Computational approaches have become indispensable tools for predicting the stability, electronic structure, and reactivity of fullerenes. These methods provide insights at the atomic level that complement experimental observations, enabling researchers to understand and manipulate fullerene properties with precision. Among the most widely used techniques are density functional theory (DFT) and molecular dynamics (MD), which offer distinct advantages in simulating different aspects of fullerene behavior.

Density functional theory is a quantum mechanical modeling method that has been extensively applied to study the electronic structure of fullerenes. DFT calculations provide accurate predictions of molecular orbitals, energy levels, and charge distribution, which are critical for understanding fullerene stability and reactivity. For instance, DFT studies on C60, the most well-known fullerene, have confirmed its high symmetry (Ih point group) and exceptional stability due to the closed-shell electronic configuration. The calculated HOMO-LUMO gap of C60, typically around 1.9 eV, aligns well with experimental measurements from photoelectron spectroscopy and optical absorption spectra. DFT also explains the stability differences among fullerene isomers by evaluating their binding energies and strain distributions. For example, isolated pentagon rule (IPR) compliant fullerenes, where no two pentagons share an edge, consistently show lower energy and higher stability than non-IPR structures, as confirmed by both calculations and synthetic yields.

Beyond C60, DFT has been used to predict the properties of larger fullerenes, such as C70, C80, and endohedral fullerenes. In endohedral systems, where atoms or molecules are encapsulated within the carbon cage, DFT accurately reproduces the charge transfer between the guest species and the fullerene shell. For instance, calculations on La@C82 show a formal charge transfer of +3 from the lanthanum atom to the cage, consistent with X-ray photoelectron spectroscopy (XPS) and nuclear magnetic resonance (NMR) data. DFT also predicts the regioselectivity of chemical reactions on fullerene surfaces, such as Diels-Alder cycloadditions or Bingel-Hirsch functionalization, by analyzing local electron density and frontier molecular orbitals. These predictions often match experimental observations of preferential addition sites, as seen in functionalized C60 derivatives.

Molecular dynamics simulations complement DFT by providing dynamic insights into fullerene behavior under various conditions. Classical MD, using empirical potentials like the reactive force field (ReaxFF) or Tersoff-Brenner potentials, can model the thermal stability and mechanical properties of fullerenes. Simulations of C60 at elevated temperatures reveal that the cage structure remains intact up to approximately 4000 K before disintegration, in agreement with experimental pyrolysis studies. MD also captures the unique "shuttlecock" motion of endohedral fullerenes, where the encapsulated atom oscillates within the cage, as later observed using transmission electron microscopy (TEM).

Reactive MD simulations have been particularly useful in studying fullerene formation mechanisms. For example, simulations of carbon vapor condensation show that fullerene growth proceeds through the coalescence of small carbon clusters and the rearrangement of carbon chains into closed cages. These pathways align with experimental observations from laser ablation and arc-discharge synthesis, where intermediate species like C2 units are detected using mass spectrometry. MD also predicts the role of catalysts, such as nickel or cobalt nanoparticles, in templating fullerene formation, which has been corroborated by high-resolution TEM studies of metal-fullerene interactions.

The mechanical properties of fullerenes, including their response to compression or deformation, have been explored using both DFT and MD. Calculations show that C60 can withstand significant elastic deformation, with a Young's modulus around 1000 GPa, matching experimental nanoindentation measurements. Under extreme pressure, DFT predicts a phase transition to a polymerized fullerite structure, which has been experimentally confirmed using X-ray diffraction under high-pressure conditions.

Computational studies also address the reactivity of fullerenes toward external agents like oxygen or free radicals. DFT-based transition state theory quantifies the activation barriers for oxidation reactions, explaining why C60 is relatively inert in ambient conditions but reacts readily under UV irradiation or in the presence of singlet oxygen. These findings correlate with experimental kinetic studies of fullerene oxidation rates. Similarly, simulations of hydrogenation reactions predict the stability patterns of C60Hx isomers, with certain addition patterns (like 1,2-addition) being energetically favored, as later verified by synthetic and spectroscopic data.

The optical properties of fullerenes, including their absorption and emission spectra, are accurately modeled using time-dependent DFT (TD-DFT). TD-DFT calculations reproduce the characteristic UV-Vis absorption peaks of C60, such as the strong band at 340 nm, which arises from allowed π-π* transitions. For functionalized fullerenes, TD-DFT predicts redshifted absorption upon addition of electron-donating groups, consistent with experimental spectrophotometry. The phosphorescence emission of fullerenes, observed at low temperatures, is also correctly attributed to triplet-state transitions by TD-DFT.

Challenges remain in computational studies of fullerenes, particularly for large or highly asymmetric systems where electron correlation effects become significant. Advanced methods like hybrid DFT functionals (e.g., B3LYP) or post-Hartree-Fock approaches (e.g., MP2, CCSD) improve accuracy but at higher computational costs. For dynamical processes occurring over long timescales, such as fullerene diffusion in solvents, accelerated MD techniques or coarse-grained models are employed to bridge the gap with experimental observations from techniques like dynamic light scattering.

The synergy between computational predictions and experimental validation has greatly advanced fullerene science. For example, early DFT studies proposed the existence of metallofullerenes with unusual charge states, such as Sc3N@C80, which were later isolated and characterized. Similarly, MD-guided synthesis protocols have optimized fullerene production yields by controlling parameters like temperature or quenching rates. As computational power grows and methods refine, the interplay between theory and experiment will continue to drive innovations in fullerene-based materials for electronics, medicine, and energy applications.