Hybridfunctional

Hybrid functional

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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 othersources (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, whichstands for Becke, 3-parameter, Lee-Yang-Parr.Contents1 Origin2 Method2.1 B3LYP2.2 PBE02.3 HSE

2.4 Meta hybrid GGA3 ReferencesOrigin

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, suchas atomization energies, bond lengths and vibration frequencies, which tend to be poorly described with simple \"ab initio\"functionals.[2]Method

A hybrid exchange-correlation functional is usually constructed as a linear combination

of the Hartree–Fock exact exchange functional, :,

and any number of exchange and correlation explicit density functionals. The parameters determining the weight of eachindividual functional are typically specified by fitting the functional's predictions to experimental or accurately calculatedthermochemical 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, three-parameter, Lee-Yang-Parr)[4][5] exchange-correlation functional is:

where , , and . and are generalized gradient

approximations: the Becke 88 exchange functional[6] and the correlation functional of Lee, Yang and Parr[7] for B3LYP, andis the VWN local-density approximation tothe 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 Hartree-Fock exchange energy in a set 3 to 1 ratio, alongwith the full PBE correlation energy:

where is the Hartree–Fock exact exchange functional, is the PBE exchange functional, and is the PBE correlation functional.[12]HSE

The HSE (Heyd-Scuseria-Ernzerhof)[13] exchange-correlation functional uses an error function screened Coulomb potentialto calculate the exchange portion of the energy in order to improve computational efficiency, especially for metallic systems.

where is the mixing parameter and is an adjustable parameter controlling the short-rangeness of the interaction. Standardvalues of and (usually referred to as HSE06) have been shown to give good results for most of systems. The HSE exchange-

correlation functional degenerates to the PBE0 hybrid functional for .

is the short range Hartree–Fock exact exchange functional, and

are the short and long range components of the PBE exchange functional, and

is the PBE [14] correlation functional.Meta hybrid GGA

The M06 suite of functionals,[15][16] are a set of four meta-hybrid GGA and meta-GGA DFT functionals. They are constructedwith empirical fitting of their parameters, but constraining to the uniform electron gas.

The family includes the functionals M06-L, M06, M06-2X and M06-HF, with a different amount of exact exchange on eachone. M06-L is fully local without HF exchange (thus it cannot be considered hybrid), M06 has 27% of HF exchange, M06-2X54% and M06-HF 100%. The advantages and utilities of each one are:M06-L: Fast, Good for transition metals, inorganic and organometallics.M06: For main group, organometallics, kinetics and non-covalent bonds.M06-2X: Main group, kinetics.

M06-HF: Charge transfer TD-DFT, 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. Thes6 scaling factor on Grimme's long range dispersion correction is 0.20, 0.25 and 0.06 for M06-L, M06 and M06-2Xrespectively. References

1. ^ A.D. Becke (1993). \"A new mixing of Hartree-Fock and local density-functional theories\". J.Chem. Phys.98 (2): 1372–1377. Bibcode:1993JChPh..98.1372B

(http://www.doczj.com/doc/462687515901020206409c5c.html /abs/1993JChPh..98.1372B). doi:10.1063/1.464304(https://http://www.doczj.com/doc/462687515901020206409c5c.html /10.1063%2F1.464304).2. ^ John P. Perdew, Matthias Ernzerhof and Kieron Burke (1996). \"Rationale for mixing exact

exchange with density functional approximations\" (http://www.doczj.com/doc/462687515901020206409c5c.html

/pubs/PEB96.pdf) (PDF). J.

Chem. Phys.105 (22): 9982–9985. Bibcode:1996JChPh.105.9982P

(http://www.doczj.com/doc/462687515901020206409c5c.html /abs/1996JChPh.105.9982P). doi:10.1063/1.472933(https://http://www.doczj.com/doc/462687515901020206409c5c.html /10.1063%2F1.472933). Retrieved 2007-05-07.3. ^ Perdew, John P.; Matthias Ernzerhof; Kieron Burke (1996-12-08). \"Rationale for mixing exactexchange with density functional approximations\"

(http://www.doczj.com/doc/462687515901020206409c5c.html /content/aip/journal/jcp/105/22/10.1063/1.472933). TheJournal of

Chemical Physics105 (22): 9982–9985. doi:10.1063/1.472933

(https://http://www.doczj.com/doc/462687515901020206409c5c.html /10.1063%2F1.472933). ISSN 0021-9606(https://http://www.doczj.com/doc/462687515901020206409c5c.html /issn/0021-9606). Retrieved 2014-09-10.4. ^ K. Kim and K. D. Jordan (1994). \"Comparison of Density Functional and MP2 Calculations onthe Water Monomer and Dimer\". J. Phys. Chem.98 (40): 10089–10094. doi:10.1021/j100091a024(https://http://www.doczj.com/doc/462687515901020206409c5c.html /10.1021%2Fj100091a024).5. ^ 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. doi:10.1021/j100096a001

(https://http://www.doczj.com/doc/462687515901020206409c5c.html /10.1021%2Fj100096a001).6. ^ A. D. Becke (1988). \"Density-functional exchange-energy approximation with correct

asymptotic behavior\" (http://www.doczj.com/doc/462687515901020206409c5c.html /abstract/PRA/v38/p3098). Phys. Rev.A38 (6): 3098–3100. Bibcode:1988PhRvA..38.3098B (http://www.doczj.com/doc/462687515901020206409c5c.html/abs/1988PhRvA..38.3098B).

doi:10.1103/PhysRevA.38.3098 (https://http://www.doczj.com/doc/462687515901020206409c5c.html/10.1103%2FPhysRevA.38.3098). PMID 9900728

(https://http://www.doczj.com/doc/462687515901020206409c5c.html /pubmed/9900728).7. ^ Chengteh Lee, Weitao Yang and Robert G. Parr (1988). \"Development of the Colle-Salvetti

correlation-energy formula into a functional of the electron density\". Phys. Rev. B37 (2): 785–789.

