The article by G. D. Mahan and J. O. Sofo explores the optimal electronic structure for thermoelectric materials to maximize their figure of merit. They derive the transport distribution function, which is a key component in the transport coefficients of electrical conductivity, thermopower, and thermal conductivity. The authors find that a delta-shaped transport distribution, where the energy distribution of electrons is narrow, maximizes the thermoelectric properties. This suggests that materials with a sharp singularity in the density of states near the chemical potential are most efficient. While an exact delta function is not achievable in real materials, materials with tightly bound electronic levels, such as rare-earth compounds, can approximate this ideal. The study also highlights the importance of minimizing the background contribution to the transport distribution to avoid significant reductions in the figure of merit.The article by G. D. Mahan and J. O. Sofo explores the optimal electronic structure for thermoelectric materials to maximize their figure of merit. They derive the transport distribution function, which is a key component in the transport coefficients of electrical conductivity, thermopower, and thermal conductivity. The authors find that a delta-shaped transport distribution, where the energy distribution of electrons is narrow, maximizes the thermoelectric properties. This suggests that materials with a sharp singularity in the density of states near the chemical potential are most efficient. While an exact delta function is not achievable in real materials, materials with tightly bound electronic levels, such as rare-earth compounds, can approximate this ideal. The study also highlights the importance of minimizing the background contribution to the transport distribution to avoid significant reductions in the figure of merit.