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## What about U? - Corrective approaches to DFT for strongly-correlated systems.

#### CECAM-HQ-EPFL, Lausanne, Switzerland

#### Organisers

Systems characterized by strong electronic correlation are the center of many applications of unquestionable technological interest (e.g., fuel cells, solar cells, batteries, homogeneous and heterogeneous catalysts, high Tc superconductors, spintronics devices, magneto/electro-active "smart" materials, etc.). However, their computational modeling is still very challenging, and no numerical approach available today is able to treat systems in this class of realistic size and complexity without incurring in overwhelming computational costs.

Current approximate implementations of Density Functional Theory are largely unable to capture the many-body character of the electronic wave function of correlated systems and result in a seriously inaccurate description of many of their physical properties. Many corrective schemes have been formulated to alleviate these problems and to improve the representation of strong electronic Coulomb interactions. These corrective functionals either downfold the problem to an effective low-energy Hamiltonian, introducing a partially screened Coulomb interaction ("Hubbard U"), or are formulated in the continuum, dealing with the bare Coulomb interaction v, which is then screened or within many-body calculations of various kind.

This workshop aims at bringing these approaches around the same table to discuss the theory and the approximations behind them, to introduce the latest extensions and applications, to compare their performances, and to highlight their specific prerogatives and their limits. The main objective is, however, to facilitate possible integrations between them and to explore the possibility to define a general, flexible, and versatile computational tool that is able to retain the most important aspects of electronic correlations without incurring in excessive computational costs.

The discussion of various methods to compute the effective electronic interactions will possibly serve as a starting point to stimulate the comparative analysis of other aspects of the corrective functionals and the definition of a common theoretical framework.

Computational approaches that will be discussed include:

1)Self-Interaction-Corrected (SIC) DFT

2)Exact Exchange and hybrid functionals

3)DFT+U

4)DFT+ Dynamical Mean-Field Theory

5)Reduced Density Matrix Functional Theory

6)DFT+Gutzwiller

7)GW and Green-function total energy functionals

8)Quantum chemistry approaches (CC, CASSCF, CI, etc Systems characterized by strong electronic correlation are the center of many applications of unquestionable technological interest (e.g., fuel cells, solar cells, batteries, homogeneous and heterogeneous catalysts, high T_{c} superconductors, spintronics devices, magneto/electro-active “smart” materials, etc.). However, their computational modeling is still very challenging, and none of the available numerical approaches is able to treat systems of realistic complexity without incurring in overwhelming computational costs.

Systems characterized by strong electronic correlation are the center of many applications of unquestionable technological interest (e.g., fuel cells, solar cells, batteries, homogeneous and heterogeneous catalysts, high T_{c} superconductors, spintronics devices, magneto/electro-active “smart” materials, etc.). However, their computational modeling is still very challenging, and none of the available numerical approaches is able to treat systems of realistic complexity without incurring in overwhelming computational costs.

Current approximate implementations of Density Functional Theory (DFT), by far the most efficient and versatile computational tool for ab-initio calculations, are largely unable to capture the many-body character of the electronic wave function of correlated systems and result in a seriously inaccurate description of many of their physical properties. Many corrective schemes have been formulated to alleviate these problems and to improve the representation of strong electronic Coulomb interactions. These corrective functionals either downfold the problem to an effective low-energy Hamiltonian, introducing a partially screened Coulomb interaction ("Hubbard U"), or are formulated in the continuum, dealing with the bare Coulomb interaction v, which is then screened or used within many-body calculations of various kind.

This workshop compares different corrective approaches developed for ab initio calculations on strongly correlated systems. In particular, it emphasizes their connections with the fundamental theory of DFT and with many-body theory, highlights their peculiar characteristics and limitations, the approximations they are based on, their range of applicability and their numerical efficiency, discusses analogies and differences between them, presents the latest advancements and applications. The main objective is to facilitate possible integrations between them and to explore the possibility to define more general, flexible, and versatile computational tools able to retain the most important aspects of electronic correlated ground states without incurring in excessive computational costs.

