Reactivity of Hydrated Electron in Finite Size System: Sodium Pickup

Jul 6, 2015 - J. Heyrovský Institute of Physical Chemistry v.v.i., The Czech Academy of Sciences, Dolejškova 3, 18223 Prague 8, Czech Republic...
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Reactivity of Hydrated Electron in Finite Size System: Sodium Pickup on Mixed N2O−Water Nanoparticles Daniela Šmídová,†,‡ Jozef Lengyel,† Andriy Pysanenko,† Jakub Med,‡ Petr Slavíček,*,‡ and Michal Fárník*,† †

J. Heyrovský Institute of Physical Chemistry v.v.i., The Czech Academy of Sciences, Dolejškova 3, 18223 Prague 8, Czech Republic Department of Physical Chemistry, University of Chemistry and Technology, Technická 5, Prague 6, Czech Republic



S Supporting Information *

ABSTRACT: We investigate the reactivity of hydrated electron generated by alkali metal deposition on small water particles with nitrous oxide dopant by means of mass spectrometry and ab initio molecular dynamics simulations. The mixed nitrous oxide/water clusters were generated in a molecular beam and doped with Na atoms in a pickup experiment, and investigated by mass spectrometry using two different ionization schemes: an electron ionization (EI), and UV photoionization after the Na doping (NaPI). The NaPI is a softionization nondestructive method, especially for water clusters provided that a hydrated electron e−s is formed in the cluster. The missing signal for the doped clusters indicates that the hydrated electron is not present in the N2O containing clusters. The simulations reveal that the hydrated electron is formed, but it immediately reacts with N2O, forming first N2O− radical anion, later O−, and finally an OH• and OH− pair.

H

close to the ionization threshold of 3.2 eV, and thus the mass spectra reflect the original neutral cluster size distributions. This was first shown for water clusters by Buck et al.9 and later on suggested by Signorell et al. as a “sizer for weakly bound ultrafine aerosol particles”,16 and demonstrated for different clusters.11,17,18 Several articles and recent reviews from Buck’s and Zeuch’s group describe the method in detail and also show its applications in vibrational spectroscopy.6,15,19,20 The sodium pickup method, therefore, represents a very powerful technique for a characterization of aerosols. This is a very important aspect, since water droplets, ice, and aerosol particles in the nanometer size range are key players in atmospheric processes,21,22 yet these small particles are difficult to investigate. In this respect, we should focus on mixed aerosols, particularly those containing nitrogen compounds. We have recently demonstrated, however, that the hydrated electron can strongly interact with molecules other than water in the mixed clusters, in particular HNO3.23 In this work, we combine photoionization (NaPI) and electron ionization (EI) techniques together with ab initio molecular dynamics simulations to reveal the nature of the interaction between hydrated electron and mixed N2O/H2O clusters. The goal of the study is twofold: first, we reveal details of the hydrated electron quenching reactions; second, we point to the limitations of sodium pickup technique for the investigation of small aerosols.

ydrated electron is an important short-lived reactive species that is formed during water photolysis and radiolysis.1 Its structure and reactivity has attracted attention ever since its identification in 1960s,2 focusing both on the electron solvated in liquid water and finite-size analogues of the hydrated electron.3 Solvated electrons are on one hand the simplest chemical species, even simpler than hydrogen atoms. On the other hand, they owe their very existence to complex interactions with solvent molecules so that their modeling becomes cumbersome; a general consensus on their structure has not yet been reached.4,5 Experimentally, the solvated electron was first prepared by Humphry Davy in 1807−1809 by dissolving alkali metals in liquid ammonia. The same approach cannot be used in liquid water, as alkali metals and water react immediately. Hydrated electron is, however, formed on the surface of finite size water particles, providing that only a single alkali atom is deposited on the cluster.6−8 With such an arrangement, we can observe the reactivity of hydrated electron in water clusters doped with the N2O quencher. Hydrated electron formed via the alkali atom deposition on molecular clusters is recently increasingly used for characterization of aerosols and crystalline particles in nanometer size range.9−11 For water clusters, the sodium pickup process results in generating Na+ and a hydrated electron e−s . Note that in finite size solid clusters the hydrated electron typically resides near the cluster surface.12−14 The hydrated electrons are weakly bound to the cluster (the binding energy converges to 3.2 eV for large clusters15) and can be detached by UV photons of relatively low energies. The remaining (H2O)N·Na+ cluster ions can be then detected by a mass spectrometer. The method (NaPI) is essentially fragmentation-free at low photon energies © 2015 American Chemical Society

