Fromager and Lasorne [Electron. Struct. 6 025002 (2024)] have recently derived an in-principle exact Kohn-Sham density functional theory (KS-DFT) of electrons and nuclei, where the nuclear density and the (so-called conditional) electronic density are mapped onto a fictitious electronically non-interacting KS molecule. In this work, we apply the exact factorization formalism to the molecular KS wavefunction, thus leading to disentangled (but coupled) marginal and conditional KS equations. We show that, while being equivalent to the original theory, these equations open new perspectives in the practical extension of regular (electronic) KS-DFT beyond the Born-Oppenheimer approximation. The importance and treatment of correlations induced in this context by second-order geometrical derivatives is also discussed.

An in-principle exact working equation to compute electronic affinity and ionization Fukui functions is derived within the N-centered (Nc) ensemble extension of density functional theory (DFT). It circumvents the kernel derivative discontinuity problem of DFT for fractional electron numbers, whose contribution is recovered through weight derivatives of the ensemble density functional potential. Thus, it allows for the design of alternative and effective approximations, such as the weight-dependent scaling of regular functionals or the interpolation between known limits of Nc ensembles.

In this work we introduce a generalized flavor, in the sense of generalized Kohn-Sham density functional theory (gKS-DFT), of the recently derived local potential functional embedding theory (LPFET) (J. Chem. Theory Comput. 2025 21 10293), where the in-principle exact formalism of DFT is combined with that of density matrix embedding theory (DMET). In generalized LPFET (gLPFET), the embedding clusters are designed from a full-size gKS system where the (in-principle non-local) Hartree-Fock exchange potential is combined with a local (in the localized orbital representation) correlation potential. The latter is optimized self-consistently such that gKS and local embedding cluster's densities match. Unlike in DMET, which uses the same (global) chemical potential value in all clusters, each embedded orbital has its own chemical potential in gLPFET. We show analytically that, when electron correlation is strongly local, the latter potential becomes a simple functional of the correlation potential. Numerical calculations on model systems confirm the high accuracy of gLPFET in this regime, in contrast to DMET. Moreover, we show that gLPFET completely fixes the flaw of LPFET in weaker correlation regimes, through its appropriate description of the Hartree-exchange potential.

Ground-state electronic structure calculations using Kohn-Sham density functional theory (KS-DFT) offer an unprecedented balance between efficiency and accuracy, now paradigmatic to the fields of quantum chemistry and condensed matter physics. KS-DFT can be extended to model electronic excitations through density mapping onto a non-interacting ensemble state in which, unlike in thermal theories, the weights assigned to the excited states vary independently. Ensemble DFT (eDFT) has lately become a vibrant area of research thanks to its numerous appeals, such as the adequate treatment of multiple excitations for which the widely used time-dependent extension of DFT struggles. Recently, an enlarged type of ensemble, referred to as a N-centered (Nc) ensemble, has been introduced to describe within the same unified formalism both neutral and charged electronic excitations. This perspective paper provides a detailed exposition of exact Nc-eDFT, with a comprehensive review of its formal developments. To cut practical computational tools out of the exact theory, three original strategies are presented, complementing existing approaches. The first one, related to the design of ensemble density-functional approximations, consists in recycling regular ground-state functionals by dressing them with a weight-dependent scaling function deduced from exact properties of eDFT. We then explore quasi-degenerate formulations of ensemble density-functional perturbation theory, suggesting alternative definitions for the ensemble Hartree, exchange, and correlation energies, individually, and paving the way toward robust orbital-dependent eDFAs. Finally, we revisit and generalize the concept of quantum bath for an ensemble of non-interacting states, laying the foundations of an in-principle exact (in the sense of lattice eDFT) quantum embedding theory of excited states.

