pith. sign in

arxiv: 2506.09960 · v1 · submitted 2025-06-11 · 🪐 quant-ph · physics.chem-ph

Refining ensemble N-representability of one-body density matrices from partial information

Julia Liebert , Anna O. Schouten , Irma Avdic , Christian Schilling , David A. Mazziotti This is my paper

Pith reviewed 2026-05-19 09:19 UTC · model grok-4.3

classification 🪐 quant-ph physics.chem-ph
keywords ensemble N-representabilityone-body reduced density matricesHorn's problemconvex polytopelattice site occupationsensemble density functional theoryexcited statespartial information
0
0 comments X

The pith

Partial information on ensemble one-body reduced density matrices can be turned into explicit N-representability constraints by linking the problem to a generalization of Horn's problem.

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

The paper introduces a hierarchy of ensemble one-body N-representability problems that incorporate partial knowledge of the 1RDMs for an ensemble of N-fermion states with fixed weights. It proposes a systematic relaxation that reduces the case of fixing full 1RDMs for selected ensemble members to constraints involving only natural occupation number vectors. This relaxed problem connects to a generalization of Horn's problem, so its solution is obtained by combining the Horn constraints with the weighted ensemble N-representability conditions. A further convex relaxation produces a convex polytope that supplies physically meaningful bounds on lattice site occupations. Readers would care because these bounds can be used inside ensemble density functional theory for excited states.

Core claim

The central claim is that the refined ensemble N-representability problem with partial 1RDM information, after systematic relaxation to natural occupation numbers, admits an explicit solution obtained by merging the constraints of a generalized Horn problem with those of the weighted ensemble N-representability conditions; an additional convex relaxation then yields a convex polytope that restricts lattice site occupations in ensemble density functional theory for excited states.

What carries the argument

The central mechanism is the systematic relaxation that converts the problem of fixing complete 1RDMs for certain ensemble elements into constraints on natural occupation number vectors alone, which is shown to be solvable via a generalized Horn problem when combined with weighted ensemble N-representability conditions.

If this is right

  • Explicit solutions are now available for the relaxed ensemble N-representability problem with partial 1RDM data.
  • A convex polytope is obtained that directly restricts lattice site occupations.
  • These restrictions can be inserted into ensemble density functional theory calculations for excited states.
  • Partial information on selected 1RDMs suffices to tighten the allowed occupation numbers without requiring complete data.

Where Pith is reading between the lines

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

  • The same relaxation technique could be tested on problems with partial information on two-body reduced density matrices.
  • Numerical implementations of the generalized Horn constraints might be benchmarked against exact diagonalization results for small lattices.
  • The convex polytope bounds may improve the accuracy of approximate functionals in excited-state ensemble DFT on real materials.

Load-bearing premise

The relaxation that reduces the refined problem of fixing full 1RDMs for selected ensemble members to a problem involving only natural occupation number vectors continues to preserve the essential physical constraints of the original N-representability problem.

What would settle it

A concrete falsifier would be an explicit set of natural occupation numbers that obey both the generalized Horn constraints and the weighted ensemble N-representability conditions yet cannot arise from any valid ensemble of N-fermion states consistent with the given partial 1RDM information.

Figures

Figures reproduced from arXiv: 2506.09960 by Anna O. Schouten, Christian Schilling, David A. Mazziotti, Irma Avdic, Julia Liebert.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic illustration of the tightening of [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Illustration of the non-convex polytope [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Illustration of the spectral polytopes [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Illustration of the residual [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Illustration of the residual [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Illustration of the construction of the spectral set [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
read the original abstract

The $N$-representability problem places fundamental constraints on reduced density matrices (RDMs) that originate from physical many-fermion quantum states. Motivated by recent developments in functional theories, we introduce a hierarchy of ensemble one-body $N$-representability problems that incorporate partial knowledge of the one-body reduced density matrices (1RDMs) within an ensemble of $N$-fermion states with fixed weights $w_i$. Specifically, we propose a systematic relaxation that reduces the refined problem -- where full 1RDMs are fixed for certain ensemble elements -- to a more tractable form involving only natural occupation number vectors. Remarkably, we show that this relaxed problem is related to a generalization of Horn's problem, enabling an explicit solution by combining its constraints with those of the weighted ensemble $N$-representability conditions. An additional convex relaxation yields a convex polytope that provides physically meaningful restrictions on lattice site occupations in ensemble density functional theory for 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.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a hierarchy of ensemble one-body N-representability problems that incorporate partial knowledge of the 1RDMs within an ensemble of N-fermion states with fixed weights w_i. It proposes a systematic relaxation that reduces the refined problem (full 1RDMs fixed for selected ensemble members) to constraints involving only natural occupation number vectors. This relaxed problem is related to a generalization of Horn's problem, whose constraints are combined with the known weighted ensemble N-representability inequalities. An additional convex relaxation produces a convex polytope that supplies restrictions on lattice-site occupations for ensemble DFT of excited states.

