Reduced Density-Matrix Functional Theory: Improving its foundation and extending its scope

ECT* - Villa Tambosi

Strada delle Tabarelle, 286
Trento - Italy

ECT* - Villa Tambosi

Strada delle Tabarelle, 286
Trento - Italy

The aim of this international workshop is to discuss and explore new aspects and challenges in Reduced Density Matrix Functional Theory (RDMFT). For this, we invite up to 25 experts in RDMFT to Trento for an intensive and informal meeting.

This in-person workshop during 3-14 October will be complemented by five mini-symposia (hybrid format), open to a broad international audience:

Symposium 1 – October 3, 14:30-18:00 CEST
Exact results in RDMFT: properties of universal functionals, role of N-representability, etc.

Symposium 2 – October 5, 9:30-13:00 CEST
RDMFT for quantum chemistry: Computational and theoretical state-of-the-art and open challenges

Symposium 3 – October 7, 9:30-13:00 CEST
Extending the scope of RDMFT: bosons, ultracold gases, superconductors, relativistic QM, polarons, etc.

Symposium 4 – October 10, 9:30-13:00 CEST
RDMFT for excited states and time-evolution

Symposium 5 – October 12, 9:30-13:00 CEST
RDMFT for translational invariant systems


In 1959, at the Colorado conference on Molecular Quantum Mechanics, Charles Coulson pointed out that the description of atomic and molecular ground states involves the two-electron reduced density matrix (2RDM), only [1,2]. Indeed, since electrons interact only by pairwise interactions, the energies and other electronic properties of atoms and molecules can be computed directly from the 2RDM. This crucial insight has defined the starting point for the development of new theoretical approaches to the ground state problem, avoiding the use of the exponentially complex N-electron wave function. Since each subfield of the Quantum Sciences typically restricts to systems all characterized by a fixed pair interaction V (for instance Coulomb interaction in quantum chemistry, contact interaction in quantum optics and Hubbard interaction in solid state physics) the ground state problem should de facto involve only the one-particle reduced density matrix (1RDM). Indeed, for Hamiltonians of the form H(h) = h + V, where h represents the one-particle terms and V the fixed pair interaction, the conjugate variable to H(h) and h, respectively, is the 1RDM. The corresponding exact one-particle theory is known as Reduced Density Matrix Functional Theory (RDMFT) which is based on the existence of an exact energy functional of the 1RDM [3]. This interaction functional is universal in the sense that it depends only on the fixed interaction V but not on the one-particle terms h [3,4].

Most notably, compared to the widely used Density Functional Theory (DFT), RDMFT has some significant conceptual advantages and is therefore expected to overcome at some point the fundamental limitations of DFT. On the one hand, the kinetic energy is described in an exact way due to the access of the full 1RDM and all the effort can be spent to derive good approximations to the interaction energy. On the other hand, RDMFT allows explicitly for fractional occupation numbers and has therefore great prospects of describing systems with strong correlations, particularly static correlations [5]. For instance, benchmark calculations revealed that the common functionals in RDMFT yield correlation energies for closed shell atoms and molecules which are by one order of magnitude more accurate than B3LYP (one of the most popular density functionals in quantum chemistry) and a precision comparable to Møller-Plesset second-order perturbation theory [5-8]. RDMFT has also succeeded in predicting more accurate gaps of conventional semiconductors than semi-local DFT and demonstrated the insulating behavior of Mott-type insulators [9-11]. At the same time, involving the full 1RDM lies, however, also at the heart of possible disadvantages of RDMFT relative to DFT. The 1RDM involves quadratically more degrees of freedom than the spatial density and enforcing the orthonormality of the natural orbitals is computationally highly demanding [5]. Furthermore, for open-shell systems, the domain of the universal functional seems to be constrained by the more involved generalized Pauli constraints. All these aspects are still hampering RDMFT from reaching its full potential and effectiveness.

Significant recent progress on reduced density matrices and the theory of fermion correlation provides new ways for overcoming the problems of the recent realization of RDMFT. For instance, it has been observed that for many solid materials, one can exploit either the spatial locality or the translational symmetry in an efficient way, leading to more accurate functionals and remarkable additional insights [12-14]. Moreover, it has been shown that the one- and two-body N-representability constraints strongly shape the exact functional [15,16]. Due to the effort of several research groups in the world the last two years, we have witnessed a remarkable progress along several lines:

  1. In the form of the DoNOF, the first RDMFT software code has been made public just about one year ago [17].
  2. In a series of very recent works [18,19], RDMFT has been extended to excited states by generalizing the work by Gross, Oliviera and Kohn on ensemble DFT. This has revealed in particular a generalization of Pauli’s exclusion principle to mixed states [20].
  3. Novel ideas were proposed to improve the numerical minimization of functionals which may help to overcome the most severe obstacle of the recent version of RDMFT [21].
  4. RDMFT has been extended to bosonic quantum systems with remarkable new insights into its fermionic counterpart [22,23].
  5. The cusp condition due to the Coulomb interaction has finally been translated into a property of the 1RDM providing new insights into the role and accurate treatment of dynamic correlation in RDMFT [24].
  6. Quite recently, modern machine learning techniques were put forward for improving functional approximations such as in [25,26].