Computational approaches that will be discussed include:

1) Self-Interaction-Corrected (SIC) DFT [1]

2) Exact Exchange and hybrid functionals [2]

3) DFT+U [3,4,5]

4) DFT+ Dynamical Mean-Field Theory [6,7]

5) Reduced Density Matrix Functional Theory [8]

6) DFT+Gutzwiller [9]

7) GW and Green-function total energy functionals

8) Quantum Chemistry approaches (e.g., CASSCF, CI, etc)

The various methods listed above are based on different theoretical approaches, aim to correct different aspects of the electronic ground state resulting from approximate DFT functionals, and present rather different levels of accuracy and numerical efficiency. Despite the notable and important progress achieved in their development, much remains to be explored and improved towards the definition and the optimization of a general computational tool able to embody the right compromise between accuracy, versatility and efficiency necessary to perform predictive calculations on systems of realistic complexity.

This workshop will provide the occasion for researchers active in the development of these corrective approaches to present the latest advancements for each of them, to discuss the approximations behind their formulation and their specific prerogatives, to compare their capabilities and performances and, most importantly, to define a common theoretical background in which all of these corrective schemes can be framed. The development of an overarching view between them will hopefully create a short-circuit between quite diverse computational communities and will stimulate new efforts to integrate different numerical approaches into new, more flexible and efficient computational tools that can be used to perform predictive calculations on the widest possible range of different materials.

The scientific discussion, framed within the general background of density functional and wave function theory, will touch several long-standing open problems: *i) *the operative definition of electronic correlation and possible strategies to fix its misrepresentation in approximate exchange-correlation functionals; *ii)* the possibility of a unified treatment of “strongly” and “weakly” correlated systems;* iii) *possible strategies to capture“dynamic” and “static” correlations and the specific features of the electronic wave function they descend from; *iv)* the approximate formulations of one- and two-body density matrices and the representability problem; *v)* the definition of effective self-energy functionals and green-functions-based approaches; *vi)* the representation of electronic localization, Mott transitions, and other consequences of electronic correlation; *vii)* the importance of a correct account of electronic correlation in describing the structural properties of solids; *viii)* invariance and computational efficiency of corrective schemes; *ix) *the definition and calculation of the effective electronic interactions entering the expression of many corrective functionals. This latter discussion, completed by the analysis of the way that Hubbard-interaction-free methods, such as GW, arrive at an implicitly “downfolded interaction”, will provide a favorable common ground to study analogies and differences between various approaches and will serve as a starting point to explore other important aspects such as, for example, their accuracy, their range of validity, and their practical implementation in actual DFT functionals.

The discussion of the above themes will be developed around the presentation of studies on various systems whose behavior is determined or controlled by electronic correlation. Systems discussed in this workshop will possibly range from transition-metal and rare-earth molecular complexes (optimal comparison ground between DFT and Quantum Chemistry) to transition-metal bulk oxides, from Mott and charge-transfer to Peierls and Slater insulators, from high T_{c} superconductors to paramagnetic insulators, from magnetic semiconductors to functional intermetallic alloys and oxides, from catalysts to materials for solar cells. Several phenomena, related to or influenced by electronic correlation, will be discussed including metal-insulator (Mott/Peierls) transitions, molecular dissociation and electron localization, superconductivity, electron-transfer processes, magnetism, etc.

To stimulate discussions and comparative analysis, the schedule of presentations will be organized, as much as possible, around common topics (e.g., transition-metal oxides, magnetic materials, superconductivity, molecular systems, electron-transfer etc) so that the specific prerogatives and limits of the different approaches employed, their accuracy and their numerical performance, will be highlighted discussing various aspects of the same problem.

This workshop particularly encourages the participation of students and junior researchers and will provide them an excellent first-hand experience with the theoretical foundation of DFT and with the construct of a quite broad spectrum of various corrective approaches to improve the description of electronic correlation. The workshop will be also useful to precisely assess the different capabilities of these computational methods, their limits and their computational efficiency. In addition, the direct contact with some of the main developers active in the field will hopefully encourage junior researchers to contribute actively to the field and to become involved in the theoretical definition and practical implementation of more general, versatile and efficient computational tools for the study of correlated systems.

## References

**France**

Silke Biermann ( Ecole Polytechnique ) - Organiser

**Switzerland**

Matteo Cococcioni ( Swiss Federal Institute of Technology, Lausanne ) - Organiser