# Density Functional Theory

Density functional theory (DFT) is a method of studying the electronic structure of multi-electron systems. Density functional theory has a wide range of applications in physics and chemistry, especially to study the properties of molecules and condensed matter. It is one of the most commonly used methods in the fields of condensed matter physical computational materials science and computational chemistry.

### Introduction

The classical methods of electronic structure theory, especially the Hartree-Fock method and the post-Hartree-Fock method, are based on complex multi-electron wave functions. The main goal of density functional theory is to replace the wave function with electron density as the basic quantity of research. Because the multi-electron wave function has 3N variables (N is the number of electrons, and each electron contains three spatial variables), and the electron density is only a function of three variables, it is more convenient to handle both conceptually and in practice.

Although the concept of density functional theory originated from the Thomas-Fermi model, it did not have a solid theoretical basis until the Hohenberg-Kohn theorem was put forward. The Hohenberg-Kohn first theorem states that the ground state energy of the system is only a functional of the electron density.

The second Hohenberg-Kohn theorem proves that the ground state energy is obtained after the system energy is minimized by taking the ground state density as a variable.

The original HK theory only applies to the ground state without a magnetic field, although it has now been generalized. The original Hohenberg-Kohn theorem only pointed out the existence of one-to-one correspondence, but did not provide any such precise correspondence. It is in these precise correspondences that there is an approximation (this theory can be extended to time-related fields to calculate the properties of excited states.

### Implementation

The most common application of density functional theory is achieved by the Kohn-Sham method. In the framework of Kohn-Sham DFT, the most difficult many-body problem (generated due to the interaction of electrons in an external electrostatic potential) is simplified to a movement of an uninteracting electron in the effective potential field. problem. This effective potential field includes the effects of external potential fields and Coulomb interactions between electrons, such as exchange-related effects. Dealing with exchange-related functions is a difficult point in KS DFT. At present, there is no accurate method to solve the EXC of exchange correlation energy. The simplest approximate solution method is local density approximation (LDA approximation). LDA approximately uses uniform electron gas to calculate the exchange energy of the system (the exchange energy of uniform electron gas can be accurately solved), and the related energy part is processed by fitting the free electron gas.

### Software

VASP (PP-PW, commercial software)
CASTEP (PP-PW, commercial software)
Abinit (PP-PW, open-source software)
Crystal (FP-LCAO, commercial software)
Quantum-ESPRESSO (PP-PW, formerly PWscf, open-source software)
Wien2k (FP-LAPW, commercial software)
Siesta (Order-N method, also known as Siesta method, based on LCAO, open-source software)
ELK (FP-LAPW, open-source software)
Exciting (PF-LAPW, open-source software)
Fleur (FP-LAPW, open-source software)
Octopus (TDDFT, used for optical property calculation, open-source software)
ATK (Siesta method, commercial software)
USPEX (crystal structure prediction, open-source software)
Calypso (prediction of crystal structure, open-source software)

### Applications

Classical DFT has found many applications, for example:

• Developing new functional materials in materials science, in particular nanotechnology.
• Studying the properties of fluids at surfaces and the phenomena of wetting and adsorption.
• Understanding life processes in biotechnology.
• Improving filtration methods for gases and fluids in chemical engineering.
• Fighting pollution of water and air in environmental science.
• Generating new procedures in microfluidics and nanofluidics.
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