Hybrid functionals are a class of approximations to the exchange–correlation energy functional in density functional theory (DFT) that incorporate a portion of exact exchange from Hartree–Fock theory with exchange and correlation from other sources (ab initio or empirical). The exact exchange energy functional is expressed in terms of the Kohn–Sham orbitals rather than the density, so is termed an implicit density functional. One of the most commonly used versions is B3LYP, which stands for Becke, 3parameter, LeeYangParr.
Origin
The hybrid approach to constructing density functional approximations was introduced by Axel Becke in 1993.^{[1]} Hybridization with Hartree–Fock (exact) exchange provides a simple scheme for improving many molecular properties, such as atomization energies, bond lengths and vibration frequencies, which tend to be poorly described with simple "ab initio" functionals.^{[2]}
Method
A hybrid exchangecorrelation functional is usually constructed as a linear combination of the Hartree–Fock exact exchange functional, E_x^{\rm HF}:

E_{\rm x}^{\rm HF}=\frac{1}{2}\sum_{i,j}\int\int\psi_i^*(\mathbf{r_1})\psi_j^*(\mathbf{r_1})\frac{1}{r_{12}}\psi_i(\mathbf{r_2})\psi_j(\mathbf{r_2})d\mathbf{r_1}d\mathbf{r_2},
and any number of exchange and correlation explicit density functionals. The parameters determining the weight of each individual functional are typically specified by fitting the functional's predictions to experimental or accurately calculated thermochemical data, although in the case of the "adiabatic connection functionals" the weights can be set a priori.^{[3]}
B3LYP
For example, the popular B3LYP (Becke, threeparameter, LeeYangParr)^{[4]}^{[5]} exchangecorrelation functional is:

E_{\rm xc}^{\rm B3LYP} = E_{\rm x}^{\rm LDA} + a_0 (E_{\rm x}^{\rm HF}  E_{\rm x}^{\rm LDA}) + a_{\rm x} (E_{\rm x}^{\rm GGA}  E_{\rm x}^{\rm LDA}) + E_{\rm c}^{\rm LDA} + a_{\rm c} (E_{\rm c}^{\rm GGA}  E_{\rm c}^{\rm LDA}),
where a_0=0.20 \,\;, a_{\rm x}=0.72\,\;, and a_{\rm c}=0.81\,\;. E_{\rm x}^{\rm GGA} and E_{\rm c}^{\rm GGA} are generalized gradient approximations: the Becke 88 exchange functional^{[6]} and the correlation functional of Lee, Yang and Parr^{[7]} for B3LYP, and E_{\rm c}^{\rm LDA} is the VWN localdensity approximation to the correlation functional.^{[8]}
Contrary to popular belief, B3LYP was not fit to experimental data. The three parameters defining B3LYP have been taken without modification from Becke's original fitting of the analogous B3PW91 functional to a set of atomization energies, ionization potentials, proton affinities, and total atomic energies.^{[9]}
PBE0
The PBE0 functional^{[10]} ^{[11]} mixes the PBE exchange energy and HartreeFock exchange energy in a set 3 to 1 ratio, along with the full PBE correlation energy:

E_{\rm xc}^{\rm PBE0} = \frac{1}{4} E_{\rm x}^{\rm HF} + \frac{3}{4} E_{\rm x}^{\rm PBE} + E_{\rm c}^{\rm PBE},
where E_{\rm x}^{\rm HF} is the Hartree–Fock exact exchange functional, E_{\rm x}^{\rm PBE} is the PBE exchange functional, and E_{\rm c}^{\rm PBE} is the PBE correlation functional.^{[12]}
HSE
The HSE (HeydScuseriaErnzerhof)^{[13]} exchangecorrelation functional uses an error function screened Coulomb potential to calculate the exchange portion of the energy in order to improve computational efficiency, especially for metallic systems.

