Design and prediction of high optical density photovoltaic polymers using machine learning-DFT studies

Mamduh J. Aljaafreh; Abrar U. Hassan

doi:10.1515/phys-2025-0172

Artikel Open Access

Design and prediction of high optical density photovoltaic polymers using machine learning-DFT studies

Mamduh J. Aljaafreh und Abrar U. Hassan

Veröffentlicht/Copyright: 14. Juli 2025

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Aus der Zeitschrift Open Physics Band 23 Heft 1

Abstract

This research provides a machine learning (ML) method to predict the optical density of photovoltaic (PV) polymers. The results show that extra gradient boosting regressor and random forest regressor are the best-performing models among all the tested ML models, whose good R-squared (R ²) values reveal their prediction accuracy. Furthermore, their SHapley Additive exPlanations analysis demonstrates that the polar surface area is the most impactful feature to influence its model outputs for highlighting its essential role in their design process. The results also reveal that the evaluated models exhibit consistent performance across different K values, with their mean squared error values spanning from 0.001 to 0.009 and R ² values of 0.96–0.98. In addition, their synthetic accessibility likelihood index scores show a value as high as 15 for the top polymers to imply favorable conditions for their practical synthesis. The current study not only expands the understanding of polymer design but also furthers the field of PVs by establishing a good framework to design efficient solar energy materials.

Keywords: polymer design; machine learning; DFT; SHAP; transformer assisted orientation; SALI

1 Introduction

Clean photovoltaic (PV) systems are becoming important for shifting toward sustainable energy solutions, contributing directly to reducing greenhouse gas emissions [1]. Additionally, incorporating clean technology-related practices like biodegradable cleaning agents and water-efficient methods can support broader sustainability goals to minimize their ecological footprint of solar energy production [2]. As the demand for renewable energy sources is growing, the maintenance of such clean PV systems is becoming important for improving their reliability and efficiency, along with strengthening public trust as a sustainable alternative to fossil fuels [3]. Ultimately, the emphasis on clean PV systems is critical for harnessing the full potential of solar energy in addressing climate change, along with fostering energy independence and ecological balance [4]. The design/development of organic dyes for light harvesting is important to advancing in PV technologies and their applications [5]. Such dyes have gained considerable attention owing to their tunable optical characteristics, with potential for cost-effective production [6]. By engineering their molecular configurations, it is possible to obtain a wide absorption spectrum range to capture solar irradiance energy [7]. Optical density (OD) measures the amount of light absorbed by a dye solution and is important to evaluate its performance in light-harvesting applications [8]. Such organic dyes exhibit optimal OD to enhance light absorption and minimize transmission losses [9]. Their high OD signifies their capacity to absorb a substantial amount of incident light to result in enhanced exciton generation to convert that energy efficiently [10].

The machine learning (ML) technique is gaining a pivotal role in discovering and optimizing new organic dyes to significantly expedite their design process after forecasting their properties from the existing data [11]. By leveraging their large datasets, their algorithms identify patterns and relationships not readily detected through experimental methods [12] to swiftly screen and assess potential candidates for target applications. The objective of the current study is to assemble a dataset of organic dyes along with their corresponding features to predict their OD properties. This study also aims to form the basis for designing new polymers with their optimized optical properties to ultimately improve their light absorption efficiency (Figure 1).

Figure 1

Workflow of the current work analysis to design new polymers with targeted OD.

2 Theory

The OD quantifies the amount of light absorbed by a sample in its solution form, such as organic dyes [13], and it can be mathematically defined by Eq. (1):

(1) OD = log 10 I I 0 ,

where I ₀ represents the intensity of the incident light before passing through the sample and I is the intensity of the transmitted light after it has passed through that sample. The relationship of OD with reference to its dye concentration is governed by the Beer–Lambert Law, which is expressed as Eq. (2):

(2) OD = ε ⋅ c ⋅ l ,

where ε is the molar absorptivity of a dye (L/(mol cm)), c is its concentration, and l is the path length of the light through the sample (cm).

3 Methodology

The study collected and analyzed a dataset of 67,567 organic dyes from different sources. Their chemical structures were converted into a simplified molecular input line entry system to make their files machine-readable for further analysis. To train various regression models, its dataset was divided into training (75%) and test (25%) sub-datasets.

