arxiv: 2604.11191 · v1 · submitted 2026-04-13 · ⚛️ physics.chem-ph

Recognition: unknown

Ensemble density functional theory of excited states: Exact N-centered formalism and practical opportunities

Lucien Dupuy , Toni Chiti , J\'er\'emy Morere , Emmanuel Fromager

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Pith reviewed 2026-05-10 15:30 UTC · model grok-4.3

classification ⚛️ physics.chem-ph

keywords ensemble density functional theoryexcited statesN-centered ensembleKohn-Sham DFTdensity functional approximationsquantum embeddingcharged excitations

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The pith

N-centered ensembles extend ensemble density functional theory to treat both neutral and charged electronic excitations in one unified formalism.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper sets out an exact formalism for ensemble density functional theory based on N-centered ensembles. These ensembles combine states with different numbers of electrons under independent weights, allowing neutral and charged excitations to be handled together rather than through separate calculations. A sympathetic reader would care because the approach targets cases where time-dependent DFT struggles, such as multiple excitations, while retaining the efficiency of Kohn-Sham methods. The authors review the formal properties and then lay out three concrete routes to turn the exact theory into usable approximations and embedding schemes for molecules and materials.

Core claim

The central claim is that an enlarged N-centered ensemble, which includes states with N, N+1, and N-1 electrons, supplies a single exact framework for ensemble DFT of excited states. This formalism yields precise relations that support the construction of practical density-functional approximations, perturbation expansions, and quantum embedding methods without losing the ability to describe both neutral and charged excitations.

What carries the argument

The N-centered ensemble, which generalizes ordinary ensembles by centering on a reference particle number N and incorporating states with adjacent electron counts under freely varying weights.

If this is right

Ground-state functionals can be reused for ensembles after multiplication by a weight-dependent scaling factor taken from exact ensemble relations.
Quasi-degenerate ensemble density-functional perturbation theory supplies separate, well-defined expressions for the ensemble Hartree, exchange, and correlation contributions.
A generalized quantum-bath construction for non-interacting ensemble states yields an in-principle exact lattice embedding theory for excited states.
Multiple excitations become accessible within the same calculation, avoiding the single-excitation bias of time-dependent DFT.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same run could produce ionization potentials and electron affinities alongside neutral excitation energies without extra computational overhead.
The embedding strategy may translate to periodic boundary conditions and thereby address excited states in solids.
Orbital-dependent functionals derived from the quasi-degenerate perturbation route could improve accuracy for systems with near-degenerate states.

Load-bearing premise

Exact relations derived from the N-centered ensemble can be turned into density-functional approximations and embedding schemes that remain accurate for real molecules without introducing large uncontrolled errors.

What would settle it

A side-by-side comparison, on a test set of small molecules, of neutral and charged excitation energies obtained from the proposed approximations against exact many-body reference values or experiment; large systematic deviations would falsify the practicality claim.

Figures

Figures reproduced from arXiv: 2604.11191 by Emmanuel Fromager, J\'er\'emy Morere, Lucien Dupuy, Toni Chiti.

**Figure 2.** Figure 2: FIG. 2: Prediction of the energy gap from the scaling [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Comparison of the exact ensemble Hxc density functional with the ensemble scaled one (SF) for different [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Comparison of the exact ensemble Hxc [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗

read the original abstract

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 a density mapping onto a non-interacting ensemble state in which, unlike in thermal theories, the weights assigned to the excited states vary independently. Thanks to its numerous appeals, like the adequate treatment of multiple excitations for which the widely-used time-dependent extension of DFT struggles, ensemble DFT (eDFT) has lately become a vibrant area of research. Recently, an enlarged type of ensemble, referred to as 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a perspective that reviews the N-centered ensemble DFT formalism and sketches three concrete strategies for turning it into approximations, but stays at the conceptual level with no tests or detailed benchmarks.

read the letter

The core contribution is a unified treatment of neutral and charged excitations via N-centered ensembles, plus three original routes to practical work: weight-dependent scaling of ground-state functionals, quasi-degenerate ensemble DFPT variants, and a generalized quantum-bath embedding. The paper does a clean job laying out the exact properties and showing how they motivate these steps without obvious internal contradictions. That unification angle is useful if the formal properties survive approximation. The exposition of the exact Nc-eDFT theory is clear and pulls together recent formal developments in one place. The three strategies follow logically from the exact relations and complement existing eDFT work rather than repeating it. Soft spots are the lack of any numerical evidence or even simple model calculations to check whether the scaling function or the embedding scheme actually controls errors in practice. The derivations for the new strategies are sketched but not fully expanded here, so it is hard to judge how much uncontrolled approximation creeps in when moving from exact theory to usable functionals. No free parameters or circular fitting appear in the abstract, which is good. This is for people already working on ensemble DFT or excited-state methods in quantum chemistry who want to see formal directions they could implement. A reader looking for ready-to-use tools or validated results will find less. It deserves a serious referee because the ideas are specific, grounded in prior exact results, and address a real gap in handling multiple excitations and charged states together. I would send it for review but would ask the authors to add at least one concrete example or error analysis for the proposed strategies.

Referee Report

2 major / 2 minor

Summary. This perspective paper introduces and expounds the N-centered (Nc) ensemble within ensemble density functional theory (eDFT) as a unified formalism for neutral and charged electronic excitations. It reviews the exact properties of Nc-eDFT, then outlines three strategies for practical use: weight-dependent scaling of ground-state functionals derived from exact eDFT relations, quasi-degenerate ensemble DFPT with alternative definitions for the ensemble Hartree, exchange, and correlation energies, and a generalized quantum-bath embedding for non-interacting ensemble states that is in-principle exact in the lattice eDFT sense.

