Periodic Reporting for period 4 - MDFT (Mathematics of Density Functional Theory)
Reporting period: 2022-03-01 to 2023-02-28
The project aims at establishing rigorous mathematical results about Density Functional Theory, which is the main method used by chemists and physicists to approximate the many-body Schrödinger equation. The project is divided into three main tasks: foundations of DFT, derivation of DFT approximate theories, and properties of DFT models.
Several important goals have been achieved in the project, most of them planned in the project but some totally unexpected.
We can specifically mention the first rigorous justification of the Local Density Approximation (LDA) and the study of properties of the Uniform Electron Gas. This is the most important model in DFT since its birth in 1964. Also, the project mentioned the possibility of improving the Lieb-Oxford bound. Using heavy numerics on the cluster financed through the project, the best constant in this inequality could be decreased. The first bound on the exchange energy could also be shown.
A long review with more than 500 references was written by the PI about classical Riesz and Coulomb gases, in order to disseminate the findings of the project and discuss possible future developments. This article published in the Journal of Mathematical Physics could become a reference in the subject.
Concerning Bose gases, the first derivation of renormalized nonlinear Gibbs measures was provided in an article published in the journal Inventiones Mathematicae.
Unexpected results were obtained concerning the best constant in the Lieb-Thirring inequality, for which a conjecture could also be disproved in two space dimensions. These results arose after the discovery, for some nonlinear DFT Kohn-Sham type models, that quantum tunneling can induce an exponentially small attraction for some nonlinear translation-invariant models.
Important results were obtained concerning Jellium in the Hartree-Fock approximation. It could be proved that the minimizer is never translation-invariant, although the energy gain due to the symmetry breaking is only exponentially small in the density.
A study of the first positive eigenvalue of Dirac-Coulomb operators was initiated. It was proved that one can minimize over the nuclear shape distribution and that the resulting measure is necessarily singular. An interesting conjecture that the minimizer is a delta measure was proposed. The first proof of the Scott correction was provided for the nonlinear Dirac-Fock model.
In an article in the Communications on Partial Differential Equations, the Vlasov equation was derived in a semi-classical limit from the Hartree equation, for an infinite system close to a homogeneous gas.
We can specifically mention the first rigorous justification of the Local Density Approximation (LDA) and the study of properties of the Uniform Electron Gas. This is the most important model in DFT since its birth in 1964. Also, the project mentioned the possibility of improving the Lieb-Oxford bound. Using heavy numerics on the cluster financed through the project, the best constant in this inequality could be decreased. The first bound on the exchange energy could also be shown.
A long review with more than 500 references was written by the PI about classical Riesz and Coulomb gases, in order to disseminate the findings of the project and discuss possible future developments. This article published in the Journal of Mathematical Physics could become a reference in the subject.
Concerning Bose gases, the first derivation of renormalized nonlinear Gibbs measures was provided in an article published in the journal Inventiones Mathematicae.
Unexpected results were obtained concerning the best constant in the Lieb-Thirring inequality, for which a conjecture could also be disproved in two space dimensions. These results arose after the discovery, for some nonlinear DFT Kohn-Sham type models, that quantum tunneling can induce an exponentially small attraction for some nonlinear translation-invariant models.
Important results were obtained concerning Jellium in the Hartree-Fock approximation. It could be proved that the minimizer is never translation-invariant, although the energy gain due to the symmetry breaking is only exponentially small in the density.
A study of the first positive eigenvalue of Dirac-Coulomb operators was initiated. It was proved that one can minimize over the nuclear shape distribution and that the resulting measure is necessarily singular. An interesting conjecture that the minimizer is a delta measure was proposed. The first proof of the Scott correction was provided for the nonlinear Dirac-Fock model.
In an article in the Communications on Partial Differential Equations, the Vlasov equation was derived in a semi-classical limit from the Hartree equation, for an infinite system close to a homogeneous gas.
We provided the first mathematically rigorous derivation of the Local Density Approximation in DFT. This is the most important model in DFT, which was introduced in the foundational paper by Hohenberg and Kohn in 1964 that gave the Nobel prize to Walter Kohn in 1998. The study of the LDA was planned in the project but the results went beyond expectation since we could provide local error bounds valid for all densities.