Introduction

Now a days, organic photovoltaic (OPV) solar cells are considered as light-weight, flexible, thin and efficient energy source which possess various applications in electronic fields1. The combined efforts of experimental and computational researchers have boosted their power conversion efficiency (PCE) up to 19.39% due to which they have been acknowledged as advanced solar energy devices2. Among the OPV devices, fullerenes are known as highly conjugated systems which heighten the mobility of charges. However they showed certain drawbacks such as structural instability, limited tuning of energy levels, undesirable morphology and poor light-absorption capability due to which their use becomes limited3,4,5. Consequently, the researchers have showed their interest toward the non-fullerene acceptors (NFAs) which are more efficient6,7.

In comparison with fullerene-based counterparts, the NFAs-based materials exhibit a balance between optimal efficiency and cost stability. Moreover, they possess interesting molecular structures. Therefore, designing NFA compounds with low cost and synthetic complexity is a competitive choice for researchers8. Li et al. reported two NFAs (MO-IDIC and MO-IDIC-2F) which showed high photovoltaic performance for commercial applications9. Among them, MO-IDIC-2F exhibited 13.46% power conversion efficiency (PCE) which is favorable. The NFAs molecules with A–D–Aʹ–D–A configuration are widely recognized as efficient light-harvesting devices and show high charge mobility. The side-chain engineering in them facilitates the intermolecular packing as well as maintains the solubility10. Efficient side-chain engineering of thieno-imidazole based compounds was reported to obtain profitable materials for organic solar cells by Areeba Asif et al. The compounds efficiently showed maximum absorbance (492–532 nm) and narrower energy gaps (1.76–1.99 eV). By comparative analysis, all the newly designed molecules showed enhanced photovoltaic attributes11.

Indolonaphthyridine (IND) is one of the emerging electron-deficient chromophores which is highly planar. It is also known in literature as the bay-annulated indigo (BAI) as it was initially prepared by blocking the excited state deactivation of indigo dye via new rings formation at bay positions of the dye. This double bay-annulation reaction leads to formation of a compound named as cibalackrot12. The indolonaphthyridine-based derivatives showed several applications as they are utilized in organic lasers, photoacoustic imaging13, transistors14, singlet fission reactions15, photodetectors and photovoltaics16. Their broad absorption in the near infrared (NIR) region of the electromagnetic spectrum rendered them good candidates to be utilized in transparent solar cell devices. Therefore, this research is highly grounded on the utilization of IND molecule.

It is observed in previous studies that the planar aromatic IND chromophore leads to poor solubility. To enable the solution processability, the solubilizing alkyl or aryl groups can be incorporated into this moiety. For this purpose, smaller groups based on thiophene and selenophene rings are considered as best choice which can effectively enable the solution processability17,18. Among the two (thiophene and selenophene rings), selenophene is the most appropriate choice for enhancing the device’s performance. This is due to its interesting polar nature, quinoidal character which induces stronger Se-Se interactions and longer conjugation length19.

Herein, new NFAs compounds (INDTR and INDTD1–INDTD8) are engineered utilizing the IND as the central core and accompanying the side chains composed of fused thiophenes (in case of INDTR) and selenophenes (in case of INDTD1–INDTD8 derivatives). Further, the proposed molecules are distinguished via the proficient end-capping with extended malononitrile-type acceptors. The reason for the incorporation of acceptors and side chains (π–spacer) is to extend the conjugation length which leads to high charge mobility in the designed derivatives. To gain insights into their structure–property relationships, the density functional theory (DFT) is utilized and it is a computational approach which uses the electronic density of a system to determine its geometry and key electronic properties20. However, the excited-state properties are calculated by applying the time-dependent DFT (TD-DFT)21. Many common features such as charge mobility, reduced band gap and broader absorption are observed in all candidates. Their photovoltaic aspects are analyzed via detailed computational analyses to find out the best choice among all for organic solar cell (OSC) applications. To augment the features like open-circuit voltage (Voc), a donor–acceptor interface is generated by using PTB7 donor polymer. It is anticipated that the newly designed NFAs compounds will show remarkable potential for solar cells.

