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| '''Hybrid functionals''' are a class of approximations to the [[Exchange interaction|exchange]]-[[Electron correlation|correlation]] energy [[Functional (mathematics)|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 quantum chemistry methods|ab initio]]'' or empirical). The exact exchange energy functional is expressed in terms of the [[Kohn-Sham equations|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, 3-parameter, Lee-Yang-Parr.
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| ==Origin==
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| The hybrid approach to constructing density functional approximations was introduced by [[Axel Becke]] in 1993.<ref>{{cite journal|author=A.D. Becke|title=A new mixing of Hartree-Fock and local density-functional theories|journal=J. Chem. Phys.|volume=98|pages=1372–1377|year=1993|doi=10.1063/1.464304 |bibcode = 1993JChPh..98.1372B|issue=2 }}</ref> Hybridization with Hartree-Fock (exact) exchange provides a simple scheme for improving many molecular properties, such as [[Enthalpy of atomization|atomization energies]], [[bond length]]s and [[Infrared spectroscopy|vibration frequencies]], which tend to be poorly described with simple "ab initio" functionals.<ref>{{cite journal|author=John P. Perdew, Matthias Ernzerhof and Kieron Burke|title=Rationale for mixing exact exchange with density functional approximations|journal=J. Chem. Phys.|volume=105|pages=9982–9985|year=1996|url=http://dft.uci.edu/pubs/PEB96.pdf|format=PDF|doi=10.1063/1.472933|accessdate=2007-05-07|bibcode = 1996JChPh.105.9982P|issue=22 }}</ref>
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| ==Method==
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| A hybrid exchange-correlation functional is usually constructed as a [[linear combination]] of the Hartree-Fock exact exchange functional, <math>E_x^{\rm HF}</math>:
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| :<math>E_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}</math>,
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| 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.
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| === B3LYP ===
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| For example, the popular B3LYP (Becke, three-parameter, Lee-Yang-Parr)<ref>{{ cite journal | author = K. Kim and K. D. Jordan | title = Comparison of Density Functional and MP2 Calculations on the Water Monomer and Dimer | journal = J. Phys. Chem. | volume = 98 | issue = 40 | pages = 10089–10094 | year = 1994 | doi = 10.1021/j100091a024 }}</ref><ref>{{ cite journal | author = P.J. Stephens, F. J. Devlin, C. F. Chabalowski and M. J. Frisch | title = ''Ab Initio'' Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields | journal = J. Phys. Chem. | volume = 98 | pages = 11623–11627 | year = 1994 | doi = 10.1021/j100096a001 | issue = 45 }}</ref> exchange-correlation functional is:
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| :<math>E_{xc}^{\rm B3LYP} = E_{x}^{\rm LDA} + a_0 (E_x^{\rm HF} - E_x^{\rm LDA}) + a_x (E_x^{\rm GGA} - E_x^{\rm LDA}) + a_c (E_c^{\rm GGA} - E_c^{\rm LDA}),</math> | |
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| where <math>a_0=0.20 \,\;</math>, <math>a_x=0.72\,\;</math>, and <math>a_c=0.81\,\;</math>. <math>E_x^{\rm GGA}</math> and <math>E_c^{\rm GGA}</math> are [[generalized gradient approximation]]s: the Becke 88 exchange functional<ref>{{cite journal | author = A. D. Becke | title = Density-functional exchange-energy approximation with correct asymptotic behavior | journal = Phys. Rev. A | volume = 38 | pages = 3098–3100 | year = 1988 | url = http://link.aps.org/abstract/PRA/v38/p3098 | doi = 10.1103/PhysRevA.38.3098 | pmid = 9900728 | issue = 6 |bibcode = 1988PhRvA..38.3098B }}</ref> and the correlation functional of Lee, Yang and Parr<ref>{{cite journal|author= Chengteh Lee, Weitao Yang and Robert G. Parr|title=Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density|journal=Phys. Rev. B|volume=37|pages=785–789|year=1988|doi=10.1103/PhysRevB.37.785|bibcode = 1988PhRvB..37..785L|issue= 2 }}</ref> for B3LYP, and <math>E_c^{\rm LDA}</math> is the VWN [[Local-density_approximation#Correlation_functional|local-density approximation]] to the correlation functional.<ref>{{cite journal|title = Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis| author = S. H. Vosko, L. Wilk and M. Nusair|journal = Can. J. Phys.|volume = 58|pages = 1200–1211|year = 1980|doi = 10.1139/p80-159|bibcode = 1980CaJPh..58.1200V|issue = 8 }}</ref>
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| Contrary to popular belief, B3LYP was not fit to experimental data.
