Hybrid functional
Encyclopedia
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
, such as LDA
, 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, 3-parameter, Lee-Yang-Parr.
and vibration frequencies
, which tend to be poorly described with simple "ab initio" functionals.
of the Hartree-Fock exact exchange functional (
) 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.
where
, 
, and 
are the three empirical parameters determined by fitting the predicted values to a set of atomization energies, ionization potentials, proton affinities, and total atomic energies; 
and 
are generalized gradient approximations: the Becke 88 exchange functional and the correlation functional of Lee, Yang and Parr, and 
is the VWN local-density approximation to the correlation functional.
where
is the mixing parameter and 
is an adjustable parameter controlling the short-rangeness of the interaction. 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 correlation functional.
Exchange interaction
In physics, the exchange interaction is a quantum mechanical effect without classical analog which increases or decreases the expectation value of the energy or distance between two or more identical particles when their wave functions overlap...
-correlation energy functional
Functional (mathematics)
In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
in density functional theory
Density functional theory
Density functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...
(DFT) that incorporate a portion of exact exchange from Hartree-Fock
Hartree-Fock
In computational physics and chemistry, the Hartree–Fock method is an approximate method for the determination of the ground-state wave function and ground-state energy of a quantum many-body system....
theory with exchange and correlation from other sources (ab initio
Ab initio quantum chemistry methods
Ab initio quantum chemistry methods are computational chemistry methods based on quantum chemistry. The term ab initiowas first used in quantum chemistry by Robert Parr and coworkers, including David Craig in a semiempirical study on the excited states of benzene.The background is described by Parr...
, such as LDA
Local-density approximation
Local-density approximations are a class of approximations to the exchange-correlation energy functional in density functional theory that depend solely upon the value of the electronic density at each point in space . Many approaches can yield local approximations to the XC energy...
, or empirical). The exact exchange energy functional is expressed in terms of the Kohn-Sham orbitals
Kohn-Sham equations
In quantum chemistry, specifically density functional theory, the Kohn–Sham equation is the Schrödinger equation of a fictitious system of non-interacting particles that generate the same density as any given system of interacting particles...
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.
Origin
The hybrid approach to constructing density functional approximations was introduced by Axel Becke in 1993. Hybridization with Hartree-Fock (exact) exchange provides a simple scheme for improving many molecular properties, such as atomization energies, bond lengthsBond length
- Explanation :Bond length is related to bond order, when more electrons participate in bond formation the bond will get shorter. Bond length is also inversely related to bond strength and the bond dissociation energy, as a stronger bond will be shorter...
and vibration frequencies
Infrared spectroscopy
Infrared spectroscopy is the spectroscopy that deals with the infrared region of the electromagnetic spectrum, that is light with a longer wavelength and lower frequency than visible light. It covers a range of techniques, mostly based on absorption spectroscopy. As with all spectroscopic...
, which tend to be poorly described with simple "ab initio" functionals.
Method
A hybrid exchange-correlation functional is usually constructed as a linear combinationLinear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
of the Hartree-Fock exact exchange functional (
) 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.B3LYP
For example, the popular B3LYP (Becke, three-parameter, Lee-Yang-Parr) exchange-correlation functional is:where
, , and are the three empirical parameters determined by fitting the predicted values to a set of atomization energies, ionization potentials, proton affinities, and total atomic energies; and are generalized gradient approximations: the Becke 88 exchange functional and the correlation functional of Lee, Yang and Parr, and is the VWN local-density approximation to the correlation functional.HSE
The HSE (Heyd-Scuseria-Ernzerhof) exchange-correlation functional uses the screened coulomb potential to calculate the exchange portion of the energy in order to improve computationally efficiency, especially for metallic systems.where
is the mixing parameter and is an adjustable parameter controlling the short-rangeness of the interaction. 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 correlation functional.