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## GW quasiparticle calculations in condensed matter physics and nanoscience

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

#### Organisers

During the past two decades computational methods for describing electronic excitations based on many-body perturbation theory have gained great relevance within the broad field of first-principles computational materials science. Such methods can obviate the limitations of standard density-functional theory for the calculation of electronic and optical properties of complex materials.

The GW method[11] where the electron self-energy is obtained as the product of the one-particle Green’s function and of the screened Coulomb interaction, enables the accurate determination of quasiparticle excitation energies and lifetimes. These physical quantities are key to interpreting a number of experiments, ranging from angle-resolved photoemission spectroscopy to charge transport in molecular electronics. In addition the GW method serves as a starting point to calculate from first-principles optical spectra by solving the Bethe-Salpeter equation.

However, the use of the GW method has been limited, with respect to that of standard density functional theory, by the much higher computational cost. Only the past few years have witnessed a more widespread use of the GW method, following the introduction of more efficient algorithms[2-5], the increased performance and availability of supercomputing clusters, and the availability of reliable and efficient software for GW calculations[6-11].

During June 2011 the CECAM Headquarters in Lausanne hosted the first international workshop entirely focused on the current status and the prospects of the GW method, entitled “Challenges and solutions in GW calculations for complex systems”. This workshop gathered over 50 researchers in the area of GW calculations from 13 Countries across the globe, and demonstrated that many groups are committing significant resources into the development and further improvement of the GW method. In the words of a pioneer in the area of GW calculations Prof. R. Godby (York), “the atmosphere reminded very much of the excellent GW workshops from the old CECAM in Paris 20 years ago”.

One lesson that we learned from this workshop is that there is a high demand for standardization and unification of the techniques and software tools adopted in this area, as well as a need for clarifying what are the advantages and the limitations of the GW method. Another lesson learned in Lausanne is that the original choice of focusing on one specific method as opposed to several techniques for excited states proved successful, insofar it favored depth over breadth in a research area where technical details and algorithmic solutions are very complex and need to be carefully addressed.

Based on these lessons we feel that, as a natural follow-up of this initiative, a tutorial on “GW quasiparticle calculations in condensed matter physics and nanoscience” would be extremely timely and well received. Such a tutorial would fit perfectly with the mission of CECAM to not only promote exchange between established researchers, but also educate younger generations, and more generally to make advanced techniques in computational materials science available to the widest possible community.

In fact, a factor which is still limiting the widespread use of the GW method is its intrinsically higher complexity with respect to standard density-functional theory. For example, the description of the frequency dependence of the screened Coulomb interaction[12] and the convergence of the quasiparticle energies with respect to the basis set cutoff or Brillouin-zone sampling, are nontrivial and can be the cause of serious incompatibilities between results obtained by different research groups [13].

