Coumarins are the phytochemicals, which belongs to the family of benzopyrone, that display interesting pharmacological properties[1]. Coumarins possess anti-inflammatory, antiallergic, hepatoprotective, antiviral, anticarcinogenic and anticoagulant activities. They also constitute an important group of organic compounds that are used as bio/ chemo sensors, laser dyes, pesticides, additives to cosmetics, as optical brightening agents, fluorescent markers and dispersed fluorescent and laser dyes[2–7]. Due to their inherent physicochemical and photophysical characteristics such as reasonable relative ease of synthesis, coumarin derivatives have been extensively investigated for electronic and photonic applications such as charge-transfer complexes, laser dyes, fluorescence whiteners, solar energy collectors, and non-linear optical materials[8, 9]. Coumarins, in particular the hydroxycoumarins (HCs), are well-known natural products, but they also occur in areas as diverse as sun screen formulations, laser dyes, pesticides, etc[6].

Hydroxycoumarin has several advantages due to its versatile chemical structure and biological properties. 4-hydroxycoumarin is a key precursor in organic synthesis, used in various pharmacological and physiological applications and can be modified to create a wide range of bioactive compounds such as anticoagulants, anticancer agents, antimicrobial agents, anti-inflammatory agent. 4HC and its derivatives possess fluorescent properties and used as fluorescent probes, optical brighteners and laser dyes[10].

Due to the diverse application of hydroxycoumarins several researchers have studied the photophysical behaviour of positional substituents of hydroxy-coumarin. Aaron et al. [11] observed the absorption and emission of some hydroxy substituted coumarins including 4HC and evaluated the dipole moment of the ground and first excited singlet-state using Bakhshiev, Kawski-Chamma-Viallet, McRae, and Suppan correlations. The first excited singlet-state dipole moments of the coumarins are noticeably higher than the corresponding ground-state values, indicating a substantial redistribution of the w-electron densities resulting in a more polar excited state. Similarly S. Kumar et al. [12] also used Solvatochromic shift method to determine excited state dipole moment experimentally for a set of 7-hydroxycumarine derivatives. The increase in dipole moment upon excitation has been explained in terms of the nature of emitting state and resonance structure. Fatima et al. [13] studied the effect of solvent polarity on the HOMO-LUMO gap of 7-hydroxycumarine which is reflected on the absorption and emission spectra of the compound. Moriya et al. [14] studied 7-hydroxycumarin absorption and emission in different solvents. On the basis of the solvent-dependent fluorescence of the fluorophores, they classified solvents into several groups: hydroxylic solvents, non-hydroxylic solvents with low dielectric constants, and those with high dielectric constants. In the hydroxylic solvent, protonation, deprotonation, and tautomerization were the main reactions, while in the non-hydroxylic solvent the formation of hydrogen bonding and ion pairs was essential. Some research groups have performed Quantum chemical studies on to evaluate excited state geometries of several hydroxycoumarins[13, 14] Jacquemin et al. [15] have performed the configuration interaction singles (CIS) and TD-DFT study of excited-state structures of several hydroxy coumarin dyes. Cerón-Carrasco and his research group [16], investigated the solvatochromic effects on the optical spectra of a typical hydroxy coumarin using TDDFT approach, considering its enol, keto, anionic and cationic forms. It has been the solvent response due which there was large increase in dipole moment upon excitation while hydrogen bonds tune both absorption and emission energies.

In recent years many scholars studied NLO and NBO properties of many compounds [17–19] as these properties are important for their practical uses. NLO properties like hyperpolarizability is curtail for various application including telecommunication, and optical computing, for materials used in optical devices. NBO properties provides a detailed understanding of molecular structure and electronic interactions. It helps identify charge transfer, hyperconjugative interactions and other factors that contribute to NLO behaviour. Organic materials are found to possess superior second order nonlinear optical properties compared to the more traditional inorganic materials. This property together with the inherent ultrafast response time and enumerable structural variations of organic materials have drawn sizeable amount of research interest in organic nonlinear optical (NLO) materials. Molecules that show asymmetric polarization induced by electron donor and acceptor groups in pi-electron conjugated molecules are candidates for electro optic and NLO applications, such as frequency doubling or second harmonic generation (SHG) [20]. Theoretical calculations, like DFT are used to predict these properties, helping researchers design and optimize materials for specific applications.

