DFT Functional

Density functional theory (DFT) is a computational method that is used to calculate the electronic structure of molecules and materials. In DFT, the total energy of a system is expressed as a functional of the electron density, and the aim is to find the electron density that minimizes the energy.

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A wide range of methods and functionals have been developed for DFT, each with its strengths and weaknesses, depending on the type of system being studied and the properties of interest. Some of the most commonly used methods and functionals in DFT include:

1. Local Density Approximation (LDA): This is the simplest and oldest DFT functional, and assumes that the exchange-correlation energy depends only on the local electron density at each point in space.
2. Generalized Gradient Approximation (GGA): This is a more sophisticated DFT functional that includes the gradient of the electron density, in addition to the local electron density, in the calculation of the exchange-correlation energy.
3. Hybrid Functionals: These are DFT functionals that include a mixture of exact exchange and GGA exchange-correlation energy terms. Examples of hybrid functionals include B3LYP, PBE0, and HSE06.
4. Meta-GGA Functionals: These are DFT functionals that include a further derivative of the electron density in the calculation of the exchange-correlation energy, in addition to the gradient and local electron density terms.
5. Range-Separated Hybrid Functionals: These are a type of hybrid functional that separate the exchange energy into short-range and long-range components, and treat them differently. Examples of range-separated hybrid functionals include CAM-B3LYP and LC-BLYP.
6. Double-Hybrid Functionals: These are DFT functionals that include two types of correlation energy, and are particularly useful for predicting the energetics of reactions and transition states. Examples of double-hybrid functionals include B2PLYP and DSD-BLYP.
7. Many-Body Perturbation Theory (MBPT): This is an alternative to DFT that is based on the solution of the many-body Schrödinger equation, and includes effects beyond the exchange-correlation energy. Examples of MBPT methods include GW and MP2.

There are many other DFT methods and functionals that have been developed, and the choice of which one to use depends on the specific system being studied and the properties of interest. In general, it is important to choose a method or functional appropriate for theaccuracy required for the calculation, while also being computationally feasible.

Local Density Approximation (LDA)

The Local Density Approximation (LDA) is a widely used approximation to the exchange-correlation functional in density functional theory (DFT). It is a simple approach that assumes that the exchange-correlation energy depends only on the electron density at a given point, rather than the gradient or other derivatives of the density.

Local Density Approximation (LDA) is one of the simplest and oldest exchange-correlation functionals used in Density Functional Theory (DFT) calculations. It assumes that the exchange-correlation energy depends only on the local electron density at each point in space. Some examples of the LDA functional include:

1. Perdew-Wang 91 (PW91): This is an LDA functional developed by Perdew and Wang in 1991. It uses a gradient expansion of the electron density to calculate the exchange-correlation energy.
2. Ceperley-Alder (CA): This is the original LDA functional developed by Ceperley and Alder in 1980. It is based on a simple parameterization of the electron gas and uses the local density approximation for the exchange-correlation energy.
3. von Barth-Hedin (vBH): This is an LDA functional developed by von Barth and Hedin in 1972. It uses a local approximation for the exchange-correlation energy and a gradient expansion for the kinetic energy.
4. Perdew-Zunger (PZ): This is an LDA functional developed by Perdew and Zunger in 1981. It uses a local approximation for the exchange-correlation energy and a gradient expansion for the kinetic energy.
5. Slater (S): This is an LDA functional developed by John Slater in 1951. It uses a simple approximation for the exchange energy based on the electronic charge density.

While the LDA functional is the simplest form of DFT, it can still provide reasonably accurate results for many systems, especially when used in combination with good basis sets and accurate geometry optimization methods. However, it is often not sufficient for systems with strong electron correlation, and higher-level functionals such as GGA or hybrid functionals may be needed.

The LDA functional has the form:

E_xc[n] = ∫ n(r) ε_xc[n(r)] dr

where E_xc[n] is the exchange-correlation energy of the system, n(r) is the electron density at point r, and ε_xc[n(r)] is the exchange-correlation energy per electron at that point.