Significance. If the proposed strategies prove implementable, the work could meaningfully advance ensemble DFT for excited-state calculations, particularly for multiple excitations where TDDFT is limited. The grounding in exact properties, the parameter-free character of the scaling approach, and the in-principle exact embedding framework are clear strengths that distinguish it from purely empirical approximations.

major comments (2)

[Practical strategies (weight-dependent scaling and quasi-degenerate DFPT)] The section outlining the three strategies claims they 'cut practical computational tools out of the exact theory,' but the weight-dependent scaling and quasi-degenerate DFPT proposals require explicit demonstration (via at least one worked equation or limiting case) that they preserve key exact conditions such as the ensemble derivative discontinuity when applied beyond model systems.
[Generalized quantum-bath embedding] In the quantum-bath embedding discussion, the generalization to ensembles of non-interacting states is presented as laying foundations for an in-principle exact theory, yet the manuscript should clarify how the bath construction extends from lattice eDFT to continuous molecular systems without introducing uncontrolled approximations that would undermine the unification of neutral and charged excitations.

minor comments (2)

[Abstract and introduction] The abstract and introduction would benefit from a short table or explicit list contrasting Nc-eDFT with standard thermal ensembles and TDDFT to highlight the unification of neutral/charged excitations.
[Exact Nc-eDFT formalism] Notation for the independent weights in the Nc ensemble should be introduced with a clear definition early in the text to avoid ambiguity when discussing the scaling function.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for the recommendation of minor revision. We address the two major comments point by point below.

read point-by-point responses

Referee: [Practical strategies (weight-dependent scaling and quasi-degenerate DFPT)] The section outlining the three strategies claims they 'cut practical computational tools out of the exact theory,' but the weight-dependent scaling and quasi-degenerate DFPT proposals require explicit demonstration (via at least one worked equation or limiting case) that they preserve key exact conditions such as the ensemble derivative discontinuity when applied beyond model systems.

Authors: We agree that an explicit demonstration strengthens the presentation. Although the strategies are derived directly from exact eDFT relations (ensuring preservation of properties such as the ensemble derivative discontinuity by construction), we will add a concrete worked example in the revised manuscript. Specifically, we will include a limiting-case analysis for the weight-dependent scaling approach applied to a two-state ensemble, verifying that the ensemble derivative discontinuity is preserved. For quasi-degenerate ensemble DFPT, we will add a brief derivation in the degenerate limit demonstrating consistency with the exact conditions. revision: yes
Referee: [Generalized quantum-bath embedding] In the quantum-bath embedding discussion, the generalization to ensembles of non-interacting states is presented as laying foundations for an in-principle exact theory, yet the manuscript should clarify how the bath construction extends from lattice eDFT to continuous molecular systems without introducing uncontrolled approximations that would undermine the unification of neutral and charged excitations.

Authors: The quantum-bath embedding is formulated to be in-principle exact within the lattice eDFT setting, as stated in the manuscript. For continuous molecular systems, any practical realization necessarily involves approximations (e.g., basis-set discretization or projection onto a finite lattice), comparable to those in conventional embedding methods. These approximations do not affect the formal unification of neutral and charged excitations, which follows directly from the N-centered ensemble definition. In the revision we will insert a clarifying paragraph that explicitly distinguishes the exact lattice framework from approximate molecular implementations while confirming that the unification remains intact at the theoretical level. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces the N-centered ensemble as an enlargement of prior eDFT formalisms, then derives exact properties of Nc-eDFT and outlines three practical strategies (weight-dependent scaling of ground-state functionals, quasi-degenerate ensemble DFPT, and generalized quantum-bath embedding) that are explicitly deduced from those exact properties. No load-bearing step reduces a claimed prediction or result to a fitted parameter, self-definition, or unverified self-citation chain; the central unification of neutral and charged excitations follows directly from the ensemble definition without circular reduction. The derivation remains self-contained against external benchmarks of ensemble DFT.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on prior formal developments in ensemble DFT; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption The mapping from an interacting ensemble to a non-interacting ensemble state is valid for excited-state modeling
Standard foundation of ensemble DFT referenced in the abstract

pith-pipeline@v0.9.0 · 5609 in / 1161 out tokens · 52479 ms · 2026-05-10T15:30:36.592761+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

159 extracted references · 148 canonical work pages

[1]

author author P Hohenberg \ and\ author W Kohn ,\ title title Inhomogeneous electron gas , \ https://doi.org/10.1103/PhysRev.136.B864 journal journal Phys. Rev. \ volume 136 ,\ pages B864 ( year 1964 ) NoStop

work page doi:10.1103/physrev.136.b864 1964
[2]

Self-consistent equations including exchange and correlation effects.Physical Review, 140(4A):A1133–A1138, 1965

author author W. Kohn \ and\ author L.J. \ Sham ,\ title title Self-consistent equations including exchange and correlation effects , \ 10.1103/PhysRev.140.A1133 journal journal Phys. Rev. \ volume 140 ,\ pages A1133 ( year 1965 ) NoStop

work page doi:10.1103/physrev.140.a1133 1965
[3]

