Emmanuel FROMAGER

Professor

✉️ fromagere[at]unistra.fr

Teaching

STUDENTS AND POSTDOC FELLOWS: Wafa MAKHLOUF (PhD), Lucien DUPUY (Postdoctoral Researcher)

Complete list of my publications: .docx (with links to the papers) or .pdf

Research Topics

Ensemble density functional theory of electronic excitations 

Standard density functional theory (DFT) is a pure-ground-state electronic structure theory [1]. It can be extended, in principle exactly, to both neutral and charged excited states via the so-called extended N-centered ensemble formalism [2,3], where the basic variable is a weighted sum of ground- and excited-state electronic densities. The ensemble Hartree-exchange-correlation (Hxc) density functional (from which the electronic repulsion energy in both ground and excited states can be evaluated [4]) varies with the ensemble weights. This weight dependence, whose modelling is still highly challenging, plays a central role in the description of the discontinuities that the Hxc density-functional derivative (the so-called Hxc potential) exhibits when an electronic excitation occurs. This fundamental feature is directly related to the exactification of Koopmans’ theorem in DFT [5,6].     

Quantum embedding theory of strongly correlated electrons

Density matrix embedding theory (DMET) [1,2] is a quantum embedding electronic structure theory where the locality of strong electron correlation is exploited. In DMET, the (extended or molecular) system under study is fragmented. A localised orbital basis is used for that purpose. Each fragment, which can be seen as an open quantum subsystem (because of the entanglement with its environment), is then embedded into a quantum bath which consists of orbitals (as many as in the fragment) that are delocalised over the fragment’s environment. The Schrödinger equation is then solved for the closed and reduced-in-size fragment+bath system (the so-called embedding cluster) from which the properties of the fragment can be deduced. While the procedure is exact for noninteracting (or mean-field) electrons [3], it is still unclear how to optimally construct a systematically improvable embedding for interacting electrons, ideally from an in-principle exact formulation of the theory. For that purpose, connections with density functional theory [4], one-electron reduced density matrix functional theory [3] , as well as quantum information theory are currently explored, for example. 

Kohn—Sham density functional theory of electrons and nuclei

Regular Kohn—Sham density functional theory (KS-DFT) is a ground-state electronic structure theory where the nuclei are treated within the Born-Oppenheimer (BO) approximation. In other words, electrons are described quantum mechanically in the presence of clamped nuclei. In this context, the electronic density is parameterized by the molecular geometry. When the BO approximation breaks down, in the vicinity of a conical intersection, for example, KS-DFT becomes inadequate. In the past, the (in-principle exact) extension of DFT to both electrons and nuclei, which is needed in this case, has been explored either from a multi-component perspective [1-5] or from an exactly-factorized expression of the molecular (electrons+nuclei) wavefunction [6,7]. More recently, an alternative molecular KS-DFT, which combines ingredients from both approaches, has been derived [8]. Its novelty lies in the exact mapping of both (geometry-dependent) electronic and nuclear densities onto an electronically noninteracting molecule, referred to as KS molecule. As a result, the theory allows standard DFT functionals to be recycled straightforwardly, thus leading to practical (crude, though) adiabatic density-functional approximations (DFAs), i.e. local, geometry wise [8]. Designing semi-local DFAs in the present context is a challenging task. How to optimally benefit from the density-functional treatment of electronic repulsions when solving the full KS molecular problem is also open to discussion. Applying the exact factorization formalism to the KS molecule is an appealing and promising path to explore [9]. Work is currently in progress in these directions.