Considering the importance of 4HC as biological and photochemical active probe, it is necessary to investigate the information about its structural and photophysical behaviour. Therefore, this research work emphasizes on understanding the molecular properties of 4-hydroxycumarine and the effect of solvents on its electronic transitions. This study utilizes the density functional theory (DFT) and time-dependent density functional theory (TD-DFT) to determine the spectral behaviour and photophysical properties such as the natural bonding orbital (NBO) and the NLO (non-linear optical) properties (polarizability, first-order hyperpolarizability and dipole moment) of 4-hydroxycumarin.

Geometry optimization is an important step in computational chemistry that involves finding the most stable structure or lower-energy configuration of a molecule. The molecule 4-hydroxycumarine (4HC) was optimized by utilizing the DFT method with B3LYP functional and 6-311 + + G(d,p) basis set. The optimized geometric form corresponding to the local minima without imaginary frequency were confirmed using frequency calculations. Figure 1 represents the optimized structure of 4HC in gas phase. The bond lengths of 4HC (gas phase) in ground and excited state are tabulated in Table 1. In excited state, the bond O1-C5, O1-C12, C5-C8, C6-C9, C7-C10, C8-C11 get elongated while the bond C9-C12, O3-C12 and C10-C11 get shorten. Georgieva et al. [31] also reported the same behaviour of 7-hydroxy-4-methylcoumarin (7H4MC) in the gas phase when it shows deprotonation behaviour. Mir [32] reported the ground state bond lengths of coumarin in gas phase and compared them with experimental values and found that they are in close agreement with each-other.

The theoretically computed HOMO-LUMO energy levels are represented in Fig. 2. The HOMO, LUMO energies and band gap (ΔE) of the 4HC molecule are calculated as -6.82 eV, -2.08 eV and 4.75 eV respectively in the gas phase. Energy Gap increases with increasing solvents polarity. Charge density of 4HC in HOMO is located near oxygen of carbonyl group (C = O), whereas in LUMO charge density shifts to the center of adjacent rings of coumarin moiety.Mir et al. [32] reported the experimental band gap of coumarin in DMSO is 4.755 eV.

Thermodynamic properties are the characteristics that describe and ensure the thermal stability of a molecule. The thermodynamic parameters of 4HC such as total energy ()), specific heat capacity (), entropy (S) etc. were estimated using frequency calculations by employing B3LYP/6-311 + + G(d,p) theory at 298 K in the ground state (GS) and tabulated in Table 2.

The heat capacity at constant volume (), total thermal energy () and entropy (S) at different temperatures have been calculated by the same approach to reveal the temperature dependency of these thermodynamic properties and are reported in Table 3. The temperature dependence of for 4HC is shown in Fig. 3 along with polynomial (quadratic) fitting. Figure 3 shows that on increasing the temperature from 50 to 1000 K, all are increasing because the molecular vibrational intensities increase with temperature. The obtained thermodynamic parameters vs. temperature were fitted using Quadratic formulas and the resulting fitted regression parameters ( ) is all equal to 0.99 for all three parameters i.e. . The corresponding fitting equations for 4HC are as follows,

Global reactivity parameters such as ionization potential (IP), chemical potential (µ), electron affinity (EA), electronegativity (χ), electrophilicity index (ω), electron-donating (ɷ−) and electron-accepting (ɷ+) power and hardness (η) are crucial in exploring the chemical reactivity of molecules in different surroundings and getting certain features associated with the reactions. Reactivity parameters are calculated using HOMO and LUMO energies as per Koopman’s theorem[33]. The value of ionization potential (IP) indicates that 4HC have electrophilic behavior while chemical potential (µ) indicate it is chemically active molecule.

The Molecular Electrostatic Potential (MEP) is an important concept in computation chemistry to analyze the nucleophilic and electrophilic sites of the molecule and to understand the reactivity of the molecule [34]. The MEP map is usually represented as a colour-coded map, where different colours represent different electrostatic potentials. Orange to red is the region with the most negative electrostatic potential, and sky blue to blue is the region with the most positive potential. The MEP map of 4HC was generated using the DFT/B3LYP/6-311 + + G(d,p) level of theory and is shown in Fig. 4 with a color range of -7.871 a.u. to 7.871 a.u. In Fig. 4, the maximum yellow color region (maximum negative electrostatic potential) mainly around the oxygen atom (O3) of carbonyl group, it is electron-rich and preferred region for an electrophilic attack [35]. The blue color region; positive electrostatic potential is mainly over the hydrogen atom (H18) of OH group, which is electron- predominant area of a nucleophilic attack. The MEP map is very convenient in the exploration of biological recognition mechanisms and intermolecular hydrogen bonding interactions [36].