Methodology

The quantum chemical investigations were performed using the Gaussian 09 suite of program22. Initially, the structural optimization of the reference and newly designed molecules (INDTR and IDTD1–INDTD8) were performed at M06 level23 of DFT in combination with 6-31G(d,p) basis set24. Their molecular configurations were viewed using the Gauss View 6.0 software25. The FMOs energies were viewed using the Avogadro software26 to calculate their energy gaps. Further, the GRPs of the studied compounds were determined by utilizing their HOMO–LUMO band gaps. The time-dependent DFT (TD-DFT) analysis at the afore-said level was performed to calculate their UV–Visible spectra, FMOs and TDM. Moreover, the absorption spectra was obtained via the Origin 8.5 software27. The DOS analysis played significant role in showing the behavior of investigated compounds among the possible electronic states were drawn using the PyMOlyze software28. Similarly, the TDM maps which represented the excitation mobility among the atoms were obtained using the Multiwfn software29 at the afore-said level. Other software programs such as Gauss Sum30 and Chemcraft31 were used to interpret the results from the output files.

Results and discussion

Designing of indolonaphthyridine-based chromophores

This study employs the quantum chemical investigation of novel fullerene-free acceptor compounds (INDTR and INDTD1–INDTD8) which aims to improve their photovoltaic performance by the end-capped modification32. The studied compounds, composed of A1–π–A2–π–A1 configuration, are designed by utilizing the indolonaphthyridine (IND) moiety as the central core (A2). Symmetric NFAs are proposed which possess the fused thiophene rings i.e., dithieno[3,2-b:2ʹ,3ʹ-d]thiophene-methane (1/1) as the π–spacer on both sides of IND core in case of INDTR, while in derivatives (INDTD1–INDTD8), more efficient selenophene rings i.e., diselenopheno[3,2-b:2ʹ,3ʹ-d]selenophene-methane (1/1) are incorporated as π-linkers instead of thiophenes. Their terminals are accommodated with extended acceptors (A1) i.e., MNM in INDTD1; FNM in INDTD2; CNM in INDTD3; BNM in INDTD4; CNT in INDTD5; CNC in INDTD6; CNS in INDTD7 and NNM in INDTD8. Figure 1 illustrates the strategy adopted for designing these chromophores. Table S11 shows their structures and IUPAC names. While, the IUPAC names of the overall molecules are shown in the Table S12 and their molecular structures are displayed in the Fig. S1. The planarity parameters of the studied compounds are depicted by calculating their dihedral angles in various fragments. Their values largely depend on the nature of substituents involved in the compounds. The results are shown in Table S1. Four dihedral angles are calculated for each investigated molecule i.e., θ1 is found between A1 and π–spacer; θ2 is reported between the A2 and π–spacer; θ3 is mentioned between  A2 and π–spacer; θ4 is found between A1 and π–spacer as shown in the Fig. 2 by using their optimized geometries. However, the side-view of their optimized structures are shown in the Fig. S2. The majority of derivatives except INDTD6 show more negative and smaller values of θ1 and θ4 as compared to that of reference compound (INDTR). This depicts that the smaller and negative values of dihedral angles enhance the intramolecular charge transfer (ICT) in the designed molecules which shows their planar nature. However, the INDTD6 compound has twisted or non-planar molecular nature as it shows positive and large values of θ1 and θ433. Further, their Cartesian coordinates are collected in the Tables S2–S10.

Figure 1
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Schematic representation of designing of INDTD1–INDTD8 from INDTR.

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Figure 2
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Optimized structures of the entitled chromophores (INDTR and INDTD1–INDTD8).

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Frontier molecular orbitals (FMOs) analysis

In frontier molecular orbital analysis, the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are considered significant orbitals for understanding of the electronic properties. It has been widely used by organic chemists to understand the reactivity and regio-selectivity of various molecules34,35. The foremost application of FMOs analysis is the understanding of the Diels–Alder (DA) reaction which showed interaction between the FMOs of a diene and dienophile36. The HOMO–LUMO study is vital in predicting the reactive sites in conjugated frameworks37. Moreover, the energy gap (Egap) between these orbitals characterizes the chemical stability and electronic conductivity as well as signifies the intramolecular charge transfer (ICT) amidst the electron donor and acceptor groups via the π–conjugated pathway38. Therefore, FMOs interpretation is crucial in quantum chemical analysis.

In the present work, the results of FMOs analysis are calculated at the M06/6-31G(d,p) level which are displayed in the Table 1. It is known that the efficiency of an OSC is determined from its energy band gap39. The proposed molecules successfully exhibit less energy gaps which show their potential to be utilized as OSC in future. The EHOMO and ELUMO values for the reference compound (INDTR) are − 5.793 and − 3.505 eV, respectively. Moreover, it shows energy gap as 2.288 eV which is the highest among all compounds. Similarly, in case of the designed derivatives (INDTD1–INDTD8), EHOMO values are − 5.766, − 5.794, − 5.816, − 5.814, − 5.892, − 5.817, − 5.898 and − 5.905 eV, correspondingly. While, their ELUMO are − 3.506, − 3.539, − 3.579, − 3.576, − 3.705, − 3.583, − 3.715 and − 3.729 eV, respectively. Further, their Egap values are 2.260, 2.255, 2.237, 2.238, 2.187, 2.234, 2.183 and 2.190 eV, respectively. The some other orbital energies (HOMO-1/LUMO + 1 and HOMO-2/LUMO + 2) are also calculated which are presented in the Table S13.