E_{\rm xc}^{\rm \omega PBEh} = a E_{\rm x}^{\rm HF,SR}(\omega) + (1a) E_{\rm x}^{\rm PBE,SR}(\omega) + E_{\rm x}^{\rm PBE,LR}(\omega) + E_{\rm c}^{\rm PBE},
where a is the mixing parameter and \omega is an adjustable parameter controlling the shortrangeness of the interaction. Standard values of a=\frac14 and \omega=0.2 (usually referred to as HSE06) have been shown to give good results for most of systems. The HSE exchangecorrelation functional degenerates to the PBE0 hybrid functional for \omega = 0. E_{\rm x}^{\rm HF,SR}(\omega) is the short range Hartree–Fock exact exchange functional, E_{\rm x}^{\rm PBE,SR}(\omega) and E_{\rm x}^{\rm PBE,LR}(\omega) are the short and long range components of the PBE exchange functional, and E_{\rm c}^{\rm PBE}(\omega) is the PBE ^{[14]} correlation functional.
Meta hybrid GGA
The M06 suite of functionals,^{[15]}^{[16]} are a set of four metahybrid GGA and metaGGA DFT functionals. They are constructed with empirical fitting of their parameters, but constraining to the uniform electron gas.
The family includes the functionals M06L, M06, M062X and M06HF, with a different amount of exact exchange on each one. M06L is fully local without HF exchange (thus it cannot be considered hybrid), M06 has 27% of HF exchange, M062X 54% and M06HF 100%.
The advantages and utilities of each one are:

M06L: Fast, Good for transition metals, inorganic and organometallics.

M06: For main group, organometallics, kinetics and noncovalent bonds.

M062X: Main group, kinetics.

M06HF: Charge transfer TDDFT, systems where self interaction is pathological.
The suite has a very good response under dispersion forces, improving one of the biggest deficiencies in DFT methods. The s6 scaling factor on Grimme's long range dispersion correction is 0.20, 0.25 and 0.06 for M06L, M06 and M062X respectively.
References

^ A.D. Becke (1993). "A new mixing of HartreeFock and local densityfunctional theories". J. Chem. Phys. 98 (2): 1372–1377.

^ John P. Perdew, Matthias Ernzerhof and Kieron Burke (1996). "Rationale for mixing exact exchange with density functional approximations" (PDF). J. Chem. Phys. 105 (22): 9982–9985.

^ Perdew, John P.; Matthias Ernzerhof; Kieron Burke (19961208). "Rationale for mixing exact exchange with density functional approximations". The Journal of Chemical Physics 105 (22): 9982–9985.

^ K. Kim and K. D. Jordan (1994). "Comparison of Density Functional and MP2 Calculations on the Water Monomer and Dimer". J. Phys. Chem. 98 (40): 10089–10094.

^ P.J. Stephens, F. J. Devlin, C. F. Chabalowski and M. J. Frisch (1994). "Ab Initio Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields". J. Phys. Chem. 98 (45): 11623–11627.

^ A. D. Becke (1988). "Densityfunctional exchangeenergy approximation with correct asymptotic behavior". Phys. Rev. A 38 (6): 3098–3100.

^ Chengteh Lee, Weitao Yang and Robert G. Parr (1988). "Development of the ColleSalvetti correlationenergy formula into a functional of the electron density". Phys. Rev. B 37 (2): 785–789.

^ S. H. Vosko, L. Wilk and M. Nusair (1980). "Accurate spindependent electron liquid correlation energies for local spin density calculations: a critical analysis". Can. J. Phys. 58 (8): 1200–1211.

^ Becke, Axel D. (1993). "Densityfunctional thermochemistry. III. The role of exact exchange". J. Chem. Phys. 98 (7): 5648–5652.

^ Perdew, John P.; Matthias Ernzerhof; Kieron Burke (1996). "Rationale for mixing exact exchange with density functional approximations". The Journal of Chemical Physics 105: 9982.

^ Adamo, Carlo; Vincenzo Barone (19990401). "Toward reliable density functional methods without adjustable parameters: The PBE0 model". The Journal of Chemical Physics 110 (13): 6158–6170.

^ Perdew, John P.; Kieron Burke; Matthias Ernzerhof (19961028). "Generalized Gradient Approximation Made Simple". Physical Review Letters 77 (18): 3865–3868.

^ Jochen Heyd, Gustavo E. Scuseria, and Matthias Ernzerhof (2003). "Hybrid functionals based on a screened Coulomb potential". J. Chem. Phys. 118 (18): 8207.

^ Perdew, John P.; Kieron Burke; Matthias Ernzerhof (19961028). "Generalized Gradient Approximation Made Simple". Physical Review Letters 77 (18): 3865–3868.

^ Zhao, Yan; Donald G. Truhlar. Theor. Chem. Account 120: 215.

^ Zhao, Yan; Donald G. Truhlar. J. Phys. Chem. 110: 13126.
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.