Most of these novel results and ideas are rather challenging, fundamental, and of potentially high impact. It will be one of the crucial challenging for the next few years to combine all those new ideas to improve the foundation of RDMFT, overcome its recent limitations and extent its scope by including also non-singlet, finite-temperature, and time-evolving systems [27,28].


[1] C. A. Coulson, “Present state of molecular structure calculations”, Rev. Mod. Phys. 32, 170 (1960).
[2] D. A. Mazziotti (ed.), Reduced Density-Matrix mechanics: with applications to many-electron atoms and molecules, John Wiley & Sons (2007).
[3] T. L. Gilbert, “Hohenberg-Kohn theorem for nonlocal external potentials”, Phys. Rev. B 12, 2111 (1975).
[4] M. Levy, “Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem”, PNAS 76, 6062 (1979).
[5] K. Pernal and K. J. Giesbertz, “Reduced Density Matrix Functional Theory (RDMFT) and Linear Response Time-Dependent RDMFT (TD-RDMFT)”, Top. Curr. Chem. 368, 125 (2016).
[6] N. Lathiotakis and M. A. L. Marques, “Benchmark calculations for reduced density-matrix functional theory”, J. Chem. Phys. 128, 184103 (2008).
[7] M. Piris, “A natural orbital functional based on an explicit approach of the two-electron cumulant”, Int. J. Quantum Chem. 113, 620 (2012).
[8] M. Piris, “Global Method for Electron Correlation”, Phys. Rev. Lett. 119, 063002 (2017).
[9] S. Sharma, J. K. Dewhurst, S. Shallcross, and E. K. U. Gross, “Spectral Density and Metal-Insulator Phase Transition in Mott Insulators Within Reduced Density Matrix Functional Theory”, Phys. Rev. Lett. 110, 116403 (2013).
[10] Y. Shinohara, S. Sharma, J. K. Dewhurst, S. Shallcross, N. N. Lathiotakis, and E. K. U. Gross, “Doping induced metal-insulator phase transition in NiO. A reduced density matrix functional theory perspective”, New J. Phys. 17, 093038 (2015).
[11] K. Pernal, “Turning reduced density matrix theory into a practical tool for studying the Mott transition”, New J. Phys. 17, 111001 (2015).
[12] M. Saubanère, M. B. Lepetit, and G. M. Pastor, “Interaction-energy functional of the Hubbard model: Local formulation and application to low-dimensional lattices”, Phys. Rev. B 94, 045102 (2016).
[13] C. Schilling and R. Schilling, “Diverging exchange force and form of the exact density matrix functional”, Phys. Rev. Lett. 122, 013001 (2019).
[14] J. Schmidt, C. L. Benavides-Riveros, and M. A. L. Marques, “Reduced density matrix functional theory for superconductors”, Phys. Rev. B 99, 224502 (2019).
[15] M. Piris, “The Role of the N-Representability in One-Particle Functional Theories”, chapter in “Many-body Approaches at Different Scales” (Springer, New York, 2016) Chapter 22.
[16] C. Schilling, “Communication: Relating the pure and ensemble density matrix functional”, J. Chem. Phys. 149, 231102 (2018).
[17] M. Piris, I. Mitxelena, Comput. Phys. Commun. 259, 107651 (2021).
[18] C. Schilling, S. Pittalis, Phys. Rev. Lett. 127, 023001 (2021)
[19] J. Liebert, F. Castillo, J.-P. Labbé, C. Schilling, arXiv:2106.03918, to appear in J. Chem. Theory Comput.
[20] F. Castillo, J.-P. Labbé, J. Liebert, A.Padrol, E.Philippe, C. Schilling, arXiv:2105.06459
[21] D. Gibney, J.N. Boyn, D.A. Mazziotti, J. Phys. Chem. Lett. 12, 385 (2021)
[22] C.L. Benavides-Riveros, J. Wolff, M.A.L. Marques, C. Schilling, Phys. Rev. Lett. 124, 180603 (2020)
[23] J. Liebert, C. Schilling, Phys. Rev. Research 3, 013282 (2021)
[24] J. Cioslowski, J. Chem. Phys. 153, 154108 (2020)
[25] J. Schmidt, C.L. Benavides-Riveros, and M.A.L. Marques, J. Phys. Chem. Lett. 10, 6425 (2019)
[26] J. Schmidt, M. Fadel, C.L. Benavides-Riveros, Phys. Rev. Research 3, L032063 (2021).
[27] K. Giesbertz, O. Gritsenko, and E. Baerends, “Response calculations with an independent particle system with exact one-particle density matrix”, Phys. Rev. Lett. 105, 013002 (2010).
[28] K. Giesbertz and M. Ruggenthaler, “One-body reduced density-matrix functional theory in finite basis sets at elevated temperatures”, Phys. Rep. 806, 1 (2019).



  • Carlos L. Benavides-Riveros (Max-Planck Institute for Complex Systems)
  • Eberhard K. U. Gross (The Hebrew University of Jerusalem)
  • Christian Schilling (LMU Munich)


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