Table 1 Energies of the frontier molecular orbitals of INDTR and INDTD1–INDTD8.
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It is known from the literature analysis that a photovoltaic material which possesses less band gap will show high PCE as an OSC material40. Herein, the electronic nature of the proposed molecules is estimated which shows that all derivatives exhibit closer energy gaps which are slightly lesser than INDTR reference. The compound (INDTD7) shows the least energy gap as 2.183 eV which is attributed to the strong terminal acceptor i.e., CNS. The introduction of sulphonic acid group (–SO3H) minimizes the energy gap, promote the charge transfer, separates the interface charges and inhibit the hole-electron recombination41. Due to its significance, the acceptor of INDTD7 is regarded highly efficient and overall molecule seems to be the reasonable candidate for OSC devices. Moreover, the compound (INDTD5) which has CNT end-capped acceptor shows slightly higher band gap (2.187 eV) than INDTD7. This band gap is further elevated in case of INDTD8 (2.190 eV) which contains NNM acceptor at its terminal. Since, –SO3H, –CN and –NO2 groups show negative mesomeric effect (–M effect) as well as inductive effect (–I effect), hence, closer values of Egap are observed in INDTD7, INDTD5 and INDTD8, respectively.

The compounds (INDTD1–INDTD4 and INDTD6) show higher band gaps in the range of 2.234–2.288 eV. In case of INDTD6, the band gap of 2.234 eV is observed which is due to the presence of CNC acceptor moiety. Similarly, in case of INDTD3 and INDTD4, higher band gaps of 2.237 and 2.238 eV are observed owing to the presence of CNM and BNM acceptors, respectively. Lastly, the derivatives (INDTD1 and INDTD2) show comparable Egap as 2.260 and 2.255 eV as they possess MNM and FNM terminal acceptors, respectively. Overall, a following increasing order is observed for Egap: INDTD7 < INDTD5 < INDTD8 < INDTD6 < INDTD3 < INDTD4 < INDTD2 < INDTD1 < INDTR.

From the FMOs diagrams, it is observed that the electronic clouds in case of HOMOs are majorly concentrated over the π–linkers and 1,5-dihydro-1,5-naphthyridine-2,6-dione (NTD) portion of the central acceptor core and NTD is widely known as a component of n-type organic semiconductors42. Moreover, a minor HOMO density is also seen over the terminal acceptors. In case of LUMOs, the electron density is highly concentrated over the central acceptor (IND) and end-capped acceptors in all the titled chromophores. While, a few clouds are also seen over the π–linkers. Figure 3 demonstrates the FMOs surface plots for INDTR and INDTD1–INDTD8 compounds. Similarly, the orbital surfaces of HOMO-1/LUMO + 1 and HOMO-2/LUMO + 2 are shown in the Fig. S3.

Figure 3
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Illustration of FMOs (HOMOs and LUMOs) of INDTR and INDTD1–INDTD8.

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Hence, it is inferred from the above discussion that the major charge density for HOMO is concentrated over the bridging units and for LUMO, it is located over the acceptor components. Moreover, the lower band gaps of the titled chromophores indicate a massive ICT in them. So, they are regarded as effective for solar cell applications.

Global reactivity parameters (GRPs)

To assess the stability and reactivity of the investigated chromophores, their global reactivity parameters (GRPs) are evaluated. These are electronegativity (X)43, chemical potential (μ)44, chemical hardness (η)45, chemical softness (σ)46, electrophilicity index (ω)47, ionization potential (IP)48, electron affinity (EA)49 and electronic charge transfer (∆Nmax). For the sake of their analysis, HOMO–LUMO band gap is utilized. The Eqs. (S1)–(S8) aid in theoretical calculations of GRPs for the INDTR and INDTD1–INDTD8 compounds. The values obtained in eV are displayed in the Table 2.

