Effect of Molecular Structure on the B3LYP-Computed HOMO–LUMO Gap: A Structure −Property Relationship Using Atomic Signatures

Abstract

Compounds possessing a small highest occupied molecular orbital–lowest unoccupied molecular orbital (HOMO–LUMO) gap (E_gap) are highly desirable due to their instability and reactivity, making them useful for a wide range of applications. However, the search for new organic compounds with a low E_gap is an expensive endeavor due to the exponentially increasing pool of virtual compounds. Accordingly, in this study, atomic Signatures were utilized as molecular descriptors to investigate the correlation between the molecular structure and the B3LYP-computed E_gap, thus aiding in the development of a quantitative structure–property relationship (QSPR). An easy-to-use robust model was constructed using forward-stepping multilinear regression with leave-one-out cross validation, resulting in a regression coefficient (r²) of 0.86 and a predictability (q²) of 0.76. The use of atomic Signatures as molecular descriptors successfully inferred correlations between different structural motifs and E_gap. The atomic fragments containing π-bonds in various aromatic compounds were found to be the most significant atomic Signatures, explaining nearly 50% of the variance in the data, with regression coefficients that decreased E_gap. This is attributed to π-electron delocalization, making this molecular fragment a reactive site in a molecule. Finally, an external test set was used to further evaluate the model’s predictive performance. The developed QSPR can be utilized as a reliable initial screening tool to identify potential candidates possessing low E_gap values.

Introduction

The highest occupied molecular orbital–lowest unoccupied molecular orbital (HOMO–LUMO) gap (E_gap) is a quantum chemical property defined as the difference between the energy of the highest occupied and lowest unoccupied molecular orbitals, HOMO and LUMO, respectively. In recent years, this property has been found to be instrumental in different chemical research areas, where, in most cases, researchers are trying to alter the configuration of organic molecules, either experimentally or computationally, in order to minimize this property.¹⁻⁴ This is desirable as molecules with low E_gap are unstable (i.e., highly reactive), making E_gap important in the design and search for new molecules with desired optoelectronic and electron-transfer properties (properties used in the development of organic photovoltaic cells,⁵⁻⁷ organic film transistors,^8,9 and organic light-emitting diodes).^2,10−14 Photovoltaic cells and organic semiconductors with low E_gap are more efficient as they are easily excited by absorbed light, producing electricity by the excitation of electrons from the HOMO to LUMO orbitals.^15,16 Similarly, molecules with lower E_gap are highly reactive, increasing their tendency to participate in chemical reactions with their surrounding environment. This phenomenon is critical for the adsorption of chemical compounds onto a substrate, which finds diverse applications in both the fields of catalysis and corrosion inhibition. Specifically, in catalysis, the E_gap of reactants can influence the adsorption process and, therefore, the rate and selectivity of the reaction.¹⁷ Additionally, in corrosion inhibition research, it was concluded that E_gap is a critical property in designing efficient corrosion inhibitors.¹⁸⁻²¹ The reactivity and kinetic instability of inhibitors with low E_gap increase their affinity to adsorb and complex with the metal substrate, protecting the surface from the corrosive environment. Furthermore, the E_gap has relevant applications in successfully correlating toxicological end points, chemical reactions, spectroscopic data, and biological activities.²²⁻²⁵ Accordingly, E_gap is a critical intrinsic quantum chemical property that is utilized in different industries to make design decisions regarding the development of new organic molecules to optimize a property of interest.

The E_gap can be calculated computationally, often using density functional theory (DFT) for the electrical band gap or time-dependent DFT (TD-DFT) for optical properties.²⁶⁻²⁹ In this study, the focus is on the HOMO–LUMO gap, which better approximates the electrical band gap.²⁹ Regardless of the calculation method, the search for new organic compounds with a low E_gap is an expensive endeavor, both computationally and experimentally, due to the exponentially increasing pool of virtual compounds. Hence, a computer-assisted method of approximating the E_gap can find value as an early stage filter guiding the researcher to a data set of molecules with relevant properties. After that, a more expensive method can be utilized to refine those selections that emerge from this early stage filter.