Teale, Trygve Helgaker, Andreas Savin, Carlo Adamo, Bálint Aradi, Alexei V

author author Andrew M. \ Teale , author Trygve \ Helgaker , author Andreas \ Savin , author Carlo \ Adamo , author B \'a lint \ Aradi , author Alexei V. \ Arbuznikov , author Paul W. \ Ayers , author Evert Jan \ Baerends , author Vincenzo \ Barone , author Patrizia \ Calaminici , \ and\ author et al. ,\ title title Dft exchange: sharing perspectives on t...

work page doi:10.1039/d2cp02827a 2022
[4]

author author John P \ Perdew , author Kieron \ Burke , \ and\ author Matthias \ Ernzerhof ,\ title title Generalized gradient approximation made simple , \ @noop journal journal Phys. Rev. Lett. \ volume 77 ,\ pages 3865 ( year 1996 ) NoStop

1996
[5]

author author Kieron \ Burke ,\ title title Perspective on density functional theory , \ 10.1063/1.4704546 journal journal The Journal of Chemical Physics \ volume 136 ,\ pages 150901 ( year 2012 ) ,\ http://arxiv.org/abs/https://pubs.aip.org/aip/jcp/article-pdf/doi/10.1063/1.4704546/19858608/150901\_1\_1.4704546.pdf https://pubs.aip.org/aip/jcp/article-p...

work page doi:10.1063/1.4704546 2012
[6]

author author Lars \ Goerigk , author Andreas \ Hansen , author Christoph \ Bauer , author Stephan \ Ehrlich , author Asim \ Najibi , \ and\ author Stefan \ Grimme ,\ title title A look at the density functional theory zoo with the advanced gmtkn55 database for general main group thermochemistry , kinetics and noncovalent interactions , \ 10.1039/C7CP0491...

work page doi:10.1039/c7cp04913g 2017
[7]

author author Atsushi \ Tanaka , author Atsuki \ Saito , author Takashi \ Murata , author Ayako \ Nakata , \ and\ author Tsuyoshi \ Miyazaki ,\ title title Large-scale dft calculations of multi-component glass systems (sio2)0.70(al2o3)0.13(xo)0.17 (x = mg, ca, sr, ba) : Accuracy of classical force fields , \ https://doi.org/10.1016/j.jnoncrysol.2023.12271...

work page doi:10.1016/j.jnoncrysol.2023.122714 2023
[8]

author author Erich \ Runge \ and\ author Eberhard KU \ Gross ,\ title title Density-functional theory for time-dependent systems , \ https://doi.org/10.1103/PhysRevLett.52.997 journal journal Phys. Rev. Lett. \ volume 52 ,\ pages 997 ( year 1984 ) NoStop

work page doi:10.1103/physrevlett.52.997 1984
[9]

\ Casida ,\ title title Time-dependent density functional response theory for molecules , \ in\ @noop booktitle Recent Advances in Density Functional Methods ,\ Vol

author author Mark E. \ Casida ,\ title title Time-dependent density functional response theory for molecules , \ in\ @noop booktitle Recent Advances in Density Functional Methods ,\ Vol. volume 1 ,\ editor edited by\ editor D. P. \ Chong \ ( publisher World Scientific ,\ year 1995 )\ pp.\ pages 155--192 NoStop

1995
[10]

\ Casida \ and\ author M

author author M.E. \ Casida \ and\ author M. Huix-Rotllant ,\ title title Progress in time-dependent density-functional theory , \ https://doi.org/10.1146/annurev-physchem-032511-143803 journal journal Annu. Rev. Phys. Chem. \ volume 63 ,\ pages 287 ( year 2012 ) NoStop

work page doi:10.1146/annurev-physchem-032511-143803 2012
[11]

\ Maitra ,\ title title Perspectives on double-excitations in tddft , \ https://doi.org/10.1016/j.chemphys.2011.03.020 journal journal Chem

author author Peter \ Elliott , author Sharma \ Goldson , author Chris \ Canahui , \ and\ author Neepa T. \ Maitra ,\ title title Perspectives on double-excitations in tddft , \ https://doi.org/10.1016/j.chemphys.2011.03.020 journal journal Chem. Phys. \ volume 391 ,\ pages 110--119 ( year 2011 ) ,\ note open problems and new solutions in time dependent d...

work page doi:10.1016/j.chemphys.2011.03.020 2011
[12]

Lacombe \ and\ author N.T

author author L. Lacombe \ and\ author N.T. \ Maitra ,\ title title Non-adiabatic approximations in time-dependent density functional theory: progress and prospects , \ https://doi.org/10.1038/s41524-023-01061-0 journal journal npj Comput Mater \ volume 9 ,\ pages 124 ( year 2023 ) NoStop

work page doi:10.1038/s41524-023-01061-0 2023
[14]

\ Dar \ and\ author Neepa T

author author Davood B. \ Dar \ and\ author Neepa T. \ Maitra ,\ title title Oscillator strengths and excited-state couplings for double excitations in time-dependent density functional theory , \ 10.1063/5.0176705 journal journal The Journal of Chemical Physics \ volume 159 ,\ pages 211104 ( year 2023 ) ,\ http://arxiv.org/abs/https://pubs.aip.org/aip/jc...

work page doi:10.1063/5.0176705 2023
[15]

\ Dar \ and\ author Neepa T

author author Davood B. \ Dar \ and\ author Neepa T. \ Maitra ,\ title title Capturing the elusive curve-crossing in low-lying states of butadiene with dressed tddft , \ 10.1021/acs.jpclett.4c03167 journal journal The Journal of Physical Chemistry Letters \ volume 16 ,\ pages 703--709 ( year 2025 ) ,\ http://arxiv.org/abs/https://doi.org/10.1021/acs.jpcle...

