#
Electronic and Optical Properties of Alkaline Earth Metal Fluoride Crystals with the Inclusion of Many-Body Effects: A Comparative Study on Rutile MgF_{2} and Cubic SrF_{2}

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Computational Methods and Resulting Ground-State Properties

_{2}, the experimental values are from Reference [51], while for c-SrF

_{2}, experimental values were obtained from References [52,53].

_{2}, calculated with PBEsol, shows a deviation in the order of 0.1%, while in the case of c-SrF${}_{2}$, the deviation from the experiment for the same observable is of the order of 0.4%. The AM05 results are approximately of the same quality. For the PBE XC scheme (with the typical under-binding effect due to gradient corrections), the results show more significant deviations from the experiment; for LDA calculations (strong over-binding due to the local approximation), the comparison with experiment is worse. For these reasons, structures obtained with PBEsol were used for the present study for the calculation of energy bands and optical properties.

## 3. Electronic Excitations in r-M$\mathrm{g}$F${}_{\mathbf{2}}$ and c-S$\mathrm{r}$F${}_{\mathbf{2}}$

#### 3.1. Energy Gaps for r-MgF${}_{2}$ and c-SrF${}_{2}$

#### 3.2. Quasiparticle Energy Bands for r-MgF${}_{2}$ and c-SrF${}_{2}$

## 4. Dielectric Function and Optical Absorption Spectrum of M$\mathrm{g}$F${}_{\mathbf{2}}$ and S$\mathrm{r}$F${}_{\mathbf{2}}$