3.1 Descriptor design

The dataset set was fed to the RDKit toolkit [14] to design its different topological, electronic, and connectivity-related numeric descriptors. Among their most common descriptors, their examples are included as follows (Eq. (3)).

(3) Molar weight ( MW ) = ∑ i = 1 n m i .

Here, m _i is the atomic mass of an atom (i).

The topological polar surface area (TPSA) was calculated from the contribution of atom (A _i) to the polar surface area (PSA) and the polar contribution of atom i (P _i) (Eqs. (3)–(6)).

(4) TPSA = ∑ A i × P i ,

where A _i is the contribution of its PSA and P _i is the polar contribution of atom i.

(5) Partition coefficient ( Log P ) = Log 10 C octanol C water ,

(6) Electronegativity ( EN ) = 1 n ∑ i = 1 n x i .

In this, χ _i indicates the EN of atom i _i.

Their valence connectivity indices ( χ n v ) [15] were calculated after assuming their hydrogen-suppressed skeletons with Eq. (7) their non-hydrogen atom assigned with their unique δ ^v as denoted by their atomic valence delta (δ ^v) [16]. For zero-order connectivity, it can be written as Eq. (7).

(7) χ o v = ∑ i = 10 n ( δ v ) − 0.6 .

Their δ ^v was calculated from their atomic number (Z) and valence electrons (Z ^v) attached to their hydrogen atoms (h) (Eq. (8)):

(8) δ = Z h − h Z − Z h − 1 .

3.2 Correlation analysis

To quantify the linear relationship between variables like XX and YY, their Pearson correlation coefficients were quantified by Eq. (9):

(9) r = n ∑ X Y − ∑ X ∑ Y n ∑ X 2 − ∑ X 2 n ∑ Y 2 − ∑ Y 2 .

Here, n shows the number of data points. Their feature importance scores [17] were calculated by following the equation for their evaluated models (Eq. (10)).

(10) FI j = 1 T ∑ t = 1 T Δ I j t ,

where FI represents their feature importance score for feature j, T is the total number of trees in the model, and Δ I j t shows a decrease in its impurity for feature j in tree t. For example, a tree-based model can be evaluated for its decrease in impurity, like Gini impurity or entropy. It can be brought by each feature of their trees.

3.3 ML analysis

For ML-related calculations, the latest version of the Python programming language and its various libraries were imported to use. The machine-readable data files were imported by using Pandas [18], while the NumPy module [19] and RDKit [14] modules were imported to design their descriptors. All the calculations were visualized by importing the Matplotlib module, along with all of its scientific calculations to be performed by the Scikit-learn module [20].

All of the relevant quantum chemical calculations were performed by an open-source quantum chemical package, PSI4 [21]. It can perform various electronic structure calculations, including those needed to estimate TPSA values (Eq. (11)). The predicted target (y _∗) for new input X ∗ was the mean of the posterior distribution [22].

(11) y ⁎ = μ ⁎ = μ ( X ⁎ ) + K ( X ⁎ ) ( X ) [ K ( X , X ) + σ n 2 I ) − 1 ( y − μ ( X ) ) .

Their output was characterized by the mean of the posterior distribution, which incorporated their training data with a covariance structure, as defined by their kernel function.

3.4 Computational studies

For relevant density functional theory calculations, Gaussian 09 (Revision D.01) [23] software was used. Initially, their molecular geometries were optimized at their ground state energy minima by using the WB97XD [24] functional with LanL2DZ [25] basis sets. At the CAM-B3LYP/6-31G+(d,p) level of theory, their subsequent calculations of electronic spectra and density of states were performed, revealing valuable insights regarding their electronic properties. By using GaussView 6.1 software, their geometrical structures were visualized, leading to an understanding of their molecular features. Their global chemical reactivity parameters as calculated by Koopmans’ theorem [26], which included their ionization potential, EN (x), electron affinity, softness (σ), hardness (η), electrophilicity index (ω), and chemical potential (μ) (Eqs. (12)–(17)).

