**CECAM-HQ-EPFL, Lausanne, Switzerland, 26-29 September 2017**

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

**Sponsors**: CECAM, Psi-k and Max Planck Institute of Microstructure Physics.

**Summary**

*This international workshop discussed and explored new aspects and challenges in Reduced Density Matrix Functional Theory (RDMFT). The main aim was to bring together leading experts in the field to address and carefully discuss open challenges in RDMFT such as implementations of 1-particle symmetries, extensions to open-shell atoms and molecules, time-evolution, temperature dependency and new insights about RDMFT from recent progress on the 1- and 2-body N-representability problems and density matrix renormalization group. **The list of speakers was carefully chosen to include experts in various disciplines required for the accomplishment of the proposed scientific program. To maximize the success of the workshop, we asked all speakers to provide rather informal and interactive presentations. We also asked them to share their slides and other supplemental materials with all the participants in advance, allowing them to prepare the workshop accordingly. This ‘homework’ enabled not only fruitful and stimulating scientific discussions, but also more involved questions.*

**INTRODUCTION**

Since electrons interact pairwise by Coulomb repulsion, the energies and other electronic properties of fermionic systems can be computed directly from the two-electron reduced density matrix (2-RDM), only. ”Banishing” this way the N-electron wave function with its exponentially many degrees of freedom, the entire quantum problem can be recast in the form of a very simple linear functional on the 2-RDM. However, the description of the 2-electron picture involves highly nontrivial representability conditions, and new approaches of the reduced electron pictures deserve to be explored. The most successful one so far is Density Functional Theory (DFT), an approach based on the Hohenberg-Kohn and Kohn-Sham theorems. Although DFT has seen a tremendous success in many-body physics, the search for highly accurate functionals has suffered from an intrinsic difficulty. Not only the exact functional for the exchange-correlation is unknown but also the one for the kinetic energy.

In 1975, Gilbert established in the form of ”Reduced Density Matrix Functional Theory” (RDMFT) a natural extension of DFT. RDMFT exploits the 1-electron picture by seeking a functional on the whole 1-electron reduced density matrix (1-RDM). Compared to DFT, the big advantage is that the kinetic energy can be described in an exact way. Any scientific effort can be solely spent on approximating the exchange-correlation functional. In contrast to the 2-electron picture, the representability conditions for the 1- RDM are known. While DFT resorts to a large zoo of engineered density-functionals, only about a dozen of 1-RDM-functionals have been proposed so far. Remarkably, those few and less developed functionals already allowed one to describe closed-shell systems with accuracies higher by one order of magnitude than DFT. Moreover, RDMFT has succeeded in predicting more accurate gaps of conventional semiconductors than DFT does and correctly captures the physics of the insulator-metal phase transition of transition metal oxides.

Yet, the theory has been hampered by the absence of a set of single particle equations. Unlike Kohn-Sham DFT or Hartree-Fock theory, RDMFT implies a set of coupled self-consistency conditions for the natural orbitals. Therefore, it is one of the big challenges in RDMFT to find ways to improve the efficiency of the current 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 directions for the future.

So far, advances in 1-RDM and 2-RDM theory have 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 (static and dynamic) electronic correlations and entanglement.

###### DESCRIPTION

This international interactive workshop discussed and explored new aspects and open challenges in RDMFT, such as:

(1) new insights about RDMFT from recent progress on the 1- and 2-body N-representability problems,

(2) new insights about RDMFT from finite uniform electron gases, random phase approximation, and exact systems,

(3) implementation of 1-particle symmetries and translational invariance,

(4) extension of RDMFT to open-shell molecules, finite temperatures, and time evolution,

(5) electronic correlations, entanglement, and recent developments on fermionic orbital optimization, and

(6) separation and quantification of dynamic and static electronic correlations.

###### PROGRAM

The program included 20 talks, given by the invited speakers, and 5 contributed talks, by young scientists. In the mornings the invited presentations were 75 minutes long (including 30 minutes for questions and discussions). In the afternoons they were 60 minutes long (with 15 minutes for questions and discussions). This way, the speakers had the time to give generous and interactive introductions, addressed to non-specialists, and also deep discussions to a more specialized audience. The contributed talks last 25 minutes.

*September 26*

The program started on **Tuesday** with a short introduction, given by two of the organizers. Carlos L. Benavides-Riveros and Christian Schilling introduced the scope and the expected methodology of the workshop. Above all, they encouraged the participants to participate actively in the scientific discussions. In the first session, Peter Gill (Australian National University) discussed the mathematical treatment of finite uniform electron gases and showed in which way they are a source of information for the construction and testing of new RDMFT models. Kasia Pernal (Technical University of Lodz) pointed out a couple of novel approaches to constructing new functionals within the framework of random phase approximation (RPA). Kasia gave a short introduction to the derivation of the MacLachlan and Ball expression and RPA functionals in DFT; afterward, she explained how to exploit this theoretical framework within RDMFT.

