# Workshops

## International Workshop on 'New challenges in Reduced Density Matrix Functional Theory: Symmetries, time-evolution and entanglement'

September 26, 2017 to September 29, 2017
Location : CECAM-HQ-EPFL, Lausanne, Switzerland
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### Organisers

• Carlos L. Benavides-Riveros (Martin-Luther-Universität Halle-Wittenberg, Germany)
• Miguel Marques (Martin-Luther-Universität Halle-Wittenberg, Germany)
• Christian Schilling (University of Oxford, United Kingdom)
• Eberhard K.U. Gross (Max Planck Institute of Microstructure Physics, Halle, Germany)

### Description

In 1959, at the Colorado conference on Molecular Quantum Mechanics, Charles Coulson pointed out that the description of atoms and molecules involves the two-electron reduced density matrix (2-RDM), only [1]. Indeed, since electrons interact only pairwise by Coulomb repulsion, the energies and other electronic properties of atoms and molecules can be computed directly from the 2-RDM. This remarkable conceptual insight has defined the starting point for the development of new theoretical approaches in quantum chemistry, ''banishing'' the N-electron wave function with its exponentially many degrees of freedom. The entire quantum problem can then be recast in the form of a very simple linear functional on the 2-RDM. Yet, since this description in the 2-electron picture involves highly-nontrivial N-representability conditions (constraining to 2-RDMs representing N-electron wave functions) new approaches in the reduced electron picture were explored [2-4]. The most successful one so far is Density Functional Theory (DFT). It is based on the Hohenberg-Kohn theorem revealing a 1-1-correspondence between the external potential, corresponding ground state wave function and its electron density. This correspondence implies the existence of a ''magic'' functional on the electron density whose minimization yields the correct ground state energy and the corresponding density. Although DFT has seen a tremendous success in quantum chemistry, material science and condensed matter physics the search for more accurate functionals has suffered from a lack of systematic improvements: Not only the exact functional for the exchange-correlation is unknown but also the one for the kinetic energy.

It is quite promising that Gilbert in 1975 has established in the form of Reduced Density Matrix Functional Theory’ (RDMFT) a natural extension of DFT [5]. RDMFT exploits the 1-electron picture by seeking a magic’ functional on the whole 1-electron reduced density matrix (1-RDM). The big advantage compared to DFT is therefore that the kinetic energy can be described in an exact way and any scientific effort can be solely spent on improving the exchange-correlation functional. In addition, it is worth noting that in contrast to the 2-electron picture, the representability conditions for the 1-RDMs are known, coinciding with the Pauli exclusion principle. While DFT is resorting to a large zoo of hundreds of engineered density-functionals only about a dozen 1-RDM-functionals were proposed so far [6,7]. Remarkably, those few and less developed functionals already allowed one to describe closed-shell atoms and molecules with accuracies higher by one order in magnitude than DFT [8]. Moreover, RDMFT has succeed in predicting more accurate gaps of conventional semiconductors than DFT does and has demonstrated insulating behavior for Mott-type insulators [9]. On the other hand, the theory has been hampered by the absence of a set of single particle equations. Unlike DFT or Hartree-Fock theory, RDMFT implies a set of coupled self-consistency conditions for the natural orbitals [10]. Therefore, it is one of the big challenges in RDMFT to find ways to improve the efficiency of the computational methods. Since the natural orbitals are known from the very beginning for translationally invariant 1-band lattice models the condensed matter regime in particular and the concept of 1-electron symmetries in general are promising direction for the future [11].

Due to tremendous recent progress on the 1-body pure and the 2-body ensemble N-representability problem, in the form of a mathematical breakthrough by A. Klyachko on the quantum marginal problem [12], new analytic tools were provided within the last ten years which can help to systematically construct more accurate 1-RDM functionals. This includes the important case of open-shell systems as well and would provide the starting point for time-dependent RDMFT. In that context, another important challenge is to explore the bridge between RDMFT and DFT- or 2-RDM-based methods including the concepts of orbital entanglement or intracular functional theory [13]. So far, advances in 1-RDM and 2-RDM theory has been fostering the development of a plethora of new paradigms in theoretical physics. Those promise to promote unprecedented growth in our ability to explore computationally a vast number of chemical questions from condensed matter and quantum chemistry to electronic correlations and entanglement. The immediate impact of these programs of research has been the development of new electronic structure methods with improved accuracy for small-to-medium-sized atoms and molecules.

