An Introduction to Orbital-Free Density Functional Theory

Abstract

Given a quantum mechanical system of N electrons and an external potential (which typically consists of the potential due to a collection of nuclei), the traditional approach to determining its ground-state energy involves the optimization of the corresponding wavefunction, a function of 3N dimensions, without considering spin variables. As the number of particles increases, the computation quickly becomes prohibitively expensive. Nevertheless, electrons are indistinguishable so one could intuitively expect that the electron density — N times the probability of finding any electron in a given region of space — might be enough to obtain all properties of interest about the system. Using the electron density as the sole variable would reduce the dimensionality of the problem from 3N to 3, thus drastically simplifying quantum mechanical calculations. This is in fact possible, and it is the goal of orbital-free density functional theory (OF-DFT). For a system of N electrons in an external potential Vext, the total energy E can be expressed as a functional of the density ρ [1], taking on the following form: 1 E\left[ \rho \right] = F\left[ \rho \right] + \mathop \smallint \limits_\Omega V_{ext} \left( {\overrightarrow r } \right)\rho \left( {\overrightarrow r } \right)d\overrightarrow r Here, Ω denotes the system volume considered, while F is the universal functional that contains all the information about how the electrons behave and interact with one another. The actual form of F is currently unknown and one has to resort to approximations in order to evaluate it. Traditionally, it is split into kinetic and potential energy contributions, the exact forms of which are also unknown.