The density functional function, within the context of Density Functional Theory (DFT), is a mathematical expression that relates the ground state energy of a quantum mechanical system to its electron density.
Understanding the Role of Density Functional in DFT
DFT aims to determine the ground state properties of a many-body system (like a molecule or a solid) by focusing on the electron density, ρ(r), rather than the many-body wavefunction. This is a significant simplification, as the electron density is a function of only three spatial coordinates, while the wavefunction depends on the coordinates of all the electrons. The density functional is a key component that connects the electron density to the total energy of the system.
The Energy Functional
In DFT, the total energy of the system is expressed as a functional of the electron density, denoted as E[ρ]. This functional can be broken down into several terms:
- Kinetic Energy Functional (T[ρ]): Represents the kinetic energy of the electrons. Accurately calculating this term is challenging and often relies on approximations.
- External Potential Energy Functional (Vext[ρ]): Represents the potential energy due to the external potential, typically the attraction between the electrons and the nuclei. This term is usually straightforward to calculate.
- Hartree Energy Functional (EH[ρ]): Represents the classical electrostatic interaction energy between the electrons.
- Exchange-Correlation Energy Functional (Exc[ρ]): This is the most complex and crucial term. It accounts for all the many-body effects, including electron exchange and correlation, which are not captured by the Hartree term. Finding accurate approximations for Exc[ρ] is a major focus of DFT research.
The total energy functional is then:
E[ρ] = T[ρ] + Vext[ρ] + EH[ρ] + Exc[ρ]
Functionals in Practice
In practice, the exact form of the kinetic energy functional and especially the exchange-correlation functional are unknown. Therefore, approximations must be used. Commonly used approximations for Exc[ρ] include:
- Local Density Approximation (LDA): Assumes the exchange-correlation energy density at a given point in space depends only on the electron density at that point.
- Generalized Gradient Approximation (GGA): Considers the gradient of the electron density, in addition to the density itself, to account for inhomogeneities in the electron density.
- Meta-GGA: Includes the kinetic energy density or the Laplacian of the density as additional variables.
- Hybrid Functionals: Mix the exact exchange energy from Hartree-Fock theory with DFT exchange and correlation functionals.
Importance of Minimization
The Hohenberg-Kohn variational principle states that the ground state electron density minimizes the total energy functional. Therefore, by minimizing E[ρ] with respect to ρ, we can, in principle, determine the ground state electron density and, consequently, other ground state properties of the system. This minimization is typically performed using the Kohn-Sham equations.
Conclusion
In summary, the density functional function is the mathematical formulation within DFT that connects the electron density to the total energy of a system, enabling the calculation of ground state properties. The accuracy of DFT calculations relies heavily on the approximations used for the kinetic energy and, most importantly, the exchange-correlation energy functional.