work page doi:10.1021/acs.jpclett.4c03167 2025
[16]

author author Anna \ Baranova \ and\ author Neepa T. \ Maitra ,\ title title Excited-state densities from time-dependent density functional response theory , \ 10.1021/acs.jctc.5c00909 journal journal Journal of Chemical Theory and Computation \ volume 21 ,\ pages 10437--10451 ( year 2025 ) NoStop

work page doi:10.1021/acs.jctc.5c00909 2025
[17]

author author Spiridoula \ Matsika ,\ title title Electronic structure methods for the description of nonadiabatic effects and conical intersections , \ 10.1021/acs.chemrev.1c00074 journal journal Chemical Reviews \ volume 121 ,\ pages 9407--9449 ( year 2021 ) ,\ http://arxiv.org/abs/https://doi.org/10.1021/acs.chemrev.1c00074 https://doi.org/10.1021/acs....

work page doi:10.1021/acs.chemrev.1c00074 2021
[18]

author author Neepa T \ Maitra ,\ title title Charge transfer in time-dependent density functional theory , \ @noop journal journal J. Phys. Condens. Matter \ volume 29 ,\ pages 423001 ( year 2017 ) NoStop

2017
[19]

author author John P \ Perdew , author Robert G \ Parr , author Mel \ Levy , \ and\ author Jose L \ Balduz Jr ,\ title title Density-functional theory for fractional particle number: derivative discontinuities of the energy , \ https://doi.org/10.1103/PhysRevLett.49.1691 journal journal Phys. Rev. Lett. \ volume 49 ,\ pages 1691 ( year 1982 ) NoStop

work page doi:10.1103/physrevlett.49.1691 1982
[20]

author author Paula \ Mori-S \'a nchez , author Aron J \ Cohen , \ and\ author Weitao \ Yang ,\ title title Localization and delocalization errors in density functional theory and implications for band-gap prediction , \ @noop journal journal Phys. Rev. Lett. \ volume 100 ,\ pages 146401 ( year 2008 ) NoStop

2008
[21]

author author Aron J \ Cohen , author Paula \ Mori-S \'a nchez , \ and\ author Weitao \ Yang ,\ title title Fractional charge perspective on the band gap in density-functional theory , \ https://doi.org/10.1103/PhysRevB.77.115123 journal journal Phys. Rev. B \ volume 77 ,\ pages 115123 ( year 2008 a ) NoStop

work page doi:10.1103/physrevb.77.115123 2008
[22]

author author Aron J \ Cohen , author Paula \ Mori-S \'a nchez , \ and\ author Weitao \ Yang ,\ title title Insights into current limitations of density functional theory , \ @noop journal journal Science \ volume 321 ,\ pages 792--794 ( year 2008 b ) NoStop

2008
[23]

author author Tamar \ Stein , author Helen \ Eisenberg , author Leeor \ Kronik , \ and\ author Roi \ Baer ,\ title title Fundamental gaps in finite systems from eigenvalues of a generalized kohn-sham method , \ @noop journal journal Phys. Rev. Lett. \ volume 105 ,\ pages 266802 ( year 2010 ) NoStop

2010
[24]

author author Xiao \ Zheng , author Aron J \ Cohen , author Paula \ Mori-S \'a nchez , author Xiangqian \ Hu , \ and\ author Weitao \ Yang ,\ title title Improving band gap prediction in density functional theory from molecules to solids , \ https://doi.org/10.1103/PhysRevLett.107.026403 journal journal Phys. Rev. Lett. \ volume 107 ,\ pages 026403 ( year...

work page doi:10.1103/physrevlett.107.026403 2011
[25]

author author Aron J \ Cohen , author Paula \ Mori-S \'a nchez , \ and\ author Weitao \ Yang ,\ title title Challenges for density functional theory , \ 10.1021/cr200107z journal journal Chem. Rev. \ volume 112 ,\ pages 289--320 ( year 2011 ) NoStop

work page doi:10.1021/cr200107z 2011
[26]

author author Eli \ Kraisler \ and\ author Leeor \ Kronik ,\ title title Piecewise linearity of approximate density functionals revisited: implications for frontier orbital energies , \ https://doi.org/10.1103/PhysRevLett.110.126403 journal journal Phys. Rev. Lett. \ volume 110 ,\ pages 126403 ( year 2013 ) NoStop

work page doi:10.1103/physrevlett.110.126403 2013
[27]

author author Eli \ Kraisler \ and\ author Leeor \ Kronik ,\ title title Fundamental gaps with approximate density functionals: The derivative discontinuity revealed from ensemble considerations , \ @noop journal journal J. Chem. Phys. \ volume 140 ,\ pages 18A540 ( year 2014 ) NoStop

2014
[28]

,\ title title Understanding band gaps of solids in generalized kohn--sham theory , \ @noop journal journal Proc

author author John P \ Perdew , author Weitao \ Yang , author Kieron \ Burke , author Zenghui \ Yang , author Eberhard K U \ Gross , author Matthias \ Scheffler , author Gustavo E \ Scuseria , author Thomas M \ Henderson , author Igor Ying \ Zhang , author Adrienn \ Ruzsinszky , et al. ,\ title title Understanding band gaps of solids in generalized kohn--...