## 5. Summary and Conclusions

## 6. Additional Material: Bulk Systems versus Clusters

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Rubloff, G.W. Far-Ultraviolet Reflectance Spectra and the Electronic Structure of Ionic Crystals. Phys. Rev. B
**1972**, 5, 662–684. [Google Scholar] [CrossRef] - Zemann, J. Crystal structures, 2nd edition. Vol. 1, by R. W. G. Wyckoff. Acta Crystallogr.
**1965**, 18, 139. [Google Scholar] [CrossRef] [Green Version] - Bechstedt, F. Many-Body Approach to Electronic Excitations; Springer: Berlin/Heidelberg, Germany, 2015. [Google Scholar] [CrossRef]
- Bordonaro, G.J. DUV Photolithography and Materials. In Encyclopedia of Nanotechnology; Bhushan, B., Ed.; Springer: Dordrecht, The Netherlands, 2012; pp. 590–604. [Google Scholar] [CrossRef]
- Bertoni, C.M.; Cappellini, G.; Finocchi, F.; Monachesi, P. 7.3.4 CaF
_{2}and other fluorides surfaces. In Physics of Solid Surfaces; Springer: Berlin/Heidelberg, Germany, 2015; pp. 387–391. [Google Scholar] [CrossRef] - Mattila, T.; Pöykkö, S.; Nieminen, R.M. Ab initio study of point defects in CdF
_{2}. Phys. Rev. B**1997**, 56, 15665–15671. [Google Scholar] [CrossRef] [Green Version] - Shi, H.; Eglitis, R.I.; Borstel, G. Ab initio calculations of the BaF
_{2}bulk and surface F centers. J. Phys. Condens. Matter**2006**, 18, 8367–8381. [Google Scholar] [CrossRef] [Green Version] - Jia, R.; Shi, H.; Borstel, G. The atomic and electronic structure of CaF
_{2}and BaF_{2}crystals with H centers: A hybrid DFT calculation study. J. Phys. Condens. Matter**2010**, 22, 055501. [Google Scholar] [CrossRef] - Tousey, R. Optical Constants of Fluorite in the Extreme Ultraviolet. Phys. Rev.
**1936**, 50, 1057–1066. [Google Scholar] [CrossRef] - Samara, G.A. Temperature and pressure dependences of the dielectric properties of PbF2and the alkaline-earth fluorides. Phys. Rev. B
**1976**, 13, 4529–4544. [Google Scholar] [CrossRef] - Scrocco, M. Satellites in X-ray photoelectron spectroscopy of insulators. I. Multielectron excitations in CaF
_{2}, SrF_{2}, and BaF_{2}. Phys. Rev. B**1985**, 32, 1301–1305. [Google Scholar] [CrossRef] - Weesner, F.J.; Wright, J.C.; Fontanella, J.J. Laser spectroscopy of ion-size effects on point-defect equilibria in PbF
_{2}:Eu^{3+}. Phys. Rev. B**1986**, 33, 1372–1380. [Google Scholar] [CrossRef] - Kosacki, I.; Langer, J.M. Fundamental absorption edge of PbF2 and Cd
_{1-x}PbxF_{2}crystals. Phys. Rev. B**1986**, 33, 5972–5973. [Google Scholar] [CrossRef] - Hull, S.; Keen, D.A. Effect of hydrostatic pressure on the crystal structure and superionic behavior of lead (II) fluoride. Phys. Rev. B
**1998**, 58, 14837–14844. [Google Scholar] [CrossRef] - Fujita, M.; Itoh, M.; Bokumoto, Y.; Nakagawa, H.; Alov, D.L.; Kitaura, M. Optical spectra and electronic structures of lead halides. Phys. Rev. B
**2000**, 61, 15731–15737. [Google Scholar] [CrossRef] - Burnett, J.; Levine, Z.; Shirley, E. Intrinsic birefringence in calcium fluoride and barium fluoride. Phys. Rev. B
**2001**, 64, 241102–1–241102–4. [Google Scholar] [CrossRef] - Inagaki, H.; Saito, A.; Sugiyama, H.; Okabayashi, T.; Fujimoto, S. Rapid inactivation of SARS-CoV-2 with deep-UV LED irradiation. Emerg. Microbes Infect.
**2020**, 9, 1744–1747. [Google Scholar] [CrossRef] - Hessling, M.; Hoenes, K.; Vatter, P.; Lingenfelder, C. Ultraviolet irradiation doses for coronavirus inactivation - review and analysis of coronavirus photoinactivation studies. GMS Hyg. Infect. Control.
**2020**, 15, Doc08. [Google Scholar] [CrossRef] [PubMed] - Kowalski, W. UVGI Disinfection Theory. In Ultraviolet Germicidal Irradiation Handbook; Springer: Berlin/Heidelberg, Germany, 2009; pp. 17–50. [Google Scholar] [CrossRef]
- Budowsky, E.I.; Bresler, S.E.; Friedman, E.A.; Zheleznova, N.V. Principles of selective inactivation of viral genome. Arch. Virol.
