Member-only story

Chapter 015: Unveiling the Mysteries of Density Functional Theory with Wil Schilders

Derek Cook

·9.2k Followers· Follow

Published in Chapter 015 Density Functional Theory Wil Schilders

5 min read · 2 weeks before

88 View Claps

5 Respond

Save

Listen

Chapter 015, Density Functional Theory

by Wil Schilders

4.5 out of 5

Language	:	English
File size	:	3853 KB
Text-to-Speech	:	Enabled
Enhanced typesetting	:	Enabled
Print length	:	56 pages
Screen Reader	:	Supported

Density Functional Theory (DFT) is a powerful computational technique used to calculate the electronic structure of atoms, molecules, and materials. It is based on the Hohenberg-Kohn theorem, which states that the ground-state energy of a system is a unique functional of the electron density. This means that all the properties of a system can be determined by knowing the electron density.

DFT is a computationally demanding technique, but it is much less expensive than other methods, such as Hartree-Fock theory. This makes it a popular choice for studying large systems, such as proteins and materials.

The Hohenberg-Kohn Theorem

The Hohenberg-Kohn theorem is the foundation of DFT. It states that the ground-state energy of a system is a unique functional of the electron density. This means that all the properties of a system can be determined by knowing the electron density.

The Hohenberg-Kohn theorem can be proven using the variational principle. The variational principle states that the ground-state energy of a system is the lowest energy that can be obtained by any trial wave function. A trial wave function is a function that satisfies the Schrödinger equation.

The Hohenberg-Kohn theorem can be used to derive the Kohn-Sham equations. The Kohn-Sham equations are a set of self-consistent equations that can be used to calculate the electron density of a system.

The Kohn-Sham Equations

The Kohn-Sham equations are a set of self-consistent equations that can be used to calculate the electron density of a system. The Kohn-Sham equations are:

[-\frac{1}{2}\nabla^2 + V_\text{eff}(\mathbf{r})]\phi_i(\mathbf{r}) = \varepsilon_i\phi_i(\mathbf{r})

where:

* $\phi_i(\mathbf{r})$ is the i-th Kohn-Sham orbital * $\varepsilon_i$ is the i-th Kohn-Sham orbital energy * $V_\text{eff}(\mathbf{r})$ is the effective potential

The effective potential is a functional of the electron density. It includes the external potential, which is the potential due to the nuclei, and the exchange-correlation potential, which is the potential due to the interactions between the electrons.

The Kohn-Sham equations can be solved self-consistently. This means that the electron density is used to calculate the effective potential, which is then used to calculate the electron density. This process is repeated until the electron density and the effective potential are consistent.

Applications of DFT

DFT is used to study a wide variety of systems, including atoms, molecules, and materials. DFT is used to calculate the ground-state energy, electron density, and other properties of these systems.

DFT is used in many different fields of science, including chemistry, physics, and materials science. DFT is used to study a wide variety of problems, including the following:

The electronic structure of atoms and molecules
The properties of materials
The reactions of molecules
The design of new materials

Strengths and Limitations of DFT

DFT is a powerful tool for studying the electronic structure of atoms, molecules, and materials. However, DFT has some strengths and limitations.

The strengths of DFT include the following:

DFT is a computationally efficient technique.
DFT can be used to study large systems.
DFT can be used to calculate a wide variety of properties.

The limitations of DFT include the following:

DFT is not always accurate.
DFT can be difficult to apply to systems with strong correlation effects.

DFT is a powerful computational technique used to study the electronic structure of atoms, molecules, and materials. DFT is based on the Hohenberg-Kohn theorem, which states that the ground-state energy of a system is a unique functional of the electron density. This means that all the properties of a system can be determined by knowing the electron density.

Chapter 015, Density Functional Theory

by Wil Schilders

4.5 out of 5