Density Functional Theory — DFT

Density Functional Theory (DFT) is a quantum mechanical theory used to describe the electronic structure of materials. It is a computational method that allows for the calculation of the electronic properties of a system by solving the Schrödinger equation for the electron density, rather than for each individual electron.

The basic theory behind density functional theory (DFT) is the Hohenberg-Kohn theorem, which states that the ground-state properties of a system can be determined by the electron density alone. This means that the many-electron wavefunction does not need to be explicitly determined, but can be approximated by the electron density.

The central equation in DFT is the Kohn-Sham equation, which is a set of equations that describe the electronic structure of a system. The Kohn-Sham equations are similar in form to the Hartree-Fock equations, but instead of a single-particle wavefunction, they use a set of auxiliary wavefunctions called Kohn-Sham orbitals.

The Kohn-Sham orbitals are determined by minimizing the total energy of the system, which includes the kinetic energy of the electrons, the Coulombic repulsion between the electrons, and the potential energy due to the interaction between the electrons and the external potential. The external potential includes the potential from the nuclei of the atoms in the system, as well as any additional potentials, such as those due to an external electric field or solvation effects.

The electron density is then calculated from the Kohn-Sham orbitals, and the total energy of the system is determined from the electron density using a functional, which is a mathematical expression that relates the electron density to the total energy of the system.

The most commonly used functional is the exchange-correlation functional, which describes the exchange and correlation interactions between the electrons. The exchange interaction arises from the antisymmetry of the electron wavefunction, while the correlation interaction arises from the fact that the electrons repel each other due to their Coulombic interactions.

The exchange-correlation functional is typically split into two parts: the exchange functional, which describes the exchange interaction between the electrons, and the correlation functional, which describes the correlation interaction. There are many different forms of exchange-correlation functionals, each with different levels of accuracy and computational cost.

Overall, DFT is a widely used method for calculating the electronic structure and properties of molecules and materials. Its ability to handle a wide range of systems, from small molecules to large proteins and materials, has made it an invaluable tool for chemists, physicists, and materials scientists.

The DFT method is generally considered a level of theory between the simple molecular orbital theory and the more accurate methods such as coupled cluster or full configuration interaction. DFT calculations can provide a good balance between accuracy and computational cost, making it a widely used approach for studying the electronic structure of molecules, surfaces, and solids.

DFT is a ground-state theory, which means it can only describe the electronic properties of the system in its lowest energy state. However, it can be used to calculate a wide range of properties, including electronic energies, structures, bonding, reactivity, and spectroscopic properties. DFT can also be used to study properties such as electron transport, magnetism, and optical properties.

There are many different types of DFT methods and functionals available, each with its own strengths and weaknesses. The choice of DFT method depends on the specific properties of the system being studied and the level of accuracy required. In general, more accurate methods require more computational resources, which can limit the size of the system that can be studied.

In Density Functional Theory (DFT) calculations, the basis set is a set of mathematical functions used to describe the electronic wave function of the system being studied. The choice of basis set is important because it can significantly affect the accuracy of DFT calculations.

There are two main types of basis sets commonly used in DFT calculations: Gaussian-type basis sets and plane wave basis sets.

1. Gaussian-type basis sets: These are typically used in molecular calculations and are defined by a set of Gaussian functions centered on each atom in the molecule. Gaussian basis sets are often chosen based on their size and polarization, which can affect the accuracy of the calculation.
2. Plane wave basis sets: These are typically used in periodic solid-state calculations and are defined by a set of plane waves with specific energies and directions. The choice of plane wave basis set is often based on the size of the wave vectors, which can affect the accuracy of the calculation.

In addition to the type of basis set, the size of the basis set is also an important consideration in DFT calculations. Larger basis sets typically provide more accurate results, but require more computational resources to perform the calculation.

There are many different basis sets available for use in DFT calculations, and the choice of basis set depends on the system being studied and the level of accuracy required. Some common basis sets used in DFT calculations include 6–31G, 6–311G, and 6–311++G(d,p) for Gaussian-type basis sets, and plane wave basis sets with kinetic energy cutoffs ranging from 20 to 100 Ry.

It is important to note that the choice of basis set can significantly affect the accuracy of DFT calculations, and it is often necessary to benchmark the chosen basis set against experimental data or higher-level calculations to ensure accuracy.

In Density Functional Theory (DFT) calculations, the exchange-correlation (XC) functional is a mathematical expression that describes the relationship between the electron density of a system and its total energy. The XC functional is a key component of DFT, as it provides a way to account for the many-body interactions of electrons in a system.

The XC functional is typically divided into two components: the exchange energy and the correlation energy. The exchange energy is the energy gained or lost when electrons are exchanged between two orbitals, while the correlation energy accounts for the many-body interactions between electrons in the system.