These issues are especially severe for researchers who are beginning to use the GW method. During the past two decades computational methods for describing electronic excitations based on many-body perturbation theory have gained great relevance within the broad field of first-principles computational materials science. Such methods can obviate the limitations of standard density-functional theory for the calculation of electronic and optical properties of complex materials. The GW method[11] where the electron self-energy is obtained as the product of the one-particle Green’s function and of the screened Coulomb interaction, enables the accurate determination of quasiparticle excitation energies and lifetimes. These physical quantities are key to interpreting a number of experiments, ranging from angle-resolved photoemission spectroscopy to charge transport in molecular electronics. In addition the GW method serves as a starting point to calculate from first-principles optical spectra by solving the Bethe-Salpeter equation. However, the use of the GW method has been limited, with respect to that of standard density functional theory, by the much higher computational cost. Only the past few years have witnessed a more widespread use of the GW method, following the introduction of more efficient algorithms[2-5], the increased performance and availability of supercomputing clusters, and the availability of reliable and efficient software for GW calculations[6-11]. During June 2011 the CECAM Headquarters in Lausanne hosted the first international workshop entirely focused on the current status and the prospects of the GW method, entitled “Challenges and solutions in GW calculations for complex systems”. This workshop gathered over 50 researchers in the area of GW calculations from 13 Countries across the globe, and demonstrated that many groups are committing significant resources into the development and further improvement of the GW method. In the words of a pioneer in the area of GW calculations Prof. R. Godby (York), “the atmosphere reminded very much of the excellent GW workshops from the old CECAM in Paris 20 years ago”. One lesson that we learned from this workshop is that there is a high demand for standardization and unification of the techniques and software tools adopted in this area, as well as a need for clarifying what are the advantages and the limitations of the GW method. Another lesson learned in Lausanne is that the original choice of focusing on one specific method as opposed to several techniques for excited states proved successful, insofar it favored depth over breadth in a research area where technical details and algorithmic solutions are very complex and need to be carefully addressed. Based on these lessons we feel that, as a natural follow-up of this initiative, a tutorial on “GW quasiparticle calculations in condensed matter physics and nanoscience” would be extremely timely and well received. Such a tutorial would fit perfectly with the mission of CECAM to not only promote exchange between established researchers, but also educate younger generations, and more generally to make advanced techniques in computational materials science available to the widest possible community. In fact, a factor which is still limiting the widespread use of the GW method is its intrinsically higher complexity with respect to standard density-functional theory. For example, the description of the frequency dependence of the screened Coulomb interaction[12] and the convergence of the quasiparticle energies with respect to the basis set cutoff or Brillouin-zone sampling, are nontrivial and can be the cause of serious incompatibilities between results obtained by different research groups [13]. These issues are especially severe for researchers who are beginning to use the GW method. With this tutorial we want the students not only to learn the use of some of the GW codes freely accessible to the scientific community but also to be aware of the most important technical and theoretical aspects which are behind such methods.

Therefore we will address several non-standard applications of the GW method like the calculation of the electronic lifetimes or total-energies. In addition the numerical procedures that define the basis of a successful GW calculation will be extensively discusses. We will dedicate specific lectures to choice of an accurate sampling of the Brillouin Zone, especially in low-dimensional systems like nano-structures that represent an important class of systems studied in the scientific community. We will also discuss the impact of the used basis (plane-waves versus localized basis) and the impact of pseudo-potentials versus all-electrons calculations.

More specific lectures will be developed to the problem of self-consistency, vertex corrections and total-energies evaluation in the GW method. These theoretical aspects will provide the students a more general view on the formal justifications, as well on the most up-to-date developments of the GW theory.

Besides theoretical and technical lectures (preferably in the morning) the tutorial will characterized by long hands-on sessions (in the afternoon) where the students will perform calculations on realistic materials following pre-organized tutorials[14]. During and after these hands-on sessions the students will have the possibility of discussing with the teachers about any aspect of the tutorial. These interactions will be particularly meaningful as the teachers of the hands-on sessions will be the main developers of the code used. We also plan to encourage the students, after the completion of the proposed hands-on exercises, to initiate small projects from their own.

Indeed a particularly valuable aspect of this tutorial is that two out of the four organizers (A. Marini and P. Umari) are the developers of two of the code used: Yambo7 and GWL6. These codes adopt a starting formalism based on plane-waves and pseudopotentials, used to describe the core and valence electrons. Moreover, both codes can perform MBPT calculations starting from DFT calculations performed using the Quantum-Espresso package [4] that we will use to to calculate the ground-state wave-functions using Density-functional Theory.

Yambo is a widely used GW code. Based on a plane-have expansion for wave-functions and operators, it offers a large range of capabilities. It is freely available through the GNU license. GWL is a more recent code which is based on optimized basis sets for representing operators in order to accelerate calculations in particular for isolated and non-crystalline systems. Within 2011 it will be fully available through the GNU license.

In addition to Yambo and GWL we will also use the FHI-aims[11] code.

## References

**Italy**

Andrea Marini Marini (CNR) - Organiser

Paolo Umari (University of Padova) - Organiser

**Spain**

Angel Rubio (Universidad del Pais Vasco) - Organiser

**United States**

Feliciano Giustino (University of Texas, Austin) - Organiser