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# General Meeting of the GDR REST

The meeting is intended to bring together the members of the GDR but also other scientists working in theoretical spectroscopy and all related areas.

The meeting is a virtual event, to be held in June 2021, via Zoom. It takes place on the afternoons of Thursdays and Fridays 3,4 and 10,11 June.

## Program

Friday 4

Thursday 10

Friday 11

## 13:55

### Welcome

13:55 - 14:00

## 14:00

### Optical properties of hexagonal boron nitride

14:00 - 14:30

Claudio Attaccalite

### New Approximation to the Exchange Correlation Potential from Connector Theory

14:00 - 14:30

Ayoub Aouina

### TDDFT Calculations with Localized Basis Set for Ion Projectiles in Gas and Solid Matters

14:00 - 14:30

Xixi Qi

### Dark state spectral signatures in Methylene blue: A tensor network approach

14:00 - 14:30

Angus Dunnet

## 14:30

### Ab-initio investigation of electronic excitations in bulk V2O5

14:30 - 15:00

Vitaly Gorelov

### Exact (ensemble) density-functional theory for energy gaps without derivative discontinuities

14:30 - 15:00

Emmanuel Fromager

### Ab Initio Study of Helicity-Dependent Light-Induced Demagnetization: From the Optical Regime to the Extreme Ultraviolet Regime

14:30 - 15:00

Philippe Scheid

### Natural-orbital representation(s) of molecular electronic transitions: insights from matrix perturbation theory

14:30 - 15:00

Thibaud Etienne

## 15:00

### Take a break

## 15:15

### Helium atom and hydrogen molecule Hylleraas-like exact solutions as benchmark for many-body theories

15:15 - 15:45

Valerio Olevano

### Diagrammatic Monte-Carlo approach to many-flavour unitary fermions

15:15 - 15:45

Gunnar Möller

### Simulating x-ray optical activity in quartz using first principle calculations

15:15 - 15:45

Solal Lellouche

### Numerical approximation of the dynamical response properties of molecules

15:15 - 15:45

Antoine Levitt

## 15:45

### Green’s functions formalism applied to strongly correlated systems

15:45 - 16:15

Quentin Marécat

### Optical properties of photo-switching metal-organic frameworks

15:45 - 16:15

Roberta Poloni

### Combined role of electronic interaction on oxygen and metal in correlated oxides

15:45 - 16:15

Robinson Outerovitch

### Theoretical description of linear and non-linear processes in magnetic materials

15:45 - 16:15

Shalu Rani

## 16:15

### One-dimensional quantum model with δ-function interaction for studying basis-set correction to wave-function methods

16:15 - 16:45

Diata Traore

### Surface spectroscopy of functionalized Si(001) surfaces

16:15 - 16:45

Stefano Mazzei

## 16:45

### Closing Remarks

16:45 - 17:00

## 17:00

## Optical properties of hexagonal boron nitride

### Claudio Attaccalite

## Numerical approximation of the dynamical response properties of molecules

### Antoine Levitt

## Exact (ensemble) density-functional theory for energy gaps without derivative discontinuities

### Emmanuel Fromager

[1] B. Senjean and E. Fromager, Phys. Rev. A 98, 022513 (2018).

[2] M. J. P. Hodgson, J. Wetherell, and E. Fromager, Phys. Rev. A 103, 012806 (2021).

[3] T. Gould and S. Pittalis, Phys. Rev. Lett. 123, 016401 (2019).

[4] E. Fromager, Phys. Rev. Lett. 124, 243001 (2020).

## Helium atom and hydrogen molecule Hylleraas-like exact solutions as benchmark for many-body theories

### Valerio Olevano

We will show a comparison of Hylleraas He and James & Coolidge H$_2$ exact solutions against several many-body theories from condensed-matter and nuclear physics, as well as quantum chemistry: configuration interaction (CI), quantum Monte Carlo (QMC, both variational VMC and diffusion DMC), density-functional theory (DFT) and time-dependent DFT (TDDFT), direct RPA and full RPA with exchange (aka time-dependent Hartree-Fock, TDHF), Bethe-Salpeter equation (BSE) on top of the GW approximation, and finally a renormalized RPA (r-RPA) in the framework of the self-consistent RPA (SCRPA) originally developed in nuclear physics.