One way to implement this filtering step is by constructing a quantitative structure–property relationship (QSPR), which creates a statistically optimized regression relation between a property of interest and a set of structurally related descriptors. Different works have been published in the literature to construct an effective and computationally inexpensive QSPR to predict the relationship between the E_gap and various descriptors. For example, Xu et al. utilized the smooth overlap atomic position (SOAP) method to provide descriptors, encoding regions of atomic geometries by using local expansion of a Gaussian atomic density based on harmonic and radial basis functions.³⁰ The data set was comprised of 323 polycyclic aromatic hydrocarbons with E_gap values ranging from 0.64 to 6.59 eV.³⁰ A model was constructed using machine learning (ML) to understand the impact of different functional groups on the E_gap; the mean absolute error (MAE) was reported to be 0.19 eV.³⁰ It was concluded that the inclusion of a ketone group in a structure had the greatest impact on the reduction of the E_gap.³⁰ Another study was conducted by Pereira et al., who utilized different ML methods to precite the HOMO, LUMO, and E_gap of 111,725 organic structures (88,537 for training, 9989 for testing, and 13,199 for validation) using a hybrid B3YLP functional with 6-31G* basis set.³¹ Different descriptors were tested, and the best results were achieved by using modified distances descriptors with random forest models, achieving an r², MAE, and root-mean-square error (RMSE) of 0.88, 0.23, and 0.33 eV, respectively. It was found that the inclusion of the orbital energy calculated using PM7 (a semiempirical method for continuous potential energy surfaces) as an additional descriptor was able to enhance the model’s performance.³¹

Furthermore, Montavon et al. constructed multiple QSPRs using a deep learning artificial neural network on thousands of organic molecules to predict various electronic properties including the HOMO and LUMO.³² The descriptors used were molecular representations derived from stoichiometry and configurational information consisting of nuclear charges and three-dimensional (3D) interatomic distances represented in a matrix.³² The constructed QSPRs were considered valid as the authors had achieved a MAE of 0.16 and 0.13 eV for the HOMO and LUMO, respectively.³² Additionally, Mazouin et al. recently used quantum machine learning to accurately predict the E_gap using the QM9 and QM7b data sets, which consist of thousands of organic compounds containing up to 7 heavy atoms.³³ It was concluded that earlier classification based on structural features prior to model training enhanced the model accuracy.³³ To achieve this, three organic classifications were introduced: (1) molecules with aromatic rings or carbonyl groups, (2) molecules with single unsaturated bonds, and (3) molecules with saturated bonds.³³ Once this partitioning was applied, a kernel ridge regression was used to construct the model, achieving a 0.1 eV MAE.³³

As identified above in some previous studies, the classification of organic compounds based on structural features was able to significantly enhance the model accuracy, illustrating the impact of the molecular structure on the E_gap. Mozum et al. further illustrated this relationship by demonstrating that three compounds (cyclohexanol, cyclohex-2-ethanol, and phenol) with identical skeletal structures but different bond saturation resulted in a significant change in E_gap.³³ This latter phenomenon is hard to quantify since the descriptors used in the literature, specifically the previously mentioned studies, do not directly articulate specific structural features, motifs, and bonding in a simple, easy-to-understand manner.

As a way to address this shortcoming, we propose a new QSPR study for E_gap that will be constructed using a computationally inexpensive fragmental structural descriptor called Signature, a descriptor not utilized in this area of research before.³⁴ By utilizing Signatures, atomic features will be encoded, helping to illuminate the impact of specific atomic fragments on the E_gap.

The Signature molecular descriptor was first introduced by Faulon for structural elucidation studies³⁵ but eventually found utility in computer-aided molecular design,³⁶ which continues to today.³⁷ A Signature is defined as a systematic codification system over an alphabet of atom types, describing the extended valence (i.e., neighborhood) of the atoms in a molecule.³⁶ In other words, Signatures describe the connectivity of an atom in a molecule to a predefined extent of branching, called height. A height-1 atomic Signature would describe the “root” atom’s connectivity to its nearest neighbor(s), while height-2 describes the atom’s connectivity to its second-nearest neighbor(s) in a molecule, etc. Summation of atomic Signatures in a structure at a specified height yields the molecular Signature of that structure at the specified height. The occurrences of atomic Signatures (typically the independent variables) in a data set can be correlated to a property of interest using a QSPR, helping to elucidate the impact of structural features.³⁸