The LDA functional assumes that the exchange energy per electron is that of a homogeneous electron gas, which is given by the Dirac exchange energy. The correlation energy is then determined by fitting to the properties of the uniform electron gas.

Despite its simplicity, the LDA functional has been found to provide reasonably accurate results for many properties of atoms, molecules, and solids, particularly for systems with slowly varying densities. However, it may not perform well for strongly correlated systems or for systems with significant density gradients.

More sophisticated functionals, such as the Generalized Gradient Approximation (GGA), have been developed to improve the accuracy of the exchange-correlation functional beyond the LDA. However, the LDA remains a useful and computationally efficient tool in DFT calculations.

As an example, consider the LDA exchange-correlation functional for a simple non-interacting system of spin-polarized electrons in a uniform magnetic field:

E_xc[n] = ∫ n(r) ε_xc[n(r)] dr

where n(r) is the electron density at point r, and ε_xc[n(r)] is the exchange-correlation energy per electron at that point.

The LDA assumes that the exchange energy per electron is that of a homogeneous electron gas, which is given by the Dirac exchange energy:

ε_x[n(r)] = -C_x n(r)^(1/3)

where C_x is a constant and n(r) is the electron density at point r.

The correlation energy per electron is then determined from fitting to the properties of the uniform electron gas:

ε_c[n(r)] = -A ln(1 + B/n(r))

where A and B are constants.

The total exchange-correlation energy of the system can then be obtained by integrating over the electron density:

E_xc[n] = ∫ n(r) ε_xc[n(r)] dr

This LDA functional can be used to calculate the total energy, electron density, and other properties of this simple system. However, for more complex systems, a more sophisticated functional may be needed to obtain accurate results.

Generalized Gradient Approximation (GGA)

Generalized Gradient Approximation (GGA) is a more advanced form of Density Functional Theory (DFT) that includes a dependence on both the electron density and its gradient. This leads to a more accurate description of the exchange-correlation energy compared to the simpler Local Density Approximation (LDA). Some examples of the GGA functional include:

1. Perdew-Burke-Ernzerhof (PBE): This is one of the most widely used GGA functionals, developed by Perdew, Burke, and Ernzerhof in 1996. It uses a gradient expansion of the exchange-correlation energy and includes a correction for the self-interaction error.
2. Revised Perdew-Burke-Ernzerhof (RPBE): This is a modified version of the PBE functional that includes an additional gradient correction to improve the description of molecules and solids.
3. Tao-Perdew-Staroverov-Scuseria (TPSS): This is a GGA functional developed by Tao, Perdew, Staroverov, and Scuseria in 2003. It includes a dependence on the kinetic energy density and was designed to improve the description of both chemical bonding and non-bonding interactions.
4. BLYP: This is an early GGA functional developed by Becke and Lee in 1988. It includes a mixing of the exchange energy from the LDA functional and the correlation energy from the P86 functional.
5. PBESOL: This is a modified version of the PBE functional that was designed to improve the description of solids. It includes a correction for the self-interaction error and has been shown to provide accurate results for a wide range of materials.

These are just a few examples of the many GGA functionals that have been developed over the years. While GGA functionals can provide more accurate results than LDA, they are still not always sufficient for systems with strong electron correlation, and higher-level functionals such as hybrid functionals or meta-GGA functionals may be needed.

