Hybrid functionals

Origin

The hybrid approach to constructing density functional approximations was introduced by Axel Becke in 1993. Hybridization with Hartree–Fock (HF) exchange (also called exact exchange) provides a simple scheme for improving the calculation of many molecular properties, such as atomization energies, bond lengths and vibration frequencies, which tend to be poorly described with simple "ab initio" functionals.

Method

A hybrid exchange–correlation functional is usually constructed as a linear combination of the Hartree–Fock exact exchange functional

: E_\text{x}^\text{HF} = -\frac{1}{2} \sum_{i,j} \iint \psi_i^(\mathbf r_1) \psi_j^(\mathbf r_2) \frac{1}{r_{12}} \psi_j(\mathbf r_1) \psi_i(\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.

B3LYP

For example, the popular B3LYP (Becke, 3-parameter, Lee–Yang–Parr) exchange-correlation functional is

: E_\text{xc}^\text{B3LYP} =(1-a) E_\text{x}^\text{LSDA} + aE_\text{x}^\text{HF} + b\vartriangle E_\text{x}^\text{B} + (1-c)E_\text{c}^\text{LSDA} + c E_\text{c}^\text{LYP} ,

where a = 0.20, b = 0.72, and c = 0.81. E_\text{x}^\text{B} is a generalized gradient approximation: the Becke 88 exchange functional and the correlation functional of Lee, Yang and Parr for B3LYP, and E_\text{c}^\text{LSDA} is the VWN local spin density approximation to the correlation functional.

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.

PBE0

The PBE0 functional{{cite journal mixes the Perdew–Burke–Ernzerhof (PBE) exchange energy and Hartree–Fock exchange energy in a set 3:1 ratio, along with the full PBE correlation energy:

: E_\text{xc}^\text{PBE0} = \frac{1}{4} E_\text{x}^\text{HF} + \frac{3}{4} E_\text{x}^\text{PBE} + E_\text{c}^\text{PBE},

where E_\text{x}^\text{HF} is the Hartree–Fock exact exchange functional, E_\text{x}^\text{PBE} is the PBE exchange functional, and E_\text{c}^\text{PBE} is the PBE correlation functional.

HSE

The HSE (Heyd–Scuseria–Ernzerhof) exchange–correlation 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_\text{xc}^{\omega\text{PBEh}} = a E_\text{x}^\text{HF,SR}(\omega) + (1 - a) E_\text{x}^\text{PBE,SR}(\omega) + E_\text{x}^\text{PBE,LR}(\omega) + E_\text{c}^\text{PBE},

where a is the mixing parameter, and \omega is an adjustable parameter controlling the short-rangeness of the interaction. Standard values of a = 1/4 and \omega = 0.2 (usually referred to as HSE06) have been shown to give good results for most systems. The HSE exchange–correlation functional degenerates to the PBE0 hybrid functional for \omega = 0. E_\text{x}^\text{HF,SR}(\omega) is the short-range Hartree–Fock exact exchange functional, E_\text{x}^\text{PBE,SR}(\omega) and E_\text{x}^\text{PBE,LR}(\omega) are the short- and long-range components of the PBE exchange functional, and E_\text{c}^\text{PBE}(\omega) is the PBE correlation functional.

Meta-hybrid GGA

The M06 suite of functionals is a set of four meta-hybrid GGA and meta-GGA DFT functionals. These functionals are constructed by empirically fitting their parameters, while being constrained to a uniform electron gas.

The family includes the functionals M06-L, M06, M06-2X and M06-HF, with a different amount of exact exchange for each one. M06-L is fully local without HF exchange (thus it cannot be considered hybrid), M06 has 27% HF exchange, M06-2X 54% and M06-HF 100%.

The advantages and usefulness of each functional 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 gives good results for systems containing dispersion forces, one of the biggest deficiencies of standard DFT methods.

A recent review of DFT functionals concludes: "Despite their excellent performance for energies and geometries, we must suspect that modern highly parameterized functionals need further guidance from exact constraints, or exact density, or both"

References

References

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2. (1996). "Rationale for mixing exact exchange with density functional approximations". J. Chem. Phys..
3. (1994). "Comparison of Density Functional and MP2 Calculations on the Water Monomer and Dimer". J. Phys. Chem..
4. (1994). "''Ab Initio'' Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields". J. Phys. Chem..
5. (2004). "Essentials of Computational Chemistry: Theories and Models, 2nd Edition {{!}} Wiley".
6. A. D. Becke. (1988). "Density-functional exchange-energy approximation with correct asymptotic behavior". Phys. Rev. A.
7. (1988). "Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density". Phys. Rev. B.
8. (1980). "Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis". Can. J. Phys..
9. Becke, Axel D.. (1993). "Density-functional thermochemistry. III. The role of exact exchange". J. Chem. Phys..
10. Perdew, John P.. (1996-10-28). "Generalized Gradient Approximation Made Simple". Physical Review Letters.
11. (2003). "Hybrid functionals based on a screened Coulomb potential". J. Chem. Phys..
12. Zhao, Yan. (2008). "The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals". Theoretical Chemistry Accounts.
13. Medvedev, Michael G.. (2017). "Density functional theory is straying from the path toward the exact functional". Science.