In previous work, atomic Signatures were used to identify the most significant molecular fragments affecting the adsorption of corrosion inhibitors on carbon steel in chloride-contaminated simulated concrete pore solution.³⁹ A data set of organic corrosion inhibitors consisting of amines, alkanolamines, and mono- and polycarboxylates were electrochemically tested to study the effect of molecular structure on corrosion inhibition.³⁹ It was concluded, by utilizing atomic Signatures, that the presence of π-bonds and nitrogen atoms (with lone pair of electrons) in the structure enhanced the adsorption process, positively impacting the dependent variable (i.e., pitting potential) and corroborating electrochemical tests.³⁹ Additionally, a QSPR model was constructed to predict the pitting potential using stepwise multilinear regression (MLR), achieving an r² of 0.75 and 0.87 at heights 1 and 2, respectively.³⁹ Similarly, Kayello et al. used atomic Signatures as descriptors to infer the importance of certain structural features on the performance of surface tension-reducing agents.³⁸ From the constructed QSPR (r² = 0.97), it was concluded that the addition of alkyl groups reduced the surface tension while an amine group increased it.³⁸ Accordingly, a computer-aided molecular design (CAMD) was initiated using atomic Signatures as molecular descriptors. This model then helped to identify new, novel structures with optimum surface tension-reducing capabilities.³⁸ Other studies exist using Signatures for CAMD, including work in the area of drug delivery,⁴⁰ pharmaceuticals,⁴¹ solvent selection,⁴² admixture design,³⁸ ionic liquids,⁴³ bio-oil additives,³⁷ organic photovoltaic materials,⁴⁴ and foam blowing agents.⁴⁵

Considering the utility of the Signature molecular descriptor and the importance that not only the structure but also the nature of the bonding plays in the calculation of E_gap, it would seem reasonable to select Signature as a good candidate for use in designing a QSPR for E_gap. Accordingly, in this work, we attempt to develop an accurate, predictive, and computationally inexpensive QSPR to correlate the E_gap using atomic Signatures as molecular descriptors. To that end, a total of 112 organic compounds were used as a training data set composed of amines, alkanolamines, carboxylic acids, amino acids, cyclic/aromatic compounds, and hybrid structures. DFT was utilized to identify the E_gap of the structurally optimized organic compounds using Becke’s three parameter hybrid functional and Lee–Yang– Parr correlation (B3LYP)/6-31G* basis set. Details of the method and results are provided in the next sections. We note that the influence of certain structural fragments is also discussed, which can aid in the design of organic compounds for different applications and environments.

Methods

Data Set and DFT Calculations

A total of 112 organic compounds were chosen to construct a structurally balanced yet diverse data set consisting of amines, alkanolamines, carboxylic acids, amino acids, cyclic compounds, and “hybrid” compounds (a structure containing two or more functional groups). These functional groups were chosen for the data set as they include a range of unique molecular structures consisting of H, C, O, and N. Elements such as these are critical as they are abundant, aiding in the creation of a more inclusive and general model to be applied to a wide range of structures. A small subset of the database is shown in Figure 1; the entire data set along with the calculated E_gap values are available as the Supporting Information.

Figure 1.

Subset of different organic molecules used as a training set.

All quantum chemical properties presented in this study were calculated using DFT, which was performed using Gaussian 16 software. The molecule geometry was fully optimized employing Becke’s three parameter hybrid functional and the Lee–Yang–Parr correlation (B3LYP) using a 6-31G* basis set. B3LYP is one of the most popularly used functionals to find different quantum properties including E_gap, making it a standard benchmark for comparing different and newly constructed models.^{23,30,31,33,46,47} All optimizations were performed in the aqueous phase using water as a solvent. The E_HOMO and E_LUMO were then extracted from the output file using an in-house script before calculating the E_gap (E_LUMO – E_HOMO). A histogram presenting the E_gap distribution is illustrated in Figure 2. It should be noted that the E_gap distribution range from 3.13 to 10.40 eV is larger than what has been typically reported in the literature on E_gap studies.^28,30

Figure 2.

Histogram showing the range and frequency of E_gap values from the 112 data set.

Model Training and Validation

Forward-stepping multilinear regression (FSMLR) with leave-one-out cross validation (LOOCV) was used to construct a QSPR using atomic Signatures as molecular descriptors. FSMLR is a simple yet powerful regression approach that was used successfully in previous studies to infer the relation between molecular fragments and a variety of properties of interest.^38,39 One of the main advantages of FSMLR is its interpretability, easily identifying significant atomic Signatures affecting, either positively or negatively, the E_gap through the values of the regression coefficients. The most common way to measure the model’s correlative strength is through r², which is the ability to correlate the training set’s property of interest. Although a high r² indicates a correlative model, this does not necessarily guarantee a predictive one. However, a QSPR capable of accurately predicting the property of interest for compounds outside of the training set is critical to the model’s usefulness, especially for design. Accordingly, cross validation can be applied to gauge the predictive power of the constructed model, quantifying it through a parameter called q². Ultimately, the model must be able to correlate experimental data in the training set and achieve an acceptable level of prediction outside the training set.