Hybrid Functionals

Hybrid functionals are a class of Density Functional Theory (DFT) functionals that mix a portion of the exact Hartree-Fock (HF) exchange energy with a portion of the exchange energy from a semi-local functional, such as the Generalized Gradient Approximation (GGA) or Local Density Approximation (LDA) functionals. This mixing allows for a more accurate description of the electronic structure, particularly for systems with strongly correlated electrons. Some examples of hybrid functionals include:

1. B3LYP: This is one of the most widely used hybrid functionals, and was developed by Becke in 1993. It uses a mixing of the Becke three-parameter exchange functional, the Lee-Yang-Parr correlation functional, and a portion of the exact Hartree-Fock exchange energy.
2. PBE0: This is a hybrid functional developed by Perdew, Burke, and Ernzerhof in 1996. It uses a mixing of the PBE functional and a portion of the exact Hartree-Fock exchange energy.
3. M06: This is a series of hybrid functionals developed by Truhlar and co-workers in 2005. The functionals are optimized for different types of systems, such as main-group elements or transition metals.
4. CAM-B3LYP: This is a hybrid functional developed by Yanai, Tew, and Handy in 2004, and is particularly useful for the study of organic molecules. It uses a mixing of the Becke three-parameter exchange functional and a portion of the exact Hartree-Fock exchange energy, along with a modified correlation functional.
5. HSE06: This is a hybrid functional developed by Heyd, Scuseria, and Ernzerhof in 2003, and is particularly useful for the study of semiconductors and insulators. It uses a mixing of the PBE functional and a portion of the exact Hartree-Fock exchange energy, along with a screened Coulomb potential.

These are just a few examples of the many hybrid functionals that have been developed over the years. Hybrid functionals are generally more computationally expensive than semi-local functionals such as GGA or LDA, but can provide much more accurate results for certain types of systems.

Meta-GGA Functionals

Meta-GGA functionals are a class of Density Functional Theory (DFT) functionals that include the kinetic energy density as an additional variable, making them more accurate for systems with non-uniform electron densities. Some examples of Meta-GGA functionals include:

1. TPSS: This is a Meta-GGA functional developed by Tao, Perdew, Staroverov, and Scuseria in 2003. It uses a non-empirical functional that includes a gradient expansion of the kinetic energy density.
2. M06-L: This is a Meta-GGA functional developed by Truhlar and co-workers in 2006. It uses a non-empirical functional that includes a gradient expansion of the kinetic energy density, along with a long-range correction.
3. B97M-V: This is a Meta-GGA functional developed by Grimme in 2006. It uses an empirical functional that includes a gradient expansion of the kinetic energy density, along with a damping function to account for long-range interactions.
4. SCAN: This is a Meta-GGA functional developed by Sun, Ruzsinszky, and Perdew in 2015. It uses a non-empirical functional that includes a gradient expansion of the kinetic energy density, along with a self-interaction correction term.
5. MVS: This is a Meta-GGA functional developed by Csonka and co-workers in 2009. It uses a non-empirical functional that includes a gradient expansion of the kinetic energy density, along with a modified version of the Perdew-Burke-Ernzerhof correlation functional.

These are just a few examples of the many Meta-GGA functionals that have been developed over the years. Meta-GGA functionals are generally more computationally expensive than GGA or LDA functionals, but can provide more accurate results for systems with non-uniform electron densities.

Range-Separated Hybrid Functionals

Range-Separated Hybrid (RSH) functionals are a type of Density Functional Theory (DFT) functional that combines Hartree-Fock exchange at long range and DFT exchange at short range. This approach improves the description of both strongly correlated and weakly correlated systems, and has been applied to a wide range of chemical systems. Some examples of RSH functionals include:

1. CAM-B3LYP: This is a RSH functional developed by Yanai, Tew, and Handy in 2004. It uses a range separation parameter to separate the long-range and short-range contributions to the exchange-correlation functional.
2. LC-ωPBE: This is a RSH functional developed by Iikura and co-workers in 2001. It uses a range separation parameter to separate the long-range and short-range contributions to the exchange-correlation functional, and includes a weighted density approximation to improve the description of the correlation energy.
3. ωB97X-D: This is a RSH functional developed by Chai and Head-Gordon in 2008. It uses a range separation parameter to separate the long-range and short-range contributions to the exchange-correlation functional, and includes a dispersion correction to improve the description of long-range van der Waals interactions.
4. PBE0: This is a RSH functional developed by Ernzerhof and Perdew in 1998. It uses a range separation parameter to separate the long-range and short-range contributions to the exchange-correlation functional, and includes a fraction of Hartree-Fock exchange at long range.
5. HSE06: This is a RSH functional developed by Heyd, Scuseria, and Ernzerhof in 2003. It uses a range separation parameter to separate the long-range and short-range contributions to the exchange-correlation functional, and includes a fraction of Hartree-Fock exchange at long range.