Table 2  Ionization potential (IP), electron affinity (EA), electronegativity (X), chemical potential (μ), global hardness (ƞ), global softness (σ) and global electrophilicity (ω).
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The results referred in the Table 2 indicate that among all candidates, INDTD5, INDTD7 and INDTD8 are the most electronegative compounds with values of X as 4.799, 4.807 and 4.810 eV, respectively. Moreover, their acceptor nature is further confirmed via minimum chemical potential (μ) values as − 4.799, − 4.807 and − 4.810 eV, respectively which also confirms the highest charge transfer (CT)50. In order to predict their reactivity, global hardness (ƞ) and global softness (σ) are significant parameters which are inversely related to each other. Molecules with higher values of σ and lower values of ƞ show least HOMO–LUMO energy gap, least stability and utmost reactivity51. Interestingly, maximum values of σ are obtained for the above-mentioned chromophores as 0.457, 0.458 and 0.457 eV−1, accordingly. Contrarily, minimum ƞ values for INDTD5, INDTD7 and INDTD8 are 1.094, 1.092 and 1.095 eV, respectively.

Other intriguing properties are electrophilicity index (ω) and electronic charge transfer (∆Nmax) both of which determine the molecules’ ability to absorb electronic charge from the environment52. Molecules with higher values of these parameters are likely to act as an electrophile and vice versa53. It is observed for the currently studied chromophores that highest values of ω and ∆Nmax are obtained for INDTD5, INDTD7 and INDTD8 among all. Their ω = 10.528, 10.583 and 10.564 eV and ∆Nmax = 4.387, 4.405 and 4.393 eV, respectively. Furthermore, the electron affinity (EA) and ionization potential (IP) are indicative of their redox potentials and show highest values for the afore-said designed molecules IP = 5.892, 5.898 and 5.905 eV, while EA = 3.705, 3.715 and 3.715 eV, respectively54. Concluding the discussion, the proposed molecules tend to be the reasonable solar cell candidates.

UV–Visible analysis

To estimate the optical properties of the studied chromophores (INDTR and INDTD1–INDTD8), their UV–Visible analysis is performed at the afore-mentioned level of TD-DFT i.e., M06/6-31G(d,p) level in gas and solvent (chloroform) phases. This includes the interpretation of absorption wavelength (λmax), oscillator strength (fos), excitation energy (E) and molecular orbital assessments of the titled compounds55. The molecules which depict high oscillation frequency along with intense absorption bands are rendered as suitable candidates for solar cell utility owing to their efficient charge transfer56. Table 3 shows the representative values of absorption wavelengths (λmax) along with their corresponding features for all the proposed molecules. While, Tables S14–S31 mentioned in the supplementary information show other values for individual compounds. Figure 4 demonstrates the UV–Visible spectra for the titled chromophores.

Table 3 Wavelength (λmax), excitation energy (E), oscillator strength (fos) and major molecular orbital assessments of compounds (INDTR and INDTD1–INDTD8) in gaseous and solvent phases.
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Figure 4
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UV–Vis plots for the investigated chromophores.

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The data mentioned above clearly shows that the absorption maxima (λmax) of designed compounds (INDTD1–INDTD8) are significantly higher than the reference compound (INDTR). The reason is the end-capped substitution in the derivatives with extended acceptors which possess electron-withdrawing substituents. The absorption maxima of designed derivatives lie in the visible region in chloroform (735.937–762.318 nm) as well as in the gaseous phase (710.384–729.571 nm).

The key optical characteristics such as absorption maxima, oscillator strengths and overall optical properties are greatly influenced by the solvent’s environment. Polar solvents tend to stabilize the excited states of chromophores more than non-polar solvents, causing a red shift, where the absorption peak shifts to longer wavelengths. Conversely, a blue shift is dictated by non-polar solvents where the absorption peak shifts to shorter wavelengths due to lesser stabilization of the excited states. The oscillator strengths (fos) are influenced by the interaction of solvent’s viscosity and polarity which plays a key role in determining the transition probabilities in a molecule. A higher viscosity changes the solvent's dielectric constant and refractive index which may prolong the optical gain, altering the linear optical properties57. Thus, the solvent characteristics can either enhance or diminish transition probabilities in the absorbing molecules. The overall optical properties of solute molecules including the absorption spectra and emission characteristics are greatly influenced by the solvent used. Specific solute–solvent interactions can also modify the solute's electronic states, resulting in varied absorption characteristics. For example, the absorption peak intensities increase via the hydrogen bonding between polar solvents and solutes58.

Chloroform is known as the moderately polar solvent in theoretical UV–Vis spectroscopy for several reasons such as low UV absorption. This ensures its minimal interference with the absorption bands of the studied compounds. Moreover, chloroform is transparent to UV–Visible light over a wide range of wavelengths, which allows for accurate measurement of absorbance for the studied compounds without background interference from the solvent. It also shows good solubility for many organic compounds and is relatively stable. Furthermore, it is non-reactive towards many organic molecules and is easy to handle which makes it a preferred choice for this study59. Overall, the chloroform phase demonstrates more prominent results as compared to the gas phase.

