Editing Electronic band structure (section)

=== Density-functional theory ===
{{Main|Density functional theory}}
{{See also|Kohn–Sham equations}}
In recent physics literature, a large majority of the electronic structures and band plots are calculated using [[density-functional theory]] (DFT), which is not a model but rather a theory, i.e., a microscopic first-principles theory of [[condensed matter physics]] that tries to cope with the electron-electron many-body problem via the introduction of an [[Exchange interaction|exchange-correlation]] term in the functional of the [[electronic density]]. DFT-calculated bands are in many cases found to be in agreement with experimentally measured bands, for example by [[angle-resolved photoemission spectroscopy]] (ARPES). In particular, the band shape is typically well reproduced by DFT. But there are also systematic errors in DFT bands when compared to experiment results. In particular, DFT seems to systematically underestimate by about 30-40% the band gap in insulators and semiconductors.<ref>{{Cite journal|last1=Assadi|first1=M. Hussein. N.|last2=Hanaor|first2=Dorian A. H.| date=2013-06-21|title=Theoretical study on copper's energetics and magnetism in TiO<sub>2</sub> polymorphs|journal=Journal of Applied Physics|volume=113|issue=23| pages=233913–233913–5 | arxiv=1304.1854 |doi=10.1063/1.4811539|bibcode=2013JAP...113w3913A|s2cid=94599250|issn=0021-8979}}</ref>

It is commonly believed that DFT is a theory to predict [[ground state]] properties of a system only (e.g. the [[total energy]], the [[atomic structure]], etc.), and that [[excited state]] properties cannot be determined by DFT. This is a misconception. In principle, DFT can determine any property (ground state or excited state) of a system given a functional that maps the ground state density to that property. This is the essence of the Hohenberg–Kohn theorem.<ref>{{cite journal | last=Hohenberg|first=P | author2=Kohn, W.|title=Inhomogeneous Electron Gas|journal=Phys. Rev.|date=Nov 1964| volume=136 | issue=3B | pages=B864–B871 | doi=10.1103/PhysRev.136.B864 | bibcode = 1964PhRv..136..864H |doi-access=free}}</ref> In practice, however, no known functional exists that maps the ground state density to excitation energies of electrons within a material. Thus, what in the literature is quoted as a DFT band plot is a representation of the DFT [[Kohn–Sham equations|Kohn–Sham energies]], i.e., the energies of a fictive non-interacting system, the Kohn–Sham system, which has no physical interpretation at all. The Kohn–Sham electronic structure must not be confused with the real, [[quasiparticle]] electronic structure of a system, and there is no [[Koopmans' theorem]] holding for Kohn–Sham energies, as there is for Hartree–Fock energies, which can be truly considered as an approximation for [[quasiparticle energies]]. Hence, in principle, Kohn–Sham based DFT is not a band theory, i.e., not a theory suitable for calculating bands and band-plots. In principle [[Time-dependent density functional theory|time-dependent DFT]] can be used to calculate the true band structure although in practice this is often difficult. A popular approach is the use of [[hybrid functional]]s, which incorporate a portion of Hartree–Fock exact exchange; this produces a substantial improvement in predicted bandgaps of semiconductors, but is less reliable for metals and wide-bandgap materials.<ref name=Paier>{{Cite journal | last1 = Paier | first1 = J. | last2 = Marsman | first2 = M. | last3 = Hummer | first3 = K. | last4 = Kresse | first4 = G. | last5 = Gerber | first5 = I. C. | last6 = Angyán | first6 = J. G. | title = Screened hybrid density functionals applied to solids | journal = J Chem Phys | volume = 124 | issue = 15 | pages = 154709 |date=2006 | doi = 10.1063/1.2187006 | pmid = 16674253 |bibcode = 2006JChPh.124o4709P }}</ref>