The comparison will be done on both ground and excited states. On the ground state we will present the results by the trace Thouless formula which coincides to the ACFDT adiabatic-connection fluctuation-dissipation theorem) formula in direct RPA, but it appears better founded beyond RPA.

## Dark state spectral signatures in Methylene blue: A tensor network approach

### Angus Dunnet

Furthermore, we find that the shape of this shoulder is highly sensitive to the presence of correlations between the environment-induced fluctuations of the bright and dark state energies, and that by introducing such correlations into our model we are able to obtain an excellent fit with experimental data. As well as providing microscopic insight into the spectral signatures of optical dark states in organic systems, this work also provides a means of comparing different levels of theory in DFT and MD in terms of their ability to reproduce experimentally observed spectra.

## Diagrammatic Monte-Carlo approach to many-flavour unitary fermions

### Gunnar Möller

this issue for fermionic theories by introducing a generalised theory with many fermion flavours, that then becomes amenable to perturbative treatments.

We demonstrate how to implement this concept in conjunction with a diagrammatic Monte-Carlo approach, using a many fermion-flavour generalisation of the unitary Fermi gas [1] for illustration. This system provides an ideal test case thanks to existing diagrammatic Monte-Carlo techniques developed in [2,3], which yield a numerically exact solution thanks to explicit resummation of the high-order asymptotics [4].

We present results on the extended large-N generalisation of the theory [5,6], and demonstrate how to apply diagMC in this setting. Using the resummation technique developed by Rossi et al [4], we show that the convergence radius in the Borel plane is enlarged as a function of fermion flavours, thus facilitating the convergence of the series in the vicinity of the transition into the superfluid phase. We argue that the combination of large-N field theory techniques with high-order numerical resummations opens up a new avenue for investigations of strongly interacting systems more generally.

[1] W. Zwerger, Springer (2012).

[2] K. v.Houcke et al., Nat Phys 8, 366 (2012).

[3] K. v.Houcke et al., PRB 99, 035140 (2019).

[4] R. Rossi, et al., PRL 121, 130405 (2018).

[5] M.Y. Veillette, et al., PRA 75, 043614 (2007).

[6] P. Nikolic and S. Sachdev, PRA 75, 033608 (2007).

## NEW APPROXIMATION TO THE EXCHANGE CORRELATION POTENTIAL FROM CONNECTOR THEORY

### Ayoub Aouina

[1] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).

[2] M. Vanzini, A. Aouina, M. Panholzer, M. Gatti, and L. Reining, arXiv:1903.07930

[3] S. Chen, M. Motta, F. Ma and S. Zhang, Phys. Rev. B 103, 075138 (2021).

## Green's functions formalism applied to strongly correlated systems

### Quentin Marécat, Benjamin Lasorne, Emmanuel Fromager and Matthieu Saubanère

In this communication, both approches are used efficiently to extract the local GF as a functional of the 1-body reduced density matrix (1RDM). More precisely, the local GF is extracted from an embedded correlated system by applying the unitary Householder transformation to the density matrix. The self-consistent mapping between the original and the embedded system is ensure via an explicit hybridization function. However, the representability domain of the 1RDM obtained from the GF (itself extracted from the embedded system) is restricted and will be discuss within the Self-energy Functional Theory [6] framework.

[1] : A. Georges and G. Kotliar, Hubbard Model in Infinite Dimensions, Phys. Rev. B 45, 6479 (1992).

[2] : M. Caffarel and W. Krauth, Exact Diagonalization Approach to Correlated Fermions in Infinite Dimensions: Mott Transition and Superconductivity, Phys. Rev. Lett. 72, 1545 (1994).

[3] : Potthoff M. Dynamical Variational Principles for Strongly Correlated Electron Systems. In: Kramer B. (eds) Advances in Solid State Physics. Advances in Solid State Physics, vol 45. Springer, Berlin

[4] : E. Fertitta and G. H. Booth, Rigorous Wave Function Embedding with Dynamical Fluctuations, Phys. Rev. B 98, 235132 (2018).

[5] : A. A. Rusakov, S. Iskakov, L. N. Tran, and D. Zgid, Self-Energy Embedding Theory (SEET) for Periodic Systems, J. Chem. Theory Comput. 15, 229 (2019).

[6] : M. Potthoff, Self-Energy-Functional Approach to Systems of Correlated Electrons, The European Physical Journal B - Condensed Matter 32, 429 (2003).