In gaseous phase, the reference compound (INDTR) shows λmax = 698.498 nm with excitation energy (E) as 1.775 eV. The oscillation strength (fos) of INDTR is 2.607 along with H → L transition of 95%. The derivatives (INDTD1–INDTD8) show λmax as 710.384, 711.607, 717.330, 717.164, 729.056, 716.998, 729.400 and 729.571 nm, respectively. They show correspondingly lower E as 1.745, 1.742, 1.728, 1.729, 1.701, 1.729, 1.700 and 1.629 eV. Moreover, their fos are depicted as 2.617, 2.642, 2.714, 2.717, 2.742, 2.698, 2.726 and 2.694, accordingly. Furthermore, all derivatives show H → L transition as 94% except INDTD1 and INDTD2 which show 95% contribution. Overall, the investigated chromophores show the following increasing trend for the absorption maxima (λmax) in the gaseous phase in nm: INDTR < INDTD1 < INDTD2 < INDTD6 < INDTD4 < INDTD3 < INDTD5 < INDTD7 < INDTD8.

For chloroform, the λmax values for the investigated compounds are higher than gaseous phase which are as follows: 724.710, 735.937, 738.084, 745.003, 744.689, 760.354, 745.720, 760.914 and 762.318 nm for INDTR and INDTD1–INDTD8, correspondingly. Similarly, their excitation energy values are lower than those obtained in gaseous phase: 1.711, 1.685, 1.680, 1.664, 1.665, 1.631, 1.663, 1.629 and 1.626 eV for INDTR and INDTD1–INDTD8, respectively. Moreover, they show oscillation strengths as 2.851, 2.869, 2.885, 2.943, 2.940, 2.958, 2.917, 2.938 and 2.909, respectively. The H → L transitions show 94% for INDTR, 93% for INDTD1–INDTD4 and INDTD6, 92% for INDTD7 and 91% for INDTD8 compound. The λmax in case of chloroform solvent are obtained in the following increasing trend for the absorption maxima (λmax) in gaseous phase in nm: INDTR < INDTD1 < INDTD2 < INDTD4 < INDTD3 < INDTD6 < INDTD5 < INDTD7 < INDTD8. The results suggested that INDTD7 and INDTD8 compounds show the red-shifted absorption values as compared to other derivatives. Hence, they are considered as reasonable candidates for solar cell devices.

Charge transfer (CT) analysis

The charge transfer analysis is performed by utilizing the NFA compound (INDTD7) in conjugation with the selected donor polymer (PTB7) as can be seen in Fig. 5.

Figure 5
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Charge transfer analysis for PTB7 (donor): INDTD7 (acceptor) complex.

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For the present study, PTB7 donor polymer is incorporated with INDTD7 compound to form the donor:acceptor complex which is shown in the Fig. 6. Among all the derivatives, INDTD7 is utilized to observe the charge transfer phenomena owing to its least energy gap and highest bathochromic shift. The Fig. 6 shows that the charge coherence is largely concentrated over the HOMO for PTB7 and in case of INDTD7, the LUMO shows dominant charge density. Thus, it is clearly observed that the ample charge transfer is seen via the HOMO to LUMO among the donor:acceptor complex.

Figure 6
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DOS pictographs of INDTR and INDTD1–INDTD8.

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Density of states (DOS)

Density of State is a valuable tool for analyzing the charge distribution around HOMOs and LUMOs of the examined compounds and assists in verifying the findings of FMOs60. This study is often conducted to forecast the percentage contribution of each component of a molecule to the overall distribution of electronic charges and facilitates in the determination of intramolecular charge transfer61. In a DOS map, the HOMO region of a compound is represented by left  side energy values, while the LUMO region is represented by right side energy values along the x-axis. The central area between these two regions is the energy gap and is similar to the bandgap of FMOs62. To explicate the DOS analysis, the studied chromophores are fragmented into the Acceptor-1 (peripheral acceptor unit), Acceptor-2 (core unit) and the \(\pi\)–spacer. These fragments are represented by blue, red and green lines, respectively, as illustrated in the Fig. 6.