Significance. If the central relaxation is shown to be exact or a controlled outer bound, the work supplies new, explicit constraints that could improve the physical consistency of ensemble density functionals for excited states on lattices. The explicit linkage to a generalized Horn problem is a constructive strength, as it imports existing mathematical machinery rather than deriving inequalities from scratch. The resulting polytope for occupation numbers is directly usable in practical calculations.

major comments (2)
  1. [Section describing the relaxation to occupation-number vectors] The systematic relaxation from fixing full 1RDMs (including off-diagonal and phase information) to natural occupation numbers alone is load-bearing for the tractability claim. The manuscript must demonstrate explicitly that this projection preserves the feasible set of the original partial-information problem or yields a controlled outer approximation; otherwise the subsequent intersection with generalized Horn constraints may admit unphysical ensemble 1RDMs. A concrete counter-example or proof sketch in the relevant section would resolve the concern.
  2. [Section on the convex relaxation and polytope construction] The additional convex relaxation that produces the final polytope for lattice-site occupations requires a clear statement of whether the resulting set is tight with respect to the combined Horn-plus-weighted-ensemble conditions or merely an outer bound. Without this clarification or supporting numerical checks on small systems, it is difficult to assess how physically meaningful the restrictions actually are.
minor comments (2)
  1. [Introduction / notation section] Notation for the ensemble weights w_i and the partial-information sets should be introduced with a compact table or diagram early in the manuscript to aid readability.
  2. [Abstract] The abstract asserts an 'explicit solution' via the Horn connection; a brief forward reference to the specific theorem or proposition number would help readers locate the combination step.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major concerns point by point below and will revise the manuscript to incorporate clarifications and additional material where needed.

read point-by-point responses

  1. Referee: [Section describing the relaxation to occupation-number vectors] The systematic relaxation from fixing full 1RDMs (including off-diagonal and phase information) to natural occupation numbers alone is load-bearing for the tractability claim. The manuscript must demonstrate explicitly that this projection preserves the feasible set of the original partial-information problem or yields a controlled outer approximation; otherwise the subsequent intersection with generalized Horn constraints may admit unphysical ensemble 1RDMs. A concrete counter-example or proof sketch in the relevant section would resolve the concern.

    Authors: We agree that explicit justification of the relaxation step is essential. The projection onto natural occupation numbers is constructed as a systematic outer approximation: any ensemble 1RDM satisfying the original partial-information constraints necessarily satisfies the relaxed occupation-number constraints, but the converse does not hold. We will add a short proof sketch in the relevant section establishing this inclusion and a concrete counter-example (N=2 fermions on a 4-site lattice) demonstrating that the relaxed set is strictly larger and can admit unphysical points that are subsequently excluded by the generalized Horn constraints. This will make clear that the final bounds remain controlled outer approximations. revision: yes

  2. Referee: [Section on the convex relaxation and polytope construction] The additional convex relaxation that produces the final polytope for lattice-site occupations requires a clear statement of whether the resulting set is tight with respect to the combined Horn-plus-weighted-ensemble conditions or merely an outer bound. Without this clarification or supporting numerical checks on small systems, it is difficult to assess how physically meaningful the restrictions actually are.

    Authors: We accept that the tightness of the final convex polytope must be stated explicitly. The polytope is obtained by a further convex relaxation of the intersection of the generalized Horn constraints and the weighted ensemble N-representability inequalities, and is therefore an outer bound. We will revise the text to state this clearly and add numerical comparisons on small systems (2-site and 4-site Hubbard chains with N=2) that quantify the gap between the polytope and the exact feasible set obtained by direct enumeration, thereby illustrating the practical utility of the bounds for ensemble DFT. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation combines independent mathematical results with a controlled relaxation