## TDDFT Calculations with Localized Basis Set for Ion Projectiles in Gas and Solid Matters

### Xixi Qi, Fabien Bruneval, Ivan Maliyov, Jean-Paul Crocombette

Modern evaluations of the stopping power rely on time-dependent density-functional theory (TDDFT). When implemented in real-time (RT), these calculations are able to capture intriguing effects that go beyond the linear-response approximation (LR).

Our presentation will focus on two main subjects:

First, we have investigated the Barkas effect [2] in insulating materials. This beyond-linear-response effect states that the stopping power of antiprotons (H-) is always lower than that of protons (H+). Using RT-TDDFT calculations, we have explored the low projectile kinetic energy region (below 2 keV) in a standard insulating material, LiF. Surprisingly, we have observed an inverse Barkas effect: the stopping power of antiprotons becomes larger than the one of protons. To investigate further, we have performed both RT and LR calculations on isolated Li+ and F- targets. We see that the binding between F- and its p electrons is weakened in the presence of an antiproton.

Second, we will talk about our progress in stopping power calculations with moving localized basis for heavier ion projectiles that have core electrons moving along their nucleus. In this case, the time-dependent Kohn-Sham equations contain an extra term [3] analogous to the Pulay forces [4]. We have implemented this term in our code MOLGW [5] and have performed calculations for atomic collisions and gas-phase targets. Our ultimate goal will be to describe heavier ions such as transition metal cations in crystalline targets.

REFERENCES:

[1] J. Lindhard, Danske Matematisk-fysiske Meddeleiser, 28, 1 (1954).

[2] W. Barkas, J. Dyer and H. Heckman, Phys. Rev. Lett., 11, 26-28 (1963)

[3] T. Kunnert and R. Schmidt, Eur. Phys. J. D, 25, 15-24 (2003)

[4] P. Pulay, Molecular Physics, 17, 2, 197-204 (1969)

[5] www.molgw.org

## Ab Initio Study of Helicity-Dependent Light-Induced Demagnetization: From the Optical Regime to the Extreme Ultraviolet Regime

### Philippe Scheid, Sangeeta Sharma, Gregory Malinowski, Stéphane Mangin and Sébastien Lebègue

While the origin of the AO–HDS is still debated, it is usually assumed that the dominating contribution to this effect comes from the inverse Faraday effect (IFE)[3, 4], designating the generation of a magnetization proportional to the intensity of the circularly polarized light. Another effect, occurring simultaneously to the IFE in dissipative materials is the magnetization induced during light absorption[5] (MILA), whose magnitude is proportional to the fluence of the light.

Recently, real–time time–dependent density functional theory (RT–TDDFT) has been successfully applied to describe the linearly polarized laser induced demagnetization[6]. In the present work[7] we investigate the dynamics of the magnetization density of Ni, Co and Fe under the influence of optical and XUV circularly polarized pump pulses using RT–TDDFT. We find that, in both cases, the induced spin–dynamics is helicity-dependent, and, that this helicity–dependence is larger when using a XUV light. The origin of this helicity-dependence is closely assessed by examining the spin–resolved time–dependent density of states. Then, we separate contribution of the IFE and the MILA in the helicity–dependent part of the magnetization dynamics and show that the latter has a greater impact, especially in the XUV regime and when the light pulse vanishes.

This work hints at the yet experimentally unexplored territory of the XUV light-induced helicity-dependent dynamics, which, according to our prediction, could magnify the helicity-dependent dynamics already exhibited in the optical regime.

[1] D. Sander, S. O. Valenzuela, D. Makarov, C. H. Marrows, E. E. Fullerton, P. Fischer, J. McCord, P. Vavassori, S. Mangin, P. Pirro, B. Hillebrands, A. D. Kent, T. Jungwirth, O. Gutfleisch, C. G. Kim, and A. Berger, “The 2017 Magnetism Roadmap,” Journal of Physics D: Applied Physics, vol. 50, p. 363001, sep 2017.

[2] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, “All-optical magnetic recording with circularly polarized light,” Physical Review Letters, vol. 99, p. 47601, jul 2007.

[3] L. P. Pitaevskii, “Electric Forces in a Transparent Dispersive Medium,” J. Exptl. Theoret. Phys. (U.S.S.R.), vol. 12, no. 5, p. 1450, 1961.

