# Density Functional Treatment of Photoionization of Sodium Clusters: Effects of Cluster Size and Exchange–Correlation Framework

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Methodology

#### 2.1. DFT Exchange–Correlation Functionals

#### 2.2. LR-TDDFT Dynamical Response

## 3. Results and Discussion

#### 3.1. Ground State Structure

#### 3.2. Total Photoionization Cross-Section

#### 3.3. Comparison with Experiments

#### 3.4. Self-Consistent Induced Potential

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Reinhard, P.G.; Suraud, E.; Dinh, P.M. An Introduction to Cluster Science; John Wiley & Sons: Hoboken, NJ, USA, 2013. [Google Scholar]
- Jena, P.; Sun, Q. Super atomic clusters: Design rules and potential for building blocks of materials. Chem. Rev.
**2018**, 118, 5755–5870. [Google Scholar] - Castleman, A., Jr.; Khanna, S. Clusters, superatoms, and building blocks of new materials. J. Phys. Chem. C
**2009**, 113, 2664–2675. [Google Scholar] [CrossRef] - Kawabata, A.; Kubo, R. Electronic properties of fine metallic particles. II. Plasma resonance absorption. J. Phys. Soc. Jpn.
**1966**, 21, 1765–1772. [Google Scholar] [CrossRef] - Kreibig, U.; Vollmer, M. Optical Properties of Metal Clusters; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013; Volume 25. [Google Scholar]
- Xia, C.; Yin, C.; Kresin, V.V. Photoabsorption by volume plasmons in metal nanoclusters. Phys. Rev. Lett.
**2009**, 102, 156802. [Google Scholar] [CrossRef] [PubMed] - Madjet, M.E.A.; Chakraborty, H. Collective resonances in the photoresponse of metallic nanoclusters. J. Phys. Conf. Ser.
**2009**, 194, 022103. [Google Scholar] [CrossRef] - Ekardt, W. Size-dependent photoabsorption and photoemission of small metal particles. Phys. Rev. B
**1985**, 31, 6360. [Google Scholar] [CrossRef] - Sun, W.G.; Wang, J.J.; Lu, C.; Xia, X.X.; Kuang, X.Y.; Hermann, A. Evolution of the structural and electronic properties of medium-sized sodium clusters: A honeycomb-like Na20 cluster. Inorg. Chem.
**2017**, 56, 1241–1248. [Google Scholar] [CrossRef] [PubMed] - Miroshnichenko, A.E.; Flach, S.; Kivshar, Y.S. Fano resonances in nanoscale structures. Rev. Mod. Phys.
**2010**, 82, 2257. [Google Scholar] [CrossRef] - Luk’Yanchuk, B.; Zheludev, N.I.; Maier, S.A.; Halas, N.J.; Nordlander, P.; Giessen, H.; Chong, C.T. The Fano resonance in plasmonic nanostructures and metamaterials. Nat. Mater.
**2010**, 9, 707–715. [Google Scholar] [CrossRef] [PubMed] - Cederbaum, L.; Zobeley, J.; Tarantelli, F. Giant intermolecular decay and fragmentation of clusters. Phys. Rev. Lett.
**1997**, 79, 4778. [Google Scholar] [CrossRef] - Marburger, S.; Kugeler, O.; Hergenhahn, U.; Möller, T. Experimental evidence for interatomic Coulombic decay in Ne clusters. Phys. Rev. Lett.
**2003**, 90, 203401. [Google Scholar] [CrossRef] - De, R.; Magrakvelidze, M.; Madjet, M.E.; Manson, S.T.; Chakraborty, H.S. First prediction of inter-Coulombic decay of C
_{60}inner vacancies through the continuum of confined atoms. J. Phys. B At. Mol. Opt. Phys.**2016**, 49, 11LT01. [Google Scholar] [CrossRef] - Shaik, R.; Varma, H.R.; Madjet, M.E.A.; Zheng, F.; Frauenheim, T.; Chakraborty, H.S. Plasmonic Resonant Intercluster Coulombic Decay. Phys. Rev. Lett.
**2023**, 130, 233201. [Google Scholar] [CrossRef] [PubMed] - Jänkälä, K.; Tchaplyguine, M.; Mikkelä, M.H.; Björneholm, O.; Huttula, M. Photon energy dependent valence band response of metallic nanoparticles. Phys. Rev. Lett.
**2011**, 107, 183401. [Google Scholar] [CrossRef] - Frank, O.; Rost, J.M. From collectivity to the single-particle picture in the photoionization of clusters. Phys. Rev. A
**1999**, 60, 392. [Google Scholar] [CrossRef] - Frank, O.; Rost, J.M. Diffraction effects in the photoionization of clusters. Chem. Phys. Lett.
**1997**, 271, 367–371. [Google Scholar] [CrossRef] - Mirin, N.A.; Bao, K.; Nordlander, P. Fano resonances in plasmonic nanoparticle aggregates. J. Phys. Chem. A
**2009**, 113, 4028–4034. [Google Scholar] [CrossRef] - Cole, J.R.; Halas, N. Optimized plasmonic nanoparticle distributions for solar spectrum harvesting. Appl. Phys. Lett.
**2006**, 89, 153120. [Google Scholar] [CrossRef] - Dragan, A.I.; Geddes, C.D. Metal-enhanced fluorescence: The role of quantum yield, Q, in enhanced fluorescence. Appl. Phys. Lett.
**2012**, 100, 093115. [Google Scholar] [CrossRef] - Liao, H.; Nehl, C.L.; Hafner, J.H. Biomedical Applications of Plasmon Resonant Metal Nanoparticles; Nanomedicine: London, UK, 2006; Volume 1, pp. 201–208. [Google Scholar]
- Huang, X.; Jain, P.K.; El-Sayed, I.H.; El-Sayed, M.A. Plasmonic photothermal therapy (PPTT) using gold nanoparticles. Lasers Med. Sci.