The effect of solvent polarity on the absorption and emission of the chosen was examined by calculating absorption and emission using IEFPCM model along with the TD-DFT calculations utilizing CAM-B3LYP functional at 6-311 + + G(d,p) basis. In the present study, water, ethanol (EtOH), methanol (MeOH), dimethyl-sulfoxide (DMSO), acetonitrile (ACN), tetrahydrofuran (THF), benzene, toluene, and cyclohexane solvents were selected for the spectral analysis. Theoretically calculated vertical transitions, corresponding excitation energy, oscillator strength (OS) and contributions in gas phase are summarized in Table 5. The strongest absorption band with S0→S1 as most probable transition appears at 269 nm in the gas phase.

Figure 5 displays the simulated UV–Vis spectra of 4HC in the selected solvents as well as in the gas phase. The main electronic transition band of 4HC in the UV–vis region is found to be between 268 nm to 271 nm depending on the solvents. The absorption spectra depend on the polarity of the solvent used.

To authenticate the computed spectral properties, the theoretically calculated results were compared with the previously reported work done by Aaron et al. [11]. They reported that the ground state dipole moment of 4HC is 5.0 D while that of excited state is 7.04 D and 4HC shows multiband absorption in the region 260–314 nm in different solvents. In all solvents there is a peak around 290 nm. The theoretically calculated results correlate well with the experimental results.

For the emission spectra, the optimization of the lowest excited state (S1) of 4HC in the gas phase and in solvents was performed using the TD-DFT approach. The obtained results are listed in Table 6. The change in transition dipole moments () between the excited singlet and ground state of 4HC in various solvents were calculated using the following relation [37] and are tabulated in Table 6.

Where is the maximum energy of absorption in cm^-1 and is the oscillator strength.

The transition dipole moment dictates whether a transition between two quantum states is possible and, if so, how likely (intense) that transition will be. It quantifies the coupling between a molecule (or atom) and the electric field of incident electromagnetic radiation (light). a transition dipole moment of 2.9⁓3.0 D signifies a strong, highly allowed spectroscopic transition due to a large, effective quantum mechanical change in charge distribution.

The wavelength of emission maximum was found to depend on the solvent polarity as represented in Table 6 and vary from 311 to 319 nm on increasing the solvent polarity. In gaseous phase it is found to be 312 nm. The first excited singlet-state dipole moment of 4HC in different solvents is higher than the corresponding ground-state values (Table 6), indicating a substantial redistribution of the π-electron densities resulting in a more polar excited state.

Aaron et al. [11] reported the experimental emission of 4HC at 370 nm, 390 nm and 395 nm in ethanol, ACN and in DMSO respectively. The emission peak calculated in the present work shows a variation from the experimentally reported results. The variation in experimental and theoretical result of emission spectra may be because of experimental observations depend on various physical parameters such as solute-solvent interaction, temperature of the surroundings, concentration of solute and hydrogen bonding ability of solvents, etc., whereas in computational calculations solute–solvent interactions are not considered.

The theoretically calculated Stokes-Shift is related to solvent polarity function. For present study Lippert-Mataga polarity function () [38] was utilized.

Where ε being the static dielectric constant and n the refractive index of the solvent. The larger the polarity of solvent, larger the Stokes-shift i.e. spectra show red shift with increasing polarity (Fig. 6).

The fluorophores that possess charge transfer characteristics are very useful in optoelectronic devices and are greatly influenced by the radiative (or excited state) lifetime. It is expected that fluorophores with considerably longer lifetimes will demonstrate effective electron injection and charge transfer. The radiative lifetime of the fluorophore is calculated by using the Eq. [39]-

where is the absorption wavenumber and is the oscillator strength.

The radiative life-time of 4HC in different solvent vary from 4.2 to 4.6 ns. This result is close to that reported by Silva et al. [40], the 4-HC has lifetime of the order of ns (0.026 ns to 10 ns).