**1981**, 68, 239–247. [Google Scholar] [CrossRef] - Molteni, E.; Fratesi, G.; Cappellini, G.; Onida, G. Optical Properties of Free and Si(001)-Adsorbed Pyrimidinic Nucleobases. Phys. Status Solidi (b)
**2017**, 255, 1700497. [Google Scholar] [CrossRef] - Molteni, E.; Cappellini, G.; Onida, G.; Fratesi, G. Optical properties of organically functionalized silicon surfaces: Uracil-like nucleobases on Si(001). Phys. Rev. B
**2017**, 95, 05437–1–05437–8. [Google Scholar] [CrossRef] [Green Version] - Cadelano, E.; Cappellini, G. Electronic structure of fluorides: General trends for ground and excited state properties. Eur. Phys. J. B
**2011**, 81, 115–120. [Google Scholar] [CrossRef] - Cadelano, E.; Furthmüller, J.; Cappellini, G.; Bechstedt, F. One- and two-particle effects in the electronic and optical spectra of barium fluoride. J. Phys. Condens. Matter
**2014**, 26, 125501. [Google Scholar] [CrossRef] - Cappellini, G.; Furthmüller, J.; Cadelano, E.; Bechstedt, F. Electronic and optical properties of cadmium fluoride: The role of many-body effects. Phys. Rev. B
**2013**, 87, 075203–1–075203–9. [Google Scholar] [CrossRef] [Green Version] - Filippetti, A.; Fiorentini, V. A practical first-principles band-theory approach to the study of correlated materials. Eur. Phys. J. B
**2009**, 71, 139–183. [Google Scholar] [CrossRef] - Ferreira, L.G.; Pelá, R.R.; Teles, L.K.; Marques, M., Jr.; Furthmülller, J. The LDA-1/2 technique: Recent Developments. In Proceedings of the 31st International Conference on the Physics of Semiconductors (ICPS), Zurich, Switzerland, 29 July–3 August 2012; AIP: Long Island, NY, USA, 2013; Volume 1566, pp. 27–28. [Google Scholar] [CrossRef]
- Matusalem, F.; Marques, M.; Teles, L.K.; Filippetti, A.; Cappellini, G. Electronic properties of fluorides by approximated quasiparticle DFT-1/2 and PSIC methods: BaF
_{2}, CaF_{2}and CdF_{2}as test cases. J. Phys. Condens. Matter**2018**, 30, 365501. [Google Scholar] [CrossRef] [PubMed] - Frandon, J.; Lahaye, B.; Pradal, F. Spectra of Electronic Excitations in CaF
_{2}, SrF_{2}, and BaF_{2}in the 8 to 150 eV Range. Phys. Status Solidi (b)**1972**, 53, 565–575. [Google Scholar] [CrossRef] - Raisin, C.; Berger, J.M.; Robin-Kandare, S. UPS and XPS spectra of CdF
_{2}and SrF_{2}and interpretation of optical properties of these compounds. J. Phys. C Solid State Phys.**1980**, 13, 1835–1844. [Google Scholar] [CrossRef] - Kudrnovský, J.; Christensen, N.E.; Maek, J. Electronic structure of fluorite-type compounds and mixed crystals. Phys. Rev. B
**1991**, 43, 12597–12606. [Google Scholar] [CrossRef] - Khenata, R.; Daoudi, B.; Sahnoun, M.; Baltache, H.; Rérat, M.; Reshak, A.H.; Bouhafs, B.; Abid, H.; Driz, M. Structural, electronic and optical properties of fluorite-type compounds. Eur. Phys. J. B
**2005**, 47, 63–70. [Google Scholar] [CrossRef] - Ivanovskikh, K.; Pustovarov, V.; Shulgin, B. Time-resolved luminescent VUV-spectroscopy of pure and doped by rare earth ions crystals of strontium fluoride. Nucl. Instruments Methods Phys. Res. Sect. A Accel. Spectrometers, Detect. Assoc. Equip.
**2005**, 543, 229–233. [Google Scholar] [CrossRef] [Green Version] - Jaiswal, S.R.; Sawala, N.S.; Nagpure, P.A.; Barde, W.S.; Omanwar, S. The Highly Efficient Inorganic SrF
_{2}:Gd^{3+},Eu^{3+}Phosphor for Mercury Free Fluorescence Lamps. Adv. Mater. Res.**2022**, 1171, 17–24. [Google Scholar] [CrossRef] - Chaney, R.C. Self-consistent energy band structure of magnesium fluoride using the LCAO method. J. Phys. C Solid State Phys.
**1980**, 13, 5691–5699. [Google Scholar] [CrossRef] - Vassilyeva, A.; Eglitis, R.; Kotomin, E.; Dauletbekova, A. Ab initio calculations of the atomic and electronic structure of MgF2 (011) and (111) surfaces. Open Phys.
**2011**, 9, 515–518. [Google Scholar] [CrossRef] - Yi, Z.; Jia, R. Quasiparticle band structures and optical properties of magnesium fluoride. J. Phys. Condens. Matter
**2012**, 24, 085602(5pp). [Google Scholar] [CrossRef] [PubMed] - Cappellini, G.; Bosin, A.; Serra, G.; Furthmüller, J.; Bechstedt, F.; Botti, S. Electronic and Optical Properties of Small Metal Fluoride Clusters. ACS Omega
**2020**, 5, 13268–13277. [Google Scholar] [CrossRef] [PubMed] - Levy, J.B.; Hargittai, M. Unusual Dimer Structures of the Heavier Alkaline Earth Dihalides: A Density Functional Study. J. Phys. Chem. A
**2000**, 104, 1950–1958. [Google Scholar] [CrossRef] - Koput, J.; Roszczak, A. CaF
_{2}As a Quasilinear Molecule: The Vibrational-Rotational Energy Levels Predicted by Ab Initio Quantum Chemistry Approach. J. Phys. Chem. A**2004**, 108, 9267–9273. [Google Scholar] [CrossRef] - Pandey, R.K.; Waters, K.; Nigam, S.; He, H.; Pingale, S.S.; Pandey, A.C.; Pandey, R. A theoretical study of structural and electronic properties of alkaline-earth fluoride clusters. Comput. Theor. Chem.
**2014**, 1043, 24–30. [Google Scholar] [CrossRef] - Calder, V.; Mann, D.E.; Seshadri, K.S.; Allavena, M.; White, D. Geometry and Vibrational Spectra of Alkaline-Earth Dihalides. II. CaF
_{2}, SrF_{2}, and BaF_{2}. J. Chem. Phys.**1969**, 51, 2093–2099. [Google Scholar] [CrossRef] - Kresse, G.; Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci.