2017
[29]

author author Maria \ Hellgren \ and\ author E. K. U. \ Gross ,\ title title Effect of discontinuities in Kohn-Sham-based chemical reactivity theory , \ 10.1063/1.3694103 journal journal The Journal of Chemical Physics \ volume 136 ,\ pages 114102 ( year 2012 ) NoStop

work page doi:10.1063/1.3694103 2012
[30]

author author Matthew JP \ Hodgson , author Eli \ Kraisler , author Axel \ Schild , \ and\ author Eberhard KU \ Gross ,\ title title How interatomic steps in the exact kohn--sham potential relate to derivative discontinuities of the energy , \ https://doi.org/10.1021/acs.jpclett.7b02615 journal journal J. Phys. Chem. Lett. \ volume 8 ,\ pages 5974--5980 (...

work page doi:10.1021/acs.jpclett.7b02615 2017
[31]

author author Eli \ Kraisler , author M. J. P. \ Hodgson , \ and\ author E. K. U. \ Gross ,\ title title From kohn–sham to many-electron energies via step structures in the exchange-correlation potential , \ 10.1021/acs.jctc.0c01093 journal journal Journal of Chemical Theory and Computation \ volume 17 ,\ pages 1390--1407 ( year 2021 ) NoStop

work page doi:10.1021/acs.jctc.0c01093 2021
[32]

Cernatic , author B

author author F. Cernatic , author B. Senjean , author V. Robert , \ and\ author E. Fromager ,\ title title Ensemble density functional theory of neutral and charged excitations , \ https://doi.org/10.1007/s41061-021-00359-1 journal journal Top Curr Chem (Z) \ volume 380 ,\ pages 4 ( year 2022 ) NoStop

work page doi:10.1007/s41061-021-00359-1 2022
[33]

ensemblization

author author Tim \ Gould , author Leeor \ Kronik , \ and\ author Stefano \ Pittalis ,\ title title “ensemblization” of density functional theory , \ 10.1063/5.0274509 journal journal The Journal of Chemical Physics \ volume 164 ,\ pages 040901 ( year 2026 ) NoStop

work page doi:10.1063/5.0274509 2026
[34]

author author A K \ Theophilou ,\ title title The energy density functional formalism for excited states , \ https://iopscience.iop.org/article/10.1088/0022-3719/12/24/013 journal journal J. Phys. C: Solid State Phys. \ volume 12 ,\ pages 5419 ( year 1979 ) NoStop

work page doi:10.1088/0022-3719/12/24/013 1979
[35]

author author A K \ Theophilou ,\ title The single particle density in physics and chemistry , \ \ ( publisher Academic Press ,\ year 1987 )\ pp.\ pages 210--212 NoStop

1987
[36]

author author E. K. U. \ Gross , author L. N. \ Oliveira , \ and\ author W. Kohn ,\ title title Rayleigh-ritz variational principle for ensembles of fractionally occupied states , \ 10.1103/PhysRevA.37.2805 journal journal Phys. Rev. A \ volume 37 ,\ pages 2805 ( year 1988 a ) NoStop

work page doi:10.1103/physreva.37.2805 1988
[37]

author author E. K. U. \ Gross , author L. N. \ Oliveira , \ and\ author W. Kohn ,\ title title Density-functional theory for ensembles of fractionally occupied states. i. basic formalism , \ 10.1103/PhysRevA.37.2809 journal journal Phys. Rev. A \ volume 37 ,\ pages 2809 ( year 1988 b ) NoStop

work page doi:10.1103/physreva.37.2809 1988
[38]

author author Francisca \ Sagredo \ and\ author Kieron \ Burke ,\ title title Accurate double excitations from ensemble density functional calculations , \ 10.1063/1.5043411 journal journal J. Chem. Phys. \ volume 149 ,\ pages 134103 ( year 2018 ) NoStop

work page doi:10.1063/1.5043411 2018
[39]

\ volume 224 ,\ pages 402--423 ( year 2020 ) NoStop

author author Clotilde \ Marut , author Bruno \ Senjean , author Emmanuel \ Fromager , \ and\ author Pierre-Fran c ois \ Loos ,\ title title Weight dependence of local exchange--correlation functionals in ensemble density-functional theory: double excitations in two-electron systems , \ 10.1039/D0FD00059K journal journal Faraday Discuss. \ volume 224 ,\ p...

work page doi:10.1039/d0fd00059k 2020
[40]

author author Tim \ Gould , author Leeor \ Kronik , \ and\ author Stefano \ Pittalis ,\ title title Double excitations in molecules from ensemble density functionals: Theory and approximations , \ 10.1103/PhysRevA.104.022803 journal journal Phys. Rev. A \ volume 104 ,\ pages 022803 ( year 2021 ) NoStop

work page doi:10.1103/physreva.104.022803 2021
[41]

author author C. A. \ Ullrich \ and\ author W. Kohn ,\ title title Kohn-sham theory for ground-state ensembles , \ 10.1103/PhysRevLett.87.093001 journal journal Phys. Rev. Lett. \ volume 87 ,\ pages 093001 ( year 2001 ) NoStop

work page doi:10.1103/physrevlett.87.093001 2001
[42]

author author Killian \ Deur , author Laurent \ Mazouin , author Bruno \ Senjean , \ and\ author Emmanuel \ Fromager ,\ title title Exploring weight-dependent density-functional approximations for ensembles in the hubbard dimer , \ https://doi.org/10.1140/epjb/e2018-90124-7 journal journal Eur. Phys. J. B \ volume 91 ,\ pages 162 ( year 2018 ) NoStop

