The studied system consisted of one FHC molecule solvated in a cubic box of methanol with an initial side length of 20 Å, containing approximately 313 solvent molecules [9] . The final box size was determined during the NPT equilibration phase to achieve the correct experimental density of methanol (0.791 g/cm³ at 300 K) [10] .

Molecular mechanics parameters (bonds, angles, dihedrals) for FHC were assigned using the GAFF2 (Generalized Amber Force Field 2) within the AMBER force field family [1] . The GAFF2 parameters were validated against ab initio calculations at the PBE0/cc-pVTZ level to ensure accuracy for hydrogen bonding interactions [2] . Dispersion corrections were incorporated using the DFT-D3 method to account for van der Waals interactions accurately [3] .

The atomic partial charges were derived quantum mechanically to ensure accuracy for electrostatic interactions. The electrostatic potential (ESP) was computed at the PBE0/cc-pVTZ level of theory on the ground-state (S₀) optimized geometry [4] . Atomic partial charges were then obtained by fitting the ESP using the restrained electrostatic potential (RESP) procedure [5] [6] .

The topology (prmtop) and initial coordinate (inpcrd) files were generated using the tleap module of AmberTools22 [7] . The solvent box was packed around the solute using Packmol, ensuring uniform solvation. Prior to dynamics, the system underwent energy minimization using the steepest descent algorithm to relax steric clashes and optimize the initial geometry, which is essential for numerical stability. Periodic boundary conditions were applied in all three spatial directions to mimic a bulk liquid environment [8] .

After energy minimization, the system was equilibrated in two phases, NPT Ensemble with 1 ns equilibration at 300 K and 1 atm to allow the density to relax to the correct experimental value and NVT Ensemble with subsequent 1 ns equilibration at 300 K using the average volume from the NPT phase [9] .

Production dynamics were then carried out in the NVT ensemble for >20 ns to ensure adequate sampling for dynamical properties.

All simulations were performed using the PMEMD.CUDA engine of the AMBER20 software suite [6] . Long-range electrostatic interactions were treated using the Particle Mesh Ewald (PME) method [7] with a 9.0 Å direct space cutoff. A Langevin thermostat with a collision frequency (gamma_ln) of 2.0 ps⁻¹ was used for temperature control, which provides robust thermal regulation suitable for dynamical properties [8] . The equations of motion were integrated with a 2 fs timestep, with all bonds involving hydrogen atoms constrained using the SHAKE algorithm [9] .

PIMD was employed to incorporate nuclear quantum effects (NQEs), which are critical for accurately modeling hydrogen bonding and proton delocalization in this system [2] [10] . The methodology is based on Feynman’s path integral formulation, where each quantum nucleus is represented by a ring polymer of P classical beads connected by harmonic springs, effectively mapping the quantum system onto an extended classical system [11] [12] .

In this study, we used P = 32 beads per atom, which provides a satisfactory compromise between quantum accuracy and computational cost for properties at 300 K [13] . All other simulation conditions (box size, number of solvent molecules, force field) were kept identical to the classical MD simulations to ensure a direct and consistent comparison.

A critical aspect of PIMD is efficient sampling of the quantum canonical ensemble. We applied the Path Integral Langevin Equation (PILE) thermostat to the normal modes of the ring polymers [14] [15] . This approach optimally thermostats the internal modes without perturbing the physical dynamics of the centroid, which is essential for accurate dynamical properties [16] .

The equations of motion were integrated using the multi-timestep RESPA algorithm [17] . The timestep (dt) was selected based on the fastest vibrational mode of the ring polymer. The fundamental period Tp is given by Tp = h/(kB * T * P). To ensure numerical stability and proper resolution of quantum fluctuations, the timestep was set to dt ≈ Tp/20 ≈ 0.5 fs [18] .

Before production, the PIMD system underwent careful equilibration under NPT conditions (300 K, 1 atm) using the same protocol as classical MD [19] . The system was considered equilibrated once potential energy and density had stabilized [20] .

The production simulation was conducted in the NVT ensemble using the average volume from PIMD-NPT equilibration [21] . The PILE thermostat and 0.5 fs timestep were maintained. The total production time was >4 ns (representing >128 ns cumulative bead dynamics), allowing for statistically robust estimates of key observables [20] .

The resulting trajectories from both classical MD and PIMD were analyzed to quantify the impact of NQEs on structural, dynamical, and spectroscopic behavior of FHC in methanol [22] .

The structural stability of 2-(2-furyl)-3-hydroxychromone (FHC) in methanol was evaluated by monitoring key thermodynamic parameters. Figure 2 shows the evolution of temperature and density during the equilibration and production phases, which stabilized at 300 K and 0.79 g/cm³, respectively [23] . The potential and kinetic energy components remained stable throughout the production phase. Statistical analysis of the final 10 ns of the trajectory showed no systematic drift and Gaussian distributions for these quantities ( Figure 3 ), confirming the system had reached robust thermal and structural equilibrium [24] .