[4] M. Berritta, R. Mondal, K. Carva, and P. M. Oppeneer, “Ab Initio Theory of Coherent Laser-Induced Magnetization in Metals,” Physical Review Letters, vol. 117, p. 137203, sep 2016.

[5] P. Scheid, G. Malinowski, S. Mangin, and S. Lebègue, “Ab initio theory of magnetization induced by light absorption in ferromagnets,” Phys. Rev. B, vol. 100, no. 21, p. 214402, 2019.

[6] K. Krieger, J. K. Dewhurst, P. Elliott, S. Sharma, and E. K. Gross, “Laser-Induced Demagnetization at Ultrashort Time Scales: Predictions of TDDFT,” Journal of Chemical Theory and Computation, vol. 11, no. 10, pp. 4870–4874, 2015.

[7] P. Scheid, S. Sharma, G. Malinowski, S. Mangin, and S. Lebègue, “Ab Initio Study of Helicity-Dependent Light-Induced Demagnetization: From the Optical Regime to the Extreme Ultraviolet Regime.,” Nano letters, vol. 21, pp. 1943–1947, mar 2021.

## Natural-orbital representation(s) of molecular electronic transitions: insights from matrix perturbation theory

### Enzo Monino, Gabriel Breuil, Elise Lognon, Benjamin Lasorne, and Thibaud Etienne

Besides, we will reveal the ambiguity of the picture of a molecular electronic transition considering the canonical tools that are still currently used for visualizing a molecular electronic transition and quantifying its nature. Without the disambiguation we suggest to introduce, and considering the possibility to imply auxiliary many-body wavefunctions as a possible picture, the representation of an unrelaxed TDDFT electronic transition would be either incomplete, arbitrary, or equivocal.

[1] A. Krylov, From orbitals to observables and back, JCP 153, 080901 (2020)

[2] S. Bäppler, F. Plasser, M. Wormit, A. Dreuw, Exciton analysis of many-body wave functions: Bridging the gap between the quasiparticle and molecular orbital pictures, PRA 90, 052521 (2014)

[3] T. Etienne, A comprehensive, self-contained derivation of the one-body density matrices from single-reference excited-state calculation methods using the equation-of-motion formalism, IJQC 120, e26110 (2020)

[4] T. Etienne, M. Pastore, Charge separation: From the topology of molecular electronic transitions to the dye/semiconductor interfacial energetics and kinetics, in Dye-sensitized solar cells: Mathematical modeling, and materials design and optimization, Masoud Soroush and Kenneth K.S. Lau Eds., Elsevier Academic Press, pp. 121-170 (2019)

[5] F. Plasser, M. Wormit, A. Dreuw, New tools for the systematic analysis and visualization of electronic excitations. I. Formalism, JCP 141, 024106 (2016)

## Simulating x-ray optical activity in quartz using first principle calculations

### Solal LELLOUCHE, Nadejda BOULDI, Lorenzo PAULATTO, Amélie JUHIN, Philippe SAINCTAVIT

Natural circular dichroism, first observed in 1895 by Aimé Cotton, is one of the most powerful tools for obtaining stereochemical information, and is the only method outside of X-ray crystallography capable of revealing the absolute configuration of a chiral system. Circular dichroism measurements have first been developed in the UV-Visible range and have recently been extended to X-ray absorption spectroscopy (XAS, where it received the name of X-ray Natural Circular Dichroism, XNCD). Up to now, XNCD has only been observed in crystals. Due to the intrinsic chemical selectivity of XAS, XNCD mixes information from the local environment of the absorbing atom to the symmetry of the crystal itself. XNCD can only be observed for crystals for which the point of the space group belongs to a family of 13 point groups. Although the theoretical basis seems rather straightforward, there are very few examples of the calculation of XNCD spectra and so far there is not much understanding on how the observed shape of XNCD signals relate to the local point group of the absorbing atom or the point group of the space group of the crystal. Following previous work by Nadejda Bouldi [1], we have performed a theoretical investigation of the XNCD signals in the chiral crystal of α-quartz.