**2008**, 23, 217–228. [Google Scholar] [CrossRef] - De Heer, W.A. The physics of simple metal clusters: Experimental aspects and simple models. Rev. Mod. Phys.
**1993**, 65, 611. [Google Scholar] [CrossRef] - Brack, M. The physics of simple metal clusters: Self-consistent jellium model and semiclassical approaches. Rev. Mod. Phys.
**1993**, 65, 677. [Google Scholar] [CrossRef] - Kohn, W.; Sham, L.J. Self-consistent equations including exchange and correlation effects. Phys. Rev.
**1965**, 140, A1133. [Google Scholar] [CrossRef] - Perdew, J.P.; Wang, Y. Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B
**1992**, 45, 13244. [Google Scholar] [CrossRef] [PubMed] - Klüpfel, P.; Dinh, P.M.; Reinhard, P.G.; Suraud, E. Koopmans’ condition in self-interaction-corrected density-functional theory. Phys. Rev. A
**2013**, 88, 052501. [Google Scholar] [CrossRef] - Saito, S.; Bertsch, G.F.; Tománek, D. Collective electronic excitations in small metal clusters. Phys. Rev. B
**1991**, 43, 6804. [Google Scholar] [CrossRef] - Van Leeuwen, R.; Baerends, E. Exchange-correlation potential with correct asymptotic behavior. Phys. Rev. A
**1994**, 49, 2421. [Google Scholar] [CrossRef] - Choi, J.; Chang, E.; Anstine, D.M.; Madjet, M.E.A.; Chakraborty, H.S. Effects of exchange-correlation potentials on the density-functional description of C
_{60}versus C_{240}photoionization. Phys. Rev. A**2017**, 95, 023404. [Google Scholar] [CrossRef] - Shaik, R.; Varma, H.R.; Chakraborty, H.S. Collective effects in photoionization of sodium clusters: Plasmon resonance spill, induced attractive force and correlation minimum. J. Phys. B At. Mol. Opt. Phys.
**2021**, 54, 125101. [Google Scholar] [CrossRef] - Madjet, M.E.; Chakraborty, H.S.; Rost, J.M.; Manson, S.T. Photoionization of C
_{60}: A model study. J. Phys. B At. Mol. Opt. Phys.**2008**, 41, 105101. [Google Scholar] [CrossRef] - Chandezon, F.; Bjørnholm, S.; Borggreen, J.; Hansen, K. Electronic shell energies and deformations in large sodium clusters from evaporation spectra. Phys. Rev. B
**1997**, 55, 5485. [Google Scholar] [CrossRef] - Gunnarsson, O.; Lundqvist, B.I. Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism. Phys. Rev. B
**1976**, 13, 4274. [Google Scholar] [CrossRef] - Oliver, G.; Perdew, J. Spin-density gradient expansion for the kinetic energy. Phys. Rev. A
**1979**, 20, 397. [Google Scholar] [CrossRef] - Marques, M.A.; Castro, A.; Rubio, A. Assessment of exchange-correlation functionals for the calculation of dynamical properties of small clusters in time-dependent density functional theory. J. Chem. Phys.
**2001**, 115, 3006–3014. [Google Scholar] [CrossRef] - Petersilka, M.; Gossmann, U.; Gross, E. Excitation energies from time-dependent density-functional theory. Phys. Rev. Lett.
**1996**, 76, 1212. [Google Scholar] [CrossRef] - Bertsch, G. An RPA program for jellium spheres. Comput. Phys. Commun.
**1990**, 60, 247–255. [Google Scholar] [CrossRef] - Parr, R.G.; Weitao, Y. Density Functional Theory of Atoms and Molecules; International series of monographs on Chemistry. 16; Oxford Science Publications: Oxford, UK, 1989; pp. 271–272. [Google Scholar]
- Madjet, M.; Chakraborty, H.S.; Rost, J.M. Spurious oscillations from local self-interaction correction in high-energy photoionization calculations for metal clusters. J. Phys. B At. Mol. Opt. Phys.
**2001**, 34, L345. [Google Scholar] [CrossRef] - Bachau, H.; Frank, O.; Rost, J.M. Photoionization of alkali metal clusters. Z. Phys. D At. Mol. Clust.
**1996**, 38, 59–64. [Google Scholar] - Fano, U. Effects of configuration interaction on intensities and phase shifts. Phys. Rev.
**1961**, 124, 1866. [Google Scholar] [CrossRef] - Bertsch, G.; Tománek, D. Thermal line broadening in small metal clusters. Phys. Rev. B
**1989**, 40, 2749. [Google Scholar] [CrossRef] - Koskinen, M.; Manninen, M. Photoionization of metal clusters. Phys. Rev. B
**1996**, 54, 14796. [Google Scholar] [CrossRef] [PubMed] - Bartels, C.; Hock, C.; Huwer, J.; Kuhnen, R.; Schwobel, J.; Von Issendorff, B. Probing the angular momentum character of the valence orbitals of free sodium nanoclusters. Science
**2009**, 323, 1323–1327. [Google Scholar] [CrossRef] - Wrigge, G.; Hoffmann, M.A.; Issendorff, B.V. Photoelectron spectroscopy of sodium clusters: Direct observation of the electronic shell structure. Phys. Rev. A
**2002**, 65, 063201. [Google Scholar] [CrossRef] - Solov’yov, A.V.; Polozkov, R.G.; Ivanov, V.K. Angle-resolved photoelectron spectra of metal cluster anions within a many-body-theory approach. Phys. Rev. A
**2010**, 81, 021202. [Google Scholar] [CrossRef] - Polozkov, R.; Ivanov, V.; Verkhovtsev, A.; Korol, A.; Solov’yov, A. New applications of the jellium model for the study of atomic clusters. J. Phys. Conf. Ser.
**2013**, 438, 012009. [Google Scholar] [CrossRef] - Zangwill, A.; Soven, P. Local field effects in photoabsorption. J. Vac. Sci. Technol.
**1980**, 17, 159–163. [Google Scholar] [CrossRef]