**1996**, 6, 15–50. [Google Scholar] [CrossRef] - Kresse, G.; Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B
**1996**, 54, 11169–11186. [Google Scholar] [CrossRef] - Vinet, P.; Rose, J.H.; Ferrante, J.; Smith, J.R. Universal features of the equation of state of solids. J. Phys. Condens. Matter
**1989**, 1, 1941–1963. [Google Scholar] [CrossRef] - Perdew, J.P.; Ruzsinszky, A.; Csonka, G.I.; Vydrov, O.A.; Scuseria, G.E.; Constantin, L.A.; Zhou, X.; Burke, K. Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces. Phys. Rev. Lett.
**2008**, 100, 136406–1–136406–4. [Google Scholar] [CrossRef] [Green Version] - Aroyo, M.I. (Ed.) Teaching Edition of International Tables for Crystallography: Crystallographic Symmetry; IUCr/Wilet: Chester, UK, 2021; ISBN 978-0-470-97422-3. [Google Scholar]
- Perdew, J.P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett.
**1996**, 77, 3865–3868. [Google Scholar] [CrossRef] [Green Version] - Armiento, R.; Mattsson, A.E. Functional designed to include surface effects in self-consistent density functional theory. Phys. Rev. B
**2005**, 72, 085108–1–085108–5. [Google Scholar] [CrossRef] [Green Version] - Perdew, J.P.; Zunger, A. Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B
**1981**, 23, 5048–5079. [Google Scholar] [CrossRef] [Green Version] - Haines, J.; Léger, J.M.; Gorelli, F.; Klug, D.D.; Tse, J.S.; Li, Z.Q. X-ray diffraction and theoretical studies of the high-pressure structures and phase transitions in magnesium fluoride. Phys. Rev. B
**2001**, 64, 134110–1–134110–10. [Google Scholar] [CrossRef] - Subhadra, K.; Hussain, K.A.; Hussain, W.; Sirdeshmukh, D.B. Thermal expansion of strontium fluoride. J. Mater. Sci. Lett.
**1985**, 4, 777–778. [Google Scholar] [CrossRef] - Gerlich, D. Elastic Constants of Strontium Fluoride between 4.2 and 300 °K. Phys. Rev.
**1964**, 136, A1366–A1368. [Google Scholar] [CrossRef] - Hedin, L. New Method for Calculating the One-Particle Green’s Function with Application to the Electron-Gas Problem. Phys. Rev.
**1965**, 139, A796–A823. [Google Scholar] [CrossRef] - Godby, R.W.; Schlüter, M.; Sham, L.J. Self-energy operators and exchange-correlation potentials in semiconductors. Phys. Rev. B
**1988**, 37, 10159–10175. [Google Scholar] [CrossRef] - Bechstedt, F.; Sole, R.D.; Cappellini, G.; Reining, L. An efficient method for calculating quasiparticle energies in semiconductors. Solid State Commun.
**1992**, 84, 765–770. [Google Scholar] [CrossRef] - Onida, G.; Reining, L.; Rubio, A. Electronic excitations: Density-functional versus many-body Green’s-function approaches. Rev. Mod. Phys.
**2002**, 74, 601–659. [Google Scholar] [CrossRef] [Green Version] - Sangalli, D.; Ferretti, A.; Miranda, H.; Attaccalite, C.; Marri, I.; Cannuccia, E.; Melo, P.; Marsili, M.; Paleari, F.; Marrazzo, A.; et al. Many-body perturbation theory calculations using the yambo code. J. Phys. Condens. Matter
**2019**, 31, 325902. [Google Scholar] [CrossRef] - Seidl, A.; Görling, A.; Vogl, P.; Majewski, J.A.; Levy, M. Generalized Kohn-Sham schemes and the band-gap problem. Phys. Rev. B
**1996**, 53, 3764–3774. [Google Scholar] [CrossRef] - Bechstedt, F.; Fuchs, F.; Kresse, G. Ab-initio theory of semiconductor band structures: New developments and progress. Phys. Status Solidi (b)
**2009**, 246, 1877–1892. [Google Scholar] [CrossRef] - Borlido, P.; Aull, T.; Huran, A.W.; Tran, F.; Marques, M.A.L.; Botti, S. Large-Scale Benchmark of Exchange–Correlation Functionals for the Determination of Electronic Band Gaps of Solids. J. Chem. Theory Comput.
**2019**, 15, 5069–5079. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Heyd, J.; Scuseria, G.E.; Ernzerhof, M. Hybrid functionals based on a screened Coulomb potential. J. Chem. Phys.
**2003**, 118, 8207–8215. [Google Scholar] [CrossRef] [Green Version] - Krukau, A.V.; Vydrov, O.A.; Izmaylov, A.F.; Scuseria, G.E. Influence of the exchange screening parameter on the performance of screened hybrid functionals. J. Chem. Phys.
**2006**, 125, 224106–1–224106–5. [Google Scholar] [CrossRef] [PubMed] - Stankovski, M.; Antonius, G.; Waroquiers, D.; Miglio, A.; Dixit, H.; Sankaran, K.; Giantomassi, M.; Gonze, X.; Côté, M.; Rignanese, G.M. G0W0 band gap of ZnO: Effects of plasmon-pole models. Phys. Rev. B
**2011**, 84, 241201–1–241201–5. [Google Scholar] [CrossRef] - Cappellini, G.; Sole, R.D.; Reining, L.; Bechstedt, F. Model dielectric function for semiconductors. Phys. Rev. B
**1993**, 47, 9892–9895. [Google Scholar] [CrossRef] - Schmidt, W.G.; Glutsch, S.; Hahn, P.H.; Bechstedt, F. Efficient (N
^{2}) method to solve the Bethe-Salpeter equation. Phys. Rev. B**2003**, 67, 085307–1–085307–7. [Google Scholar] [CrossRef] - Caruso, F.; Rinke, P.; Ren, X.; Scheffler, M.; Rubio, A. Unified description of ground and excited states of finite systems: The self-consistentGWapproach. Phys. Rev. B