These are just a few examples of the many RSH functionals that have been developed over the years. RSH functionals can provide more accurate results for systems with strongly correlated and weakly correlated interactions, and have found widespread use in computational chemistry.

Double-Hybrid Functionals

Double-hybrid functionals are a class of density functional theory (DFT) functionals that combine elements of both conventional DFT functionals and wave function-based methods. They include additional correlation and exchange functionals beyond conventional hybrid functionals, and often include the addition of second-order Møller-Plesset (MP2) correlation energy or coupled-cluster (CCSD(T)) correlation energy.

Some examples of double-hybrid functionals include:

1. B2-PLYP: This is a double-hybrid functional developed by Grimme and co-workers in 2006. It uses a combination of the BLYP functional and a second-order perturbative correction based on MP2 correlation energy.
2. DSD-BLYP: This is a double-hybrid functional developed by Karton and co-workers in 2006. It combines the BLYP functional with a second-order perturbative correction based on MP2 correlation energy, as well as a density-dependent dispersion correction.
3. B2GP-PLYP: This is a double-hybrid functional developed by Goerigk and Grimme in 2010. It combines the B2-PLYP functional with a Gaussian process regression-based correction to the exchange-correlation functional.
4. XYG3: This is a double-hybrid functional developed by Tawada and co-workers in 2004. It uses a combination of the PBE functional, a second-order perturbative correction based on MP2 correlation energy, and a scaled opposite-spin (SOS) correction.
5. DSD-PBEP86: This is a double-hybrid functional developed by Goerigk and Grimme in 2011. It combines the PBEP86 functional with a second-order perturbative correction based on MP2 correlation energy, as well as a density-dependent dispersion correction.

Double-hybrid functionals are generally more accurate than conventional hybrid functionals for certain classes of chemical systems, including weakly bound systems and transition metal complexes. However, they are also computationally more expensive than conventional hybrid functionals, and their accuracy can be highly dependent on the specific system being studied.

Many-Body Perturbation Theory (MBPT)

Many-body perturbation theory (MBPT) is a theoretical framework for calculating electronic properties of molecules and materials that goes beyond the density functional theory (DFT). Here are some examples of MBPT methods:

1. GW approximation: This is a widely used MBPT method that calculates the electronic self-energy of a system, which includes the effect of electron-electron interactions beyond the mean-field approximation. The GW approximation has been used to calculate electronic properties of a variety of materials, including metals, semiconductors, and insulators.
2. Random Phase Approximation (RPA): This is an MBPT method that calculates the response of the electron system to an external perturbation, such as an electric field. RPA has been used to study a wide range of electronic and optical properties of molecules and materials.
3. Bethe-Salpeter equation (BSE): This is an MBPT method that calculates excitonic properties of molecules and materials. Excitons are excited states that arise from the interaction of an electron and a hole in the system, and they play an important role in a variety of optical and electronic properties of materials. The BSE has been used to study excitonic properties of a wide range of materials, including semiconductors and organic molecules.
4. Coupled Cluster Theory (CCT): This is an MBPT method that is widely used in quantum chemistry to calculate the properties of molecular systems. CCT is particularly powerful for calculating properties of systems with strong electron correlation effects, such as transition metal complexes and systems with strong electron-electron interactions.
5. Dynamical Mean Field Theory (DMFT): This is an MBPT method that is particularly useful for studying strongly correlated electronic systems, such as transition metal oxides and rare earth materials. DMFT takes into account the effect of local electronic interactions on the electronic structure of the system, and has been used to study a wide range of electronic and magnetic properties of materials.