The Lieb–Oxford ineqaulity in Quantum Mechanics

Many important properties of atoms or molecules can be determined by their electronic structures, in particular by their ground state energy. Finding these energies using traditional methods is computationally very costly which has led to the introduction of density functional theory. While properties of quantum mechanical systems are normally described in terms of their wave-functions, density functional theory uses only their charge densities, which is computationally much more efficient. For example, to approximate the Coulomb energy of a quantum system, one can consider a classical expression that only involves the charge density. This expression does in general not yield the real Coulomb energy, which depends on the wave-function of the system. The Lieb Oxford inequality provides a lower bound on the error. Since its first rigorous proof, this inequality has played a fundamental role in quantum mechanics, and is an important tool in constructing approximations of the Coulomb energy in density functional theory. This research project investigates two improvements of the LiebOxford inequality that should lead to more precise computations of ground state energies, and to a better understanding of electronic structures. The LiebOxford inequality involves a constant for which only upper and lower bounds are known. The first part of the research aims to find improvements of the constant depending on the number of particles in the system. This is well-understood in the special case of a single particle, where the constant is explicitly known and where a system exists, for which the inequality becomes an equality. For two particles, proving the existence of such a minimising system and finding the explicit value of the constant has remained unsolved for more than thirty years. I want to provide an answer to these questions by applying recent results from optimal transport theory, which show that it is sufficient to only consider a very special class of quantum mechanical systems for the minimisation problem. Subsequently I want to extend the methods to more than two particles. Another way to improve the inequality is to incorporate specific properties of the materials under consideration. For example, most molecules have the special property of being spin-unpolarised. Density functional theory works especially well in this setting but so far this property has not been used to improve the LiebOxford inequality. I aim to prove a recently formed mathematical conjecture which states that for spin-unpolarised systems, the aforementioned constant does not depend on the number of particles. To solve this problem, the conjecture can be related to a concise inequality for a set of orthonormal functions, which has independent mathematical merit.

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