(12) IP = − E HOMO ,

(13) EA = − E LUMO ,

(14) η = 1 2 ( E LUMO − E HOMO ) ,

(15) σ = 1 2 η ,

(16) μ = − IP − EA 2 ,

(17) χ = IP + EA 2 .

Their electrophilicity (ω) was calculated from [27] Eq. (18).

(18) ω = μ 2 2 η .

4 Results and discussion

The results summarized in Table 1 reflected the performance of various ML models in predicting the properties of organic dyes, based on distinct chemical tokens. The Random Forest [28] model notably achieved an exceptional R ² value of 0.98 for both the amide and NH₂ tokens, indicating exceptional prediction accuracy, with root mean squared error (RMSE) values recorded at 0.021 and 0.002, respectively. These findings suggested that the model effectively represented these particular chemical groups, ensuring reliable predictions of dye properties containing such functionalities. Likewise, the Random Forest model exhibited strong performance for the NHOH and halogen tokens, each yielding an R ² value of 0.97, showcasing its robustness across diverse molecular structures. Conversely, the LightGBM model [29] delivered an R ² value of 0.91 for the −OH token, which, though commendable, demonstrated a marginally lower predictive performance compared to the Random Forest model for this particular attribute (Figure 2).

Table 1

A view of the model performance of the designed tokens for the collected dataset

Tokens	Model	R ²	RMSE
NHOH	Random Forst	0.97	0.012
NOC	xGBoost	0.97	0.012
−OH	LightGBM	0.91	0.012
COO−	Random Forest	0.82	0.009
COO2	Random Forest	0.67	0.012
C_O	AdaBoost	0.88	0.012
NH₂	Random Forest	0.98	0.021
Amide	Random Forest	0.98	0.002
Bicyclic	Decision Tree	0.78	0.012
Halogen	Random Forest	0.92	0.021
Hydrazine	Random Forest	0.81	0.041
Thiophene	Random Forest	0.89	0.021

Figure 2

Evaluation of the (a) graph neural tokens, (b) their t-SNE map, (c) t-SNE map of top1 generation, and (d) top 10 generation.

Also, the xGBoost model [30] exhibited good performance for its NOC token, attaining an R ² value of 0.97, which affirmed its effectiveness in capturing the dynamics associated with this particular chemical structure. However, tokens like COO₂ and bicyclic displayed lower performance with R ² values of 0.67 and 0.78, respectively, indicating difficulties in accurately modeling these intricate molecular features. Overall, the data emphasized the potential of utilizing ensemble methods such as Random Forest and xGBoost for predicting dye properties in organic synthesis, especially for well-characterized chemical groups (Table 1). Understanding the performance of these tokens in relation to different ML models not only aided in refining model selection but also directed future studies aimed at optimizing their design of new organic dyes with their desired properties. The results illustrated the significance of particular chemical functionalities in enhancing their predictive accuracy to setting the stage for more focused innovations in the field of organic photonics.

The t-distributed stochastic neighbor embedding (t-SNE) maps [31] for Generation 1 and Generation 10 offered valuable visual insights into the distribution and clustering of data [32]. In both generations, their maps depicted how their molecular features could have been represented by their tokens, had distributed spatially within their two-dimensional space, which spanned across −100 to 100. For Generation 1, the t-SNE map showed its broader and more scattered clusters to reflect greater variability in their properties. Their clustering patterns also revealed their distinct groupings based on their chemical functionalities, such as their close clustering, which was associated with their higher predictive R ² values and lower RMSEs. This clustering suggested that data in these groups exhibit similar behavior regarding their light-harvesting properties, as indicated by the performance of the Random Forest model. When progressed to Generation 10, the t-SNE map revealed its more defined clustering, as the ML models became increasingly proficient to capture the relationships between chemical structures and their performance metrics. Their improved performance observed in later generations likely reflected the incorporation of additional data and model refinements to facilitate their deeper understanding, resulting in optimized polymer design. In general, their visual examination as t-SNE maps from Generation 1–10 clearly illustrated the influence of ML on feature extraction and modeling efficiency. The progression of clustering patterns also confirmed their increasing sophistication of the ML models to highlight the significance of clean, well-structured datasets in ensuring accurate predictions for the development of new organic dyes.