Received: June 15, 2015 Accepted: July 6, 2015 Published: July 6, 2015 2865

DOI: 10.1021/acs.jpclett.5b01269 J. Phys. Chem. Lett. 2015, 6, 2865−2869

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The Journal of Physical Chemistry Letters

expansions decreased. The fact that the signal intensity decreased in Figure 1 IIa−c demonstrates that the NaPI method is sensitive only to the pure (H2O)N clusters. On the other hand, the universal EI method detects (but also fragments) all the clusters, and their abundances increased in Figure 1 IIIa−c. This explanation opens another question: Why is the NaPI method insensitive to the mixed N2O containing clusters? An obvious answer is a reaction of N2O with Na prior to the photoionization. The cluster detection with the low-energy photons in the NaPI method requires that the ion pair Na+ with solvated electron e−s is generated prior to the photoionization; then the photodetachment of the e−s weakly bound to the cluster leaves behind the molecular cluster doped by Na+. However, if a reaction with a solvated electron takes place in the cluster, the above-mentioned e−s detachment mechanism cannot occur. The experimental results thus strongly suggest that the hydrated electron is involved in the subsequent reaction and disappears from the system. To reveal the nature of this process, we have employed ab initio simulations. The pure (H2O)N mass spectra yielded the mean cluster size N̅ ≈ 17. At P0 = 7, bar almost all the generated clusters contained some N2O molecules, since we could not detect them by the NaPI method. Therefore, we performed our calculations for the Na·(H2O)15·N2O cluster as a representative system for our experiment. We observe in our simulations a cascade of reactions. The reaction starts with the formation of a hydrated electron paired with the sodium cation:

Figure 1 compares the NaPI and EI mass spectra. The top row shows the NaPI spectra of (H2O)N clusters generated in

Figure 1. Mass spectra measured at different expansion conditions (water reservoir TR = 328 K, nozzle temperature T0 = 343 K). Top row corresponds to NaPI spectra with pure He as the buffer gas at P0 = 5, 6, and 7 bar, (Ia−c). Middle and bottom rows correspond to the spectra obtained with 5% N2O/He mixture as the buffer gas (P0 as above): series II and III correspond to NaPI and EI spectra, respectively. Red bars label the mixed fragments containing some N2O molecules and their fragments.

expansions of pure He through the water reservoir using three different He pressures: 5, 6, and 7 bar, panels Ia, Ib, and Ic, respectively. The spectra are composed of (H2O)NNa+ mass peaks and reflect the neutral cluster size distributions;9,24 in all three cases the mean cluster size corresponds to N̅ ≈ 17. The middle row shows the mass spectra where the coexpansion was done with 5% mixture of N2O in He. At P0 = 5 bar, the (H2O)NNa+ mass peaks were present. The spectrum extends to even larger cluster sizes than in the case of pure He. This can be justified by N2O acting as a buffer gas at low concentrations, cooling the expansion more efficiently than He, and thus even larger (H2O)N clusters were generated than in pure helium expansions. However, the signal weakened significantly at 6 bar and disappeared at 7 bar. Does the presence of more N2O in the expansion at the higher stagnation pressure inhibit cluster generation? To answer this question, we recorded the mass spectra under the same conditions (including Na-doping) using the EI (bottom). This series clearly shows that clusters are generated in the mixed He/N2O/H2O expansions. The observed charged fragments in the mass spectra are smaller due to the large fragmentation after EI compared to NaPI.24 Nevertheless, these spectra proved that even larger clusters were generated in the mixed expansions at 7 bar than at 5 bar, yet these clusters were not detected by the NaPI method. The EI spectra also exhibit mixed N2O/H2O fragment ions, which are labeled (red) in Figure 1. The detailed analysis can be found in the Supporting Information (SI) where the mixed N2O/H2O mass peak series are assigned. However, important for the present conclusions is the qualitative observation that the spectra at 5 bar are dominated by pure protonated water (H2O)nH+ fragments (blue), while the mixed peaks (red) grow in intensity with the stagnation pressure, i.e., overall N2O concentration in the expansion. The spectra show unambiguously that the mixed (N2O)M·(H2O)N clusters are generated and their concentration and size increased with P0, while the relative concentration of pure (H2O)N clusters in the