Moreover, this study reveals that electrons are distributed from HOMO, which has a high capacity for donation, to LUMO, which prefers to accept them63. The modification of peripheral acceptor moieties alters the pattern of charge distribution, as indicated by the HOMO–LUMO percentages of DOS displayed in the Table S32. In this study, Acceptor-2 corresponds to 39.5, 38.9, 39.5, 39.9, 39.9, 41.6, 40.1, 42.1 and 42.0% for the HOMO, while 44.7, 42.9, 40.7, 38.2, 38.0, 30.2, 37.1, 29.5, and 27.9% for the LUMO in INDTR and INDTD1–INDTD8, accordingly. Similarly, the \(\pi\)–spacer contributes 45.9, 46.5, 46.0, 45.5, 45.5, 44.0, 45.4, 43.6 and 43.7% towards the HOMO and 25.4, 27.5, 27.7, 28.0, 27.9, 28.1, 27.9, 28.3, and 27.5% towards the LUMO for INDTR and INDTD1–INDTD8, respectively. Further, the terminal Acceptor-1 exhibits distribution percentages of 14.6, 14.6, 14.5, 14.6, 14.6, 14.4, 14.5, 14.3, and 14.3% for the HOMO, while 29.9, 29.6, 31.6, 33.8, 34.1, 41.7, 35.0, 42.1 and 44.6% for the LUMO in INDTR and INDTD1–INDTD8, correspondingly. According to the preceding discussion, the highest charge density in HOMOs is localized over the selenophene based \(\pi\)–spacer as well as over central Acceptor-2 of all designed chromophores, whereas the charge density in LUMOs is mostly centered over the Acceptor-1. Furthermore, these \(\pi\)–spacer play a crucial role in the charge transfer process. In a nutshell, the DOS graphs show that a significant charge from the central Acceptor-2 towards the terminal Acceptor-1 is shifted via the selenophene bridges in all the investigated compounds.

Transition density matrix (TDM)

TDM is an efficient tool which provides qualitative details about the 2-D movement of electrons, exciton dynamics and the extent of coherence within conjugated molecular structures64. It also investigates the nature of transitions particularly from the ground state (S0) to excited state (S1), degree of delocalization within the molecules as well as the interaction between charge donating and accepting moieties throughout the transition65. The TDM plots for INDTR and INDTD1–INDTD8 are generated using the TD-DFT computations at M06/6-31G(d,p) functional and the results are illustrated by using the Multiwfn software. The influence of hydrogen atoms is disregarded due to their minimal impact in transitions. In these TDM graphs, the bottom x-axis represents the number of individual atoms, while the density of individual electrons is depicted by color bars on the right y-axis.

To have a better understanding, the molecules are categorized into three fragments: end-capped acceptor (A1), core acceptor (A2) and the \(\pi\)–spacer. All molecules (INDTR and INDTD1–INDTD8) exhibit charge transformation from the central core (A2) towards the terminal charge accepting (A1) moieties. The \(\pi\)-linker unit facilitates this process without causing any interference. It is observed that charge density is primarily localized on the central A2 unit, moderately distributed on the π–spacer and shows minimal coherence at the terminal A1 unit in all studied compounds in both diagonal and off-diagonal regions. Moreover, substantial charge scattering occurs along the diagonal paths (see Fig. 7). The increasing order of interaction coefficients for charge donating and accepting units is found as: INDTD7 < INDTD5 < INDTD8 < INDTD6 < INDTD4 = INDTD3 < INDTD2 = INDTD1 < INDTR. Due to lesser hole–electron coupling, INDTD7 has a lower coefficient of interaction than other modified compounds and exhibits more substantial exciton dissociation in the excited state. This allows for faster excitation and elevated charge transmission rate. The interaction coefficient of INDTD7 is comparable to that of INDTD5. This is due to the fact that INDTD5 also has lower coupling of exciton which may lead to greater charge dissociation rate. Consequently, all designed compounds reveal momentous charge dissociation than the INDTR compound. This implies that the studied compounds (INDTR and INDTD1–INDTD8) are versatile and unique which are capable of developing good OSCs.

Figure 7
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Transition density matrix maps of INDTR and INDTD1–INDTD8 compounds.

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Exciton binding energy (Eb)

The exciton dissociation energy (EDE) is a crucial factor in determining the efficiency of photovoltaic devices. It is the amount of energy needed to separate excitons into holes and electrons. The columbic force between holes and electrons is calculated by determining the binding energies of the investigated compounds. A smaller value of Eb indicates weaker columbic interactions and a significant charge transfer66. The Eq. (1)67 is used to calculate the Eb values and the outcomes are shown in the Table 4.

$$E_{{\text{b}}} = E_{{{\text{gap}}}} - E_{{{\text{opt}}}} .$$
(1)
Table 4 Calculated binding energies of the studied compounds.
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It is noted that the exciton binding energy (Eb) is the difference between the energy gap (Egap) and the first excitation energy (Eopt). Here, Egap represents the bandgap between the HOMO and LUMO, while Eopt signifies the minimum energy required for the first excitation67,68,69.