full rationale

The paper's central step is a systematic relaxation of the partial-information ensemble N-representability problem to constraints on natural occupation numbers alone, followed by an explicit combination of those constraints with a generalization of Horn's problem and the known weighted ensemble N-representability inequalities. No equation in the provided abstract or description reduces the target polytope to a fitted parameter or to a self-citation that is itself defined by the present result. The relaxation is presented as an outer bound whose physical utility is asserted separately; the text does not claim or require that the occupation-number projection is exactly equivalent to the original feasible set. Self-citations to prior N-representability work by overlapping authors are present but function as external input rather than load-bearing justification for the new relaxation or the Horn connection. The derivation therefore remains self-contained against external benchmarks and does not collapse by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on the standard mathematical properties of reduced density matrices and the known solution set of Horn's problem; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math The N-representability conditions for weighted ensembles of fermion states are known and can be combined with Horn-type inequalities.
    Invoked when the abstract states that the relaxed problem is solved by combining generalized Horn constraints with weighted ensemble N-representability conditions.

pith-pipeline@v0.9.0 · 5714 in / 1352 out tokens · 25454 ms · 2026-05-19T09:19:06.307202+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

134 extracted references · 134 canonical work pages · 1 internal anchor

  1. [1]

    Thus, Horn’s problem corresponds to the case of|K| = 1 fixed natu- ral occupation number vectors (see Sec

    Generalized Horn problem The generalized Horn problem, which deals with the sum of principal submatrices of a Hermitian matrix [80], extends the well-known Horn problem [78, 79, 101, 102] beyond the sum of two Hermitian matrices. Thus, Horn’s problem corresponds to the case of|K| = 1 fixed natu- ral occupation number vectors (see Sec. II) and will be used...

    work page

  2. [2]

    [78–80, 101, 102], to fully charac- terize Λ(w, LK) for generic r and cardinality |K| of the index set K

    Characterisation of Λ(w, LK) In this section, we use the solution to Horn’s problem and its generalization for sums ofm Hermitian matrices, as discussed in Refs. [78–80, 101, 102], to fully charac- terize Λ(w, LK) for generic r and cardinality |K| of the index set K. The first key result is the following lemma: Lemma 1. The set Λ↓(w, LK) ≡ n λ↓ | λ ∈ Λ(w,...

    work page

  3. [3]

    Discussion The derivation of the setΛ↓(w, LK) for |K| = 1 fixed natural occupation number vectors corresponds to the original Horn problem, which deals with the sum of two Hermitian matrices with fixed, decreasingly ordered vec- tors of eigenvalues, as explained in Sec. IIIB1. Even for |K| = 1, the number of linear constraints grows signifi- cantly with t...

    work page

  4. [4]

    produces the inequalities in Eq. (36). Moreover, Eq. (36) reveals that unless all(N − 1) largest entries of λ(1) equal one (recall Fig. 3 for N = 2 ), the new exclusion principle constraints defining the spectral set λ ∈ Σ(w, λ(1))are always stricter than thew-constraints that define Σ(w). Furthermore, using the hyperplane representation of the spectral s...

    work page 1949

  5. [5]

    A. J. Coleman, Structure of fermion density matrices, Rev. Mod. Phys.35, 668 (1963)

    work page 1963

  6. [6]

    Garrod and J

    C. Garrod and J. K. Percus, Reduction of the N-Particle Variational Problem, J. Math. Phys.5, 1756 (1964)

    work page 1964

  7. [7]

    Kummer, n-Representability Problem for Reduced Density Matrices, J

    H. Kummer, n-Representability Problem for Reduced Density Matrices, J. Math. Phys.8, 2063 (1967)

    work page 2063

  8. [8]

    R. M. Erdahl, Representability, Int. J. Quantum Chem. 13, 697 (1978)

    work page 1978

  9. [9]

    A. J. Coleman and V. Yukalov,Reduced Density Matri- ces: Coulson ’s challenge (Springer Berlin, Heidelberg, 2000)

    work page 2000

  10. [10]

    Klyachko, Quantum marginal problem and N- representability, J

    A. Klyachko, Quantum marginal problem and N- representability, J. Phys. Conf. Ser.36, 72 (2006)

    work page 2006

  11. [11]

    Altunbulak and A

    M. Altunbulak and A. Klyachko, The Pauli principle revisited, Commun. Math. Phys.282, 287 (2008)

    work page 2008

  12. [12]

    D. A. Mazziotti, Structure of fermionic density ma- trices: Complete N-representability conditions, Phys. Rev. Lett.108, 263002 (2012)

    work page 2012

  13. [13]