(H 2O)N

Na ⎯⎯⎯⎯⎯⎯⎯→ Na + + es−

(1)

The solvated electron then almost immediately reacts with N2O dopant:

N2O + es− → N2O−

(2)

This step can easily be observed visually in the simulations because the N2O is linear molecule, while the N2O− anion has a broken structure. In the next step, the radical anion releases the nitrogen molecule, forming an O− radical anion: N2O− → N2 + O−

(3)

The nascent radical anion is rather reactive, and it further reacts with water in the cluster to generate OH− anion and OH• radical pair: O− + H 2O → OH•+OH−

(4)

Figure 2 shows the ionization energies (IEs) and Mulliken charges on several atoms along a selected trajectory, starting with unsolvated Na atom near the cluster surface. We observe both from the Mulliken charges and spin densities that at the beginning, the unpaired electron is connected to the sodium atom. The IE is close to the one of the sodium atom, yet in less than a picosecond, the IE falls down to about 3 eV. This evolution corresponds to a sodium solvation and the formation of an e−s ·Na+ pair.12 At this moment, the unpaired electron is still close to the sodium atom. In the next step, the charge transfer process (eq2) takes places. This reaction is reflected by a sharp increase of the IE to ≈8 eV. Already at this step, the NaPI experiment becomes blind to the clusters. The dynamical processes, however, continue. The formation of the oxygen radical anion (eq3) leads to a further increase in the IE (≈11 eV); in our sample trajectory this process takes 2866

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reaction steps. We observe that the whole reaction series is on average almost completed within the 5 ps duration of our simulations. The sodium solvation and formation of equilibrated hydrated electron is rather fast, while the subsequent reaction steps can take a longer time. It ought to be mentioned that the kinetics can be somewhat different, depending on the electronic structure method used (see also SI). The robust observation here is that even the first product formed is characterized by a significantly increased ionization energy. The observed reaction products are generally in accord with the known facts on reactivity of N2O in the condensed phase. The nitrous oxide is being used as a hydrated electron scavenger, with the rate constant close to its diffusion limit.25 The reaction of an electron with N2O in liquid water was recently studied theoretically using dielectric continuum model, and only a small barrier was found for this reaction.26,27 In finite size systems, the reaction seems to proceed essentially without any barrier. The mechanism of the subsequent reactions was a subject of some discussion. Originally, the N2O− was considered as a relatively stable intermediate. Later its fast dissociation was established. 28 Both (H 2 O) n O − and (H2O)nOH− products were observed from reactions of negatively charged water clusters with N2O.29,30 The oxygen radical anion should be relatively stable in the aqueous phase.31 Our simulations, on the other hand, suggest that the oxygen radical anion quickly accepts a proton from a neighboring water molecule (see SI for the possible role that a self-interaction error of the GGA BLYP functional may play in the enhanced reactivity of the oxygen radical anion). The reactivity of an oxygen radical anion in liquid water and in finite size clusters can be different for physical reasons. The oxygen radical anion has a very high solvation energy32 due to the long-range polarization of the solvent. In small water particles, however, the oxygen radical anion is much less stabilized, which leads to decreasing of the reaction barrier for the reaction of the radical anion with water. We have combined different mass spectrometry techniques with ab initio dynamical simulations to investigate the mechanism of hydrated electron quenching by N2O. While the hydrated electron is formed upon a deposition of sodium atom on small water clusters, no hydrated electron is detected by the NaPI method for the N2O containing clusters. This indicates fast reactions forming species with high ionization energies. Ab initio molecular dynamics simulation have revealed the chain of reactions forming subsequently N2O−, O−, and OH• and OH− pair. The whole sequence occurs on a picosecond time scale for the finite size clusters investigated in our simulations. Already the first reaction step leads to the formation of species with more tightly bound electrons and therefore an increase in the ionization energy. In the future, experiments should be designed to experimentally detect the OH− or OH• radical products that have been observed theoretically in this work.

