Density functional theory has become a cornerstone for calculating thermal properties of nanomaterials due to its ability to predict electronic and vibrational behaviors from first principles. At the nanoscale, thermal transport deviates significantly from bulk materials due to quantum confinement, increased surface-to-volume ratios, and modified phonon dispersion relations. These effects necessitate specialized computational approaches to accurately model heat transfer mechanisms in nanostructures such as graphene, nanowires, and quantum dots.

Phonon dispersion relations form the foundation for thermal property calculations in DFT frameworks. The harmonic approximation treats atomic vibrations as simple harmonic oscillators, allowing computation of force constants through density functional perturbation theory. For a graphene monolayer, DFT calculations reveal a linear phonon dispersion near the Γ point for acoustic modes, with out-of-plane (ZA) modes showing quadratic behavior. This contrasts sharply with bulk graphite where interlayer coupling modifies the ZA branch. Nanowires exhibit flattened phonon branches compared to bulk crystals due to dimensional confinement, with silicon nanowires showing up to 40% reduction in acoustic phonon group velocities below 5 nm diameters.

Grüneisen parameters quantify anharmonic effects by measuring phonon frequency shifts under strain. DFT calculations enable mode-resolved Grüneisen parameter determination through finite-difference approaches where the crystal is subjected to small deformations. In silicon nanowires, DFT reveals Grüneisen parameters exceeding bulk values by factors of 2-3 for surface-localized modes. Graphene exhibits negative Grüneisen parameters for ZA modes near the K point, explaining its negative thermal expansion coefficient at low temperatures. These anharmonicity measurements directly inform thermal conductivity predictions through the Boltzmann transport equation.

Thermal conductivity calculations combine phonon dispersion data with scattering rates using iterative solutions to the Boltzmann equation. Boundary scattering dominates in nanostructures, implemented through Matthiessen's rule with scattering lengths limited by physical dimensions. For a 10 nm diameter silicon nanowire at 300 K, DFT-predicted thermal conductivity shows 80% reduction compared to bulk silicon due to surface scattering. Graphene nanoribbons demonstrate strong edge scattering effects, with zigzag edges producing 30% lower thermal conductivity than armchair edges in sub-5 nm ribbons. Phonon confinement effects emerge when nanostructure dimensions approach phonon wavelengths, causing mode quantization that DFT captures through supercell approaches.

The harmonic approximation fails to describe several key nanoscale thermal phenomena. It cannot predict finite thermal conductivity in perfect crystals or temperature-dependent phonon frequencies. Coupling DFT with molecular dynamics simulations addresses these limitations through direct computation of anharmonic force constants. Ab initio molecular dynamics simulations of silicon nanowires reveal three-phonon scattering rates 2-3 orders of magnitude higher than bulk values due to surface mode coupling. Graphene simulations show anomalous four-phonon scattering becoming dominant above 500 K, a phenomenon inaccessible to harmonic DFT.

Nanostructure thermal properties exhibit strong size dependence that DFT helps quantify. For silicon nanowires, diameter reduction from 100 nm to 3 nm causes thermal conductivity to drop from 80 W/mK to 10 W/mK at room temperature according to DFT-based calculations. The crossover from diffusive to ballistic transport occurs around 100 nm lengths for acoustic phonons in silicon. Graphene shows opposite size dependence, with thermal conductivity increasing from 2000 W/mK to 4000 W/mK as flake size grows from 1 μm to 10 μm due to reduced boundary scattering of long-wavelength phonons.

Surface reconstruction significantly impacts nanoscale thermal transport, requiring careful DFT treatment. Silicon nanowires reconstruct to form dimers on {100} surfaces, creating localized phonon modes that scatter heat carriers. DFT calculations predict these reconstructions reduce thermal conductivity by an additional 15-20% compared to ideal surfaces. In gold nanowires, surface states hybridize with bulk modes to create new scattering channels that lower thermal conductivity by 30% relative to classical size-effect predictions.

Defects introduce additional scattering mechanisms that DFT can characterize at atomic resolution. Single vacancies in graphene reduce thermal conductivity by 20% at 0.1% concentration according to DFT-based Green's function methods. Silicon nanowires with 1% isotopic disorder show 40% thermal conductivity reduction compared to pristine wires. Dislocations in nanowires create strain fields that DFT reveals can scatter phonons more effectively than point defects at equivalent densities.

Interfaces between nanomaterials present complex thermal transport challenges for DFT. Graphene-substrate interactions induce flexural phonon hardening that reduces out-of-plane mode contributions to thermal conductivity by 50%. Heterostructure interfaces like MoS2-WSe2 exhibit interfacial thermal conductance values ranging from 20-50 MW/m2K depending on stacking angle, as predicted by nonequilibrium Green's function methods combined with DFT force constants.

Recent advances in DFT methodologies improve nanoscale thermal property predictions. Temperature-dependent effective potential methods account for anharmonicity without full molecular dynamics simulations. Machine learning force fields trained on DFT data enable larger-scale simulations while preserving quantum accuracy. Wavelet-based approaches accelerate phonon calculations for complex nanostructures with thousands of atoms.

Despite these advances, DFT thermal property calculations face several limitations. Computational cost restricts system sizes typically below 1000 atoms for full phonon calculations. Electron-phonon coupling contributions to thermal transport require expensive GW or hybrid functional calculations. Disordered systems like alloys or amorphous nanomaterials demand large supercells that strain computational resources. These challenges motivate ongoing development of multiscale methods that combine DFT accuracy with larger-scale modeling techniques.

Future directions include improved treatments of four-phonon scattering, more accurate van der Waals interactions for layered materials, and better integration with device-scale thermal models. As computational power increases and methods refine, DFT will continue providing fundamental insights into nanoscale thermal phenomena that guide the design of thermoelectric materials, thermal management solutions, and novel nanoelectronic devices.