Quantum Mechanics Without Wave Functions: Advancing Orbital-Free Methods for Materials Research

Abstract

Improvements in our theoretical understanding of the material world, frequently aided by computer simulations, have led to innumerable scientific advances. While, in principle, it is advantageous to base these virtual experiments on the fundamental laws of quantum mechanics, naïve application of this strategy would quickly overwhelm humanity’s computing resources. Density functional theory (DFT) offers a way forward, enabling explicitly quantum-mechanical simulations that, in practice, retain much of the accuracy of the naïve approach. Hundred-atom simulations utilizing standard DFT are now ubiquitous, and larger systems can be studied with more effort.

Orbital-free DFT, the main subject of this dissertation, is uniquely suited to cases when large, or very fast, calculations are required. The orbital-free approach facilitates, to give some examples, study of complex Mg-Al alloys, many-atom defects in solids, and properties of liquid metals. In orbital-free DFT, the kinetic energy of an electron system is approximated from the electron density alone—and, for a given class of materials, the dramatic benefits of the orbital-free approach can only be realized if accurate approximations are found.

This dissertation delivers new insights relevant to the nearly-century-long quest for a universally accurate kinetic energy approximation for orbital-free DFT. Such an approximation would revolutionize materials simulation, yielding co-benefits to many scientific disciplines.

The dissertation work is unified by consideration of a nonnegative kinetic energy density (KED), analogous to the electron density, which integrates to the full kinetic energy. The first new results are summarized by first- and second-order response functionals for the KED based on a free-electron reference system. Using only the electron density as input, these response functionals can approximate the KED of nearly-free-electron systems with unprecedented accuracy, and the evidence provided suggests they will help expand the reach of orbital-free DFT. Another new result is an upper bound for the KED involving orbital-free ingredients; mathematical constraints of this sort have long proven useful for developing improved approximations. Finally, the dissertation demonstrates new capabilities for a modified version of orbital-free DFT intended to improve accuracy for materials with large variations in electron density, such as transition metals.