Figure 2. Reactions taking place in the Na·(H2O)15·N2O complex for a selected trajectory. Upper panel shows ionization energies along the BLYP-based MD trajectory recalculated at the BMK/6-31++g** level together with spin densities at four different time instants. The lower panel displays Mulliken atomic charges for different atoms along the trajectory.

place at the time of 1 ps. The final step (eq4) of the whole cascade, the formation of the OH− anion and OH• radical, leads to certain decrease in the IE. The formation of the OH• radical is not without interest, as the powerful reductive species, hydrated electron, is transformed into a strong oxidative agents, hydroxyl radical, which plays an important role both in the chemistry of the atmosphere and in radiation chemistry. Figure 3 shows the evolution of the IEs for all simulated trajectories together with the probabilities for the distinct



EXPERIMENTAL AND THEORETICAL METHODS The present experiments were performed on our versatile CLUster Beam apparatus (CLUB) apparatus.33−35 In this study we have exploited the reflectron time-of-flight mass spectrometer (RTOF) using two different ionization methods, EI and NaPI described recently.24 The RTOF was described in details elsewhere.36,37 The expansion conditions and some measurement details are outlined in the SI. The theoretical simulations of the reaction between the sodium atom and the N2O(H2O)15

Figure 3. Upper panel shows ionization energies along the BLYPbased MD trajectories recalculated at the BMK/6-31++g** level for 20 trajectories. The lower panel displays the time evolution of probability for the formation of different species. 2867