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| 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.<ref>{{cite journal|last=Becke|first=Axel D.|year=1993|title=Density-functional thermochemistry. III. The role of exact exchange|journal=J. Chem. Phys.|volume=98|issue=7|pages=5648–5652|doi=10.1063/1.464913 |bibcode = 1993JChPh..98.5648B }}</ref>
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| === HSE ===
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| The HSE (Heyd-Scuseria-Ernzerhof)<ref>{{ cite journal | author = Jochen Heyd, Gustavo E. Scuseria, and Matthias Ernzerhof | title = Hybrid functionals based on a screened Coulomb potential | journal = J. Chem. Phys. | volume = 118 | issue = 18 | pages = 8207 | year = 2003 | doi = 10.1063/1.1564060 }}</ref> exchange-correlation functional uses an [[error function]] screened [[Coulomb potential]] to calculate the exchange portion of the energy in order to improve computationally efficiency, especially for metallic systems.
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| :<math>E_{xc}^{\rm \omega PBEh} = a E_x^{\rm HF,SR}(\omega) + (1-a) E_x^{\rm PBE,SR}(\omega) + E_x^{\rm PBE,LR}(\omega) + E_c^{\rm PBE},</math>
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| where <math>a</math> is the mixing parameter and <math>\omega</math> is an adjustable parameter controlling the short-rangeness of the interaction. Standard values of <math>a=\frac14</math> and <math>\omega=0.2</math> (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 <math>\omega = 0</math>. <math>E_x^{\rm HF,SR}(\omega)</math> is the short range Hartree-Fock exact exchange functional, <math>E_x^{\rm PBE,SR}(\omega)</math> and <math>E_x^{\rm PBE,LR}(\omega)</math> are the short and long range components of the PBE exchange functional, and <math>E_C^{\rm PBE}(\omega)</math> is the PBE <ref>{{ cite journal | doi = 10.1103/PhysRevLett.77.3865| volume = 77 | issue = 18 | pages = 3865–3868 | last = Perdew | first = John P. | coauthors = Kieron Burke, Matthias Ernzerhof | title = Generalized Gradient Approximation Made Simple | journal = Physical Review Letters | accessdate = 2011-09-28 | date = 1996-10-28 | url = http://link.aps.org/doi/10.1103/PhysRevLett.77.3865 | pmid=10062328}}</ref> correlation functional.
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| === Meta hybrid GGA ===
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| The M06 suite of functionals,<ref>{{ cite journal | doi = 10.1007/s00214-007-0310-x| volume = 120 | pages = 215 | last = Zhao | first = Yan | coauthors = Donald G. Truhlar | journal = Theor. Chem. Account | accessdate = 2012-10-25 }}</ref><ref>{{ cite journal | doi = 10.1021/jp066479k | volume = 110 | pages = 13126 | last = Zhao | first = Yan | coauthors = Donald G. Truhlar | journal = J. Phys. Chem. | accessdate = 2012-10-25 }}
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| </ref> are a set of four meta-hybrid GGA DFT functionals. They are constructed with empirical fitting of their parameters, but constraining to the uniform electron gas.
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| The family includes the functionals M06-L, M06, M06-2X and M06-HF, with a different amount of exact exchange on each one. M06-L is fully local without HF exchange (thus it cannot be considered hybrid), M06 has 27% of HF exchange, M06-2X 54% and M06-HF 100%.
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| The advantages and utilities of each one are:
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| * M06-L: Fast, Good for transition metals, inorganic and organometallics.
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| * M06: For main group, organometallics, kinetics and non-covalent bonds.
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| * M06-2X: Main group, kinetics.
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| * M06-HF: Charge transfer TD-DFT, systems where self interaction is pathological.
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| 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 M06-L, M06 and M06-2X respectively.
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| ==References==
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| <references />
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| [[Category:Density functional theory]]
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