Furthermore, their Pearson correlation coefficients [33] of descriptors shown in the lower half of the correlation matrix revealed meaningful relationships between their descriptors associated with them. A positive correlation signified that an increase in one feature was generally accompanied by an increase in the other, whereas a negative correlation indicated an inverse relationship between two features. The correlation between NOCount and NHOHCount of 0.40 revealed a moderate positive correlation to imply an interdependence between the presence of nitrogen-containing groups and hydroxyl functional groups (Figure 3).

Figure 3

A correlation heatmap of the OD with top correlated RDKit based graph neural network descriptors.

This relationship suggested that polymers containing hydroxyl groups might also exhibit their higher count of nitrogen functionalities to influence their optical characteristics. The correlation between fr-Ar-OH and NHOHCount (0.23) revealed a weak positive association, indicating that although there was some relationship between these two features, it was not particularly strong. On the other hand, the correlation between fr-C-O and NOCount (0.74) stood out, demonstrating a strong positive correlation. This showed a close relationship between the presence of carbonyl functionalities and the number of nitrogen functionalities in the organic dyes, which could have influenced their performance as light-harvesting materials. Additionally, several features did not show strong correlations. For instance, the correlation between fr-COO and NHOHCount (0.12) and that between fr-COO and fr-Ar-OH (0.03) revealed weak relationships, indicating negligible interdependence among these features. Likewise, the correlation of 0.98 between fr-COO2 and fr-COO indicated an almost perfect positive relationship, demonstrating that these functionalities were likely to coexist closely within the molecular structures being investigated. The most significant relationships were observed with fr-NH₂, which showed a strong correlation with NHOHCount (0.71) and a moderate correlation with fr-C-O (0.11). Additionally, the high correlation of fr-amide and NHOHCount (0.70) alongside its moderate correlation with NOCount (0.22) underscored the interplay between nitrogenous and hydroxyl groups in shaping the characteristics of dyes. Also, these correlations unveiled the complex network of interactions among functional groups, which likely had a substantial influence on the electronic and optical characteristics of the organic dyes. Understanding such relationships through analyzing their Pearson correlations provided important guidance for further study to aid in their design and optimization. This approach particularly highlighted the importance of specific functional group combinations that enhanced performance in light-harvesting applications.

A comparison of performance metrics proved that both the random forest and xGBoost regression models exhibited proficiency in predicting their OD. Specifically, the random forest model produced an R ² value of 0.87 to suggest that it accounted for ∼87% of the variance in the target variable. Furthermore, it attained a notably low RMSE of 0.023, indicating that its predictions were both exceptionally consistent and accurate. However, the xGBoost model showed an R ² of 0.84 to reflect its good performance for implying that it could have explained ∼84% of the variance in the target variable. Although its RMSE of 0.12 was slightly higher than that of the random forest model, it still represented a reproducible result. Both models surpassed other regression models in predicting OD, with the random forest model slightly outperforming the xGBoost model in terms of overall effectiveness (Figure 4). This could likely be due to its ability to manage complex feature interactions and its ability to manage noise/outliers.

Figure 4

A view of the scatter plots of (a) xGBoost and (b) Random Forest and (c) xGBoost based SHapley Additive exPlanations (SHAP) value beeswarm plot.