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(14) Signorell, R.; Yoder, B.; West, A.; Ferreiro, J. J.; Saak, C. AngleResolved Valence Shell Photoelectron Spectroscopy of Neutral Nanosized Molecular Aggregates. Chem. Sci. 2014, 5, 1283−1295. (15) Zeuch, T.; Buck, U. Sodium Doped Hydrogen Bonded Clusters: Solvated Electrons and Size Selection. Chem. Phys. Lett. 2013, 579, 1− 10. (16) Yoder, B. L.; Litman, J. H.; Forysinski, P. W.; Corbett, J. L.; Signorell, R. Sizer for Neutral Weakly Bound Ultrafine Aerosol Particles Based on Sodim Doping and Mass Spectrometric Detection. J. Phys. Chem. Lett. 2011, 2, 2623−2628. (17) Litman, J. H.; Yoder, B. L.; Schläppi, B.; Signorell, R. SodiumDoping as a reference to study the influence of intracluster chemistry on the Fragmentation of Weakly-Bound Clusters Upon Vacuum Ultraviolet Photionization. Phys. Chem. Chem. Phys. 2013, 15, 940− 949. (18) Schläppi, B.; Ferreiro, J. J.; Litman, J. H.; Signorell, R. SodiumSizer for Neutral Nanosized Molecular Aggregates: Quantitative Correction of Size-Dependence. Int. J. Mass Spectrom. 2014, 372, 13−21. (19) Buck, U.; Pradzynski, C. C.; Zeuch, T.; Dieterich, J. M.; Hartke, B. A Size Resolved Investigation of Large Water Clusters. Phys. Chem. Chem. Phys. 2014, 16, 6859−6871. (20) Pradzynski, C. C.; Forck, R. M.; Zeuch, T.; Slavíček, P.; Buck, U. A Fully Size-Resolved Perspective on the Crystallization of Water Clusters. Science 2012, 337, 1529−1532. (21) Finlayson-Pitts, B. J.; Pitts, J. N. Chemistry of the Upper and Lower Atmosphere; Academic Press: San Diego, CA, 2000. (22) Kulmala, M.; Kontkanen, J.; Junninen, H.; Lehtipalo, K.; Manninen, H. E.; Nieminen, T.; Petäjä, T.; Sipilä, M.; Schobesberger, S.; Rantala, P.; et al. Direct Observations of Atmospheric Aerosol Nucleation. Science 2013, 339, 943−946. (23) Lengyel, J.; Pysanenko, A.; Kočišek, J.; Poterya, V.; Pradzynski, C.; Zeuch, T.; Slavíček, P.; Fárník, M. Nucleation of Mixed Nitric Acid-Water Ice Nanoparticles in Molecular Beams that Starts with a HNO3 Molecule. J. Phys. Chem. Lett. 2012, 3, 3096−3101. (24) Lengyel, J.; Pysanenko, A.; Poterya, V.; Kočišek, J.; Fárník, M. Extensive Water Cluster Fragmentation After Low Energy Electron Ionization. Chem. Phys. Lett. 2014, 612, 256−261. (25) Takahashi, K.; Ohgami, S.; Koyama, Y.; Sawamura, S.; Marin, T. W.; Bartels, D. M.; Jonah, C. D. Reaction Rates of the Hydrated Electron with N2O in High Temperature Water and Potential Surface of the N2O− Anion. Chem. Phys. Lett. 2004, 383, 445−450. (26) Kryachko, E.; Vinckier, C.; Nguyen, M. Another Look at the Electron Attachment to Nitrous Oxide. J. Chem. Phys. 2001, 114, 7911−7917. (27) Uhlig, F.; Jungwirth, P. Embedded Cluster Models for Reactivity of the Hydrated Electron. Z. Phys. Chem. 2013, 227, 1583−1593. (28) Zehavi, D.; Rabani, J. Pulse Radiolytic Investigation of Oaq− Radical Ions. J. Phys. Chem. 1971, 75, 1738−1744. (29) Arnold, S. T.; Morris, R. A.; Viggiano, A. A.; Johnson, M. A. Thermal Energy Reactions of Size-Selected Hydrated Electron Clusters (H2O)−n . J. Phys. Chem. 1996, 100, 2900−2906. (30) Balaj, O. P.; Siu, C.-K.; Balteanu, I.; Beyer, M. K.; Bondybey, V. E. Free Electrons, the Simplest Radicals of Them All: Chemistry of Aqueous Electrons as Studied by Mass Spectrometry. Int. J. Mass Spectrom. 2004, 238, 65−74. (31) Buxton, G.; Greenstock, C.; Helman, W.; Ross, A. CriticalReview of Rate Constants for Reactions of Hydrated Electrons, Hydrogen-Atoms and Hydroxyl Radicals (·OH/·O−) in Aqueous Solution. J. Phys. Chem. Ref. Data 1988, 17, 513. (32) Marcus, Y. Thermodynamics of Solvation Ions, Part 5.-Gibbs Free Energies of Hydration at 298.15 K. J. Chem. Soc., Faraday Trans. 1991, 87, 2995−2999. (33) Poterya, V.; Lengyel, J.; Pysanenko, A.; Svrčková, P.; Fárník, M. Imaging of Hydrogen Halides Photochemistry on Argon and Ice Nanoparticles. J. Chem. Phys. 2014, 141, 074309. (34) Poterya, V.; Kočišek, J.; Pysanenko, A.; Fárník, M. Caging of Cl Atoms from Photodissociation of CF2Cl2 in Clusters. Phys. Chem. Chem. Phys. 2014, 16, 421−429.

clusters was modeled with ab initio molecular dynamics approach using the CP2K program package.38,39 The energies and forces were calculated at the density functional (DFT) level, using the BLYP functional.40,41 All the details and procedures are given in the SI.



ASSOCIATED CONTENT

* Supporting Information S

Details of the experiment and theoretical calculations and further results. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/ acs.jpclett.5b01269.



AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *Phone: +420 (2)6605 3206. Fax: +420 (2)8658 2307. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS This work has been supported by the Grant Agency of the Czech Republic project No.: 14-08937S. REFERENCES

