Density Functional Theory (DFT) is a widely used computational method in theoretical chemistry and materials science. It is based on the concept of the electron density, which is the probability of finding an electron at a specific point in space. The electron density is a fundamental property of a system, as it determines the interaction between atoms and molecules. DFT allows us to calculate the electronic structure of molecules, solids, and surfaces solely from their electron density, without having to consider the wave functions of individual electrons. This makes it a powerful tool for predicting the properties of materials and designing new materials for energy conversion and storage. The theoretical foundations of DFT lie in the Hohenberg-Kohn theorems, which state that the ground-state electron density uniquely determines the external potential of a system. Since its first successful application to molecules by Kohn and Sham in 1965, DFT has become a fundamental tool in theoretical chemistry and materials science, with numerous applications in the prediction of molecular properties, the study of solid-state physics, and the design of new materials. In this book, we will explore the theoretical foundations, practical aspects, and applications of DFT in detail.
Density Functional Theory (DFT) is a computational method used in theoretical chemistry and materials science to calculate the electronic structure of molecules, solids, and surfaces. The basic principle of DFT is that the electron density, which is the probability of finding an electron at a specific point in space, is a fundamental property of a system, as it determines the interaction between atoms and molecules. DFT allows us to determine the electronic structure of a system solely from its electron density, without having to consider the wave functions of individual electrons. This is in contrast to wave-function-based methods, such as Hartree-Fock theory, which require the solution of the Schrödinger equation for each electron. The theoretical foundations of DFT lie in the Hohenberg-Kohn theorems, which state that the ground-state electron density uniquely determines the external potential of a system. DFT has numerous applications in the prediction of molecular properties, the study of solid-state physics, and the design of new materials for energy conversion and storage.
Density Functional Theory (DFT) is a theoretical framework that is used to calculate the electronic structure of molecules, solids, and surfaces. The basic principle of DFT is that the electron density of a system uniquely determines its external potential. This means that the electronic structure of a system can be obtained by determining its electron density, without having to solve the Schrödinger equation for each electron.
The theoretical foundations of DFT lie in the Hohenberg-Kohn theorems, which were formulated in 1964 by Walter Kohn and Lu Jeu Sham. The first theorem states that the ground-state energy of a system is a unique functional of its electron density. This means that for a given electron density, there is only one ground-state energy. The second theorem states that the external potential of a system is also a unique functional of its electron density. This means that for a given electron density, there is only one external potential that will produce that density.
The Hohenberg-Kohn theorems provide a powerful theoretical framework for DFT, as they allow us to calculate the electronic structure of a system directly from its electron density. The electron density can be obtained by solving the Kohn-Sham equations, which are based on the concept of a fictitious system of non-interacting electrons that have the same electron density as the real system. The ground-state energy of the real system is then obtained by solving the Kohn-Sham equations for this fictitious system.
The Kohn-Sham equations are a set of equations that describe the electronic structure of a system. They are based on the idea of a fictitious system of non-interacting electrons, which have the same electron density as the real system. The Kohn-Sham equations are similar to the Schrödinger equation, but they include an additional potential term that accounts for the electron-electron interactions that are not included in the non-interacting electron model.
The exchange-correlation functional is a key component of DFT, as it accounts for the electron-electron interactions that are not included in the non-interacting electron model. The exchange-correlation functional is typically divided into two components: the exchange functional, which accounts for the exchange of electrons between different orbitals, and the correlation functional, which accounts for the repulsion between electrons in the same orbital.
So, the theoretical background of DFT is based on the concept of the electron density, which uniquely determines the external potential of a system. The Hohenberg-Kohn theorems provide a theoretical framework for DFT, and the Kohn-Sham equations and exchange-correlation functional are used to calculate the electronic structure of a system.
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| Walter Kohn |
Density Functional Theory (DFT) has its roots in the 1960s, when Walter Kohn and Lu Jeu Sham published a landmark paper in Physical Review titled "Self-Consistent Equations Including Exchange and Correlation Effects". In this paper, they introduced the Kohn-Sham equations, which form the theoretical foundation of DFT.
Before the development of DFT, theoretical chemistry relied on wave-function-based methods, such as Hartree-Fock theory, to describe the electronic structure of molecules. These methods required the solution of the Schrödinger equation for each electron, which made them computationally expensive and limited their ability to study large systems.
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| Pierre Hohenberg |
In contrast, DFT is based on the concept of the electron density, which is the probability of finding an electron at a specific point in space. The electron density is a fundamental property of a system, as it determines the interaction between atoms and molecules. The Hohenberg-Kohn theorems, which were also published in the 1960s, demonstrated that the ground-state electron density uniquely determines the external potential of a system. This allowed Kohn and Sham to develop the Kohn-Sham equations, which describe the electronic structure of a system solely from its electron density, without having to consider the wave functions of individual electrons.