To evaluate the impact of descriptors on model performance, their SHAP value [34] beeswarm plot showed that their physicochemical characteristics could be important to their model performance, with the PSA having the greatest influence on predicting OD. This was expected, as the PSA represented the molecular surface area accessible to solvent molecules, which could influence how light interacts with the molecule. The polymers with their high PSA values could typically exhibit a larger surface area exposed to the solvent, which could result in their increased light absorption, eventually influencing their OD. Their molecular weight ranked the second most influential factor affecting their model performance. This significance could be attributed to the tendency of larger molecules to exhibit more complex molecular structures, which, in turn, might alter their optical properties. Additionally, the logarithmic partition coefficients, such as SlogP, XLogP, and iLogP, significantly contributed to the model performance. These values signified the hydrophobicity of molecules, which can impact their interaction with light. Molecules with higher logP values were more lipophilic, potentially leading to increased light absorption and scattering, which in turn can affect their OD. The model also took into account the calculated log octanol–water partition coefficient (cLogP), which served as an additional indicator of their polymer hydrophobicity. Their hydrogen bond acceptors (HBA) and donors (HBD) also played a key role in determining the model performance. This was due to the fact that hydrogen bonding could alter the molecular structure, subsequently impacting its optical properties. The polymers with a greater number of HBA and HBD might exhibit different absorption and scattering patterns, which can affect their OD. Also, their Pearson correlations revealed a strong positive correlation between OD and several lipophilicity measures, such as LogP, XLogP3, cLogP, and MLogP. This suggested that an increase in the lipophilicity of their polymer could be associated with their increase in its OD. This correlation was likely attributed to the fact that more lipophilic molecules tend to exhibit a greater affinity for non-polar solvents, which could lead to alterations in their optical properties. The strong correlation with lipophilicity measures suggested that the model was capable of effectively capturing the relationships between these properties and OD, leading to more precise predictions (Figure 5). Conversely, the analysis also showed a negative correlation between OD and the number of HBD. This suggested that the polymers with their higher number of HBD were likely to exhibit lower optical densities.

Figure 5

A view of the (a) Pearson correlation and (b) feature importance heatmaps of solubility-related top descriptors.

That negative correlation might be due to the ability of HBD to affect their molecular structure, which could alter how they interact with light to ultimately influence their OD. On the other hand, a weak correlation was observed between OD and PSA. This suggested that the effect of PSA on OD was less significant than that of the lipophilicity measures. Similarly, the correlation with HBA was weak, indicating that it might not be a reliable predictor of OD.

5 Model evaluation

The performance of the best applied models was also validated by using its two distinct cross-validation strategies of 5-Fold Cross-Validation (CV) [35] and Leave One Group Out (LOGO) CV [36]. Their outcomes were presented through mean squared error (MSE) and R ² values. The 5-Fold CV results showed that the model performed exceptionally well, achieving a low MSE of 1.2 and a remarkable R ² value of 0.96 (Figure 6).

Figure 6

K-Fold CV and LOGO-based model evaluation results of the xGBoost regression model.

This indicated whether the model has successfully captured substantial amount of variance in the target variable to demonstrate the accuracy and precision. The low MSE value further suggested that the model predictions closely aligned with the actual values, with minimal discrepancies. In contrast, the LOGO CV results were quite unexpected, which yielded its MSE and R ² of 0.0. It is important to note that the 5-Fold CV results could better represent its model performance, as that approach was less susceptible to overfitting and offered a robust evaluation of model generalizability. The K-Fold CV results also provided its more detailed and refined assessment of model performance. That approach consisted of dividing the data into K subsets, training the model using K – 1 subsets, and evaluating its performance on the remaining subset. This process was repeated K times, and the average performance metrics were then calculated and reported. The results demonstrated that the model performed consistently well across different K values, with MSE values spanning from 0.001 to 0.009 and R ² values between 0.96 and 0.98. Among them, their lowest MSE value (0.001) was observed with K = 3 and K = 5 to suggest that the model could have exceptionally accurate predictions in these cases. The highest R ² value of 0.98 was achieved with K = 2, indicating that the model successfully captured nearly all of the variance in the target variable. The consistency of the results across various K values indicated that the model was robust and generalized new data effectively. The low MSE values and high R ² values suggested that the model can make accurate and reliable predictions, a finding further reinforced by the 5-Fold CV results.

5.1 Application of transformer assisted orientation

Leveraging transformer models to design new polymers could offer their structured framework to predict their material properties for encoding their molecular structures for enhancing their PV characteristics [37]. The mathematical formulations outlined below could serve as their foundation to apply their transformer-based architectures for their polymer design to explore their wide range of molecular architectures. In their transformer layer, their combination of attention and feed-forward transformations could be achieved through their residual connections and layer normalizations (Eq. (19)):

(19) ∂ φ ∂ t = D ∇ 2 φ + ε k BT φ ∇ ² μ .

Considering their final output of the transformer model (H), which could encode their properties of their input polymer structures, their property predictions are given by Eq. (20):

(20) y = f ( x ; θ ) .

