Orbital-free density functional theory

In computational chemistry, orbital-free density functional theory is a quantum mechanical approach to electronic structure determination which is based on functionals of the electronic density. It is most closely related to the Thomas–Fermi model. Orbital-free density functional theory is, at present, less accurate than Kohn–Sham density functional theory models, but has the advantage of being fast, so that it can be applied to large systems.

The Kinetic Energy of Electrons

The Hohenberg-Kohn theorems^[1] guarantee that, for a system of atoms, there exists a functional of the electron density that yields the total energy. Minimization of this functional with respect to the density gives the ground-state density from which all of the system's properties can be obtained. Although the Hohenberg-Kohn theorems tell us that such a functional exists, they do not give us guidance on how to find it. In practice, the density functional is known exactly except for two terms. These are the electronic kinetic energy and the exchange-correlation energy. The lack of the true exchange-correlation functional is a well known problem in DFT and there exists a huge variety of approaches to approximate this crucial component.

The fact that there is no known density functional for the electron kinetic energy is generally circumvented in another way. The traditional approach of density functional theory is to assume that the system can be treated as electrons residing in single-particle states (called orbitals). The total wave function can then be written as a Slater determinant of these single-particle orbitals. The orbitals themselves are found by diagonalizing the effective Kohn-Sham Hamiltonian. The kinetic energy of an electron in a single-particle state can be written exactly in terms of the orbital, $|\phi_i\rangle$, as (in atomic units)

$E_{kinetic}=-\frac{1}{2}\langle\phi_i|\nabla^2|\phi_i\rangle$.

The problem with this approach is that it requires diagonalization of the Kohn-Sham Hamiltonian to find the single-particle orbitals. Further, since the Hamiltonian itself depends on these orbitals, the problem must be solved self-consistently. This is, in general, a computationally expensive process. If one could write the electron kinetic energy as a density functional, the problem of diagonalizing a large matrix could be replaced with a relatively straightforward functional optimization problem. Thus, finding an accurate density functional for the kinetic energy is the key focus of the so-called "orbital free" methods.

One of the first attempts to do this was the Thomas–Fermi model, which wrote the kinetic energy as^[2]

$E_{TF}=\frac{3}{10}\left(3\pi^2\right)^{\frac{2}{3}}\int{\left[n\left(\vec{r}\right)\right]^{\frac{5}{3}}d^3r}$.

This expression is based on the homogeneous electron gas and, thus, is not very accurate for most physical systems. Finding more accurate and transferable kinetic energy density functionals is the focus of ongoing research.

References

1. Hohenberg, P; Kohn, W (1964). "Inhomogeneous Electron Gas". Physical Review 136: B864–B871. Bibcode:1964PhRv..136..864H. doi:10.1103/PhysRev.136.B864.
2. Ligneres, Vincent L.; Emily A. Carter (2005). "An Introduction to Orbital Free Density Functional Theory". In Syndey Yip. Handbook of Materials Modeling. Springer Netherlands. pp. 137–148.