In the second session of Tuesday, there were presented new ideas and new applications of reduced density matrices. Andreas Savin (University Pierre and Marie Curie) argued that combining wave function methods with density functionals can be seen from a reduced density matrix perspective. In a joint work with Nektarios Lathiotakis, Nikitas Gidopoulos (Durham University) exhibited an approach to determine effective potentials for RDMFT. Jamal Berakdar (MLU Halle-Wittenberg) discussed applications of reduced densities for studying the thermodynamic behavior of nanoscale quantum structures (e.g. quantum spin systems with non-collinear magnetic order or quantum heat engines with a working substance being a helical multiferroic structure). Finally, Iris Theophilou (MPI for Structure and Dynamics of Matter) showed a mapping to a non-interacting system to solve the Hubbard model within RDMFT. Tuesday sessions ended with the poster session. Seven PhD students and postdocs from Poland, Germany, India, Japan, Iran, and Czechia presented their research results in the main hall of CECAM-HQ.

*September 27*

In the first session of **Wednesday**, we discussed the importance of 2-body representability conditions for constructing functionals in RDMFT. In practical applications, the energy functional of the theory employs the exact energy functional for the 2-RDM. However, an approximate expression for the 2-RDM is built from the natural occupation numbers and the natural orbitals of the 1-RDM. The so-called natural orbital functionals (NOF) are designed to satisfy some of the known 2-RDM representability conditions. Accordingly, in the first part of his talk, Mario Piris (University of the Basque Country) talked about the role of the N-representability in approximate 1-particle functional theories. In the second part, Mario presented a new method to achieve dynamic and static electronic correlations. Since in real computations one uses finite basis sets, in his talk Klass Giestbertz (Vrije Universiteit Amsterdam) pointed out some mathematical aspects concerning such a low dimensionality of the Hilbert spaces. He also discussed how crucial is a rigorous foundation of RDMFT.

In the afternoon, Pina Romaniello (University Paul Sabatier) stressed the relation of RDMFT with spectroscopy and provided some examples. Eduard Matito (Donostia International Physics Center) discussed how to separate dynamic and static electronic correlations using the relations between 2-RDM and 1-RDM. The expressions reached this way depend only on the natural occupation numbers. Peter Knowles (Cardiff University) exhibited an example of two wavefunctions with identical one-electron spectrum, but strongly different two-electron density matrices. To Peter, it means that algebraic density-matrix functionals cannot be exact. The question for the role of the generalized Pauli exclusion principle (with its constraints on the occupation numbers) in RDMFT was addressed by Nicole Helbig (Forschungszentrum Juelich).

*September 28*

On **Thursday** morning, Oleg Gritsenko (Vrije Universiteit Amsterdam) pointed out the crucial role of fractional occupation numbers in the description of double excitations obtained in time-dependent RDMFT. Nektarios Lathiotakis (Theoretical and Physical Chemistry Institute, Athens) started his talk with an overview of RDMFT and its main differences to DFT. Inspired by local DFT, Nektarios discussed the recent progress in developing effective Hamiltonians in RDMFT. In the afternoon, we had two talks on electronic correlations and tensor networks. Norbert Mauser (Wolfgang Pauli Institute) presented the so-called “nonfreeness”, the measure he and Alex Gottlieb introduced some years ago to quantify electronic correlations. A vivid discussion on how to quantify correlations in statistical mixtures followed. In his talk, Örs Legeza (Hungarian Academy of Sciences) overviewed tensor-network techniques and discussed how such techniques treat high-dimensional optimization tasks. No doubt, its versatility is one of the reasons why the theory is widely used in many-body quantum physics with long-range interactions and ab initio quantum chemistry.

We finished the day with the contributed-talk session. Our conference banquet was held at the restaurant Port de Pully, very close to Lake Geneva with —Lord Byron once wrote— its “stillness” and “stern delights”:

*September 29*

The conference finished on **Friday** with four talks. Sangeeta Sharma (MPI of Microstructure Physics) discussed her work in finding the correct ground-state properties of transition metal oxides (TMO). The good news is: RDMFT is able to capture the correct insulating state of TMO under ambient conditions. As well, away from ambient pressure, RDMFT correctly captures the physics of the insulator-metal phase transition. The bad news is that the minimization process in RDMFT must be upgraded in order to be used in infinite systems, for it is not fast enough to beat DFT or Hartree-Fock. Eugene de Prince (Florida State University) overviewed 2-RDM theory. The current variational 2-RDM approach can be used to realize polynomially-scaling complete active space self-consistent field. Paul Ayers (McMaster University) displayed some recent techniques to refine non-representable RDM. The workshop finished with the talk of Joshua Hollett (University of Winnipeg). Joshua stressed the importance of taking different approaches to electron correlations. He also discussed recent progress in deriving an “on-top” density functional for dynamic correlation. A closing word wrapped up our workshop.