Gross-Oliveira-Kohn density-functional theory (GOK-DFT) for ensembles is the DFT analog of state-averaged wavefunction-based (SA-WF) methods. In GOK-DFT, the SA (so-called ensemble) exchange-correlation (xc) energy is described by a single functional of the density which, for a fixed density, depends on the weights assigned to each state in the ensemble. We show that if a many-weight-dependent xc functional is employed, then it becomes possible to extract, in principle exactly, all individual energy levels from a single GOK-DFT calculation, exactly like in a SA-WF calculation. More precisely, starting from the Kohn-Sham energies, a global Levy-Zahariev-type shift as well as a state-specific (ensemble-based) xc derivative correction must be applied in order to reach the energy level of interest. We illustrate with the asymmetric Hubbard dimer the importance and substantial weight dependence of both corrections. A comparison with more standard extraction procedures, which rely on a sequence of ensemble calculations, is made at the ensemble exact exchange level of approximation.

Following a recent work [E. Fromager, J. Phys. Chem. A 2025, 129, 4, 1143-1155] on the ensemble density functional theory (DFT) of excited electronic energy levels, we derive in this paper the ensuing static linear response theory, thus allowing for an in-principle exact evaluation of excited-state density-density linear response functions in a completely frequency-independent setting. Once individual-state components of the inverse ensemble linear response function have been introduced, a working Dyson-type equation naturally emerges for each state, individually. By considering the zero-weight limit of the theory, which infinitesimally deviates from standard Kohn--Sham DFT, exact excited-state corrections to ground-state linear response DFT can be identified. They involve the first-order weight derivatives of the ensemble Hartree-exchange-correlation (Hxc) potential and kernel, thus confirming the importance in ensemble DFT of both weight and density-functional derivatives of the ensemble Hxc energy functional.

Localized orbital-based quantum embedding, as originally formulated in the context of density matrix embedding theory (DMET), is revisited from the perspective of lattice density functional theory (DFT). An in-principle exact (in the sense of full configuration interaction) formulation of the theory, where the occupations of the localized orbitals play the role of the density, is derived for any (model or ab initio) electronic Hamiltonian. From this general formalism we deduce an exact relation between the local Hartree-exchange-correlation (Hxc) potential of the full-size Kohn-Sham (KS) lattice-like system and the embedding chemical potential that is adjusted on each embedded fragment, individually, such that both KS and embedding cluster systems reproduce the exact same local density. When well-identified density-functional approximations (that find their justification in the strongly correlated regime) are applied, a practical self-consistent local potential functional embedding theory (LPFET), where the local Hxc potential becomes the basic variable, naturally emerges from the theory. LPFET differs from previous density embedding approaches by its fragment-dependent embedding chemical potential expression, which is a simple functional of the Hxc potential. Numerical calculations on prototypical systems show the ability of such an ansatz to improve substantially the description of density profiles (localized orbitals occupation numbers in this context) in strongly correlated systems.

In recent studies by Yalouz et al. [J. Chem. Phys. 157, 214112 (2022)] and Sekaran et al. [Phys. Rev. B 104, 035121 (2021) and Computation 10, 45 (2022)], density matrix embedding theory (DMET) has been reformulated through the use of the Householder transformation as a novel tool to embed a fragment within extended systems. The transformation was applied to a reference non-interacting one-electron reduced density matrix to construct fragments’ bath orbitals, which are crucial for subsequent ground state calculations. In the present work, we expand upon these previous developments and extend the utilization of the Householder transformation to the description of multiple electronic states, including ground and excited states. Based on an ensemble noninteracting density matrix, we demonstrate the feasibility of achieving exact fragment embedding through successive Householder transformations, resulting in a larger set of bath orbitals. We analytically prove that the number of additional bath orbitals scales directly with the number of fractionally occupied natural orbitals in the reference ensemble density matrix. A connection with the regular DMET bath construction is also made. Then, we illustrate the use of this ensemble embedding tool in single-shot DMET calculations to describe both ground and first excited states in a Hubbard lattice model and an ab initio hydrogen system. Finally, we discuss avenues for enhancing ensemble embedding through self-consistency and explore potential future directions.

