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Integrated classical/quantum-classical neural networks, DFT, and experiments for predicting bandgap and color of CrxSbxTi1−2xO2 yellow-orange pigments

This study presents an integrated framework combining experimental synthesis, density functional theory calculations, and classical/hybrid quantum-classical neural networks to accurately predict the bandgap and color of Cr/Sb co-doped TiO₂ ceramic pigments, demonstrating the potential of quantum machine learning for materials design.

Original authors: Seyed Yousof Vaselnia, Mohsen Khajeh Aminian, Reza Dehghan banadaki

Published 2026-07-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Seyed Yousof Vaselnia, Mohsen Khajeh Aminian, Reza Dehghan banadaki

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a chef trying to bake the perfect yellow-orange ceramic pigment. You want to know exactly how much energy it takes to make the color "pop" (the bandgap) and what the final shade will look like, but baking trial-and-error is slow and messy. This paper is like a team of scientists building a super-smart kitchen assistant that combines three different tools: a real-world oven (experiments), a microscopic recipe book (DFT calculations), and a futuristic crystal ball (neural networks).

The Real-World Test
First, the team actually baked a specific pigment called $Cr_{0.0625}Sb_{0.0625}Ti_{0.875}O_2$ . When they looked at it under a microscope, they saw it wasn't a perfect sphere; it looked like tiny, irregular clumps of cauliflower, about 83 nanometers in size. When they shined light on it, it absorbed blue and green light, giving it a vibrant yellow-orange glow. They measured its "energy gap" (the energy needed to change its color properties) to be about 2.2 eV.

The Microscopic Recipe Book (DFT)
Next, they used a computer simulation called Density Functional Theory (DFT) to read the atomic recipe. However, the standard recipe book had a glitch: it predicted the energy gap was only 1.4 eV, which was too low. It was like the recipe saying the cake needs 1 cup of sugar when it actually needs 2. To fix this, they added a special "correction spice" called the Hubbard U. With this spice, the simulation jumped up to 1.9 eV, getting much closer to the real oven results. They also found that when they added Chromium (Cr) and Antimony (Sb) to the mix, the standard recipe made the material act like a metal (zero energy gap), but with the correction spice, it behaved like a proper semiconductor again with a gap of 1.3 eV.

The Crystal Ball (Neural Networks)
Here is where the paper gets really fancy. The team built two types of "crystal balls" to predict these properties without baking every single time:

The Classical Ball: A standard computer brain trained on data from thousands of other materials.
The Hybrid Quantum-Classical Ball: A mix of a standard computer and a simulated quantum computer (a type of computer that uses the weird rules of quantum physics to solve problems).

They trained these balls on a dataset of 1,527 different materials to predict the energy gap, and on 195 different pigments to predict the color.

What the Crystal Balls Found
The results were a bit like a race. Both types of crystal balls were pretty good at guessing the energy gap and the color, but the Hybrid Quantum-Classical models were slightly better at matching the real-world experiments.

For the Energy Gap: When the models had all the detailed atomic information (like the exact height of the energy steps), they were incredibly accurate. One hybrid model guessed the energy gap of pure Rutile $TiO_2$ with an error of just 0.001 eV! Even when they removed the detailed step information and only gave the models a general "balance" number, the models still guessed the trend correctly, though with a bit more wiggle room.
For the Color: The models were better at predicting the "y" coordinate of the color (how yellow it is) than the "x" coordinate. The hybrid models consistently got closer to the actual yellow-orange shade of the new pigment than the standard computer models did.

What They Didn't Find (and What They Ruled Out)
The paper is careful to say that while the hybrid models were slightly better, they didn't completely crush the standard models in every single case. Sometimes the standard model was just as good. Also, the paper explicitly notes that they did not find a way to perfectly predict the color for every single pigment composition; the accuracy varied depending on how much Chromium was in the mix. For instance, for the sample with the highest amount of Chromium, the hybrid model called HQCNN-HEA was the best, but for other amounts, a different hybrid model (HQCNN-SE) took the lead.

How Sure Are They?
The authors are confident in their measurements of the real pigment—they physically made it and measured it. They are also confident in their computer simulations, which they ran with specific settings to ensure they were accurate. However, the "quantum" part of their work was a simulation of a quantum computer running on a regular computer. They didn't run this on a real, physical quantum machine yet. They suggest that this hybrid approach is a promising path forward, but they don't claim it's a solved problem for all materials yet. They also point out a limitation: they had to rely on a relatively small dataset for the color predictions because there aren't many published records that list the exact chemical mix, the baking temperature, and the final color all together.

In short, the team showed that mixing real experiments, advanced physics simulations, and a touch of quantum-inspired computing creates a powerful toolkit for predicting how ceramic pigments will look and behave, with the quantum-inspired tools offering a slight edge in accuracy.

Technical Summary: Integrated Classical/Quantum–Classical Neural Networks, DFT, and Experiments for Predicting Bandgap and Color of CrxSbxTi1−2xO2 Pigments

Problem Statement
The accurate prediction of the bandgap ( $E_g$ ) and color of ceramic pigments is a critical challenge in materials science. While Density Functional Theory (DFT) is a standard tool, it often struggles with computational demands and accuracy in describing electronic structures of complex systems, particularly regarding the underestimation of bandgaps in standard DFT calculations. Furthermore, while machine learning (ML) has accelerated materials discovery, the application of hybrid quantum–classical neural networks to the direct numerical prediction of $E_g$ and chromaticity in crystalline pigments remains largely unexplored. This study addresses the gap in predicting these properties for the yellow-orange rutile-based pigment system $Cr_xSb_xTi_{1-2x}O_2$ , a material of practical importance that lacks comprehensive theoretical analysis in its rutile phase.