work page doi:10.1140/epjb/e2018-90124-7 2018
[43]

author author Killian \ Deur \ and\ author Emmanuel \ Fromager ,\ title title Ground and excited energy levels can be extracted exactly from a single ensemble density-functional theory calculation , \ https://doi.org/10.1063/1.5084312 journal journal J. Chem. Phys. \ volume 150 ,\ pages 094106 ( year 2019 ) NoStop

work page doi:10.1063/1.5084312 2019
[44]

author author Emmanuel \ Fromager ,\ title title Individual correlations in ensemble density functional theory: State- and density-driven decompositions without additional kohn-sham systems , \ 10.1103/PhysRevLett.124.243001 journal journal Phys. Rev. Lett. \ volume 124 ,\ pages 243001 ( year 2020 ) NoStop

work page doi:10.1103/physrevlett.124.243001 2020
[45]

author author Thais R. \ Scott , author John \ Kozlowski , author Steven \ Crisostomo , author Aurora \ Pribram-Jones , \ and\ author Kieron \ Burke ,\ title title Exact conditions for ensemble density functional theory , \ 10.1103/PhysRevB.109.195120 journal journal Phys. Rev. B \ volume 109 ,\ pages 195120 ( year 2024 ) NoStop

work page doi:10.1103/physrevb.109.195120 2024
[46]

author author Sara \ Giarrusso \ and\ author Pierre-Fran c ois \ Loos ,\ title title Exact excited-state functionals of the asymmetric hubbard dimer , \ 10.1021/acs.jpclett.3c02052 journal journal J. Phys. Chem. Lett. \ volume 14 ,\ pages 8780--8786 ( year 2023 ) NoStop

work page doi:10.1021/acs.jpclett.3c02052 2023
[47]

author author Pierre-François \ Loos \ and\ author Sara \ Giarrusso ,\ title title Excited-state-specific kohn–sham formalism for the asymmetric hubbard dimer , \ 10.1063/5.0255324 journal journal The Journal of Chemical Physics \ volume 162 ,\ pages 144104 ( year 2025 ) NoStop

work page doi:10.1063/5.0255324 2025
[48]

author author Pierre-François \ Loos ,\ title title Excited states of the uniform electron gas , \ 10.1063/5.0263799 journal journal The Journal of Chemical Physics \ volume 162 ,\ pages 204105 ( year 2025 ) ,\ http://arxiv.org/abs/https://pubs.aip.org/aip/jcp/article-pdf/doi/10.1063/5.0263799/20528988/204105\_1\_5.0263799.pdf https://pubs.aip.org/aip/jcp...

work page doi:10.1063/5.0263799 2025
[49]

author author Tim \ Gould \ and\ author Stefano \ Pittalis ,\ title title Hartree and exchange in ensemble density functional theory: Avoiding the nonuniqueness disaster , \ https://doi.org/10.1103/PhysRevLett.119.243001 journal journal Phys. Rev. Lett. \ volume 119 ,\ pages 243001 ( year 2017 ) NoStop

work page doi:10.1103/physrevlett.119.243001 2017
[50]

author author Tim \ Gould , author Stefano \ Pittalis , author Julien \ Toulouse , author Eli \ Kraisler , \ and\ author Leeor \ Kronik ,\ title title Asymptotic behavior of the hartree-exchange and correlation potentials in ensemble density functional theory , \ https://doi.org/10.1039/C9CP03633D journal journal Phys. Chem. Chem. Phys. \ volume 21 ,\ pag...

work page doi:10.1039/c9cp03633d 2019
[51]

author author Tim \ Gould , author Gianluca \ Stefanucci , \ and\ author Stefano \ Pittalis ,\ title title Ensemble density functional theory: Insight from the fluctuation-dissipation theorem , \ 10.1103/PhysRevLett.125.233001 journal journal Phys. Rev. Lett. \ volume 125 ,\ pages 233001 ( year 2020 ) NoStop

work page doi:10.1103/physrevlett.125.233001 2020
[52]

author author Tim \ Gould \ and\ author Stefano \ Pittalis ,\ title title Density-driven correlations in many-electron ensembles: Theory and application for excited states , \ 10.1103/PhysRevLett.123.016401 journal journal Phys. Rev. Lett. \ volume 123 ,\ pages 016401 ( year 2019 ) NoStop

work page doi:10.1103/physrevlett.123.016401 2019
[53]

author author Tim \ Gould ,\ title title Variational principles in ensemble and excited-state density- and potential-functional theories , \ 10.1103/PhysRevA.111.032806 journal journal Phys. Rev. A \ volume 111 ,\ pages 032806 ( year 2025 ) NoStop

work page doi:10.1103/physreva.111.032806 2025
[54]

author author Zeng-hui \ Yang , author John R \ Trail , author Aurora \ Pribram-Jones , author Kieron \ Burke , author Richard J \ Needs , \ and\ author Carsten A \ Ullrich ,\ title title Exact and approximate kohn-sham potentials in ensemble density-functional theory , \ https://doi.org/10.1103/PhysRevA.90.042501 journal journal Phys. Rev. A \ volume 90 ...