Cross validation (CV) is a statistical technique used to evaluate a model’s predictive ability through partitioning available dependent variables into different folds or subsets.⁴⁸ From this, the model will use one subset as the test set while the other subsets will be used for model training purposes. This process will be systematically repeated until all subsets have been used to train/test the model, and accordingly, the average performance of the model is recorded, and an optimum predictive model is created. LOOCV is a type of CV training technique that treats every data point as a test set, while the remaining ones are used as a training set.⁴⁹ It should be noted that this method is typically used on relatively small data sets.

To avoid overfitting, a LOOCV was utilized to assess the model’s predictability in this study. All of the atomic Signatures used in the QSPR had an α-to-enter/remove significance level (α) of 0.1 and a p-value of less than 0.05, indicating the rejection of the null hypothesis. Furthermore, the MAE and RMSE were calculated and used as statistical metrics to assess the model’s performance, as seen in eqs 1 and 2

1

2

where m is the number of molecules in the data set, Egapⁱ_pred is the model-predicted E_gap of compound i, and Egapⁱ_DFT is the DFT-calculated E_gap of compound i.

Results and Discussion

Atomic Molecular Signature

A Signature is a systematic codification system over an alphabet of atom types, describing the extended valence (i.e., neighborhood) of the atoms of a molecule.^34,36 In other words, Signature describes the connectivity of atoms in a molecule to an extent of branching up to a predefined distance called the height. The summation of atomic Signatures at a certain height in a molecule yields the molecular Signature at the predetermined height.

As an example, Figure 3 represents the construction of the molecular Signature (heights 0, 1, and 2) of acetic acid, and the root atom is indicated by a star, which is C in this specific example. As seen from Figure 3, at height = 0, the atomic Signature is simply the atom itself [C]. In contrast, at height = 1, the atomic Signature describes all atoms bonded to the root atom without backtracking. The atomic Signature for this C atom at height = 1 is [C]([C][H][H][H]), meaning that the carbon atom in acetic acid is single-bonded to a carbon and three hydrogens. The parentheses indicate how far an atom is from the root, and the bonds between atoms are assumed to be single unless otherwise specified (“=” for a double bond, “t” for a triple bond, and “p” for the aromatic bond). At height = 2, the atomic Signature becomes more specific and detailed as it describes the connectivity of the root atom to neighboring atoms two bonds away without backtracking. As a result, the atomic Signature at height = 2 is [C]([C]([=O][O])[H][H][H]), illustrating that there are two oxygen atoms bonded (one is single-bonded, while the other is double-bonded) to the carbon atom that is attached to the root carbon atom. Summing up all of the atomic Signatures at a given height will result in the molecular Signature of the molecule, as seen in Figure 3. The number preceding each atomic Signature represents the occurrences of this atomic Signature in a structure.

Figure 3.

Construction of the molecular Signature of acetic acid at heights 0, 1, and 2. The selected root atom is the carbon indicated by a “*”. Acetic acid is presented in 3D and 2D to show the branching of the root atom at the aforementioned heights. The atomic Signature at heights 0, 1, and 2 for this starred root atom are [C], [C]([C][H][H][H]), and [C]([C](=[O][O])[H][H][H]), respectively.

QSPR Construction

An in-house script was used to deconstruct the molecular data files (MOL files) of each molecular structure in the data set into height-1 atomic Signatures. The script generated 67 unique height-1 atomic Signatures, which are presented in Table 1. These 67 atomic Signatures were used as the independent variables to correlate with the E_gap (dependent variable) in a QSPR. As mentioned, an FSMLR with LOOCV was employed to construct the model by selecting the most significant atomic Signatures affecting the property of interest based on the previously defined selection criteria. To illustrate the occurrence of each atomic Signature in a specific molecular structure, a descriptor matrix was constructed where the rows represent the molecules, columns represent the atomic Signatures, and the value of the matrix is the occurrence of each height-1 atomic Signature within a molecule. Figure 4 illustrates a schematic of the descriptor matrix used to construct the QSPR.