In their transformer layer, the combination of attention and feed-forward transformations could be achieved by using their residual connections and layer normalization. So, the polymer density update could be achieved as follows (Eq. (21)):

(21) φ ( t + 1 ) = φ ( t ) + Δ t D ∇ 2 φ ( t ) + ε k BT φ ( t ) ∇ 2 μ ( t ) .

Similarly, their predicted orientation was updated as follows (Eq. (22)):

(22) y ( t + 1 ) = f ( x ( t ) ; θ ) + Δ t ∂ f ∂ x ( x ( t ) ; 0 ) ∇ x ( x ( t ) ) .

The theory of transformer-assisted orientation in polymer design is marking an exciting convergence of advanced ML techniques. By employing such transformer architectures to analyze and predict polymer properties, the discovery and optimization of new materials can be optimized to drive innovations. The design of 50 new polymers with their OD range of 0.68–0.98 marked its significant advancement by incorporating their chemical composition and optical properties. Through their design and characterization, the goal is to design materials with their tailored optical properties to meet the needs of diverse high-tech applications. Such polymers with this OD could exhibit their broad and versatile significance. Their ability to interact with light could effectively make them suitable for a variety of applications in optoelectronics and other renewable energy technologies [38]. In its second round of polymer generation, the introduction of 100 new polymers with an OD range of 1.06–2.26 could signify a notable development compared to their previous range (Figure 7). This increased OD suggested that such polymers could be capable of absorbing more light. Their higher OD could also be correlated to enhance their mechanical properties, such as their increased strength/toughness to make them suitable for their challenging applications.

Figure 7

A view of different generations of polymers with their predicted OD.

Additionally, their improved optical characteristics could create new opportunities for their applications in optoelectronic devices, where their precise control over light transmission and reflection was essential for facilitating their advancements in organic light-emitting diodes (OLEDs) and display technologies. Moreover, their extended OD range could enable better tailoring of their sensing applications. The polymers from this second round were also expected to be suitable for advanced smart materials, such as electrochromic devices, promoting the development of energy-efficient windows and displays that can dynamically adjust heat and light. During their third round of polymer generation, their 1,000 new polymers with an OD range of 2.26–3.51 were developed to represent their notable advancement in their material performance and versatility over the previous rounds. This significant increase in their OD could show that such polymers exhibit enhanced light absorption abilities for making them ideal for their specific applications. Their increased OD range implied that these materials were capable of absorbing more light energy, which was essential for applications that required enhanced energy conversion efficiency, such as in solar cells. The ability to effectively capture sunlight resulted in improved power generation, supporting the development of more sustainable energy solutions. Moreover, polymers with ODs in this range might exhibit enhanced mechanical properties, such as increased strength and durability, making them ideal for use in challenging environments and applications. Along with enhanced absorption characteristics, the polymers produced in this round offered substantial benefits for optoelectronic devices. Focusing on higher ODs, these polymers were poised to drive advancements in technologies like solid-state lighting, OLEDs, and high-resolution displays where accurate light management was necessary. Additionally, the considerable rise in OD made these polymers highly suitable for photonic applications, such as waveguides and optical filters, where controlling light propagation was vital. Their polymer design could be significant to provide their valuable opportunity for their increase diversity in molecular designing and functional properties. This generated library could improve their ability to identify and select optimal candidates for tailoring their specific applications, including smart materials responsive to environmental changes and advanced coatings with improved performance against environmental stressors.

5.2 Charge density patterns

These selected polymers from the designed set demonstrated their distinctive features of charge distributions for their FMOs [39]. The HOMO charge density was specifically distributed unequally across the main moiety to suggest that certain regions of their molecular structure could play significant role in stabilizing their charge (Figure 8). This uneven distribution could also highlight the areas to have concentrated electronic clouds to direct potential sites of reactive interactions for promoting their localized charge transfer processes. Such factors could also collectively enhance their light absorption ability and its suitability for photonic applications. On the other hand, their LUMO charge density displayed a uniform distribution across their main moiety to suggest that the LUMOs could accommodate their electron delocalization over a large portion of their framework. Such feature could be beneficial for electron acceptor functions, as it enhances the efficiency of charge separation processes when the polymers were utilized in devices like organic PVs or sensors. The uniform distribution of the LUMO suggested that such polymers could be well suited for their diverse electronic transitions to enhance their photophysical properties. The distinct patterns observed for FMO charge densities also indicated that such polymers could be suitable for their particular functions. Their localized HOMO nature can be leveraged for their strong light absorption and electron donating abilities, whereas their uniform LUMO could prompt their efficient electron acceptance/transfer, which could be vital for applications involving their effective charge transfer [40].