Light harvesting efficiency (LHE) predicts the ability of organic compounds to absorb photons and then inject photoexcited electrons into the conduction band of semiconductors [39]. LHE is estimated using the equation

The calculated values of LHE of 4HC (using theoretical values of OS) in different solvents have been found in the range of 0.42–0.45. The Light Harvesting Efficiency (LHE) of a molecule significantly influences the performance of organic solar cells (OSCs). A higher LHE directly increases the short-circuit current density (J_SC), which ultimately enhances the overall efficiency of the device[41–43]. For the molecule 4HC, the experimentally obtained LHE value in water is approximately 0.43. This relatively high LHE suggests that 4HC is an efficient light absorber, capable of converting 43% of the absorbed light energy into usable excited states that facilitate effective charge transfer. This characteristic makes 4HC a promising component for highly efficient OSCs.

After geometry optimization, static dipole polarizability (α), first-order hyperpolarizability (β) and second-order hyperpolarizability (γ) values of 4HC was calculated using density functional theory (DFT) at B3LYP/ 6-311 + + G(d,p) level of theory. The value of hyperpolarizability is a measure of NLO activity of the molecular system. It is associated with intra-molecular charge transfer that is attributed to electron cloud movement through π-conjugated framework of electrons. The electron cloud is capable of interacting with an external electric field and is found to increase the asymmetric electronic distribution in either or both the ground and excited states, thus leading to an increased optical non-linearity [44]. First hyperpolarizability is a third rank tensor that can be described by a 3 ×3 × 3 matrix. The 27 components of the 3D matrix can be reduced to 10 components due to the Kleinman symmetry[45]. The components of hyperpolarizability are useful to understand charge delocalization in the molecule.The molecular polarizability and hyperpolarizability tensors related to the dipole moment induced in an isolated molecule by the applied electric field is given by [46],

where, are the indices referring to the molecular coordinate system, is polarizability, and are the first- and second-order hyperpolarizability and static dipole moment.

The polarizability , first-order hyperpolarizability (β) and second order hyperpolarizability are calculated using the x, y and z components from following equations [44, 47–51]-

Where,

Frequency dependent first-order hyperpolarizability(β)

Where,

Second-order hyperpolarizability (γ)

The nonlinear optical properties such as electro optic Pockels effect (EOPE) , the Second Harmonic Generation of first hyperpolarizability (SHG) , static , electric field induced second harmonic generation (ESHG) and dc Kerr effect static second hyperpolarizability are reported in Table 7, Table 8 and Table 9. Using the ESHG γ ( -2ω; ω, ω, 0) and dc Kerr static second hyperpolarizability γ (-ω; ω,0,0), the degenerate four wave mixing (DFWM) γ (-ω; ω, -ω, ω) and frequency dependent quadratic refractive index () are calculated.

frequency dependent quadratic refractive index ()[52]-

The polarizabilities and hyper polarizabilities are calculated in atomic units (a.u.), the calculated values have been converted into electrostatic units (esu) (α: 1a.u.= esu, β: 1 a.u.= esu, γ: 1 a.u.= esu)[51]. The static hyperpolarizability (β) and polarizability(α) are presented in Table 7. The frequency dependent first order hyperpolarizability (β) in Table 8 and second order hyperpolarizability is presented in Table 9.

The magnitude of the molecular hyperpolarizability (β) is one of important key factors in a NLO system. Urea is one of the prototypical molecules used in the study of the NLO properties of molecular systems and frequently used as a threshold value for comparative purposes. The computed first hyperpolarizability(β_static) of 4HC molecule is 12.25 cm⁵/esu by B3LYP methods. Theoretically, the first-order hyperpolarizability(β) of 4HC molecule is higher than the magnitude of urea (β of urea is 4.5 cm⁵/esu reported by Ledoux et al. [53]). Thus, this molecule might serve as a prospective building block for NLO materials. The dynamic γ_static values for the title molecules is 17.75esu slightly higher than the cubic hyperpolarizability of para-nitroaniline (p-NA) (γ _p-NA = 12.71×10^-36 esu) given in [54]. Thus, 4HC shows good NLO response. The degenerate four wave mixing (DFWM) γ (-ω; ω, -ω, ω) and frequency dependent quadratic refractive index () values are 32294.67 au and respectively for 4HC.

Nonlinear refraction is a key nonlinear optical mechanism in isotropic media, including all gases, liquids, and a large class of solids. In dielectric media, nonlinear refraction causes an intensity-dependent increase of the index of refraction, which gives rise to spectral broadening and is the basis for nearly all femtosecond pulse compression mechanisms[52].