**2012**, 86, 081102–1–081102–5. [Google Scholar] [CrossRef] [Green Version] - Vassilyeva, A.; Eglitis, R.; Kotomin, E.; Dauletbekova, A. Ab initio calculations of MgF
_{2}(001) and (011) surface structure. Phys. B Cond. Matt.**2010**, 405, 2125–2127. [Google Scholar] [CrossRef] - Jia, R.; Shi, H.; Borstel, G. Ab initio calculations for SrF
_{2}with F- and M-centers. Comp. Mat. Sci.**2008**, 43, 980–988. [Google Scholar] [CrossRef] - Yue, L.; Jia, R.; Shi, H.; He, X.; Eglitis, R.I. First-Principles Calculations for the H Center in SrF
_{2}Crystals. J. Phys. Chem. A**2010**, 114, 8444–8449. [Google Scholar] [CrossRef] - Jibran, M.; Murtaza, G.; Khan, M.; Khenata, R.; Muhmmad, S.; Ali, R. First principle study of MF2 ( M=Mg, Ca, Sr, Ba, Ra) compounds. Comp. Mater. Sci.
**2014**, 81, 575–581. [Google Scholar] [CrossRef] - Kolobanov, V.N.; Mikhailin, V.V.; Chernov, S.P.; Spassky, D.A.; Makhov, V.N.; Kirm, M.; Feldbach, E.; Vielhauer, S. Luminescence of singlet self-trapped excitons in MgF
_{2}. J. Phys. Condens. Matter**2009**, 21, 375501. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lisitsyn, V.; Lisitsyna, L.; Popov, A.; Kotomin, E.; Abuova, F.; Akilbekov, A.; Maier, J. Stabilization of primary mobile radiation defects in MgF
_{2}crystals. Nucl. Instruments Methods Phys. Res. Sect. B Beam Interact. Mater. Atoms**2016**, 374, 24–28. [Google Scholar] [CrossRef] - Shishkin, M.; Kresse, G. Self-consistent GW calculations for semiconductors and insulators. Phys. Rev. B
**2007**, 75. [Google Scholar] [CrossRef] - Golze, D.; Keller, L.; Rinke, P. Accurate Absolute and Relative Core-Level Binding Energies from GW. J. Phys. Chem. Lett.
**2020**, 11, 1840–1847. [Google Scholar] [CrossRef] [Green Version] - Li, J.; Jin, Y.; Rinke, P.; Yang, W.; Golze, D. Benchmark of GW Methods for Core-Level Binding Energies. J. Chem. Theory Comput.
**2022**, 18, 7570–7585. [Google Scholar] [CrossRef] - Li, J.; Golze, D.; Yang, W. Combining Renormalized Singles GW Methods with the Bethe–Salpeter Equation for Accurate Neutral Excitation Energies. J. Chem. Theory Comput.
**2022**, 18, 6637–6645. [Google Scholar] [CrossRef] - Eglitis, R.I.; Purans, J.; Jia, R. Comparative Hybrid Hartree-Fock-DFT Calculations of WO
_{2}-Terminated Cubic WO_{3}as Well as SrTiO_{3}, BaTiO_{3}, PbTiO_{3}and CaTiO_{3}(001) Surfaces. Crystals**2021**, 11, 455. [Google Scholar] [CrossRef] - Botti, S.; Marques, M.A.L. Strong Renormalization of the Electronic Band Gap due to Lattice Polarization in the GW Formalism. Phys. Rev. Lett.
**2013**, 110. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lambrecht, W.R.L.; Bhandari, C.; van Schilfgaarde, M. Lattice polarization effects on the screened Coulomb interaction W of the GW approximation. Phys. Rev. Mater.
**2017**, 1. [Google Scholar] [CrossRef] - Cao, H.; Yu, Z.; Lu, P.; Wang, L.W. Fully converged plane-wave-based self-consistent GW calculations of periodic solids. Phys. Rev. B
**2017**, 95. [Google Scholar] [CrossRef] [Green Version] - Albert, J.P.; Jouanin, C.; Gout, C. Electronic energy bands in the fluorite structure: CaF
_{2}and CdF_{2}. Phys. Rev. B**1977**, 16, 4619–4629. [Google Scholar] [CrossRef] - Fox, M. Optical Properties of Solids; Oxford University Press: Oxford, UK, 2001; Volume 384, p. 416. [Google Scholar]
- Wannier, G.H. The Structure of Electronic Excitation Levels in Insulating Crystals. Phys. Rev.
**1937**, 52, 191–197. [Google Scholar] [CrossRef] - Satta, G.; Cappellini, G.; Olevano, V.; Reining, L. Many-body effects in the electronic spectra of cubic boron nitride. Phys. Rev. B
**2004**, 70, 195212–1–195212–13. [Google Scholar] [CrossRef] [Green Version] - Kammerlander, D.; Botti, S.; Marques, M.A.L.; Marini, A.; Attaccalite, C. Speeding up the solution of the Bethe-Salpeter equation by a double-grid method and Wannier Wannier interpolation. Phys. Rev. B
**2012**, 86, 125203–1–125203–5. [Google Scholar] [CrossRef] [Green Version] - Mocci, P.; Malloci, G.; Bosin, A.; Cappellini, G. Time-Dependent Density Functional Theory Investigation on the Electronic and Optical Properties of Poly-C,Si,Ge-acenes. ACS Omega
**2020**, 5, 16654–16663. [Google Scholar] [CrossRef]