Figure 8

A view of the charge transfer patterns and their computed UV-vis spectra analysis of selected polymers from newly predicted designs.

Similarly, their energy levels offered important insights into their electronic properties for promising applications. For polymer 1, their HOMO energy had −3.1 eV value, while their LUMO energy value was at −1.5 eV, with an energy gap (E _gap) of 1.6 eV. This E _gap indicated their moderate charge transfer to make it appropriate for use in organic electronics that require efficient charge movement. For polymer 2, its E _HOMO was −3.4 eV, with the E _LUMO being −1.5 eV, resulting in their slightly larger E _gap of 1.9 eV. This larger E _gap indicated their lower conductivity as compared to polymer 1, which might have reduced its effectiveness in high-performance applications. Polymer 3 exhibited its E _HOMO of −2.6 eV and a E _LUMO of −0.7 eV, having an E _gap of 1.9 eV. Its reduced E _HOMO indicated their favorable alignment for their electron donation, whereas their higher E _LUMO could facilitate their effective electron acceptance to act as positioning a promising candidate for their PV applications [41]. In contrast, polymer 4 showed an E _HOMO of −2.4 eV and an E _LUMO of −1.4 eV to result in their narrower E _gap of 1.0 eV to suggest their improved conductivity, for potentially benefiting applications that require their rapid charge transfer patterns, such as certain organic field-effect transistors. Finally, polymer 5 presented a HOMO energy of −3.2 eV and a remarkably low LUMO of −0.45 eV, resulting in a significantly larger E _gap of 2.75 eV.

5.3 Electron excitation analysis

For selected polymers, transition density matrix (TDM) analysis [42] provided notable insights into their electronic transitions and charge transfer properties to complement their FMO charge distributions. Additionally, the TDM offered a quantitative depiction of the spatial distribution of electronic transitions between the HOMO and LUMO of polymers to demonstrate how the charge density could shift during their excitation to impact their optical and electronic behavior. For example, polymers with a considerable overlap in the TDM between their HOMO and LUMO suggested efficient and localized electron movement, which could improve their charge mobility to lead to strong photophysical responses (Figure 9). For polymer 1, the localized nature of the HOMO charge density integrated with a more uniform LUMO distribution most probably resulted in a TDM that highlighted certain regions within the polymer where charge transfer was increased. This revealed that polymer 1 could effectively utilize light absorption, making it suitable for applications in OPVs where localized charge separation was essential. Similarly, polymer 3, with its lower HOMO energy and a relatively higher LUMO, might exhibit a TDM that encouraged efficient electron delocalization during transitions. This feature enhanced its suitability for applications requiring efficient charge transport, such as sensors and electronic devices. In contrast, the smaller E _gap of 1.0 eV showed a narrower energy barrier for electron transitions, as further confirmed by the TDM analysis. Such a polymer likely demonstrates a significant charge migration, making it well-suited for applications that require fast charge transfer. In contrast, polymer 5 with its wider E _gap of 2.75 eV might result in a TDM with fewer transitions between the HOMO and LUMO, suggesting reduced charge carrier mobility. This property might limit its application in high-performance electronics but could improve its stability and effectiveness in frameworks, where reduced electron transfer was required.

Figure 9

A view of the (a) TDM heatmaps and their orbital contributions, (b) molecule electrostatic potentials, and (c) charge density difference cubes of selected polymers.