work page doi:10.1103/physreva.90.042501 2014
[55]

author author Tim \ Gould , author Leeor \ Kronik , \ and\ author Stefano \ Pittalis ,\ title title Charge transfer excitations from exact and approximate ensemble kohn-sham theory , \ https://doi.org/10.1063/1.5022832 journal journal J. Chem. Phys. \ volume 148 ,\ pages 174101 ( year 2018 ) NoStop

work page doi:10.1063/1.5022832 2018
[56]

author author Pierre-Fran c ois \ Loos \ and\ author Emmanuel \ Fromager ,\ title title A weight-dependent local correlation density-functional approximation for ensembles , \ 10.1063/5.0007388 journal journal J. Chem. Phys. \ volume 152 ,\ pages 214101 ( year 2020 ) NoStop

work page doi:10.1063/5.0007388 2020
[57]

author author Zeng-hui \ Yang ,\ title title Second-order perturbative correlation energy functional in the ensemble density-functional theory , \ 10.1103/PhysRevA.104.052806 journal journal Phys. Rev. A \ volume 104 ,\ pages 052806 ( year 2021 ) NoStop

work page doi:10.1103/physreva.104.052806 2021
[58]

author author Tim \ Gould ,\ title title Approximately self-consistent ensemble density functional theory: Toward inclusion of all correlations , \ 10.1021/acs.jpclett.0c02894 journal journal J. Phys. Chem. Lett. \ volume 11 ,\ pages 9907--9912 ( year 2020 ) NoStop

work page doi:10.1021/acs.jpclett.0c02894 2020
[59]

author author Tim \ Gould , author Derk P. \ Kooi , author Paola \ Gori-Giorgi , \ and\ author Stefano \ Pittalis ,\ title title Electronic excited states in extreme limits via ensemble density functionals , \ 10.1103/PhysRevLett.130.106401 journal journal Phys. Rev. Lett. \ volume 130 ,\ pages 106401 ( year 2023 ) NoStop

work page doi:10.1103/physrevlett.130.106401 2023
[60]

author author Tim \ Gould \ and\ author Stefano \ Pittalis ,\ title title Local density approximation for excited states , \ 10.1103/PhysRevX.14.041045 journal journal Phys. Rev. X \ volume 14 ,\ pages 041045 ( year 2024 ) NoStop

work page doi:10.1103/physrevx.14.041045 2024
[61]

author author Tim \ Gould , author Stephen G. \ Dale , author Leeor \ Kronik , \ and\ author Stefano \ Pittalis ,\ title title State-specific density functionals for excited states via a density-driven correlation model , \ 10.1103/PhysRevLett.134.228001 journal journal Phys. Rev. Lett. \ volume 134 ,\ pages 228001 ( year 2025 ) NoStop

work page doi:10.1103/physrevlett.134.228001 2025
[62]

author author Michael \ Filatov ,\ title title Spin-restricted ensemble-referenced kohn--sham method: basic principles and application to strongly correlated ground and excited states of molecules , \ 10.1002/wcms.1209 journal journal WIREs Comput. Mol. Sci. \ volume 5 ,\ pages 146 ( year 2015 ) NoStop

work page doi:10.1002/wcms.1209 2015
[63]

author author Michael \ Filatov , author Seunghoon \ Lee , \ and\ author Cheol Ho \ Choi ,\ title title Description of sudden polarization in the excited electronic states with an ensemble density functional theory method , \ 10.1021/acs.jctc.1c00479 journal journal Journal of Chemical Theory and Computation \ volume 17 ,\ pages 5123--5139 ( year 2021 ) NoStop

work page doi:10.1021/acs.jctc.1c00479 2021
[64]

author author Tim \ Gould \ and\ author Leeor \ Kronik ,\ title title Ensemble generalized Kohn--Sham theory: The good, the bad, and the ugly , \ 10.1063/5.0040447 journal journal J. Chem. Phys. \ volume 154 ,\ pages 094125 ( year 2021 ) NoStop

work page doi:10.1063/5.0040447 2021
[65]

author author Emmanuel \ Fromager ,\ title title Ensemble density functional theory of ground and excited energy levels , \ 10.1021/acs.jpca.4c06744 journal journal The Journal of Physical Chemistry A \ volume 129 ,\ pages 1143--1155 ( year 2025 ) NoStop

work page doi:10.1021/acs.jpca.4c06744 2025
[66]

author author Lucien \ Dupuy \ and\ author Emmanuel \ Fromager ,\ title title Exact static linear response of excited states from ensemble density functional theory , \ 10.1021/acs.jpca.5c04552 journal journal The Journal of Physical Chemistry A \ volume 129 ,\ pages 9095--9109 ( year 2025 ) NoStop

work page doi:10.1021/acs.jpca.5c04552 2025
[67]

author author Kimberly J. \ Daas , author Steven \ Crisostomo , \ and\ author Kieron \ Burke ,\ https://arxiv.org/abs/2507.19464 title Ensemble time-dependent density functional theory , \ ( year 2025 ),\ http://arxiv.org/abs/2507.19464 arXiv:2507.19464 [physics.chem-ph] NoStop

work page arXiv 2025
[68]

author author Bruno \ Senjean \ and\ author Emmanuel \ Fromager ,\ title title Unified formulation of fundamental and optical gap problems in density-functional theory for ensembles , \ https://doi.org/10.1103/PhysRevA.98.022513 journal journal Phys. Rev. A \ volume 98 ,\ pages 022513 ( year 2018 ) NoStop

work page doi:10.1103/physreva.98.022513 2018
[69]

author author Filip \ Cernatic , author Pierre-Fran c ois \ Loos , author Bruno \ Senjean , \ and\ author Emmanuel \ Fromager ,\ title title Neutral electronic excitations and derivative discontinuities: An extended n -centered ensemble density functional theory perspective , \ 10.1103/PhysRevB.109.235113 journal journal Phys. Rev. B \ volume 109 ,\ pages...