This work presents an alternative, general, and in-principle exact extension of electronic Kohn-Sham density functional theory (KS-DFT) to the fully quantum-mechanical molecular problem. Unlike in existing multi-component or exact-factorization-based DFTs of electrons and nuclei, both nuclear and electronic densities are mapped onto a fictitious electronically non-interacting molecule (referred to as KS molecule), where the electrons still interact with the nuclei. Moreover, in the present molecular KS-DFT, no assumption is made about the mathematical form (exactly factorized or not) of the molecular wavefunction. By expanding the KS molecular wavefunction à la Born-Huang, we obtain a self-consistent set of "KS beyond Born-Oppenheimer" electronic equations coupled to nuclear equations that describe nuclei interacting among themselves and with non-interacting electrons. An exact adiabatic connection formula is derived for the Hartree-exchange-correlation energy of the electrons within the molecule and, on that basis, a practical adiabatic density-functional approximation is proposed and discussed.

A recent work [arXiv:2401.04685] has merged $N$-centered ensembles of neutral and charged electronic ground states with ensembles of neutral ground and excited states, thus providing a general and in-principle exact (so-called extended $N$-centered) ensemble density functional theory of neutral and charged electronic excitations. This formalism made it possible to revisit the concept of density-functional derivative discontinuity, in the particular case of single excitations from the highest occupied Kohn-Sham (KS) molecular orbital, without invoking the usual "asymptotic behavior of the density" argument. In this work, we address a broader class of excitations, with a particular focus on double excitations. An exact implementation of the theory is presented for the two-electron Hubbard dimer model. A thorough comparison of the true physical ground- and excited-state electronic structures with that of the fictitious ensemble density-functional KS system is also presented. Depending on the choice of the density-functional ensemble as well as the asymmetry of the dimer and the correlation strength, an inversion of states can be observed. In some other cases, the strong mixture of KS states within the true physical system makes the assignment "single excitation" or "double excitation" irrelevant.

This work merges two different types of many-electron ensembles, namely the Theophilou-Gross-Oliveira-Kohn ensembles of ground and neutrally-excited states, and the more recent $N$-centered ensembles of neutral and charged ground states. On that basis, an in-principle exact and general, so-called extended $N$-centered, ensemble density-functional theory of charged and neutral electronic excitations is derived. We revisit in this context the concept of density-functional derivative discontinuity for neutral excitations, without ever invoking nor using the asymptotic behavior of the ensemble electronic density. The present mathematical construction fully relies on the weight dependence of the ensemble Hartree-exchange-correlation density-functional energy, which makes the theory applicable to lattice models and opens new perspectives for the description of gaps in mesoscopic systems.

In this work, we introduce an original self-consistent scheme based on the one-body reduced density matrix ($\gamma$) formalism. A significant feature of this methodology is the utilization of an optimal unitary transformation of the Hamiltonian, determined through a self-consistently determined, unitary reflection $\mathbf{R}[\gamma]$. This enables the extraction of all reduced properties of the system from a smaller, accurately solved embedding cluster, and to systematically reconstruct the reduced density matrix of the system. This process ensures that both extended and embedded systems satisfy the local virial-like relation, providing quantitative insight into the correspondence between the fragment in the extended system and its embedded analogue. The performance and convergence of the method, as well as the N-representability of the resulting correlated density matrix, are evaluated and discussed within the context of the one-dimensional Hubbard model, which provides exact results for a comprehensive comparison.

The equivalence in one-electron quantum baths between the practical implementation of density matrix embedding theory (DMET) and the more recent Householder-transformed density matrix functional embedding theory has been shown previously in the standard but special case where the reference full-size (one-electron reduced) density matrix, from which the bath is constructed, is idempotent [S. Yalouz et al., J. Chem. Phys. 157, 214112 (2022)]. We prove mathematically that the equivalence remains valid when the density matrix is not idempotent anymore, thus allowing for the construction of correlated (one-electron) quantum baths. A density-matrix functional exactification of DMET is derived within the present unified quantum embedding formalism. Numerical examples reveal that the embedding cluster can be quite sensitive to the level of density-matrix functional approximation used for computing the reference density matrix.