There are many different types of XC functionals that can be used in DFT calculations, each with its own strengths and weaknesses. Some common types of XC functionals include:

1. Local Density Approximation (LDA): This is a type of XC functional that is based on the electron density at a single point in space. LDA has a relatively simple form and is computationally efficient, but can provide inaccurate results for some types of systems.
2. Generalized Gradient Approximation (GGA): This is a type of XC functional that takes into account the gradient of the electron density in addition to the electron density itself. GGA functionals are more accurate than LDA for many systems, but may not be accurate for all systems.
3. Hybrid functionals: These are XC functionals that combine elements of LDA or GGA functionals with exact exchange, which is a more accurate but computationally expensive way to account for electron exchange. Examples of hybrid functionals include B3LYP, PBE0, and M06–2X.
4. Meta-GGA functionals: These are XC functionals that include a dependence on the Laplacian of the electron density, in addition to the electron density and its gradient. Meta-GGA functionals are more accurate than GGA functionals for some types of systems, but are also more computationally expensive.

The choice of XC functional depends on the specific properties of the system being studied and the level of accuracy required. In general, hybrid functionals are often used for organic compounds and other molecular systems, while GGA or meta-GGA functionals are often used for solid-state systems. Benchmarking against experimental data or higher-level calculations is often necessary to assess the accuracy of the chosen XC functional.

Local Density Approximation (LDA)

Local Density Approximation (LDA) is a type of exchange-correlation functional used in Density Functional Theory (DFT) calculations. The LDA is the simplest form of the XC functional, which is based on the electron density at a single point in space.

The LDA assumes that the exchange-correlation energy at any given point in space is a function only of the electron density at that point. In other words, the LDA approximates the exchange-correlation energy as a local functional of the density, meaning that it doesn’t account for any spatial variation of the density. This makes the LDA computationally efficient and easy to implement, but it can lead to inaccurate results for some types of systems.

Despite its simplicity, LDA has been successful in predicting a wide range of properties of atoms, molecules, surfaces, and solids. LDA functionals are often used as a starting point for more complex functionals, such as Generalized Gradient Approximation (GGA) functionals.

One of the main limitations of LDA is its inability to accurately describe systems with strong electron correlations, such as transition metal complexes and some semiconductors. In these cases, more complex functionals that go beyond the LDA, such as hybrid functionals or meta-GGA functionals, may be necessary to obtain accurate results.

Generalized Gradient Approximation (GGA)

Generalized Gradient Approximation (GGA) is a type of exchange-correlation (XC) functional used in Density Functional Theory (DFT) calculations. GGA is an improvement over the Local Density Approximation (LDA), as it takes into account the gradient of the electron density in addition to the density itself.

GGA functionals describe the XC energy as a functional of both the electron density and its first derivative (the gradient), resulting in a more accurate description of the electronic structure compared to LDA. The gradient-dependent terms in GGA functionals account for the spatial variation of the density and help to capture the effects of electron correlation, resulting in improved accuracy for a wide range of systems.

There are many different types of GGA functionals, with varying levels of accuracy and computational efficiency. Examples of popular GGA functionals include PBE (Perdew-Burke-Ernzerhof), BLYP (Becke-Lee-Yang-Parr), and PW91 (Perdew-Wang 1991).

GGA functionals have been successful in predicting a wide range of properties of atoms, molecules, surfaces, and solids. However, they are still limited in their ability to describe strongly correlated systems, such as transition metal complexes and some semiconductors. In these cases, more advanced functionals that include higher-order derivatives of the electron density, such as meta-GGA or hybrid functionals, may be necessary to obtain accurate results.

Hybrid functionals

Hybrid functionals are a type of exchange-correlation (XC) functional used in Density Functional Theory (DFT) calculations. They are called “hybrid” because they combine the local density approximation (LDA) or generalized gradient approximation (GGA) exchange-correlation functionals with a fraction of non-local Hartree-Fock (HF) exchange.

The non-local HF exchange term in hybrid functionals includes contributions from orbitals that are not included in the LDA or GGA exchange-correlation functionals. By combining the LDA or GGA exchange-correlation functionals with non-local HF exchange, hybrid functionals are able to account for both the static and dynamic electron correlation, leading to more accurate predictions of electronic properties.

Hybrid functionals are typically more accurate than GGA or LDA functionals for systems with strong electron correlation, such as transition metal complexes, semiconductors, and molecules with strong charge transfer. Examples of popular hybrid functionals include B3LYP (Becke’s three-parameter hybrid functional with the Lee-Yang-Parr correlation functional), PBE0 (PBE with 25% HF exchange), and HSE (Heyd-Scuseria-Ernzerhof) functionals.

One limitation of hybrid functionals is their increased computational cost compared to simpler functionals such as LDA or GGA. However, they are often necessary to obtain accurate results for certain types of systems, and are widely used in electronic structure calculations for a wide range of materials.