Initially, DFT was applied primarily to small molecules and atoms. However, over the following decades, improvements in computational resources and the development of new exchange-correlation functionals allowed DFT to be applied to larger and more complex systems, including solids, surfaces, and biomolecules.
Density Functional Theory (DFT) is a widely used method in theoretical chemistry and materials science. In this chapter, we will explore some of the applications of DFT in more detail.
- Predicting Molecular Properties
DFT is commonly used to predict the properties of molecules, such as their electronic structure, vibrational frequencies, and thermodynamic properties. These predictions can be used to understand the reactivity and stability of molecules, as well as to design new compounds with specific properties.
- Designing New Materials
DFT is also used to design new materials, such as catalysts, semiconductors, and superconductors. By predicting the properties of different materials and optimizing their structures, DFT can be used to identify materials with specific properties, such as high conductivity or selectivity.
- Studying Surfaces and Interfaces
DFT is particularly useful for studying surfaces and interfaces, where the electronic properties can be significantly different from those in the bulk material. By modeling the electronic structure of surfaces and interfaces, DFT can provide insights into catalysis, corrosion, and other surface-related phenomena.
- Investigating Electronic Transport
DFT can also be used to investigate electronic transport properties of materials, such as the conductivity and mobility of charge carriers. This information is important for designing electronic devices, such as transistors and solar cells.
- Simulating Chemical Reactions
DFT can be used to simulate chemical reactions and understand the reaction mechanisms. By calculating the energy profiles of different reaction pathways, DFT can identify the most likely reaction pathways and provide insights into the rate and selectivity of reactions.
- Studying Biological Systems
DFT is also used in the study of biological systems, such as enzymes and proteins. By modeling the electronic properties of these systems, DFT can provide insights into their function and interactions with other molecules.
DFT has a wide range of applications in chemistry, materials science, and biology, and has become an indispensable tool for understanding and designing new materials and molecules.
Density Functional Theory (DFT) is a computational method used to calculate the electronic structure of molecules, solids, and surfaces. It is based on the concept of the electron density, which is the probability of finding an electron at a specific point in space. The electron density is a fundamental property of a system, as it determines the interaction between atoms and molecules.
The theoretical foundations of DFT lie in the Hohenberg-Kohn theorems, which state that the ground-state electron density uniquely determines the external potential of a system. This allows us to determine the electronic structure of a system solely from its electron density, without having to consider the wave functions of individual electrons. This is in contrast to wave-function-based methods, such as Hartree-Fock theory, which require the solution of the Schrödinger equation for each electron.
The Kohn-Sham equations are the central equations of DFT and describe the electronic structure of a system. These equations are based on the concept of a fictitious system of non-interacting electrons, which have the same electron density as the real system. The ground-state energy of the real system is then obtained by solving the Kohn-Sham equations for this fictitious system.
The exchange-correlation functional is a key component of DFT, as it accounts for the electron-electron interactions that are not included in the non-interacting electron model. The exchange-correlation functional is typically divided into two components: the exchange functional, which accounts for the exchange of electrons between different orbitals, and the correlation functional, which accounts for the repulsion between electrons in the same orbital.
The electron density is a fundamental quantity in DFT, as it determines the external potential of a system. The electron density is obtained by solving the Kohn-Sham equations, and it is used to calculate various properties of a system, including the total energy, electron distribution, and response to external fields.
The theoretical foundations of DFT provide a powerful framework for calculating the electronic structure of molecules, solids, and surfaces. By using the electron density as the fundamental quantity, DFT allows for accurate and efficient calculations of a wide range of properties, making it an essential tool in theoretical chemistry and materials science.
The Kohn-Sham equations are the central equations of Density Functional Theory (DFT), which describe the electronic structure of a system. The equations were first proposed by Walter Kohn and Lu Sham in 1965, and they have since become a key tool in computational chemistry and materials science.
The Kohn-Sham equations are based on the concept of a fictitious system of non-interacting electrons, which have the same electron density as the real system. This means that the ground-state energy of the real system can be obtained by solving the Kohn-Sham equations for this fictitious system.
The Kohn-Sham equations take the form of a set of coupled partial differential equations, which must be solved self-consistently. The equations describe the behavior of the electron density, and they include terms for the kinetic energy of the electrons, the external potential of the system, and the exchange-correlation potential.
The exchange-correlation potential is a key component of the Kohn-Sham equations, as it accounts for the electron-electron interactions that are not included in the non-interacting electron model. The exchange-correlation potential is typically divided into two components: the exchange functional, which accounts for the exchange of electrons between different orbitals, and the correlation functional, which accounts for the repulsion between electrons in the same orbital.
The solution of the Kohn-Sham equations requires the use of numerical methods, such as finite difference or finite element methods. The self-consistent solution of the equations involves an iterative process, in which the electron density is updated until convergence is reached.