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 300 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 776 entries, the paper represents a broad snapshot of DFT, anno 2022.

Currently, there is a growing interest in the development of a new hierarchy of methods based on the concept of seniority, which has been introduced quite recently in quantum chemistry. Despite the enormous potential of these methods, the accurate description of both dynamical and static correlation effects within a single and in-principle-exact approach remains a challenge. In this work, we propose an alternative formulation of reduced density-matrix functional theory (RDMFT) where the (one-electron reduced) density matrix is mapped onto an ab initio seniority-zero wave function. In this theory, the exact natural orbitals and their occupancies are determined self-consistently from an effective seniority-zero calculation. The latter involves a universal higher-seniority density matrix functional for which an adiabatic connection (AC) formula is derived and implemented under specific constraints that are related to the density matrix. The pronounced curvature of the (constrained) AC integrand, which is numerically observed in prototypical hydrogen chains and the Helium dimer, indicates that a description of higher-seniority correlations within second-order perturbation theory is inadequate in this context. Applying multiple linear interpolations along the AC or connecting second-order perturbation theory to a full-seniority treatment via Padé approximants are better strategies. Such information is expected to serve as a guide in the future design of higher-seniority density-matrix functional approximations.

Quantum embedding is a divide and conquer strategy that aims at solving the electronic Schrödinger equation of sizeable molecules or extended systems. We establish in the present work a clearer and in-principle-exact connection between density matrix embedding theory (DMET) and density-functional theory (DFT) within the simple but nontrivial one-dimensional Hubbard model. For that purpose, we use our recent reformulation of single-impurity DMET as a Householder transformed density-matrix functional embedding theory (Ht-DMFET). On the basis of well-identified density-functional approximations, a self-consistent local potential functional embedding theory (LPFET) is formulated and implemented. Combining both LPFET and DMET numerical results with our formally exact density-functional embedding theory reveals that a single statically embedded impurity can in principle describe the density-driven Mott–Hubbard transition, provided that a complementary density-functional correlation potential (which is neglected in both DMET and LPFET) exhibits a derivative discontinuity (DD) at half filling. The extension of LPFET to multiple impurities (which would enable to circumvent the modeling of DDs) and its generalization to quantum chemical Hamiltonians are left for future work.

Recently, some of the authors introduced the use of the Householder transformation as a simple and intuitive method for embedding local molecular fragments [see Sekaran et al., Phys. Rev. B 104, 035121 (2021) and Sekaran et al., Computation 10, 45 (2022)]. In this work, we present an extension of this approach to the more general case of multi-orbital fragments using the block version of the Householder transformation applied to the one-body reduced density matrix, unlocking the applicability to general quantum chemistry/condensed matter physics Hamiltonians. A step-by-step construction of the block Householder transformation is presented. Both physical and numerical areas of interest of the approach are highlighted. The specific mean-field (noninteracting) case is thoroughly detailed as it is shown that the embedding of a given N spin-orbital fragment leads to the generation of two separated sub-systems: (1) a 2N spin-orbitals "fragment+bath" cluster that exactly contains N electrons and (2) a remaining cluster's "environment" described by so-called core electrons. We illustrate the use of this transformation in different cases of embedding scheme for practical applications. We particularly focus on the extension of the previously introduced Local Potential Functional Embedding Theory and Householder-transformed Density Matrix Functional Embedding Theory to the case of multi-orbital fragments. These calculations are realized on different types of systems, such as model Hamiltonians (Hubbard rings) and ab initio molecular systems (hydrogen rings).

I gave this informal seminar in the lab of Quantum Chemistry (Institute of Chemistry, Strasbourg).