5.4 Synthetic accessibility

In the current study, the synthetic accessibility of 973 chromophores, with a focus on their synthetic accessibility likelihood index (SALI) scores [43], was evaluated. Their highest SALI score was found to be as high as ∼15 to show their substantial potential for practical use in applications, like dye production, sensor development, and other fields where chromophores were vital. A high SALI score suggested that these chromophores can be manufactured with relative ease, making them promising candidates for further research and development (Figure 10). This finding highlighted the significance of synthetic accessibility in designing and selecting chromophores for practical use, as it directly influenced their feasibility for large-scale production and real-world applications. The high SALI score of 15 reflected both the structural simplicity and favorable properties of these chromophores, emphasizing their potential for integration into commercial products. Such information could assist in prioritizing chromophores for their further exploration to speed up the development of innovative materials and technologies. Analysis of the top 30 SALI scores among the 973 chromophores was done to reveal valuable insights into their synthetic accessibility and optical properties.

Figure 10

SALI score of newly predicted polymers.

The highest SALI score recorded was 14.9, indicating that this particular chromophore was not only easy to manufacture but also demonstrated a significant absorbance value of 0.39 (Table 2). This indicated a robust potential for practical applications in disciplines such as dye production and sensor development. The absorbance values of the top entries varied, with the eighth chromophore showing the highest value of 0.82 and a SALI score of 7.5. This showed that while many chromophores were accessible, only a few stood out notably in terms of both ease of synthesis and optical properties. The data revealed a sharp decline in SALI scores after the top few entries to suggest that there should be more focus on these high-scoring chromophores for their further advancement.

Table 2

A comparison of SALI scores with their predicted ODs

Chromophore	Absorbance	SALI score	Chromophore	Absorbance	SALI score
1	2.39	14.9	16	3.39	7.0
2	1.34	8.9	17	2.33	6.9
3	3.77	8.8	18	3.34	6.9
4	2.34	8.7	19	3.39	6.9
5	1.38	8.4	20	2.32	6.5
6	3.41	8.0	21	3.46	6.5
7	2.31	7.7	22	2.33	6.4
8	2.82	7.5	23	2.31	6.4
9	2.40	7.4	24	3.49	6.4
10	2.28	7.3	25	3.42	6.3
11	3.40	7.3	26	2.46	6.2
12	1.30	7.3	27	3.46	6.2
13	1.40	7.2	28	3.49	6.2
14	3.38	7.1	29	2.31	6.2
15	1.38	7.0	30	2.52	6.2

6 Conclusions

In conclusion, this research has led to the successful development of ML-based good predictive models to predict the OD of a collected dataset of organic polymers. For this, their polymer transformer assisted orientation approach generated their library of new chemical structures. The evaluated ML models, particularly xGBoost and Random Forest Regression, demonstrated their good predictive accuracy, with their PSA emerging as an important feature for their design. Furthermore, the current study not only advances the understanding of polymer design but also can establish their reliable framework to discover new materials for PV applications. Also, its future research directions show great promise. Their experimental synthesis and characterization of the top-performing polymers can validate their PV performance. Additionally, exploring the integration of these polymers into functional devices, such as organic PV cells, can also be essential to uncover their full potential. Their evaluated models can be further expanded to generate a good polymer library to uncover their new materials with their improved properties. Moreover, their applicability to other energy-related applications, such as energy storage and fuel cells, warrant further investigation.

Funding information: This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2502).
Author contributions: A.U. Hassan and M.J. Aljaafreh: conceptualization, methodology, investigation, software, visualization, writing of the original draft, and review and editing of the manuscript. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Received: 2025-02-24

Revised: 2025-04-30

Accepted: 2025-06-06

Published Online: 2025-07-14

This work is licensed under the Creative Commons Attribution 4.0 International License.

Supplementary material

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https://doi.org/10.1515/phys-2025-0172

Schlagwörter für diesen Artikel

polymer design; machine learning; DFT; SHAP; transformer assisted orientation; SALI

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Design and prediction of high optical density photovoltaic polymers using machine learning-DFT studies

Artikel

Abstract

1 Introduction

2 Theory

3 Methodology

3.1 Descriptor design

3.2 Correlation analysis

3.3 ML analysis

3.4 Computational studies

4 Results and discussion

5 Model evaluation

5.1 Application of transformer assisted orientation

5.2 Charge density patterns

5.3 Electron excitation analysis

5.4 Synthetic accessibility

6 Conclusions

References

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