work page doi:10.1103/physrevb.109.235113 2024
[70]

author author Bruno \ Senjean \ and\ author Emmanuel \ Fromager ,\ title title N-centered ensemble density-functional theory for open systems , \ 10.1002/qua.26190 journal journal Int. J. Quantum Chem. \ volume 120 ,\ pages e26190 ( year 2020 ) NoStop

work page doi:10.1002/qua.26190 2020
[71]

author author M. J. P. \ Hodgson , author J. Wetherell , \ and\ author Emmanuel \ Fromager ,\ title title Exact exchange-correlation potentials for calculating the fundamental gap with a fixed number of electrons , \ 10.1103/PhysRevA.103.012806 journal journal Phys. Rev. A \ volume 103 ,\ pages 012806 ( year 2021 ) NoStop

work page doi:10.1103/physreva.103.012806 2021
[72]

author author Aurora \ Pribram-Jones , author Zeng \ hui Yang , author John \ R.Trail , author Kieron \ Burke , author Richard \ J.Needs , \ and\ author Carsten \ A.Ullrich ,\ title title Excitations and benchmark ensemble density functional theory for two electrons , \ 10.1063/1.4872255 journal journal J. Chem. Phys. \ volume 140 ,\ pages 18A541 ( year 2...

work page doi:10.1063/1.4872255 2014
[73]

author author Zeng-hui \ Yang , author Aurora \ Pribram-Jones , author Kieron \ Burke , \ and\ author Carsten A \ Ullrich ,\ title title Direct extraction of excitation energies from ensemble density-functional theory , \ https://doi.org/10.1103/PhysRevLett.119.033003 journal journal Phys. Rev. Lett. \ volume 119 ,\ pages 033003 ( year 2017 ) NoStop

work page doi:10.1103/physrevlett.119.033003 2017
[74]

author author Ingvar \ Lindgren \ and\ author John \ Morrison ,\ title Perturbation theory , \ in\ 10.1007/978-3-642-61640-2_9 booktitle Atomic Many-Body Theory \ ( publisher Springer Berlin Heidelberg ,\ address Berlin, Heidelberg ,\ year 1986 )\ pp.\ pages 184--211 NoStop

work page doi:10.1007/978-3-642-61640-2_9 1986
[75]

author author J. H. \ Van Vleck ,\ title title On -type doubling and electron spin in the spectra of diatomic molecules , \ 10.1103/PhysRev.33.467 journal journal Phys. Rev. \ volume 33 ,\ pages 467--506 ( year 1929 ) NoStop

work page doi:10.1103/physrev.33.467 1929
[76]

author author Alexander Yu. \ Sokolov ,\ title title Chapter four - multireference perturbation theories based on the dyall hamiltonian , \ in\ https://doi.org/10.1016/bs.aiq.2024.04.004 booktitle Novel Treatments of Strong Correlations ,\ series Advances in Quantum Chemistry , Vol. volume 90 ,\ editor edited by\ editor Ramon A. Miranda \ Quintana \ and\ ...

work page doi:10.1016/bs.aiq.2024.04.004 2024
[77]

author author Wafa \ Makhlouf , author Bruno \ Senjean , \ and\ author Emmanuel \ Fromager ,\ title title Local potential functional embedding theory of molecular systems: Localized orbital-based embedding from an exact density-functional perspective , \ 10.1021/acs.jctc.5c01256 journal journal Journal of Chemical Theory and Computation \ volume 21 ,\ pag...

work page doi:10.1021/acs.jctc.5c01256 2025
[78]

author author Wafa \ Makhlouf , author Bruno \ Senjean , \ and\ author Emmanuel \ Fromager ,\ https://arxiv.org/abs/2602.15624 title Generalized local potential functional embedding theory of localized orbitals , \ ( year 2026 ),\ http://arxiv.org/abs/2602.15624 arXiv:2602.15624 [cond-mat.str-el] NoStop

work page arXiv 2026
[79]

author author Gerald \ Knizia \ and\ author Garnet Kin-Lic \ Chan ,\ title title Density matrix embedding: A simple alternative to dynamical mean-field theory , \ 10.1103/PhysRevLett.109.186404 journal journal Phys. Rev. Lett. \ volume 109 ,\ pages 186404 ( year 2012 ) NoStop

work page doi:10.1103/physrevlett.109.186404 2012
[80]

author author Filip \ Cernatic \ and\ author Emmanuel \ Fromager ,\ title title Extended n-centered ensemble density functional theory of double electronic excitations , \ https://doi.org/10.1002/jcc.27387 journal journal Journal of Computational Chemistry \ volume 45 ,\ pages 1945--1962 ( year 2024 ) NoStop

work page doi:10.1002/jcc.27387 1945
[81]

author author Marco Martínez \ González \ and\ author Paul W. \ Ayers ,\ title title Counterexamples to the convexity of the energy for coulomb particles , \ 10.1021/acs.jpclett.5c03648 journal journal The Journal of Physical Chemistry Letters \ volume 17 ,\ pages 2524--2529 ( year 2026 ) NoStop

work page doi:10.1021/acs.jpclett.5c03648 2026

Showing first 80 references.