Table 1. List of 67 Height-to-One Atomic Signatures for the 112 Molecular Structures.

#	height-1 signatures	#	height-1 signatures	#	height-1 signatures	#	height-1 signatures
1	[C](=[C][H][H])	21	[C]([C][O]=[O])	41	[C](p[C]p[C][N])	61	[N](p[N]p[N])
2	[C](=[C][H][N])	22	[C]([C]p[C]p[C])	42	[C](p[C]p[C][O])	62	[O](=[C])
3	[C]([C]=[C][H])	23	[C]([C]p[C]p[N])	43	[C](p[C]p[C]p[C])	63	[O]([C][C])
4	[C]([C][C]=[C])	24	[C]([C]p[C]p[O])	44	[C](p[C]p[C]p[N])	64	[O]([C][H])
5	[C]([C][C]=[N])	25	[C]([C]p[N]p[N])	45	[C](p[C]p[C]p[O])	65	[O]([C][N])
6	[C]([C][C]=[O])	26	[C]([C]p[N]p[O])	46	[C](p[C]p[N]p[N])	66	[O]([H][N])
7	[C]([C][C][C][C])	27	[C]([H][H][H][N])	47	[H]([C])	67	[O](p[C]p[C])
8	[C]([C][C][C][H])	28	[C]([H][H][H][O])	48	[H]([N])
9	[C]([C][C][C][N])	29	[C]([H][H][O][O])	49	[H]([O])
10	[C]([C][C][C][O])	30	[C]([H][N]=[O])	50	[N](=[C][H])
11	[C]([C][C][H][H])	31	[C]([H]p[N]p[N])	51	[N](=[C][O])
12	[C]([C][C][H][N])	32	[C]([H]p[N]p[O])	52	[N]([C][C][C])
13	[C]([C][C][H][O])	33	[C]([N][N]=[N])	53	[N]([C][C][H])
14	[C]([C][H]=[N])	34	[C]([N]p[N]p[N])	54	[N]([C][H][H])
15	[C]([C][H]=[O])	35	[C](p[C]=[O]p[O])	55	[N]([C][H][O])
16	[C]([C][H][H][H])	36	[C](p[C][H]p[N])	56	[N]([C]p[C]p[N])
17	[C]([C][H][H][N])	37	[C](p[C][H]p[O])	57	[N](p[C][H]p[N])
18	[C]([C][H][H][O])	38	[C](p[C][N]p[N])	58	[N](p[C]p[C])
19	[C]([C][H][O][O])	39	[C](p[C]p[C]=[O])	59	[N](p[C]p[C][H])
20	[C]([C][N]=[O])	40	[C](p[C]p[C][H])	60	[N](p[C]p[N])

Figure 4.

Schematic of the descriptor matrix used. The rows represent the 112 molecular structures, while the columns are the 67 generated atomic Signature. The occurrences of each Signature in a molecule are the values of the matrix.

The effect of the number of atomic Signatures added to the model on explaining the variance in the data (i.e., r²) is illustrated in Figure 5. The first atomic Signature (see Figure 5) was able to explain 48.6% of the variance, indicating it to be the most significant. With every subsequent addition, the effect decreased until a plateau was reached, beyond which adding more descriptors would lead to overfitting. Out of the 67 atomic Signatures, 18 were identified as significant according to a significance level (α) of 0.1 included in the QSPR. The selected atomic Signatures are listed in Table 2 in the order they were added (most significant to least significant), along with their corresponding regression coefficient and p-value. The p-values of the selected atomic Signatures are lower than 0.05, rejecting the null hypotheses, thus establishing their statistical significance.⁵⁰ Moreover, the lack of fit p-value was found to be 0.09, making it insignificant and, in turn, indicating that there is no strong evidence to support the lack of fit, further attesting to the model’s strength.⁵¹ The achieved r² and q² are 0.86 and 0.76, respectively, making the model correlative as well as predictive. To further assess the model’s statistical performance, the RMSE and MAE were calculated using eqs 1 and 2 and found to be 0.528 and 0.408 eV, respectively. The MAE and RMSE are within an acceptable range for the proposed QSPR model, which is designed to serve as an efficient initial screening tool and to establish correlations between structural motifs and E_gap. As highlighted in the introduction, while more accurate machine learning models, such as those by Mazouin et al. (MAE = 0.1 eV) and Pereira et al. (MAE = 0.23 eV), exist, this study focuses on providing a computationally inexpensive and interpretable model for early stage screening, providing researchers with a practical tool for identifying promising compounds before employing more resource-intensive methods.