The Eb values of INDTR and INDTD1–INDTD8 are 0.577, 0.575, 0.575, 0.573, 0.573, 0.556, 0.571, 0.554 and 0.564 eV, respectively. The escalating trend of Eb for all the studied compounds is as follows: INDTD7 < INDTD5 < INDTD8 < INDTD6 < INDTD4 = INDTD3 < INDTD2 = INDTD1 < INDTR. It is noted that compounds having Eb values less than 1.9 eV are regarded as highly efficient materials for NF-OSCs69. Interestingly, all tailored chromophores reveal lower exciton binding energy values than 1.9 eV and also lesser than the reference chromophore. The aforesaid chromophores with low Eb enable easier exciton dissociation and rapid charge flow which in turn increases current charge density and makes them favorable optoelectronic materials. Therefore, among all modified compounds, INDTD7 and INDTD5 have lower Eb values and exhibit efficient charge mobility and photovoltaic properties owing to their greater tendency of exciton dissociation.

Hole–electron analysis

Hole–electron analysis is effectively performed using the Multiwfn 3.7 software29 in order to determine the mobility of holes and electrons in the studied compounds70. The dark colors represent the highest density regions while the light colors indicate lowest density regions on vertical axis in the entitled graphs. Figure 8 shows that the hole intensity in the INDTR compound is highest at the C23 and C25 atoms in the π–spacer region, while the electron intensity is maximum at C45 and C46 atoms within the end-capped acceptor. Nevertheless, it is observed that in all modulated compounds hole intensity is maximum at different atoms of carbon present in the π-linker unit, while electron intensity is higher at various atoms of carbon present in the terminal acceptor region owing to the presence of robust electron withdrawing moieties attached to the terminal acceptors.

Figure 8
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Electron and hole maps of INDTR and INDTD1–INDTD8 compounds.

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Remarkably, in INDTD8 chromophore, where the sulphonic acid (–SO3H) groups are substituted with nitro (–NO2) groups, enhances electron density at the nitrogen atoms and significant charge transfer occurs due to resonance and robust electron-withdrawing effects. In general, INDTR and INDTD1–INDTD4 chromophores appear to be electron-type materials, while INDTD5–INDTD8 chromophores emerge to be hole-type materials. This is because maximum charge density is observed in both electronic as well as hole bands as presented in the Fig. 8. Furthermore, charge transfer in the studied compounds is manifested by the electron–hole coupling. The descending trend of charge displacement via electron–hole coupling in all the examined compounds is as follows: INDTD8 > INDTD7 > INDTD5 > INDTD6 > INDTD4 > INDTD3 > INDTD2 > INDTD1 > INDTR. This declining pattern indicates that electron–hole coupling is greater in INDTD8, INDTD7 and INDTD5 chromophores than the INDTR chromophore. Consequently, all designed derivatives exhibit significant charge transfer and therefore regarded as efficient optoelectronic materials.

Open circuit voltage (Voc)

The open-circuit voltage (Voc) is known as the maximum voltage delivered by a solar cell device to an external load at zero current71. One of the challenging aspects to achieve high-performance OSCs is to obtain high open-circuit voltage (Voc). It is known from the literature analysis that the most efficient OSCs have attained the Voc in the range of 0.8–0.9 V along with a large energy loss of > 0.5 eV72,73. Moreover, the Voc plays a key role in determining the power conversion efficiency (PCE). Hence, to achieve high PCE, the understanding of Voc and Eloss is indispensable74,75. Various influencing parameters are present which can affect the Voc among which the most revealing are the distribution of charge transfer (CT) states, DOS distribution, infra-structure and donor–acceptor (D–A) interface area. There are certain scaling factors for the measurement of Voc for a solar cell. These are HOMO of the donor and LUMO of an acceptor compound. Mathematically, the Voc is measured as the energy difference of the HOMO of donor and the LUMO of acceptor stated by the Scharber’s Equation76.

$${V}_{oc}=I/e(\left|{\text{E}}_{\text{HOMO}}^{\text{D}}\right|-\left|{\text{E}}_{\text{LUMO}}^{\text{A}}\right|)-0.3.$$
(2)

The 0.3 is a constant value which is known as voltage drop77.