In density functional theory (DFT), the solvent can have a significant effect on the properties of a molecule or material, especially in the context of reactions or processes that occur in solution. The solvent can interact with the solute, inducing changes in its electronic structure, thermodynamics, and reactivity.

To account for the effects of solvent in DFT calculations, a number of solvation models have been developed. These models typically fall into two broad categories: explicit and implicit solvation models.

Explicit solvation models represent the solvent molecules explicitly in the calculation, typically by adding solvent molecules to the computational cell or by constructing a separate simulation box containing the solvent. Explicit solvation models are computationally expensive, but they can be accurate and provide a detailed description of the solute-solvent interactions.

Implicit solvation models, on the other hand, model the solvent as a continuum with a dielectric constant, without explicitly representing individual solvent molecules. These models are computationally less expensive than explicit models, but they can be less accurate and may require parameterization.

Some popular implicit solvation models used in DFT calculations include the COSMO (conductor-like screening model), PCM (polarizable continuum model), and CPCM (continuum solvation model with conductor-like polarizable continuum). These models represent the solvent as a dielectric continuum with a spatially varying dielectric constant that accounts for the polarization of the solvent molecules around the solute.

It is important to note that the accuracy of a solvation model depends on the properties of the system being studied, as well as the level of theory used in the DFT calculation. As a result, it is important to validate solvation model results with experimental data and to carefully consider the choice of solvation model and level of theory when performing DFT calculations in a solvent environment.

COSMO stands for Conductor-like Screening Model. It is a theoretical method used in computational chemistry to simulate the effects of solvation on a molecule or ion in a solution.

In COSMO, the solvent is modeled as a continuum with a dielectric constant, and the solute is treated as a collection of point charges. The point charges are then screened by a continuum of counter charges that mimic the effect of solvent polarization.

The screening charges are placed on a surface that encloses the solute, and the size and shape of this surface are determined by the shape and size of the solute. The surface is then divided into discrete grid points, and the electrostatic potential at each point is calculated.

COSMO can be used with a wide range of electronic structure methods, including Hartree-Fock and density functional theory. It has been used to study a wide range of chemical systems, from small molecules to proteins and nucleic acids.

COSMO is computationally efficient, and it has been found to give accurate results for a range of solvation properties, such as solvation energies, dipole moments, and vibrational frequencies. However, like any theoretical method, COSMO has limitations and assumptions, and its accuracy may be influenced by factors such as the choice of dielectric constant, the size and shape of the solute, and the level of theory used to calculate the solute’s electronic structure. As a result, it is important to validate COSMO results with experimental data and other solvation models.

PCM stands for the Polarizable Continuum Model. It is an implicit solvation model used in quantum chemical calculations to take into account the effects of solvation on the electronic structure and properties of a molecule or system.

PCM treats the solvent as a continuous medium with a dielectric constant and calculates the electrostatic interactions between the solute and solvent using a set of effective charges and polarizabilities. The solute is described as a collection of point charges that are embedded in a dielectric continuum.

The dielectric constant represents the degree of polarization of the solvent molecules and can be assigned different values depending on the solvent. The effective charges and polarizabilities are obtained from the electrostatic potential on the surface of the solute, which is determined by the electron density calculated from the quantum mechanical calculations.

PCM is computationally efficient and can be used with a wide range of quantum chemical methods, including density functional theory (DFT) and Hartree-Fock theory. It has been applied to study a wide range of chemical systems, such as reactions in solution, protein-ligand binding, and electrochemistry.

However, like any theoretical model, PCM has limitations and assumptions, and its accuracy may be influenced by factors such as the choice of dielectric constant, the size and shape of the solute, and the level of theory used in the quantum chemical calculations. As a result, it is important to validate PCM results with experimental data and other solvation models.

CPCM stands for the continuum solvation model with the conductor-like polarizable continuum model. It is a theoretical method used in computational chemistry to simulate the effects of solvation on a molecule or ion in a solution.

In CPCM, the solvent is modeled as a continuous medium with a dielectric constant, and the solute is treated as a collection of point charges. The point charges are then surrounded by a virtual surface, which separates the solute from the solvent.

The virtual surface used in CPCM is polarizable, which means it can respond to the presence of an electric field by changing its shape and inducing charge polarization. This allows CPCM to capture some of the non-electrostatic contributions to solvation, such as the effects of hydrogen bonding and van der Waals interactions.

CPCM is a popular solvation model in computational chemistry because it is computationally efficient and can be used with a wide range of electronic structure methods, including Hartree-Fock and density functional theory. It has been used to study a wide range of chemical systems, from small molecules to proteins and nucleic acids.

However, like any theoretical method, CPCM has limitations and assumptions, and its accuracy may be influenced by factors such as the choice of dielectric constant, the size and shape of the virtual surface, and the level of theory used to calculate the solute’s electronic structure. As a result, it is important to validate CPCM results with experimental data and other solvation models.