**Figure 1.**QP energy bands for r-MgF${}_{2}$ in the GW${}_{0}$ scheme (

**top**) and within the G${}_{0}$W${}_{0}$ method (

**bottom**). See text for details.

**Figure 2.**QP energy bands for c-SrF${}_{2}$ in the GW${}_{0}$ scheme (

**top**) and the scQP-GW (

**bottom**) scheme. See text for details.

**Figure 3.**Dielectric function components for r-MgF${}_{2}$ (

**top**) and c-SrF${}_{2}$ (

**bottom**) in the BSE scheme on top of GW${}_{0}$ bands (see text). Red lines refer to the real part and black lines to the imaginary parts of the dielectric functions. In the case of c-MgF${}_{2}$, full lines refer to the $zz$ component and broken lines to the $xx$ and $yy$ components.

**Figure 4.**Dielectric functions of r-MgF${}_{2}$ (

**top**) resulting in the BSE scheme (for the two components of light polarization) and of c-SrF${}_{2}$ (

**bottom**) on top of G${}_{0}$W${}_{0}$ and scQP-GW respectively self-energy calculations (see text). Red lines refer to the real part and black lines to the imaginary parts of the dielectric functions. In the case of r-MgF${}_{2}$, full lines refer to the $zz$ component and broken lines to the $xx$ and $yy$ components.

**Figure 5.**Dielectric functions components (

**top**: imaginary part,

**bottom**: real part) of r-MgF${}_{2}$ in the BSE scheme on top of GW${}_{0}$ energies in comparison with the experiment reported in Reference [37] (see text). Red lines refer to our calculated spectra and black lines to the experimental data. Isotropic averages only are displayed.

**Figure 6.**Dielectric function of c-SrF${}_{2}$ calculated in the BSE scheme on top of GW${}_{0}$ energies (

**top**: imaginary part,

**bottom**: real part) in comparison with the experiment from Reference [29]. Red lines refer to our calculated spectra and black lines to the experimental data.

**Table 1.**Ground-state properties of the difluoride crystals r-MgF${}_{2}$ and c-SrF${}_{2}$. The lattice parameter a of both structures and parameter c for the rutile are reported together with the bulk modulus and its pressure derivative.

PBEsol | r-MgF${}_{2}$ | c-SrF${}_{2}$ |
---|---|---|

a [Å] | 4.6313 | 5.7744 |

c [Å] | 3.0558 | — |

$c/a$ | 0.6598 | — |

B${}_{0}$ [MPa] | 97.1 | 72.8 |

dB${}_{0}$/dp | 4.69 | 4.71 |

**Table 2.**Cutoff parameters and ground-state energy of the difluoride crystals under study. The total energy for the simulation cell ${E}_{0}$, the cutoff energy ${E}_{cut}$ for non-norm-conserving (partial) wave functions, and the cutoff energy ${E}_{aug}$ for the plane-wave expanded “intermediate” PAW augmentation charges are given. On the last line, the k-point mesh used for the BZ integration is listed.