Bibcode:1988PhRvB..37..785L (http://www.doczj.com/doc/462687515901020206409c5c.html /abs/1988PhRvB..37..785L).doi:10.1103/PhysRevB.37.785 (https://http://www.doczj.com/doc/462687515901020206409c5c.html/10.1103%2FPhysRevB.37.785).

8. ^ S. H. Vosko, L. Wilk and M. Nusair (1980). \"Accurate spin-dependent electron liquidcorrelation energies for local spin density calculations: a critical analysis\". Can. J. Phys.58 (8): 1200–1211. Bibcode:1980CaJPh..58.1200V

(http://www.doczj.com/doc/462687515901020206409c5c.html /abs/1980CaJPh..58.1200V). doi:10.1139/p80-159(https://http://www.doczj.com/doc/462687515901020206409c5c.html /10.1139%2Fp80-159).9. ^ Becke, Axel D. (1993). \"Density-functional thermochemistry. III. The role of exactexchange\". J. Chem. Phys.98 (7): 5648–5652. Bibcode:1993JChPh..98.5648B

(http://www.doczj.com/doc/462687515901020206409c5c.html /abs/1993JChPh..98.5648B). doi:10.1063/1.464913(https://http://www.doczj.com/doc/462687515901020206409c5c.html /10.1063%2F1.464913).

10. ^ Perdew, John P.; Matthias Ernzerhof; Kieron Burke (1996). \"Rationale for mixing exact

exchange with density functional approximations\" (http://www.doczj.com/doc/462687515901020206409c5c.html/10.1063/1.472933). The Journal of Chemical Physics105: 9982. doi:10.1063/1.472933

(https://http://www.doczj.com/doc/462687515901020206409c5c.html /10.1063%2F1.472933). ISSN 0021-9606(https://http://www.doczj.com/doc/462687515901020206409c5c.html /issn/0021-9606).

11. ^ Adamo, Carlo; Vincenzo Barone (1999-04-01). \"Toward reliable density functional methodswithout adjustable parameters: The PBE0 model\"

(http://www.doczj.com/doc/462687515901020206409c5c.html /resource/1/jcpsa6/v110/i13/p6158_s1). The Journal ofChemical Physics110

(13): 6158–6170. doi:10.1063/1.478522 (https://http://www.doczj.com/doc/462687515901020206409c5c.html/10.1063%2F1.478522). ISSN 0021-9606

(https://http://www.doczj.com/doc/462687515901020206409c5c.html /issn/0021-9606). Retrieved 2013-06-21.12. ^ Perdew, John P.; Kieron Burke; Matthias Ernzerhof (1996-10-28). \"Generalized Gradient

Approximation Made Simple\" (http://www.doczj.com/doc/462687515901020206409c5c.html

/doi/10.1103/PhysRevLett.77.3865). Physical Review Letters77 (18): 3865–3868. Bibcode:1996PhRvL..77.3865P(http://www.doczj.com/doc/462687515901020206409c5c.html /abs/1996PhRvL..77.3865P).doi:10.1103/PhysRevLett.77.3865

(https://http://www.doczj.com/doc/462687515901020206409c5c.html /10.1103%2FPhysRevLett.77.3865). PMID 10062328(https://http://www.doczj.com/doc/462687515901020206409c5c.html /pubmed/10062328). Retrieved 2011-09-28.13. ^ Jochen Heyd, Gustavo E. Scuseria, and Matthias Ernzerhof (2003). \"Hybrid functionals based

on a screened Coulomb potential\". J. Chem. Phys.118 (18): 8207. Bibcode:2003JChPh.118.8207H

(http://www.doczj.com/doc/462687515901020206409c5c.html /abs/2003JChPh.118.8207H). doi:10.1063/1.1564060(https://http://www.doczj.com/doc/462687515901020206409c5c.html /10.1063%2F1.1564060).14. ^ Perdew, John P.; Kieron Burke; Matthias Ernzerhof (1996-10-28). \"Generalized Gradient

Approximation Made Simple\" (http://www.doczj.com/doc/462687515901020206409c5c.html

/doi/10.1103/PhysRevLett.77.3865). Physical Review Letters77 (18): 3865–3868. Bibcode:1996PhRvL..77.3865P(http://www.doczj.com/doc/462687515901020206409c5c.html /abs/1996PhRvL..77.3865P).doi:10.1103/PhysRevLett.77.3865

(https://http://www.doczj.com/doc/462687515901020206409c5c.html /10.1103%2FPhysRevLett.77.3865). PMID 10062328(https://http://www.doczj.com/doc/462687515901020206409c5c.html /pubmed/10062328). Retrieved 2011-09-28.15. ^ Zhao, Yan; Donald G. Truhlar. Theor. Chem. Account120: 215. doi:10.1007/s00214-007-0310-x(https://http://www.doczj.com/doc/462687515901020206409c5c.html /10.1007%2Fs00214-007-0310-x).16. ^ Zhao, Yan; Donald G. Truhlar. J. Phys. Chem.110: 13126. doi:10.1021/jp066479k(https://http://www.doczj.com/doc/462687515901020206409c5c.html /10.1021%2Fjp066479k).Retrieved from \"http://www.doczj.com/doc/462687515901020206409c5c.html /w/index.php?title=Hybrid_functional&oldid=629131474\"Categories: Density functional theory

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