In this study, the characterization of INDTR and INDTD1–INDTD8 are performed to observe their Voc at the afore-said level of DFT. According to the molecular nature of the investigated molecules, PTB7 donor polymer is selected for establishing the donor–acceptor blend and to specifically align their energy levels. The EHOMO of PTB7 is − 6.467 eV and ELUMO is − 1.685 eV. Similarly, the energy gap between its HOMO and LUMO is 4.782 eV. The literature analysis shows that ELUMO of the donor must be higher that ELUMO of the acceptor for efficient charge mobility. Hence, ELUMO of the designed acceptors (INDTR and INDTD1–INDTD8) are as follows: − 3.505, − 3.506, − 3.539, − 3.579, − 3.576, − 3.705, − 3.583, − 3.715 and − 3.729 eV, respectively which are lower than that of PTB7. Figure S4 depicts the formation pattern of these chromophores with the donor polymer (PTB7). The calculated results show that these compounds exhibit significant Voc values. The reason for the prominent results is the end-capped tailoring with efficient acceptor moieties. The following decreasing order is obtained for Voc among the titled chromophores: INDTR > INDTD1 > INDTD3 > INDTD4 > INDTD3 > INDTD6 > INDTD5 > INDTD7 > INDTD8 (Table S33).

Literature shows that the theoretically calculated values of Voc for the non-fullerene acceptor compounds are reported higher than the experimental values of Voc as reported in the case of ternary OSCs incorporating 2TT unit i.e., thiophene [3,2-b] thiophene (TT) with Voc value of 1.04 V which is lower2 than theoretically calculated values of INDTD1–INDTD8 compounds. It is expected that this research  may facilitate in achieving improved optoelectronic and photovoltaic properties from the newly designed acceptor compounds. Moreover, this analysis may support their future synthesis for the next-generation photovoltaics.

Fill factor and power conversion efficiency

The performance of a solar cell (SC) device is largely examined by the key factors such as fill factor (FF), open circuit voltage (Voc) and short circuit current density (Jsc)78. Thus, the FF for the studied compounds is theoretically estimated by using the following Equation.

$$\text{FF}=\frac{\frac{{\text{eV}}_{\text{oc}}}{\text{KBT}}-{\text{ln}}\left[\frac{{\text{eV}}_{\text{oc}}}{\text{KBT}}+0.72\right]}{\frac{{\text{eV}}_{\text{oc}}}{\text{KBT}}+1}.$$
(3)

The parameter such as \(\frac{{\text{eV}}_{\text{oc}}}{\text{KBT}}\) signifies the normalized voltage. While, “e” is known as the elementary charge which is fixed at 1. KB and T are Boltzmann constant (1.380649 × 10−23 J per kelvin) and temperature (300 K), respectively.

Among the designed compounds, highest FF is obtained for INDTD2 (0.954) as compared to other derivatives as shown the Table S34. Similarly, another significant metric which is known as the power conversion efficiency (PCE) is calculated using the Eq. (4) which is as follows:

$$\text{PCE}=\frac{{J}_{\text{sc}}{V}_{\text{oc}}\text{FF}}{{\text{P}}_{\text{in}}}.$$
(4)

The \({\text{P}}_{\text{in}}\) is known as the incident power from approaching light source which is inversely related to the PCE of a photovoltaic material. It is known from the literature that \({J}_{\text{sc}}\) value largely influences the band gaps and charge transfer rates of a particular molecule79. From the above-mentioned Equation, a reasonable approximation of PCE is made which showed that it is found in a range of 31.77–36.91% and the highest PCE is observed for INDTD1 (36.91%). Moreover, all derivatives show comparable PCEs with the INDTR reference (36.63%). Thus, the proposed organic compounds are significant candidates for development of efficient photovoltaic materials.

Conclusion

In summary, new non-fullerene acceptors (INDTR and INDTD1–INDTD8) are proposed systematically to explore their optoelectronic and photovoltaic characteristics by utilizing the quantum chemical approach. The designed derivatives showed lower band gaps (2.183–2.269 eV) and bathochromic shifts in solvent such as 735.937–762.318 nm. Interestingly, the most significant results are obtained for INDTD5, INDTD7 and INDTD8 chromophores as compared to other compounds. The compounds (INDTD5, INDTD7 and INDTD8) showed least energy gaps (2.187, 2.183 and 2.190 eV, respectively) and binding energies (0.556, 0.554 and 0.564 eV, respectively) which are indicative of efficient charge mobility and higher exciton dissociation rates, respectively. Moreover, these chromophores also exhibited the highest absorption values in the UV–Vis spectra 760.354, 760.914 and 762.318 nm, correspondingly. Furthermore, the investigated molecules are blended with the donor polymer (PTB7) for the sake of photovoltaic insights and showed good results for Voc, FF and PCEs. The hole–electron analysis rendered effective charge mobilities in entitled chromophores. These DFT analyses for the studied compounds demonstrate that the end-capping strategy adopted for designing the indolonaphthyridine core-based NFAs is significant in attaining efficient photovoltaic features.