    D. A. Mazziotti, Significant conditions for the two- electron reduced density matrix from the constructive solution of N representability, Phys. Rev. A85, 062507 (2012)

    work page 2012

  14. [14]

    Maciazek, A

    T. Maciazek, A. Sawicki, D. Gross, A. Lopes, and C. Schilling, Implications of pinned occupation numbers for natural orbital expansions. ‘II: rigorous derivation and extension to non-fermionic systems, New J. Phys. 22, 023002 (2020)

    work page 2020

  15. [15]

    Schilling, C

    C. Schilling, C. L. Benavides-Riveros, A. Lopes, T. Ma- ciazek, and A. Sawicki, Implications of pinned occupa- tion numbers for natural orbital expansions: I. gener- alizing the concept of active spaces, New J. Phys.22, 023001 (2020). 16

    work page 2020

  16. [16]

    D. A. Mazziotti, Quantum many-body theory from a solution of the N-representability problem, Phys. Rev. Lett. 130, 153001 (2023)

    work page 2023

  17. [17]

    Y.-K. Liu, M. Christandl, and F. Verstraete, Quan- tumcomputationalcomplexityofthe N-representability problem: QMA complete, Phys. Rev. Lett.98, 110503 (2007)

    work page 2007

  18. [18]

    O’Gorman, S

    B. O’Gorman, S. Irani, J. Whitfield, and B. Fefferman, Intractability of electronic structure in a fixed basis, PRX Quantum3, 020322 (2022)

    work page 2022

  19. [19]

    Klyachko, Quantum marginal problem and represen- tations of the symmetric group, arXiv:0409113 (2004)

    A. Klyachko, Quantum marginal problem and represen- tations of the symmetric group, arXiv:0409113 (2004)

    work page 2004

  20. [20]

    Schilling, D

    C. Schilling, D. Gross, and M. Christandl, Pinning of fermionic occupation numbers, Phys. Rev. Lett. 110, 040404 (2013)

    work page 2013

  21. [21]

    Schilling, Quasipinning and its relevance for N- fermion quantum states, Phys

    C. Schilling, Quasipinning and its relevance for N- fermion quantum states, Phys. Rev. A 91, 022105 (2015)

    work page 2015

  22. [22]

    Schilling, C

    C. Schilling, C. L. Benavides-Riveros, and P. Vrana, Re- constructing quantum states from single-party informa- tion, Phys. Rev. A96, 052312 (2017)

    work page 2017

  23. [23]

    Schilling, M

    C. Schilling, M. Altunbulak, S. Knecht, A. Lopes, J. D. Whitfield, M.Christandl, D.Gross,andM.Reiher,Gen- eralized Pauli constraints in small atoms, Phys. Rev. A 97, 052503 (2018)

    work page 2018

  24. [24]

    D. A. Mazziotti, Realization of quantum chemistry without wave functions through first-order semidefinite programming, Phys. Rev. Lett.93, 213001 (2004)

    work page 2004

  25. [25]

    D. A. Mazziotti, Large-Scale Semidefinite Programming for Many-Electron Quantum Mechanics, Phys. Rev. Lett. 106, 083001 (2011)

    work page 2011

  26. [26]

    Schilling, Communication: Relating the pure and en- semble density matrix functional, J

    C. Schilling, Communication: Relating the pure and en- semble density matrix functional, J. Chem. Phys.149, 231102 (2018)

    work page 2018

  27. [27]

    Piris, Global natural orbital functional: Towards the complete description of the electron correlation, Phys

    M. Piris, Global natural orbital functional: Towards the complete description of the electron correlation, Phys. Rev. Lett.127, 233001 (2021)

    work page 2021

  28. [28]

    L. M. Sager-Smith and D. A. Mazziotti, Reducing the quantum many-electron problem to two electrons with machinelearning,J.Am.Chem.Soc. 144,18959(2022)

    work page 2022

  29. [29]

    L. M. Sager, A. O. Schouten, and D. A. Mazziotti, Beginnings of exciton condensation in coronene analog of graphene double layer, J. Chem. Phys.156, 154702 (2022)

    work page 2022

  30. [30]

    G. M. Jones, R. R. Li, A. E. I. DePrince, and K. D. Vogiatzis, Data-driven refinement of electronic ener- gies from two-electron reduced-density-matrix theory, J. Phys. Chem. Lett.14, 6377 (2023)

    work page 2023

  31. [31]

