gap, shift the absorption edge to the visible region, and enhance photocatalytic efficiency. On the other hand, excess additions of some dopants, such as N, may lead to the formation of the oxygen vacancy and Ti3+ due to charge imbalance, which increase the charge carrier recombination and hinder conversion efficiency. Hence, codoping is carried out to maintain the charge balance through charge compensation, add new electronic levels, suppress the recombination of charge carriers, and further increase photocatalytic efficiency. Besides, dopants also have an influence on the stability of the anatase phase and surface area. For example, dopants, such as Zr, Ag, W, Ce, and Nb, are found to inhibit the anatase-to-rutile phase transformation, while Mn, Cu, and Co are found to promote it. During the TiO2 synthesis, the high-temperature calcination is usually carried out to achieve high crystallinity, which, however, might lead to extensive grain coarsening and surface area reductions. Additions of dopants that inhibit the phase transformation to structure help stabilize the anatase phase at elevated processing temperature, hinder grain growth, decrease crystallite sizes, and thus increase the surface area. A high surface area indicates increased structural defects on the surface, such as unsaturated surface cations and surface hydroxyl groups, which favor the simultaneous absorption of organic molecules and enhance the photocatalytic efficiency. It should be pointed out that effects of modified lattices and surface areas on band gap tuning and photocatalytic properties are synergistic. There is no linear or monotonic relationship between lattice parameters, surface areas, and band gaps. In this work, the developed model is able to learn and capture the synergistic effects of the structure and morphology on Eg values.

The final GPR model is detailed in Figure 3, which shows a good alignment between predicted and experimental data. The correlation coefficient (CC), root-mean-square error (RMSE), and mean absolute error (MAE) are 99.99%, 0.00003442 (0.0012% of the average experimental Eg), and 0.00002872 (0.0010% of the average experimental Eg), respectively, representing good prediction performance.

Figure 3.

Figure 3

Experimental vs predicted Eg. The final GPR model is built using the whole sample with the exponential kernel, constant basis function, and standardized predictors. It has a log-likelihood of −3.5784, β̂ of 2.8382, σ̂ of 0.0029, σ̂l of 0.2896, and σ̂f of 0.2939. Detailed numerical predictions are listed in Table 1 (column 6).

3.3. Prediction Stability

Given the relatively small sample size (see Table 1) used, the prediction stability of the GPR is assessed through bootstrap analysis in Figure 4, which shows that the modeling approach maintains high CCs, low RMSEs, and low MAEs over the bootstrap samples. This result suggests that the GPR might be generalized for Eg modeling of anatase TiO2 based on larger samples.

Figure 4.

Figure 4

Bootstrap analysis of GPR prediction stability. Five thousand bootstrap samples are drawn with replacements from the whole sample. Each bootstrap sample is used to train the GPR based on the exponential kernel, constant basis function, and standardized predictors and obtain the associate model performance. The histograms show distributions of CC, RMSE, and MAE over the 5000 bootstrap samples, whose averages are 99.99%, 0.00002320, and 0.00001782, respectively.

3.4. Prediction Sensitivity

Table 2 shows that the exponential kernel is generally the optimal choice among kernels considered. With the exponential kernel, prediction results are not sensitive to choices of basis functions except for the case of the empty basis function. Given the exponential kernel, the constant basis function is selected as the final specification for its simplicity, which usually is a benefit to model generalization, and its slight better performance as compared to more complicated basis functions, such as linear and pure quadratic.

Table 2. GPR Prediction Sensitivities to Choices of Kernels and Basis Functionsa.

kernel basis function CC (%) RMSE RMSE/sample mean (%) MAE MAE/sample mean (%)
exponential constant 99.99 0.00003442 0.0012 0.00002872 0.0010
exponential empty 82.38 0.17850737 6.1343 0.15125528 5.1978
exponential linear 99.99 0.00003472 0.0012 0.00002926 0.0010
exponential pure quadratic 99.99 0.00003554 0.0012 0.00002975 0.0010
squared exponential constant 71.48 0.21057940 7.2364 0.18170821 6.2443
matern 5/2 constant 99.99 0.00007681 0.0026 0.00005651 0.0019
rational quadratic constant 99.98 0.00630021 0.2165 0.00459014 0.1577
a

The final GPR model is based on the exponential kernel and constant basis function, with standardized predictors.

