In a crystal structure of metallic solids, the free electron model (zero potential, φ e = 0) for the behavior of valence electrons is used. However, in a periodic lattice (crystal), there is periodic crystal potential, so the electron Hamiltonian becomes H e = − − ℏ ℏ 2 2 m e ∇ ∇ 2 + φ φ c ( x ) , where m e is the electron mass, and the periodic potential is expressed as φ c (x) = Σ g φ g exp[ i (g ∙ x)] (g : reciprocal lattice vector). The time-independent Schrödinger equation with this Hamiltonian is given as (the eigenvalue equation) H e ψ ψ e , x ( x ) = E e ( κ κ e ) ψ ψ e , x ( x ) , where the eigenfunction ψ e,κ is the electron wave function, and eigenvalue E e (κ e), is the electron energy (κ e : electron wavevector). The relation between wavevector, κ e and energy E e provides the electronic band structure. In practice, a lattice as many-body systems includes interactions between electrons and nuclei in potential, but this calculation can be too intricate. Thus, many approximate techniques have been suggested and one of them is density functional theory (DFT), uses functionals of the spatially dependent electron density instead of full interactions. DFT is widely used in ab initio software (ABINIT, CASTEP, Quantum ESPRESSO, SIESTA, VASP, WIEN2k, etc.). The electron specific heat is based on the energy states and occupancy distribution (the Fermi–Dirac statistics). In general, the heat capacity of electron is small except at very high temperature when they are in thermal equilibrium with phonons (lattice). Electrons contribute to heat conduction (in addition to charge carrying) in solid, especially in metals. Thermal conductivity tensor in solid is the sum of electric and phonon thermal conductivity tensors K = K e + K p.