Natural bonding orbital (NBO) study provides an appropriate framework for investigating the transfer of charge and intra- or inter-molecular bonding in molecular systems[55]. The stable interaction between donor and acceptor involves the electron density delocalization of occupied and unoccupied NBOs [44]. The NBO analysis for 4HC at DFT/B3LYP/6-311G++(d,p) level of theory was performed in order to elucidate the conjugation, hyper-conjugation and delocalization of electron density within the molecule. The bond type, occupancy, electron density, hybridization and their corresponding characters of NBOs for 4HC are tabulated in Table 10. The atom O is considered to be highly electro-negative as the Pc (polarization coefficient square) value is large. For example, the NBO σ(O1-C5) is established by the contribution of 67.86% of the electron density of the O1 and 32.14% of the C5 atom. This NBO is originated by the connection of sp^1.87 (P_C = 0.82 and p-character: 65.09%) of the O1 atom and sp^3.14 (P_C = 0.57 and p-character: 75.65%) of the C5 atom.

The stabilization energy of a molecule in NBO study is calculated using second-order perturbation theory and is defined by the given equation:

where, signifies stabilization energy, symbolizes orbital occupancy of the donor; ( ) and is the off diagonal and diagonal NBO Fock matrix elements[56]. The more the value of , the stronger is the association between electron acceptor and donor moieties and a higher degree of conjugation throughout the entire molecular system[17, 57].

The stabilization energy for few donor-acceptor interactions was tabulated in Table 11. The most significant NBOs are π(C6-C9) →π*(O3-C12), LP(O3) →π*(O1-C12), LP(O2) →π*(C6-C9) with stabilization energy 24.14, 38.23, 34.46 Kcal/mol respectively. When the donor–acceptor interactions arise within the molecular framework, the occupancies and energies of donor orbitals reduce and charge transfer interactions occur. The NBOs interactions probably arise because of the delocalization of π electrons from one NBO to another or delocalization of lone pair charge density and hence charge transfer interactions occur within the 4HC molecule, which stabilizes the molecular framework.

The detailed analysis of 4HC's structural, electronic, and optical properties suggests its promising use in many fields. The relatively high LHE value is a crucial indicator that 4HC can efficiently absorb photons and convert light energy, making it a good candidate for active materials in Organic Solar Cells (OSCs). The combination of its lifetime and band gap energy suggests that 4HC can be engineered to manage charge carriers effectively, making it suitable not only for solar cells but also for wide bandgap power devices where stable, high-efficiency energy conversion is required. The molecule exhibits properties favorable for altering light signals, which is the basis for NLO applications. The study's finding that 4HC has favorable Nonlinear Optical (NLO) parameters suggests it can be used in devices that require a non-linear response to intense light. These applications typically include optical switching, optical data storage, frequency doubling, and other advanced photonics technologies. The NBO analysis confirmed that charge transfer interactions contribute significantly to stabilizing the molecular system. This stability is desirable in functional materials where robust and repeatable performance is essential, especially in electronic or optoelectronic devices.

The small energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) suggests that 4HC is a reactive molecule, which can be beneficial in certain catalytic or synthetic pathways. The MEP map showing the O3 oxygen atom as an electron-rich center pinpoints the preferred region for electrophilic attack. This information is invaluable for chemists attempting to synthesize novel derivatives of 4HC with tailored properties.

The molecular structure of 4-hydroxycumarine (4HC) was thoroughly investigated using DFT and TD-DFT theories to evaluate its potential across several advanced technological fields. The significant structural, molecular and thermodynamic parameters were obtained. The small energy gap suggests that it is a reactive molecule. MEP map shows that oxygen atom (O3) is electron-rich and it is a preferred region for an electrophilic attack. Depending on the medium, the calculated absorption maxima of 4HC lies in the range 268–271 nm and emission maxima in the range of and 313–319 nm, respectively.

Key findings indicate that 4HC is highly promising for optoelectronic applications, primarily due to its relatively high Light Harvesting Efficiency (LHE) and favorable electronic parameters (lifetime and band gap energy), making it a strong candidate for active materials in Organic Solar Cells (OSCs) and wide bandgap power devices. Furthermore, the calculated Nonlinear Optical (NLO) parameters suggest its utility in advanced photonics technologies like optical switching and data storage. Chemically, the molecule is predicted to be reactive due to a small HOMO-LUMO energy gap, and the MEP map identifies the O3 oxygen atom as the electron-rich site for electrophilic attack, providing crucial guidance for the synthesis of new derivatives. Finally, NBO analysis confirmed that significant charge transfer interactions stabilize the molecular system, which is vital for robust performance in functional electronic and optoelectronic devices.