    Eugene DePrince III, Variational determination of the two-electron reduced density matrix: A tutorial re- view, WIREs Comput Mol Sci.14, e1702 (2024)

    A. Eugene DePrince III, Variational determination of the two-electron reduced density matrix: A tutorial re- view, WIREs Comput Mol Sci.14, e1702 (2024)

    work page 2024

  32. [32]

    N. C. Rubin, R. Babbush, and J. McClean, Application of fermionic marginal constraints to hybrid quantum al- gorithms, New J. Phys.20, 053020 (2018)

    work page 2018

  33. [33]

    S. E. Smart and D. A. Mazziotti, Efficient two-electron ansatz for benchmarking quantum chemistry on a quan- tum computer, Phys. Rev. Res.2, 023048 (2020)

    work page 2020

  34. [34]

    G. A. Quantum, Collaborators*†, F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, S. Boixo, M. Broughton, B. B. Buckley, D. A. Buell, B. Burkett, N. Bushnell, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, S. Demura, A. Dunsworth, E. Farhi, A. Fowler, B. Foxen, C. Gidney, M. Giustina, R. Graff, S. Habegger, M. P. Harrigan, A. Ho, S...

    work page 2020

  35. [35]

    A. E. Raeber and D. A. Mazziotti, Non-equilibrium steady state conductivity in cyclo[18]carbon and its boron nitride analogue, Phys. Chem. Chem. Phys.22, 23998 (2020)

    work page 2020

  36. [36]

    Avdic and D

    I. Avdic and D. A. Mazziotti, Fewer measurements from shadow tomography withN-representability conditions, Phys. Rev. Lett.132, 220802 (2024)

    work page 2024

  37. [37]

    Avdic and D

    I. Avdic and D. A. Mazziotti, Enhanced shadow tomog- raphy of molecular excited states via the enforcement of N-representability conditions by semidefinite program- ming, Phys. Rev. A110, 052407 (2024)

    work page 2024

  38. [38]

    Gritsenko, K

    O. Gritsenko, K. Pernal, and E. J. Baerends, An im- proved density matrix functional by physically moti- vated repulsive corrections, J. Chem. Phys.122, 204102 (2005)

    work page 2005

  39. [39]

    Pernal and K

    K. Pernal and K. J. H. Giesbertz, Reduced den- sity matrix functional theory (RDMFT) and lin- ear response time-dependent rdmft (TD-RDMFT), in Density-Functional Methods for Excited States , edited by N. Ferré, M. Filatov, and M. Huix-Rotllant (Springer International Publishing, Cham, 2016) p. 125

    work page 2016

  40. [40]

    Kamil, R

    E. Kamil, R. Schade, T. Pruschke, and P. E. Blöchl, Reduced density-matrix functionals applied to the Hub- bard dimer, Phys. Rev. B93, 085141 (2016)

    work page 2016

  41. [41]

    Schade, E

    R. Schade, E. Kamil, and P. Blöchl, Reduced density- matrix functionals from many-particle theory, Eur. Phys. J. Special Topics226, 2677 (2017)

    work page 2017

  42. [42]

    Schade and P

    R. Schade and P. E. Blöchl, Adaptive cluster approx- imation for reduced density-matrix functional theory, Phys. Rev. B97, 245131 (2018)

    work page 2018

  43. [43]

    C. L. Benavides-Riveros and M. A. L. Marques, Static correlated functionals for reduced density matrix func- tional theory, Eur. Phys. J. B91, 133 (2018)

    work page 2018

  44. [44]

    Piris, Natural orbital functional for multiplets, Phys

    M. Piris, Natural orbital functional for multiplets, Phys. Rev. A100, 032508 (2019)

    work page 2019

  45. [45]

    Mitxelena, M

    I. Mitxelena, M. Piris, and J. M. Ugalde, Chapter seven - advances in approximate natural orbital func- tional theory, in State of The Art of Molecular Elec- tronic Structure Computations: Correlation Methods, Basis Sets and More , Advances in Quantum Chemistry, Vol. 79, edited by L. U. Ancarani and P. E. Hoggan (Academic Press, 2019) p. 155

    work page 2019

  46. [46]

    Schmidt, C

    J. Schmidt, C. L. Benavides-Riveros, and M. A. L. Mar- ques, Reduced density matrix functional theory for su- perconductors, Phys. Rev. B99, 224502 (2019)

    work page 2019

  47. [47]