Despite its limitations, the Kohn-Sham method has proven to be a powerful tool in theoretical chemistry and materials science, and it has been used to study a wide range of systems, including molecules, solids, and surfaces. Ongoing research continues to refine and improve the accuracy of DFT calculations, with the goal of enabling the design of new materials and the prediction of their properties with greater accuracy and efficiency.
Exchange-correlation functionals are a key component of Density Functional Theory (DFT), as they account for the electron-electron interactions that are not included in the non-interacting electron model. The exchange-correlation functional is typically divided into two components: the exchange functional, which accounts for the exchange of electrons between different orbitals, and the correlation functional, which accounts for the repulsion between electrons in the same orbital. In this chapter, we will explore the different types of exchange-correlation functionals and their applications.
1 Local Density Approximation (LDA)
The local density approximation (LDA) is the simplest exchange-correlation functional, and it is based on the electron density at each point in space. In the LDA, the exchange-correlation energy is given by a functional of the electron density, and it is assumed to be uniform within a small volume element. This approximation works well for systems with slowly varying electron densities, such as simple metals and insulators.
2 Generalized Gradient Approximation (GGA)
The generalized gradient approximation (GGA) is a more sophisticated exchange-correlation functional that includes not only the electron density but also its gradient. The GGA accounts for the variation in electron density within a volume element and is more accurate than the LDA for a wider range of systems. The GGA has been successful in predicting the properties of transition metals and their compounds.
3 Hybrid Functionals
Hybrid functionals are exchange-correlation functionals that combine the LDA or GGA with a fraction of the exact exchange energy calculated using wave-function-based methods. Hybrid functionals have been shown to improve the accuracy of DFT calculations for a wide range of systems, including transition metals, semiconductors, and molecular systems.
4 Meta-GGA Functionals
Meta-GGA functionals are exchange-correlation functionals that include not only the electron density and its gradient but also the second derivative of the electron density. These functionals are more complex than the LDA, GGA, and hybrid functionals and are designed to improve the accuracy of DFT calculations for systems with strong correlation effects, such as transition metal oxides and rare-earth materials.
5 Range-Separated Functionals
Range-separated functionals are exchange-correlation functionals that separate the exchange and correlation energy into short- and long-range components. The short-range component is described using a GGA or hybrid functional, while the long-range component is described using a LDA or Hartree-Fock functional. Range-separated functionals have been shown to improve the accuracy of DFT calculations for systems with strong electron correlation effectsThe electron density plays a central role in Density Functional Theory (DFT) by providing a way to describe the electronic structure of a system in terms of a probability density rather than individual electron wave functions. The electron density is defined as the probability of finding an electron at a particular point in space.
In DFT, the electron density is used to calculate various properties of a system, including the total energy, electron distribution, and response to external fields. The electron density is obtained by solving the Kohn-Sham equations, which are a set of equations that describe the electronic structure of a system in terms of a fictitious system of non-interacting electrons that have the same electron density as the real system.
The electron density also determines the external potential of a system, which is the potential felt by an electron in the presence of all the other electrons and nuclei in the system. The external potential is calculated using the electron density, and it is used to solve the Kohn-Sham equations and obtain the ground-state electronic structure of the system.
One of the advantages of using the electron density in DFT is that it allows for the calculation of properties of complex systems that are difficult or impossible to calculate using other methods, such as wave-function-based methods. This is because the electron density provides a way to describe the electronic structure of a system in terms of a probability distribution, which can be more easily calculated than individual electron wave functions.
The choice of exchange-correlation functional is a critical aspect of Density Functional Theory (DFT), as it plays a major role in determining the accuracy of the calculations. The exchange-correlation functional accounts for the electron-electron interactions that are not included in the non-interacting electron model of the Kohn-Sham equations.
There are many different types of exchange-correlation functionals available, ranging from simple local functionals to more complex non-local functionals. The most commonly used functionals are based on the generalized gradient approximation (GGA) and the hybrid functionals that combine GGA with exact exchange terms.
The choice of functional depends on the system being studied and the properties that need to be calculated. For example, GGA functionals are generally suitable for systems with weakly interacting molecules or van der Waals interactions, while hybrid functionals are better suited for systems with strong electron correlation effects, such as transition metal complexes and organometallic compounds.
It is important to note that no single functional is universally accurate for all systems, and it is common practice to test several different functionals to determine the most appropriate one for a specific system. This process is often referred to as benchmarking, where the results of DFT calculations are compared with experimental data or higher-level theoretical methods, such as coupled-cluster theory or density matrix renormalization group (DMRG) methods.
In addition to the choice of functional, other practical aspects of DFT calculations, such as convergence criteria and numerical methods, can also affect the accuracy of the results. Therefore, it is important to carefully consider all aspects of the DFT calculation to ensure the most accurate and reliable results.
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