PBEsol | r-MgF${}_{2}$ | c-SrF${}_{2}$ |
---|---|---|

E${}_{cut}$ [eV] | 1020 | 640 |

E${}_{aug}$ [eV] | 1700 | 1640 |

E${}_{0}$ [eV] | −30.1122 | −16.3389 |

k-point set | 12 × 12 × 18 | 12 × 12 × 12 |

**Table 3.**Ground-state properties of r-MgF${}_{2}$ and c-SrF${}_{2}$. For r-MgF${}_{2}$ in the upper part of the table, the lattice parameters a, c, their ratio $c/a$, and the x parameter are reported as functions of the XC potential used for the calculations in the first four rows. In the following three rows, the bulk modulus, its pressure derivative, and the total energy of the unit cell are given. In the last column, the experimental values from Reference [51] are listed. In the lower part of the table, we report the ground-state properties of c-SrF${}_{2}$. The lattice parameter a was reported as in the first row, i.e., as a function of the different XC potentials used. In the following three rows, we provide the bulk modulus, its pressure derivative, and the total energy of the unit cell. In the last column, the experimental data after References [52,53] are given.

${\mathbf{r}-\mathbf{MgF}}_{2}$ | $\mathbf{PBEsol}$ | $\mathbf{PBE}$ | $\mathbf{AM}05$ | $\mathbf{LDA}$ | $\mathbf{EXP}$ |
---|---|---|---|---|---|

a[Å] | 4.6313 | 4.6928 | 4.6649 | 4.5638 | 4.6249 |

c[Å] | 3.0558 | 3.0875 | 3.0741 | 3.0194 | 3.0520 |

$c/a$ | 0.6598 | 0.6579 | 0.6590 | 0.6616 | 0.6599 |

x | 0.3033 | 0.3035 | 0.3037 | 0.3030 | 0.3027 |

B${}_{0}$[GPa] | 97.1 | 90.1 | 91.6 | 111.2 | 101 ± 3 |

dB${}_{0}$/dP | 4.69 | 4.74 | 4.73 | 4.64 | 4.2 ± 1.1 |

E${}_{0}$[eV] | −30.1122 | −28.7552 | −29.7466 | −33.0805 | |

${\mathbf{c}-\mathbf{SrF}}_{\mathbf{2}}$ | $\mathbf{PBEsol}$ | $\mathbf{PBE}$ | $\mathbf{AM}\mathbf{05}$ | $\mathbf{LDA}$ | $\mathbf{EXP}$ |

a[Å] | 5.7744 | 5.8712 | 5.8094 | 5.6813 | 5.7994 |

B${}_{0}$[GPa] | 72.8 | 64.5 | 67.5 | 84.9 | 67.1 − 74.6 |

dB${}_{0}$/dp | 4.71 | 4.73 | 4.74 | 4.61 | 4.2 ± 1.1 |

E${}_{0}$[eV] | −16.3389 | −15.6630 | −16.0187 | −17.8951 |

**Table 4.**Quasi-particle energies for the fundamental energy gaps of r-MgF${}_{2}$ and c-SrF${}_{2}$ calculated with different approximations are reported and compared with available experimental results (for r-MgF${}_{2}$ from Reference [37] and for c-SrF${}_{2}$ from Reference [1]). B3PW refers to hybrid exchange–correlation potential calculations: the value for r-MgF${}_{2}$ from Reference [68], the values for c-SrF${}_{2}$ after References [69,70]. The row “Other” refers to the theoretical data for r-MgF${}_{2}$ from Reference [37] and for c-SrF${}_{2}$ from References [32,71] in parentheses.

${\mathbf{r}-\mathbf{MgF}}_{2}$ | Direct Gap | ${\mathbf{\Delta}}_{\mathit{cf}}$ |
---|---|---|

[eV] | [eV] | |

PBEsol | 6.921 | −0.320 |

HSE06 | 9.433 | −0.289 |

G${}_{0}$W${}_{0}$ | 12.800 | −0.291 |

GW${}_{0}$ | 13.243 | −0.285 |

scQP-GW | 13.945 | −0.277 |

B3PW | 9.48 | - |

Other | 12.17 | - |

Exp. | 12.4 | −0.2 |

${\mathbf{c}-\mathbf{SrF}}_{\mathbf{2}}$ | Direct Gap | Indirect Gap |

[eV] | [eV] | |

PBEsol | 6.932 | 6.827 |

HSE06 | 9.172 | 9.072 |

G${}_{0}$W${}_{0}$ | 11.437 | 11.316 |

GW${}_{0}$ | 11.820 | 11.700 |

scQP-GW | 12.490 | 12.375 |

B3PW | 11.306/10.96 | - |

Other | 11.24 | 11.20(7.55) |

Exp. | 11.25 | — |

**Table 5.**Relevant optical absorption observables and dielectric constants of the difluorides r-MgF${}_{2}$ and c-SrF${}_{2}$. The first peak energy position at the onset, the binding energy of the exciton and the ${\epsilon}_{\infty}$ values are reported as obtained in different BSE and GW schemes as discussed in the text and compared with corresponding experimental values from Reference [37] and Reference [29] (fourth column). In the case of r-MgF${}_{2}$, both the values for the two principal directions of the crystal were reported.