**Figure 1.**Ground state radial potential and wavefunctions (inset) for HOMO and HOMO−1 levels calculated for Na${}_{20}$ (

**a**), Na${}_{40}$ (

**b**), and Na${}_{92}$ (

**c**) using DFT-SIC and DFT-LB94. The horizontal lines indicate occupied levels (red for DFT-LB94 and blue for DFT-SIC), the nodeless orbitals are represented with thick lines and orbitals with nodes are with dotted lines).

**Figure 2.**LR-TDDFT and LR-DFT photoionization cross-sections of Na${}_{20}$ (

**a**), Na${}_{40}$ (

**b**), and Na${}_{92}$ (

**c**) using LB94 and SIC.

**Figure 3.**LR-TDDFT cross-sections for Na${}_{20}$ (

**a**) and Na${}_{92}$ (

**b**) compared with the available experimental data from [6]. The cross-sections from experiments are fitted using a Lorentzian.

**Figure 4.**The real (

**a**) and imaginary (

**b**) components of the radial self-consistent field potential, V${}_{ind}$, using LB94 and SIC within the LR-TDDFT frame are shown for Na${}_{20}$, Na${}_{40}$, and Na${}_{92}$. For visual aid, some smoothing techniques have been applied.

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**

Shaik, R.; Varma, H.R.; Chakraborty, H.S.
Density Functional Treatment of Photoionization of Sodium Clusters: Effects of Cluster Size and Exchange–Correlation Framework. *Atoms* **2023**, *11*, 114.
https://doi.org/10.3390/atoms11080114

**AMA Style**

Shaik R, Varma HR, Chakraborty HS.
Density Functional Treatment of Photoionization of Sodium Clusters: Effects of Cluster Size and Exchange–Correlation Framework. *Atoms*. 2023; 11(8):114.
https://doi.org/10.3390/atoms11080114

**Chicago/Turabian Style**

Shaik, Rasheed, Hari R. Varma, and Himadri S. Chakraborty.
2023. "Density Functional Treatment of Photoionization of Sodium Clusters: Effects of Cluster Size and Exchange–Correlation Framework" *Atoms* 11, no. 8: 114.
https://doi.org/10.3390/atoms11080114