The international interactive workshop will discuss and explore new aspects and challenges in ''Reduced Density Matrix Functional Theory'' such as
(1) New insights about RDMFT from recent progress on the 1- and 2-body N-representability problem.
(2) Implementation of 1-particle symmetries and macroscopic regime.
(3) Extension of RDMFT to open-shell atoms and molecules.
(4) Time-evolution.

The workshop will therefore address and carefully discuss the following questions manifesting themselves in four open challenges:

[C1] Derivation of universal structural insights about the exact functional in RDMFT
- What do the 1-body pure N-representability constraints (generalized Pauli constraints) imply for the structure of the magic’ 1-RDM functional for such systems?
- How does the algebraic decay of the natural occupation numbers (NON) due to the cusp reflect itself on the level of functionals?
- According to E. Lieb the magic’ functional depends explicitly on the particle number. Can we understand this dependency in specific physical regimes?

[C2] Simplification of the orbital self-consistent field equation in case of 1-electron symmetries
- For translationally invariant 1-band lattice models the functional depends only on the NON. Can we construct a meaningful functional?
- How do such functionals generalize in case of more than one band?
- How does an orbital symmetry in atoms or molecules manifest itself in simpler natural orbital self-consistency equations?
- What are the computational complexity efforts in practice for solving the self-consistency conditions for the NON and natural orbitals, respectively?

[C3] Extension of RDMFT to open-shell atoms and molecules
- Why is the description of open-shell systems much more involved?
- Can we suggest a meaningful functional for electronic systems in strong external magnetic fields (i.e. for fully-polarized electrons)?
- Can we split the description into closed shells and valence shells?

[C4] Describing quantum dynamics within RDMFT
- How do we describe time-evolutions in DFT?
- Why can this not be done in a similar way in RDMFT?
- How does the concept of reduced orbital entanglement simplify the description of time- evolutions in RDMFT?
- Can we tackle the weak coupling regime?

The list of workshop participants is carefully chosen to include leading experts in various disciplines required for the success of our proposed scientific program. To maximize this success all speakers will be asked to provide rather informal presentations which need to be shared with all workshop participants at least six weeks prior to the workshop. This will allow all participants to prepare for the workshop accordingly. This `homework’ will enable fruitful scientific discussions as they are planned as an important part of our event. The workshop is also supposed to initiate several new collaborations tackling the four important open challenges [C1-C4] in the next few years.

### References

[1] C. A. Coulson, Present state of molecular structure calculations, Rev. Mod. Phys. 32, 170 (1960).
[2] J. Cioslowski (ed.), Many-electron Densities and Reduced Density Matrices, Plenum Printers (2000).
[3] A. J. Coleman and V. I. Yukalov, Reduced Density Matrices: the Coulson's challenge, Springer (2000).
[4] D. A. Mazziotti (ed.), Reduced Density-Matrix mechanics: with applications to many-electron atoms and molecules, John Wiley & Sons (2007).
[5] T. L. Gilbert, Hohenberg-Kohn theorem for nonlocal external potentials, Phys. Rev. B 12, 2111 (1975).
[6] S. Goedecker and C. J. Umrigar, Natural orbital functional for the many-electron problem, Phys. Rev. Lett. 81, 866 (1998).
[7] N. N. Lathiotakis and M. Marques, Benchmark calculations for reduced density matrix functional theory, J. Chem. Phys. 128, 184103 (2008).
[8] M. Piris, A natural orbital functional based on an explicit approach of the two-electron cumulant, Int. J. Quantum Chem. 113, 620 (2012).
[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] K. Pernal, "Effective Potential for Natural Spin Orbitals", Phys. Rev. Lett. 94, 233002 (2005).
[11] S. Sabatino, J. A. Berger, L. Reining and P. Romaniello, Reduced density-matrix functional theory: Correlation and spectroscopy, J. Chem. Phys. 143, 024108 (2015).
[12] M. Altunbulak and A. Klyachko, The Pauli principle revisited, Commun. Math. Phys. 282, 287 (2008).
[13] N. N. Lathiotakis, N. Helbig, A. Rubio and N. I. Gidoupoulos, Local reduced-density-matrix-functional theory: Incorporating static correlation effects in Kohn-Sham equations, Phys. Rev. A 90, 032511 (2014).