4. Conclusions

In this study, we develop the Gaussian process regression (GPR) model for predictions of anatase TiO2 photocatalysts’ energy band gaps, Eg’s, based on the lattice parameters and surface area. Our model demonstrates a high correlation coefficient of 99.99% between predicted Eg’s and their experimental values. In addition, the model shows high prediction accuracy as reflected through the prediction root-mean-square error and mean absolute error being 0.0012 and 0.0010% of the average experimental Eg, respectively. Finally, model performance is illustrated to be stable. These results suggest that the GPR should be useful to model and understand relationships between structural and morphological parameters and Eg’s. This modeling method is simple and straightforward and does not require a lot of parameters, which are advantages for applications and computations. The model can be applied to a wide range of undoped and doped-TiO2 made by various synthesis methods and utilized to facilitate design and understandings of multidoped TiO2 photocatalysts with tunable Eg’s.

5. Proposed Methodology

5.1. Brief Description of Gaussian Process Regression

GPRs are nonparametric kernel-based probabilistic models. Consider a training data set, {(xi, yi); i = 1, 2, ..., n} where Inline graphic and Inline graphic, from an unknown distribution. A trained GPR predicts values of the response variable ynew given an input matrix xnew.

Recall a linear regression model, y = xTβ + ε, where ε ∼ N(0, σ2). A GPR aims at explaining y by introducing latent variables, l(xi) where i = 1, 2, ..., n, from a Gaussian process such that the joint distribution of l(xi)s is Gaussian and explicit basis functions, b. The covariance function of l(xi) captures the smoothness of y, and basis functions project x into a feature space of dimension p.

A GP is defined by the mean and covariance. Let m(x) = E(l(x)) be the mean function and k(x, x′) = Cov [l(x), l(x′)] the covariance function and consider now the GPR model, y = b(x)Tβ + l(x), where l(x) ∼ GP(0, k(x, x′)) and Inline graphic. k(x, x′) is often parameterized by the hyperparameter, θ, and thus might be written as k(x, x′|θ). In general, different algorithms estimate β, σ2, and θ for model training and would allow specifications of b and k as well as initial values for parameters.

The current study explores four kernel functions, namely exponential, squared exponential, matern 5/2, and rational quadratic, whose specifications are listed in eqs 1eqs 1, respectively, where σl is the characteristic length scale defining how far apart x’s can be for y’s to become uncorrelated, σf is the signal standard deviation, Inline graphic, and α is a positive-valued scale-mixture parameter. Note that σl and σf should be positive. This could be enforced through θ such that θ1 = log σl and θ2 = log σf.

5.1. 1
5.1. 2
5.1. 3
5.1. 4

Similarly, four basis functions are investigated here, namely, empty, constant, linear, and pure quadratic, whose specifications are listed in eqs 5eqs 5, respectively, where B = (b(x1), b(x2), ..., b(xn))T, X = (x1, x2, ..., xn)T, and

5.1.

.

5.1. 5
5.1. 6
5.1. 7
5.1. 8

To estimate β, σ2, and θ, the marginal log-likelihood function in eq 9 is to be maximized, where K(X,X|θ) is the covariance function matrix given by Inline graphic. The algorithm first computes β̂ (θ, σ2), maximizing the log-likelihood function with respect to β given θ and σ2. It then obtains the β-profiled likelihood, log {P(y|X, β̂(θ, σ2), θ, σ2)}, which is to be maximized over θ and σ2 to compute their estimates.

5.1. 9

5.2. Performance Evaluation

Performance of the proposed GPR models is evaluated using the root-mean-square error (RMSE), mean absolute error (MAE), and correlation coefficient (CC) in eqs 10eq 9, respectively, where n is the number of data points, aiexp and ai are the ith (i = 1, 2, ..., n) experimental and estimated energy band gap, and Inline graphic and Inline graphic are their averages.

5.2. 10
5.2. 11
5.2. 12

Acknowledgments

There is no funding received for this study.

The authors declare no competing financial interest.

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