Figure 5.

Regression coefficient as a function of the number of atomic Signatures added to the QSPR.

Table 2. QSPR Regression Coefficients of Selected Atomic Signatures (in Order of Decreasing Significance) along with Their Corresponding p-Value.

variables	coefficient	p-value
constant	8.475	–
[C](p[C]p[C][H])	–0.2552	0.000
[O](=[C])	–0.7567	0.000
[C]([H]p[N]p[N])	–0.476	0.035
[C]([C]=[C][H])	–0.893	0.000
[N](=[C][O])	–1.345	0.000
[C]([C]p[N]p[N])	1.721	0.000
[H]([N])	–0.558	0.001
[C](p[C]p[C][O])	–1.172	0.000
[O]([C][C])	0.569	0.000
[C]([C][C][H][H])	0.2168	0.000
[N]([C][H][O])	1.326	0.005
[H]([C])	–0.0596	0.002
[O](p[C]p[C])	–0.926	0.000
[C](p[C][H]p[N])	–0.541	0.000
[N]([C][H][H])	0.809	0.020
[C](p[C]p[C]p[N])	–0.418	0.012
[C](=[C][H][H])	–0.937	0.003
[C](p[C]=[O]p[O])	–0.709	0.042

As a way to further evaluate the predictive power of the model created, we selected a ten-compound external test set. The ten compounds were selected to be representative of the original data set, having an E_gap range spanning from 3.8 to 8.7 eV. This test set, along with their HOMO, LUMO, and E_gap, is presented in Figure 6. The constructed QSPR (see Figure 7) was able to successfully predict the E_gap values of the test set achieving an r² of 0.91, corroborating the model’s predictive power. It should be noted that the QSPR is only used to establish prediction and correlation of the E_gap for organic compounds composed of C, H, O, and N atoms using the B3LYP functional with 6-31G* basis set in aqueous solution.

Figure 6.

HOMO and LUMO energies of the test set used to validate the constructed QSPR. The difference between both molecular orbitals is the E_gap (blue line).

Figure 7.

Predicted vs DFT-calculated E_gap values for the training and test sets using the height-1 constructed QSPR.

Impact of Molecular Structure

The QSPR of the E_gap is represented as the summation of the product of the regression coefficients and occurrence of each atomic Signature in a molecule as seen in eq 3. Although the atomic Signatures are slightly correlated, as they are not perfectly orthogonal, one can still infer the effect and impact of each atomic Signature on the E_gap through the magnitude and sign of the regression coefficient. For example, the regression coefficient of [C](p[C]p[C][H]), the most impactful atomic Signature affecting the E_gap, is negative, which indicates that the presence of aromatic bonds in a structure can lower the HOMO–LUMO gap. This is analogous to the work done by F. Aihara, as he stated that a reduced HOMO–LUMO gap by definition is closely related to aromaticity.² Furthermore, the presence of [O](=[C]), [N](=[C][O]), and [C]([H]p[N]p[N]), which also have negative regression coefficients, decreases the E_gap value, which can be attributed to double bonds, aromatics, and heteroatoms being reactive sites in a molecule. Similarly, [C]([C]=[C][H]) had a regression coefficient of −0.893, showing its importance in reducing the E_gap, which is attributed to the π-conjugated nature of the atomic Signature, agreeing with the literature.⁵² On the other hand, the presence of an internal alkane chain group, [C]([C][C][H][H]), and terminal nitrogen atomic Signature, [N]([C][H][H]), in a molecule increases the E_gap value due to a positive regression coefficient. It should be noted that the magnitude of the regression coefficients is also dependent on the number of occurrences of the atomic Signature in the descriptor matrix. For example, the regression coefficient of [H]([C]) is the smallest due to the common presence of this molecular fragment in a molecule; [H]([C]) is present 871 times in this 112-molecule data set.