One of the structures for SiO 2 is α-quartz that crystalizes in one of the two chiral space groups P3 1 21 and P3 2 21. These two space groups are compatible with the observation of XNCD. By the way, α-quartz has no magnetic ion so that its net magnetization is zero and one cannot expect to detect any XMCD (X-ray Magnetic Circular Dichroism) when recording XNCD. We have calculated XNCD and the angular dependence of the signal using the code Quantum Espresso, that is based on Density Functional Theory (DFT). Quantum Espresso uses a plane wave basis set and pseudo-potentials. For the λ and ∆ enantiomers, we shall present the XNCD signals at both Si and O K-edges and their angular dependence. The ultimate target of the calculation is to make connection between the XNCD signals and some operator, the pseudo-deviator, that is related to the intensity of the integrated XNCD signal.

[1] N. Bouldi, N. J. Vollmers, C. G. Delpy-Laplanche, Y. Joly, A. Juhin, Ph. Sainctavit, Ch. Brouder, M.

Calandra, L. Paulatto, F. Mauri, and U. Gerstmann. X-ray magnetic and natural circular dichroism from first principles: calculation of K- and L1- edge spectra. Physical Review B 96, 085123 (2017)

## Optical properties of photo-switching metal-organic frameworks

### Roberta Poloni

Recently, photoresponsive MOFs have been proposed as a potentially efficient technology for gas capture and release as light, possibly in the visible range, rather than temperature, can be used to change the gas adsorption properties of the material. Recent experimental efforts have demonstrated that this can be achieved by functionalizing MOFs using molecular photoswitches such as azobenzenes [1]. However, due to the lack of computational studies on the subject, the nature of the excitations in the MOF and their similarity with the molecular case have not been established so far and yet, this would have strong implications on the efficiency of the process.

In our work we have address this point by computing the electronic structure and the optical absorption spectra of the MOF in trans and cis configuration using the BSE/GW method. In order to predict the BSE spectra with good accuracy, an orbital-tuning approach will be employed to select the exchange and correlation functional for the initial DFT calculations [2]. Finally, periodic and non-periodic BSE and GW calculations within a QM/MM scheme will be performed and compared to assess the the validity of fragment models [3]: if justified it will allow for a more efficient design of the optimal MOF for energy-efficient gas capture-and-release.

[1] Park et al, Reversible alteration of CO 2 adsorption upon photochemical or thermal treatment in a MOF. J. Am. Chem. Soc. 2012, 134, 99–102, DOI: https://doi.org/10.1021/ja209197;

[2] Kshirsagar, D’Avino, Blase, Li, Poloni, Accurate Prediction of the S 1 Excitation Energy in Solvated Azobenzene Derivatives via Embedded Orbital-Tuned Bethe-Salpeter Calculations. J. Chem. Theory Comput. 2020, 16, 2021– 2027, DOI: https:doi.org/10.1021/acs.jctc.9b01257;

[3] Kshirsagar, Attaccalite, Blase, Li, Poloni, Bethe–Salpeter Study of the Optical Absorption of trans and cis Azobenzene-Functionalized Metal–Organic Frameworks Using Molecular and Periodic Models J. Phys. Chem. C 2021, DOI: https://doi.org/10.1021/acs.jpcc.1c00367

## Combined role of electronic interaction on oxygen and metal in correlated oxides

### R. Outerovitch and B. Amadon

This method is extensively used in f and d orbitals of transition metals, lanthanides and actinides. However, the correlation effect occurring in the p orbitals have been shown to improve the density of states and structural parameters of simple oxides. [1-3] In this paper, we use our generalized cRPA implementation (as described in [4] and based on [5]), to calculate the screened interaction on both metal (d or f ) and oxygen (p) orbitals . We also discuss the influence of those interaction on density of state and structural parameters of both charge transfer and Mott-Hubbard insulators.

1 I. A. Nekrasov, M. A. Korotin, and V. I. Anisimov, “Coulomb interaction in oxygen p-shell in LDA+U method and its influence on calculated spectral and magnetic properties of transition metal oxides”, arXiv:cond-mat/0009107 (2000).

2 L. A. Agapito, S. Curtarolo, and M. Buongiorno Nardelli, “Reformulation of DFT+U as a Pseudohybrid Hubbard Density Functional for Accelerated Materials Discovery”, Phys. Rev. X 5, 011006 (2015).

3 O. K. Orhan and D. D. O’Regan, “First-principles Hubbard U and Hund’s J corrected approximate density functional theory predicts an accurate fundamental gap in rutile and anatase TiO 2 ”, Phys. Rev. B 101, 245137 (2020).