    Piris and I

    M. Piris and I. Mitxelena, DoNOF: An open-source im- plementation of natural-orbital-functional-based meth- ods for quantum chemistry, Comput. Phys. Commun. 259, 107651 (2021)

    work page 2021

  48. [48]

    Di Sabatino, C

    S. Di Sabatino, C. Verdozzi, and P. Romaniello, Time 17 dependent reduced density matrix functional theory at strong correlation: insights from a two-site Anderson impurity model, Phys. Chem. Chem. Phys.23, 16730 (2021)

    work page 2021

  49. [49]

    Lemke, J

    Y. Lemke, J. Kussmann, and C. Ochsenfeld, Efficient integral-direct methods for self-consistent reduced den- sity matrix functional theory calculations on central and graphics processing units, J. Chem. Theory Comput. 18, 4229 (2022)

    work page 2022

  50. [50]

    Gibney, J.-N

    D. Gibney, J.-N. Boyn, and D. A. Mazziotti, Den- sity functional theory transformed into a one-electron reduced-density-matrix functional theory for the cap- ture of static correlation, J. Phys. Chem. Lett.13, 1382 (2022)

    work page 2022

  51. [51]

    Mitxelena and M

    I. Mitxelena and M. Piris, Benchmarking GNOF against FCI in challenging systems in one, two, and three di- mensions, J. Chem. Phys.156, 214102 (2022)

    work page 2022

  52. [52]

    Senjean, S

    B. Senjean, S. Yalouz, N. Nakatani, and E. Fromager, Reduced density matrix functional theory from an ab initio seniority-zero wave function: Exact and approx- imate formulations along adiabatic connection paths, Phys. Rev. A106, 032203 (2022)

    work page 2022

  53. [53]

    Liebert, A

    J. Liebert, A. Y. Chaou, and C. Schilling, Refining and relating fundamentals of functional theory, J. Chem. Phys. 158 (2023)

    work page 2023

  54. [54]

    S. M. Sutter and K. J. H. Giesbertz, One-body reduced density-matrix functional theory for the canonical en- semble, Phys. Rev. A107, 022210 (2023)

    work page 2023

  55. [55]

    Gibney, J.-N

    D. Gibney, J.-N. Boyn, and D. A. Mazziotti, Univer- sal generalization of density functional theory for static correlation, Phys. Rev. Lett.131, 243003 (2023)

    work page 2023

  56. [56]

    Gibney, J.-N

    D. Gibney, J.-N. Boyn, and D. A. Mazziotti, Enhancing density-functional theory for static correlation in large molecules, Phys. Rev. A110, L040802 (2024)

    work page 2024

  57. [57]

    C. L. Benavides-Riveros, T. Wasak, and A. Recati, Extracting many-body quantum resources within one- body reduced density matrix functional theory, Phys. Rev. Res.6, L012052 (2024)

    work page 2024

  58. [58]

    Vladaj, Q

    M. Vladaj, Q. Marécat, B. Senjean, and M. Saubanére, Variational minimization scheme for the one-particle re- duced density matrix functional theory in the ensemble N-representability domain, J. Chem. Phys.161, 074105 (2024)

    work page 2024

  59. [59]

    conjecture

    J. Cioslowski and K. Strasburger, Constraints upon functionals of the 1-matrix, universal properties of nat- ural orbitals, and the fallacy of the Collins “conjecture”, J. Phys. Chem. Lett.15, 1328 (2024)

    work page 2024

  60. [60]

    N. G. Cartier and K. J. H. Giesbertz, Exploiting the Hessian for a better convergence of the SCF-RDMFT procedure, J. Chem. Theory Comput.20, 3669 (2024)

    work page 2024

  61. [61]

    Yao and N

    Y.-F. Yao and N. Q. Su, Enhancing reduced density matrix functional theory calculations by coupling or- bital and occupation optimizations, arXiv:2402.03532 (2024)

    work page arXiv 2024

  62. [62]

    Watanabe, Über die Anwendung thermodynamis- cher Begriffe auf den Normalzustand des Atomkerns, Z

    S. Watanabe, Über die Anwendung thermodynamis- cher Begriffe auf den Normalzustand des Atomkerns, Z. Physik 113, 482–513 (1939)

    work page 1939

  63. [63]