(1) Garrett, B. C.; Dixon, D. A.; Camaioni, D. M.; Chipman, D. M.; Johnson, M. A.; Jonah, C. D.; Kimmel, G. A.; Miller, J. H.; Rescigno, T. N.; Rossky, P. J.; et al. Role of Water in Electron-Initiated Processes and Radical Chemistry: Issues and scientific advances. Chem. Rev. 2005, 105, 355−389. (2) Hart, E.; Boag, J. Absorption Spectrum of Hydrated Electron in Water and in Aqueous Solutions. J. Am. Chem. Soc. 1962, 84, 4090− 4095. (3) Young, R.; Neumark, D. Dynamics of Solvated Electrons in Clusters. Chem. Rev. 2012, 112, 5553−5577. (4) Larsen, R.; Glover, W.; Schwartz, B. Does the Hydrated Electron Occupy a Cavity? Science 2010, 329, 65−69. (5) Herbert, J.; Jacobson, L. Nature’s Most Squishy Ion: The Important Role of Solvent Polarization in the Description of the Hydrated Electron. Int. Rev. Phys. Chem. 2011, 30, 1−48. (6) Steinbach, C.; Buck, U. Reaction and Solvation of Sodium in Hydrogen Bonded Solvent Clusters. Phys. Chem. Chem. Phys. 2005, 7, 986−990. (7) Mundy, C.; Hutter, J.; Parrinello, M. Microsolvation and Chemical Reactivity of Sodium and Water Clusters. J. Am. Chem. Soc. 2000, 122, 4837−4838. (8) Bobbert, C.; Schulz, C. Solvation and Chemical Reaction of Sodium in Water Clusters. Eur. Phys. J. D 2001, 16, 95−97. (9) Bobbert, C.; Schütte, S.; Steinbach, C.; Buck, U. Fragmentation and Reliable Size Distributions of large Ammonia and Water Clusters. Eur. Phys. J. D 2002, 19, 183−192. (10) Schütte, S.; Buck, U. Strong Fragmentation of Large Rare Gas Clusters by High Energy Electron Impact. Int. J. Mass Spectrom. 2002, 220, 183−192. (11) Forysinski, P.; Zielke, P.; Luckhaus, D.; Corbett, J.; Signorell, R. Photoionization of Small Sodium-Doped Acetic Acid Clusters. J. Chem. Phys. 2011, 134, 094314. (12) Forck, R. M.; Dauster, I.; Schieweck, Y.; Zeuch, T.; Buck, U.; Ončaḱ , M.; Slavíček, P. Observation of Two Classes of Isomers of Hydrated Electrons in Sodium-Water Clusters. J. Chem. Phys. 2010, 132, 221102. (13) West, A.; Yoder, B.; Luckhaus, D.; Saak, C.; Doppelbauer, M.; Signorell, R. Angle-Resolved Photoemission of Solvated Electrons in Sodium-Doped Clusters. J. Phys. Chem. Lett. 2015, 6, 1487−1492. 2868

DOI: 10.1021/acs.jpclett.5b01269 J. Phys. Chem. Lett. 2015, 6, 2865−2869

Letter

The Journal of Physical Chemistry Letters (35) Lengyel, J.; Pysanenko, A.; Poterya, V.; Slavíček, P.; Fárník, M.; Kočišek, J.; Fedor, J. Irregular Shapes of Water Clusters Generated in Supersonic Expansions. Phys. Rev. Lett. 2014, 112, 113401. (36) Kočišek, J.; Lengyel, J.; Fárník, M.; Slavíček, P. Energy and Charge Transfer in Ionized Argon Coated Water Clusters. J. Chem. Phys. 2013, 139, 214308. (37) Kočišek, J.; Lengyel, J.; Fárník, M. Ionization of Large Homogeneous and Heterogeneous Clusters Generated in AcetyleneAr Expansions: Cluster Ion Polymerization. J. Chem. Phys. 2013, 138, 124306. (38) VandeVondele, J.; Krack, M.; Mohamed, F.; Parrinello, M.; Chassaing, T.; Hutter, J. QUICKSTEP: Fast and accurate density functional calculations using a mixed Gaussian and plane waves approach. Comput. Phys. Commun. 2005, 167, 103−128. (39) VandeVondele, J.; Hutter, J. Gaussian Basis Sets for Accurate Calculations on Molecular Systems in Gas and Condensed Phases. J. Chem. Phys. 2007, 127, 114105. (40) Becke, A. Density-Functional Exchange-Energy Approximation with Correct Asymptotic-Behavior. Phys. Rev. A: At., Mol., Opt. Phys. 1988, 38, 3098−3100. (41) Lee, C.; Yang, W.; Parr, R. Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron-Density. Phys. Rev. B: Condens. Matter Mater. Phys. 1988, 37, 785−789.

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