Kohn-Sham density functional theory is the most widely-used method of electronic-structure calculation in materials physics and chemistry, because it reduces the many-electron ground-state problem to a computationally tractable self-consistent one-electron problem. Exact in principle, it requires in practice an approximation to the density functional for the exchange-correlation energy. Common approximations fall on one of the five rungs of a ladder, with higher rungs being more complicated to construct but potentially more accurate. The first three or semi-local rungs are important, because (a) they are computationally efficient, (b) they can be constructed non-empirically or with minimal empiricism, and (c) they can serve as input to fourth-rung functionals including hybrid functionals, or as a correction to the fifth-rung Random Phase Approximation. The third-rung meta-generalized gradient approximation can recognize and describe covalent, metallic, and weak bonds [1], providing a good description of the equilibrium properties of many molecules and solids.
[1] J. Sun, B. Xiao. Y. Fang, R. Haunschild, P. Hao, A. Ruzsinszky, G.I. Csonka, G.E. Scuseria, and J.P. Perdew, Phys. Rev. Lett. 111, 106401 (2013).
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