Tru Physics

Density Functional Theory (DFT)

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Introduction

Density Functional Theory (DFT) is a computational quantum mechanical modelling method used to investigate the electronic structure of many-body systems, especially atoms, molecules, and the condensed phases.

The Basics of DFT

The fundamental premise behind DFT is the Hohenberg-Kohn theorem, which states that the ground state properties of a many-electron system are uniquely determined by the electron density distribution. The second Hohenberg-Kohn theorem introduces the concept of the universal density functional, which leads to the concept of energy as a functional of electron density.

Mathematically, Hohenberg-Kohn theorem implies that for any system of interacting particles in an external potential v(\vec{r}), the potential is determined uniquely (apart from a constant) by the ground state particle density n(\vec{r}):

v(\vec{r}) = v'(\vec{r}) + C

where C is a constant.

Kohn-Sham Equations

While the Hohenberg-Kohn theorem is the foundation of DFT, the practical implementation of DFT largely follows from the Kohn-Sham equations. These equations introduce the concept of a system of non-interacting particles that reproduce the exact ground state density of the real system.

The Kohn-Sham equations can be written as:

\left[-\dfrac{1}{2}\nabla^2 + V_{eff}(\vec{r})\right]\psi_i(\vec{r}) = \epsilon_i\psi_i(\vec{r})

where:

  • \psi_i(\vec{r}) are the Kohn-Sham orbitals,
  • \epsilon_i are the Kohn-Sham eigenvalues, and
  • V_{eff}(\vec{r}) is the effective potential, which includes the external potential and the effects of electron-electron interactions.

Exchange-Correlation Functional

The greatest challenge in DFT calculations is the accurate determination of the exchange-correlation functional, which accounts for the effects of electron exchange and correlation. The exact form of the exchange-correlation functional is not known, and various approximations are used in practice, such as the local density approximation (LDA), generalized gradient approximation (GGA), and hybrid functionals.

Applications of DFT

DFT has a wide range of applications in physics, chemistry, and materials science. It is commonly used to calculate electronic band structures and densities of states of solids, predict molecular geometries and reaction pathways in chemistry, and determine the electronic properties of materials and molecules.

Conclusion

DFT is a powerful tool in quantum mechanics and computational physics and chemistry, enabling the prediction and understanding of the properties of atoms, molecules, and solids. The success of DFT relies on the careful choice of the exchange-correlation functional, which continues to be an active area of research.

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