A direct comparison of the potential energy evolution from classical MD and PIMD simulations reveals a crucial finding, the average potential energy is consistently lower in the PIMD simulation ( Figure 4 ). This is a direct manifestation of the inclusion of nuclear quantum effects (NQEs), primarily zero-point energy (ZPE). The ZPE effectively lifts the system to a higher vibrational baseline, allowing it to sample lower regions of the classical potential energy well that are inaccessible to classical nuclei, which are frozen at their minimum at 0 K. This result underscores the necessity of accounting for quantum fluctuations when modeling systems dominated by light atoms like hydrogen [24] [25] .

Further evidence of NQEs is found in the atomic coordinate and velocity distributions. As illustrated in Figure 5 , the distributions derived from PIMD are broader than those from classical MD. This is the expected signature of quantum mechanical delocalization, where nuclei are not point particles but are instead represented by smeared probability distributions [26] . The PIMD framework, by representing each nucleus as a ring polymer, explicitly captures this delocalization, leading to a wider exploration of phase space. This fundamental difference is critical for accurately modeling properties that depend on nuclear position, such as hydrogen bond lengths and vibrational spectra [27] .

The molecular diffusion coefficient (D) was calculated from the velocity autocorrelation function (VACF) of the center of mass of FHC using the Green-Kubo relation D = (1/3) ∫⟨v(0)*v(t)⟩ dt [28] .

The VACF decay ( Figure 6 ) shows a clear difference between the two simulation methods. The VACF from the PIMD simulation decays more slowly than its

classical counterpart. A slower decay of the VACF indicates that the molecule’s velocity remains correlated for a longer time, which is directly linked to a reduction in the computed diffusion coefficient [29] .

The calculated diffusion coefficients are D_MD = 1.8 × 10⁻⁵ cm²/s and D_PIMD = 1.4 × 10⁻⁵ cm²/s. A summary of these and other key properties is provided in Table 1 . This ~22% reduction in diffusion under PIMD is a significant quantum effect [30] . It can be attributed to the interplay of zero-point energy and delocalization. Stronger Effective Hydrogen Bonding, Quantum delocalization allows protons to sample shorter O-H-O distances, effectively strengthening and tightening the hydrogen bond network between FHC and the methanol solvent [31] . The more structured solvation shell around FHC, reinforced by quantum effects, creates a more restrictive cage, hindering the translational motion of the solute molecule [32] .

This result aligns with the experimental observation that FHC’s ESIPT kinetics are slowed in protic, hydrogen-bonding environments [33] . The PIMD simulation suggests that part of this environmental modulation may stem from quantum-mechanically enhanced solvent-solute interactions that restrict mobility even in the ground state.

The infrared (IR) absorption spectrum was computed from the Fourier transform of the autocorrelation function of the total dipole moment derivative of the system [21] . As shown in Figure 7 , the PIMD-derived spectrum exhibits broader absorption bands and a general red-shift compared to the classical MD spectrum.

These spectral changes are classic signatures of nuclear quantum effects, arising from the sampling of a wider distribution of molecular geometries due to quantum nuclear delocalization, effectively incorporating anharmonicity [34] . Primarily caused by zero-point energy, which lowers the effective barrier for vibrational modes, reducing their fundamental frequencies.

These effects are most pronounced for vibrations involving hydrogen atoms due to their large quantum fluctuations. The improved agreement of the PIMD spectrum with expected experimental line shapes highlights its superiority over classical MD for predicting spectroscopic properties. It provides a more realistic representation of vibrational modes, anharmonicity, and the influence of the quantum mechanical hydrogen bond network [35] .

The incorporation of nuclear quantum effects through PIMD paints a consistent picture of FHC’s behavior in methanol, NQEs promote a tighter, more structured, and more strongly interacting solvation environment. Energetically, the system benefits from zero-point energy, sampling lower potential energies. Structurally, nuclei are delocalized, leading to smoother, broader distributions of atomic positions, particularly for hydrogen bonds. Dynamically, this translates to stronger solute-solvent interactions that restrict translational diffusion. Spectroscopically, the vibrations are softened and broadened, reflecting the anharmonic quantum nature of the potential energy surface.

This comprehensive description of the ground-state microenvironment is crucial for understanding FHC’s photophysics. The quantum-mechanically enhanced hydrogen bonding captured by PIMD likely creates a solvation shell that pre-organizes the molecule and modifies the potential energy landscape for the ESIPT reaction. This provides a foundational, atomistic explanation for the profound sensitivity of FHC’s ESIPT kinetics to its solvent environment, as observed experimentally [36] .