4 J.-B. Morée, R. Outerovitch, and B. Amadon, “First-principles calculation of the Coulomb interaction parameters U and J for actinide dioxides”, Phys. Rev. B 103, 045113 (2021).

5 P. Seth, P. Hansmann, A. van Roekeghem, L. Vaugier, and S. Biermann, “Towards a First-Principles Determination of Effective Coulomb Interactions in Correlated Electron Materials: Role of Intershell Interactions”, Phys. Rev. Lett. 119, 056401 (2017).

## Theoretical description of linear and non-linear processes in magnetic materials

### Shalu Rani and Valérie Véniard

Ultra-fast control on the magnetic state could have the strong impact on magnetic recording technology. Second harmonic generation (SHG) is a process where two photons are absorbed by a material, and a photon of twice the energy of the incoming photons is emitted. This spectroscopy is used to study the optical properties of materials because it reveals additional information, compared with linear optical spectroscopies. Due to dipole selection rules, SHG is forbidden in centrosymmetric materials, and it is possible to obtain a structural and electronic characterization for these systems. However, the absence of time-inversion and space symmetry in antiferromagnetic materials leads to new contributions in the second harmonic generation, thus revealing the arrangement of spins in the solid. SHG becomes a powerful tool to study of ultra-fast demagnetization processes.

There are few satisfactory theoretical descriptions for SHG in magnetic materials, since spin- polarization, electron-electron interactions, and local field effect must be treated on the same footing. The project is to calculate the Second order response function of Cr2O3 using Time-Dependent Density Functional Theory including spin-polarization. The linear response of Cr2O3 using different approximations (Independent Particle approximation (IPA), Random Phase approximation (RPA), and alpha kernel is calculated in the TDDFT framework. The value of band gap for the Cr2O3 and the scissor shift is calculated using GW approximation. The BSE calculations is performed using scissor shift and BSE spectra is obtained. The strongly bound excitonic peak has been observed in the BSE spectra. The presence of this peak is important, as in the experiment performed [1], the SHG signal has been measured in this range of frequency. The transition which are responsible for the excitonic peak has been studied. The excitonic peak has been described using the Wannier model. The BSE spectra using the spin-polarization has been calculated. The second-order response function including spin-polarization is calculated using Random Phase approximations (RPA). We concluded that second-order response functions of Cr2O3 including spin-polarization is non-zero.

[1] V. G. Sala et al., “Resonant optical control of the structural distortions that drive ultrafast demagnetization in Cr2 O3,” Phys. Rev. B, vol. 94, no. 1, Jul. 2016.

## Surface spectroscopy of functionalized Si(001) surfaces

### Stefano Mazzei, Valérie Véniard, Christine Giorgetti

We present in this contribution the results obtained for the Si(001) surface functionalized with two different molecules, the Thymine and the Uracil. The electronic structure has been obtained with a DFT approach, using Abinit [3]. Using DP [4] and 2Light codes [5] [6], which allows the calculation of the optical response of materials in the TDDFT framework, we computed several surface-related linear and nonlinear spectroscopic quantities.

[1] K. Seino, W.G. Schmidt, F. Bechstedt, Phys. Rev. B 69, 245309 (2004)

[2] E. Molteni, G. Cappellini, G. Onida, and G. Fratesi, Phys. Rev. B 95, 075437 (2017)

[3] X. Gonze et al., Comput. Phys. Commun. 205, 106-131 (2016)

[4] http://dp-code.org/

[5] E. Luppi, V. Véniard, Phys. Rev. B 82, 235201 (2010)

[6] N. Tancogne-Dejean, C. Giorgetti, V. Véniard, Phys. Rev. B 94, 125301 (2016)

## One-dimensional quantum model with δ-function interaction for studying basis-set correction to wave-function methods

### Diata Traore, Emmanuel Giner, Julien Toulouse

We calculate the correlation energy density of this UEG and parametrize it and compare with the corresponding infinite 1D UEG [2].

[1] E. Giner, B. Pradines, A. Ferté, R. Assaraf, A. Savin and J. Toulouse, J. Chem. Phys. 149, 194301 (2018).

[2] R. J. Magyar and K. Burke, Phys. Rev. A 70, 032508 (2004).