3

According to Table 2, the five most significant atomic Signatures explain ∼70% of the data variation and negatively impact the E_gap value prediction. These atomic Signatures consist mainly of aromatics, heteroatoms, and double bonds, indicating that their presence in a molecule is associated with a decrease in the HOMO–LUMO gap. This is attributed to high electron density, making these structural features reactive sites, specifically due to π–π interactions, lone pair electrons, and/or delocalized π-bonds. This phenomenon can be clearly seen in the example demonstrated by B. Mazouin et al., where the addition of aromatic bonds decreased the E_gap, as aromatic bonds can contribute to a delocalized π-electron system, thus increasing the tendency of a molecule to participate in electron transfer and be more reactive.³³ It should be noted that the effect of aromatics ([C](p[C]p[C][H])) and carbonyl groups ([O](=[C])) on the E_gap is analogous with other studies found in the literature.^2,30,53Figure 8 shows the five most significant atomic Signatures color-coded to fragmentally illustrate them in different molecular structures. This illustration provides evidence of the ability of Signatures to encapsulate different structural motifs, making it an efficient and chemically interpretable descriptor, an advantage missing in other available molecular descriptors.

Figure 8.

Five of the most significant atomic Signatures, negatively affecting the E_gap, are color-coded in different structures, illustrating how atomic Signatures can be precise and easily interpretable.

On the contrary, the presence of internal alkane chain groups, [C]([C][C][H][H]), in a structure causes an increase in the E_gap due to their nonconjugated nature. This causes the HOMO and LUMO orbitals to most likely not overlap, resulting in decreased reactivity. Additionally, as seen in eq 3, the presence of [N]([C][H][H]) in a molecular structure is positively correlated to the E_gap, which could be due to a steric hindrance effect that can cause a decrease in the rate reaction due to an increased HOMO LUMO gap.⁵⁴ Similarly, the increased E_gap observed in the presence of the [C]([C]p[N]p[N]) atomic Signature may be attributed to the carbon atom disrupting the electron delocalization. It should be noted that the effect of atomic Signatures depends on the molecular structure studied and the position of the molecular fragment in the structure itself.

Benefits of Signatures as Molecular Descriptors

By utilizing atomic Signature, a robust model was constructed using a relatively small data set, where different structural fragments and motifs infer relations to the property of interest (i.e., E_gap). Atomic Signatures are one of the very few invertible molecular descriptors, a descriptor that can be used to directly build new chemical structures, helping in optimizing a property of interest through a CAMD. As has been shown, one of the major benefits of atomic Signatures as molecular descriptors is their ability to add physical insights into identifying key structural fragments and motifs contributing to a property. This is due to Signatures intrinsically encapsulating specific atomic bonding, thus making it a highly transferable molecular descriptor that can be applied to structurally diverse systems, facilitating the design and screening of new compounds with optimized properties of interest. Unlike molecular fingerprints that include explicit valence, formal charge, and hybridization type, Signatures provide a clear path from descriptor to structure, making them a more advantageous molecular descriptor to be utilized in correlating, screening, and designing new desired compounds.

Conclusions

By utilizing atomic Signatures as molecular descriptors, a robust, computationally inexpensive QSPR was constructed to correlate the effect of different structural motifs on the B3LYP-computed E_gap of organic compounds. A database was constructed of 112 different amines, alkanolamines, amino acids, carboxylic acids, cyclic compounds, and hybrids of these compounds. Sixty-seven height-1 atomic Signatures were generated and then correlated to the E_gap of the compounds using forward-stepping multilinear regression with LOOCV. r² and q² values of 0.86 and 0.76 were achieved, respectively, indicating a robust model. The model was able to establish that the presence of signatures encapsulating aromatics, heteroatoms, and π-bonds in a molecule was able to significantly reduce the E_gap. This is attributed to these structural fragments’ high electron density and unpaired electrons, making them reactive sites due to π–π interactions, lone pair electrons, and/or delocalized π-bonds. In contrast, the presence of terminal nitrogen or internal alkane chain groups causes an increase in the E_gap, which can be attributed to steric hindrance and nonconjugated bonding, respectively. To further corroborate the model’s performance, the model was able to successfully predict the E_gap of an external test set, achieving an r² of 0.91.

Data Availability Statement

An excel file is available as Supporting Information. The first and second worksheets include the E_gap values and descriptor matrix of the training set, while the third and fourth includes the E_gap values and descriptor matrix of the external test set. An in-house script was employed to generate the atomic signatures, thereby creating the descriptor matrix. All molecules utilized for both training and testing purposes are provided within the aforementioned excel file.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.4c08626.

• E_gap values for the training set (Table S1); descriptor matrix for the training set (Table S2); E_gap values for the test set (Table S3); descriptor matrix for the test set (Table S4) (XLSX)

The authors declare no competing financial interest.