Methodology
The research employs a tripartite framework integrating experimental synthesis, first-principles calculations, and machine learning:

Experimental Work: The specific composition $Cr_{0.0625}Sb_{0.0625}Ti_{0.875}O_2$ was synthesized via a conventional ceramic route (solid-state reaction at 1000 °C). The material was characterized using Scanning Electron Microscopy (SEM), X-ray Diffraction (XRD), and UV-Vis spectroscopy to determine morphology, phase purity, and optical bandgap via Tauc plots.
First-Principles Calculations (DFT/DFT+U): Electronic structures of undoped and Cr/Sb co-doped rutile $TiO_2$ were analyzed using Quantum ESPRESSO. Calculations included standard DFT and DFT+U (incorporating Hubbard $U$ corrections for Ti-3d and Cr-3d orbitals) to correct for electron correlation. Charge density and Electron Localization Function (ELF) analyses were performed to understand bonding character.
Machine Learning Models:
- Datasets: Two datasets were constructed: (i) 1,527 tetragonal materials from the Materials Project database for $E_g$ prediction (1–2 eV range), and (ii) 195 rutile-based ceramic pigments from literature and current experiments for color prediction (CIE $x, y$ coordinates).
- Architectures: Four models were developed and compared:
  - C-NN: A fully classical neural network serving as a quantum-equivalent baseline.
  - HQCNN-SE: Hybrid Quantum–Classical Neural Network with a Strongly Entangling ansatz.
  - HQCNN-HEA: Hybrid model with an Enhanced Hardware-Efficient Ansatz.
  - HQCNN-QI: Hybrid model with a QAOA-inspired ansatz.
- Scenarios: For $E_g$ prediction, two scenarios were tested: Scenario 1 utilized explicit band-edge features (CBM, VBM, Fermi level), while Scenario 2 removed these in favor of an electronic asymmetry feature ($asym$) to test robustness.
- Training: Models were implemented using PyTorch and PennyLane, trained on normalized data with AdamW optimizers, and evaluated using Mean Absolute Error (MAE), Mean Squared Error (MSE), and $R^2$ . SHAP (SHapley Additive exPlanations) analysis was used for interpretability.

Key Results

Experimental Findings: The synthesized pigment exhibited micron-sized agglomerates with a yellow-orange color ( $L^*=58.14, a^*=13.35, b^*=46.77$ ). XRD confirmed a dominant rutile phase with minor anatase. The experimental direct optical bandgap was determined to be approximately 2.2 eV.
DFT and DFT+U Insights: Standard DFT underestimated the $E_g$ of rutile $TiO_2$ (1.4 eV) and predicted a quasi-metallic state (zero $E_g$ ) for the doped system. The inclusion of the Hubbard $U$ correction improved accuracy, yielding 1.9 eV for undoped and 1.3 eV for the doped system, restoring semiconducting character. PDOS analysis revealed that $E_g$ enlargement in the doped system arises from the upward shift of Ti-3d derived conduction bands.
Bandgap Prediction ( $E_g$ ):
- Scenario 1 (Full Features): All models achieved high accuracy ( $R^2 \geq 0.98$ for most). The HQCNN-HEA model provided the most accurate estimate for rutile $TiO_2$ (error of 0.001 eV). SHAP analysis confirmed that CBM and VBM were the dominant features, consistent with the physical definition of $E_g$ .
- Scenario 2 (Reduced Features): Accuracy decreased slightly, but models maintained physical consistency. The $asym$ feature became the primary driver, effectively replacing explicit band-edge data.
Color Prediction:
- Models predicted the $y$ chromaticity coordinate more accurately than $x$ . The HQCNN-SE model achieved the highest $R^2$ for $y$ (0.8731), while C-NN performed best for $x$ (0.7799).
- SHAP analysis revealed a strong statistical coupling between the $x$ and $y$ coordinates, where the complementary coordinate was the most influential feature.
- For the specific $Cr_xSb_xTi_{1-2x}O_2$ compositions tested, hybrid quantum–classical models generally provided predictions closer to experimental values than the classical baseline, with HQCNN-SE and HQCNN-HEA showing the most reliable agreement across different dopant levels.

Significance and Claims
The authors claim that this work demonstrates an effective framework linking experiment, first-principles calculations, and neural networks for predicting key properties of ceramic pigments. The primary novelty lies in the application of classical and hybrid quantum–classical neural networks to the direct prediction of pigment color and bandgap values.

The study asserts that hybrid architectures offer slightly improved predictive performance and more consistent agreement with experiments compared to classical models, particularly in capturing the trends in $E_g$ and chromaticity. The work highlights the potential of hybrid quantum–classical models to provide compact and physically consistent representations of prediction mechanisms, as evidenced by SHAP analysis. While acknowledging limitations regarding the availability of doped structural data and comprehensive experimental databases, the paper positions this integrated approach as a viable strategy for materials design, pointing toward the future utility of fully quantum machine learning models as hardware capabilities advance.

Technical Summary: Integrated Classical/Quantum–Classical Neural Networks, DFT, and Experiments for Predicting Bandgap and Color of CrxSbxTi1−2xO2 Pigments

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