**CONCLUSIONS**

Overall, this international conference was an interesting and stimulating meeting of scientists working in different areas of physics and chemistry connected with or inspired by reduced density matrices. It was, by and large, a great opportunity to discuss and point out new challenges for RDMFT. We hope this meeting will prompt the community to pursue new collaborations, boosting the versatility, development, and applications of RDMFT.

Some of the participants mentioned the possibility of another such meeting in two or so years, and we strongly hope that it can be repeated in the near future. Last but not least, we acknowledge the warm atmosphere and kind support of CECAM, Psi-k and Max Planck Institute of Microstructure Physics (Halle, Germany).

#### LIST OF INVITED TALKS

**Peter Gill**(Australian National University):*”Finite uniform electron gases”*.**Kasia Pernal**(Technical University of Lodz):*”Novel approaches towards constructing reduced density matrix functionals within random phase approximation framework”*.**Andreas Savin**(University Pierre and Marie Curie):*”Combining methods without double counting in RDMFT context”*.**Nikitas Gidopoulos**(Durham University):*”Effective potentials to minimize the total energy functional in DFT and RDMFT”*.**Jamal Berakdar**(MLU Halle-Wittenberg):*”Reduced density matrix applied to quantum thermodynamics”*.**Iris Theophilou**(MPI for Structure and Dynamics of Matter, Hamburg):*”Density and kinetic energy density -to- potential mapping on a lattice”.***Mario Piris**(University of the Basque Country):*”NOF-MP2: A global method for the electron correlation”.***Klaas Giesbertz**(Vrije Universiteit Amsterdam):*”One-body reduced density matrix to potential mapping in finite-basis sets at elevated temperatures”.***Pina Romaniello**(University Paul Sabatier, Toulouse):*”Correlation and Spectroscopy in Reduced Density-Matrix Functional Theory”.***Eduard Matito**(Donostia International Physics Center):*”Separation of dynamic and nondynamic correlation”.***Peter Knowles**(Cardiff University):*”Density matrix functionals: existence, uniqueness, and applicability”.***Nicole Helbig**(Forschungszentrum Juelich):*”Spin and pure state N-representability constraints in reduced density matrix functional theory”.***Oleg Gritsenko**(Vrije Universiteit Amsterdam):*”Natural excitation orbitals as descriptors of single excitations and fractional occupation numbers as descriptors of double excitations in time-dependent response density matrix functional theories”.***Nektarios Lathiotakis**(Theoretical and Physical Chemistry Institute, Athens):*”Effective Hamiltonians in RDMFT and single particle properties”.***Norbert Mauser**(Wolfgang Pauli Institute, Vienna):*”Measures for electron correlation: Nonfreeness”*.**Örs Legeza**(Hungarian Academy of Sciences):*”Tensor network state ansatz for systems of continuously confined fermions”*.**Sangeeta Sharma**(MPI of Microstructure Physics):*”Spectrum within Reduced Density Matrix Functional Theory: application to transition metal oxides”.***Eugene DePrince**(Florida State University): ”*Analytic energy gradients, excited states, and pure-state N-representability in v2RDM-driven CASSCF’*‘.**Paul Ayers**(McMaster University):*”Refining non-N-representable Density Matrices”*.**Joshua Hollett**(University of Winnipeg):*”A two-pronged approach to electron correlation in RDMFT”.*

#### LIST OF CONTRIBUTED TALKS

**Christian Krumnow**(Freie Universität Berlin):*”**Adapting tensor network states to the structure of interacting fermions”.***Stefan Donsa**(Technical University Wien):*”**The time-dependent two-particle reduced density matrix method”.***Mauricio Antonio Rodriguez Mayorga**(Universitat de Girona):*”**Steeplechase for 2-RDM approximations”.***Ion Mitxelena**(University of the Basque Country):*”**Recent advances in NOF theory”.***Mojdeh Banafsheh**(Universty of Geneva):*”**Analytically inverted non-additive kinetic potential functional at small density overlaps”.*

The workshop program, the abstracts of the talks and the posters, the PDF files with the slides used during the oral presentations, and some supplemental materials can be found on the official workshop’s webpage: https://www.cecam.org/workshop-1448.html.