    E. K. U. Gross, L. N. Oliveira, and W. Kohn, Rayleigh- Ritz variational principle for ensembles of fractionally occupied states, Phys. Rev. A37, 2805 (1988)

    work page 1988

  64. [64]

    Ding, C.-L

    L. Ding, C.-L. Hong, and C. Schilling, Ground and ex- cited states from ensemble variational principles, Quan- tum 8, 1525 (2024)

    work page 2024

  65. [65]

    Schilling and S

    C. Schilling and S. Pittalis, Ensemble reduced density matrix functional theory for excited states and hierar- chicalgeneralizationofPauli’sexclusionprinciple,Phys. Rev. Lett.127, 023001 (2021)

    work page 2021

  66. [66]

    Liebert, F

    J. Liebert, F. Castillo, J.-P. Labbé, and C. Schilling, Foundation of one-particle reduced density matrix func- tional theory for excited states, J. Chem. Theory Com- put. 18, 124 (2022)

    work page 2022

  67. [67]

    Liebert and C

    J. Liebert and C. Schilling, An exact one-particle the- ory of bosonic excitations: from a generalized Hohen- berg–Kohn theorem to convexified N-representability, New J. Phys.25, 013009 (2023)

    work page 2023

  68. [68]

    Liebert and C

    J. Liebert and C. Schilling, Deriving density-matrix functionals for excited states, SciPost Phys. 14, 120 (2023)

    work page 2023

  69. [69]

    L. N. Oliveira, E. K. U. Gross, and W. Kohn, Density- functional theory for ensembles of fractionally occupied states. II. Application to the He atom, Phys. Rev. A37, 2821 (1988)

    work page 1988

  70. [70]

    Z.-h. Yang, A. Pribram-Jones, K. Burke, and C. A. Ull- rich, Direct extraction of excitation energies from en- semble density-functional theory, Phys. Rev. Lett.119, 033003 (2017)

    work page 2017

  71. [71]

    E.Fromager,Individualcorrelationsinensembledensity functional theory: State- and density-driven decompo- sitions without additional Kohn-Sham systems, Phys. Rev. Lett.124 (2020)

    work page 2020

  72. [72]

    Loos and E

    P.-F. Loos and E. Fromager, A weight-dependent local correlation density-functional approximation for ensem- bles, J. Chem. Phys.152, 214101 (2020)

    work page 2020

  73. [73]

    Cernatic, B

    F. Cernatic, B. Senjean, V. Robert, and E. Fromager, Ensemble density functional theory of neutral and charged excitations, Top. Curr. Chem.280, 4 (2022)

    work page 2022

  74. [74]

    Gould and L

    T. Gould and L. Kronik, Ensemble generalized Kohn–Sham theory: The good, the bad, and the ugly, J. Chem. Phys.154, 094125 (2021)

    work page 2021

  75. [75]

    Yang, Second-order perturbative correlation en- ergy functional in the ensemble density-functional the- ory, Phys

    Z.-h. Yang, Second-order perturbative correlation en- ergy functional in the ensemble density-functional the- ory, Phys. Rev. A104, 052806 (2021)

    work page 2021

  76. [76]

    Gould, D

    T. Gould, D. P. Kooi, P. Gori-Giorgi, and S. Pittalis, Electronic excited states in extreme limits via ensem- ble density functionals, Phys. Rev. Lett.130, 106401 (2023)

    work page 2023

  77. [77]

    Giarrusso and P.-F

    S. Giarrusso and P.-F. Loos, Exact excited-state func- tionals of the asymmetric Hubbard dimer, J. Phys. Chem. Lett.14, 8780 (2023)

    work page 2023

  78. [78]

    T. R. Scott, J. Kozlowski, S. Crisostomo, A. Pribram- Jones,andK.Burke,Exactconditionsforensembleden- sity functional theory, Phys. Rev. B109, 195120 (2024)

    work page 2024

  79. [80]

    Castillo, J.-P

    F. Castillo, J.-P. Labbé, J. Liebert, A. Padrol, E. Philippe, and C. Schilling, An effective solution to convex 1-body N-representability, Ann. Henri Poincaré 24, 2241–2321 (2023)

    work page 2023

  80. [81]

    Liebert, F

    J. Liebert, F. Castillo, J.-P. Labbé, M. Maciazek, and C. Schilling, Solving one-body ensemble N- representability problems with spin, arXiv:2412.01805 (2024)

    work page arXiv 2024

Showing first 80 references.