${\mathbf{r}-\mathbf{MgF}}_{2}$ | BSE (G${}_{0}$W${}_{0}$) | BSE (GW${}_{0}$) | BSE (scQP-GW) | $\mathbf{EXP}$ |
---|---|---|---|---|

E${}_{Peak,\left|\right|}$[eV] | 11.37 | 11.81 | 12.23 | 11.6 |

E${}_{Peak,\perp}$[eV] | 11.76 | 12.21 | 12.62 | 12.1 |

E${}_{Bind,\left|\right|}$[eV] | 1.14 | 1.14 | 1.43 | 0.8 |

E${}_{Bind,\perp}$[eV] | 1.04 | 1.03 | 1.32 | 0.5 |

${\epsilon}_{\infty ,\left|\right|}$ | 1.88 | 1.85 | 1.84 | 1.67 |

${\epsilon}_{\infty ,\perp}$ | 1.91 | 1.890 | 1.87 | 1.68 |

${\mathbf{c}-\mathbf{SrF}}_{\mathbf{2}}$ | BSE (G${}_{\mathbf{0}}$W${}_{\mathbf{0}}$) | BSE (GW${}_{\mathbf{0}}$) | BSE (scQP-GW) | $\mathbf{EXP}$ |

E${}_{Peak}$[eV] | 10.01 | 10.40 | 10.83 | 10.6 |

E${}_{Bind}$[eV] | 1.43 | 1.42 | 1.66 | 0.65 |

${\epsilon}_{\infty}$ | 2.18 | 2.15 | 2.13 | 2.08 |

**Table 6.**Excited and optical properties of the clusters (MgF${}_{2}$)${}_{n}$ and (SrF${}_{2}$)${}_{n}$, $n=1,2,3$ and the crystalline solids (r-MgF${}_{2}$ and c-SrF${}_{2}$). The quasiparticle gap ${\mathrm{E}}_{gap}$, the optical onset ${\mathrm{E}}_{opt}$, and the binding energy of the exciton ${\mathrm{E}}_{\mathrm{b}}$ are displayed. In the first row, we report the data on the (MgF${}_{2}$)${}_{n}$ clusters, in the second, we present the data on the solid r-MgF${}_{2}$ from Reference [37], in the third, we present the outcomes of the present work for solid r-MgF${}_{2}$ with experimental data in parentheses. In the last two rows, the data for (SrF${}_{2}$)${}_{n}$ clusters and solid c-SrF${}_{2}$ are given. The outcomes for the clusters are from Reference [38].

${\mathbf{E}}_{\mathbf{gap}}$ | ${\mathbf{E}}_{\mathbf{Peak}}$ | ${\mathbf{E}}_{\mathbf{Bind}}$ | |
---|---|---|---|

[eV] | [eV] | [eV] | |

Clusters (MgF${}_{2}$)${}_{n}$ | 11.45–12.49 | 6.56–6.78 | 4.49–5.71 |

Solid r-MgF${}_{2}$ (Present) | 13.24 (12.4) | 11.8 (11.6) | 1.4 (0.8) |

Solid r-MgF${}_{2}$ (Other) | 12.17 | 10.90 | 1.13 |

Clusters (SrF${}_{2}$)${}_{n}$ | 9.33–10.16 | 5.10–5.26 | 4.23–4.9 |

Solid c-SrF${}_{2}$ (Present) | 11.82 (11.25) | 10.4 (10.6) | 1.4(0.65) |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cappellini, G.; Furthmüller, J.; Bechstedt, F.; Botti, S.
Electronic and Optical Properties of Alkaline Earth Metal Fluoride Crystals with the Inclusion of Many-Body Effects: A Comparative Study on Rutile MgF_{2} and Cubic SrF_{2}. *Symmetry* **2023**, *15*, 539.
https://doi.org/10.3390/sym15020539

**AMA Style**

Cappellini G, Furthmüller J, Bechstedt F, Botti S.
Electronic and Optical Properties of Alkaline Earth Metal Fluoride Crystals with the Inclusion of Many-Body Effects: A Comparative Study on Rutile MgF_{2} and Cubic SrF_{2}. *Symmetry*. 2023; 15(2):539.
https://doi.org/10.3390/sym15020539

**Chicago/Turabian Style**

Cappellini, Giancarlo, Jürgen Furthmüller, Friedhelm Bechstedt, and Silvana Botti.
2023. "Electronic and Optical Properties of Alkaline Earth Metal Fluoride Crystals with the Inclusion of Many-Body Effects: A Comparative Study on Rutile MgF_{2} and Cubic SrF_{2}" *Symmetry* 15, no. 2: 539.
https://doi.org/10.3390/sym15020539