Article

pubs.acs.org/JPCA

Ab Initio Molecular Dynamics of Dimerization and Clustering in Vapors † ‡ Vitaly V. Chaban and Oleg V. Prezhdo*, † Instituto de Cienciâ e Tecnologia, Universidade Federal de Saõ Paulo, 12231-280 Saõ Josédos Campos, SP Brazil ‡ Departments of Chemistry, Physics and Astronomy, University of Southern California, Los Angeles, California 90089, United States

ABSTRACT: Alkali metals are known to form dimers, trimers, and tetramers in their vapors. The mechanism and regularities of this phenomenon characterize the chemical behavior of the first group elements. We report ab initio molecular dynamics (AIMD) simulations of the alkali metal vapors and characterize their structural properties, including radial distribution functions and atomic cluster size distribu- tions. AIMD confirms formation of Men, where n ranges from 2 to 4. High pressure sharply favors larger structures, whereas high temperature decreases their fraction. Heavier alkali metals maintain somewhat larger fractions of Me2,Me3, and Me4, relative to isolated atoms. A single atom is the most frequently observed structure in vapors, irrespective of the element and temperature. Due to technical difficulties of working with high temperatures and pressures in experiments, AIMD is the most affordable method of research. It provides valuable understanding of the chemical behavior of Li, Na, K, Rb, and Cs, which can lead to development of new chemical reactions involving these metals.

■ INTRODUCTION moderate temperatures and pressures, suggesting that higher Clusters of metals represent a vibrant field of research, in which imperfections result primarily from the formation of stable both experimental and theoretical methods are successfully tetramer molecules Li4,K4, and Cs4. The standard enthalpy for − 1 5 the formation of K4 from four atoms in the vapor employed. All alkali metals are known to form diatomic − −1 molecules through the formation of weak metal−metal phase amounts to 146.0 kJ mol . In particular, it exceeds − ΔH bonds.6 11 The weak bond originates from the overlap between (K2) more than 2-fold. diffuse valence orbitals of two atoms, considered within the Urban and Sadlej applied a number of high-level ab initio one- approximation.12 Various heteronuclear alkali methods to investigate the basic electric properties of the alkali atom dimers, including dipole moment and dipole polar- metal dimers, such as NaLi, KNa, RbK, etc., are also possible. 12 − izability. The authors criticized density functional theory The metal metal bonding leads to a partial association of the − alkali metal vapors. The resulting structures constitute a (DFT) exchange correlation models for underestimation of − significant research interest.13 18 According to spectroscopic the parallel component of the dipole polarizability, as compared and vapor pressure studies, a significant fraction of dimers is to more accurate ab initio methods and experimental Downloaded via UNIV OF SOUTHERN CALIFORNIA on November 7, 2019 at 23:06:26 (UTC).

See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. observations. expected in alkali metal vapors. The association increases with 22 the atomic mass of the element, Li < Na < K < Rb < Cs. Sarkisyan and co-workers devised an interesting method to Optical absorbance spectra of small clusters of alkali metals experimentally determine the metal dimer dissociation energy dimers, trimers, and tetramerspoint to the existence of by observing the dependence of the relative dimer density on distinct vibrational energy levels. The spectra are qualitatively temperature in the superheated vapor of cesium. The proposed different for larger clusters. For instance, the clusters containing method is also sensitive to the amount of the investigated ff 19,20 substance. Recently, an important simulation effort was more than 8 atoms exhibit collective e ects in their spectra. 6 The heat of dimerization decreases as the size of the element described by Donoso and co-workers. Trimers Li3,Na3,K3, 21 − increases. Ewing and co-workers employed the van’tHoff Rb3, and Cs3 were studied using Born Oppenheimer molecular equation to compute the standard enthalpies of dimerization, dynamics as clusters in a vacuum; Na7 was also simulated for ↔ comparison. Using fairly short simulations (8.0 ps) with using 2Me Me2. All enthalpies derived this way are negative: ΔH − −1 ΔH − −1 10 replicas per system, the authors derived reliable structural (Li2)= 76.5 kJ mol ; (K2)= 56.4 kJ mol ; ΔH − −1 and electric properties. Pseudorotation and crossover were (Cs2)= 48.5 kJ mol . Smaller dimerization enthalpies can be understood as generally weaker Me−Me bonds due to larger bond lengths. The same authors21 performed an analysis Received: May 7, 2016 of the PVT data of the , potassium, and cesium vapors. Revised: June 10, 2016 They concluded that the degree of dimerization is significant at Published: June 13, 2016

© 2016 American Chemical Society 4302 DOI: 10.1021/acs.jpca.6b04609 J. Phys. Chem. A 2016, 120, 4302−4306 The Journal of Physical Chemistry A Article considered as two coupled phenomena determining movement Table 1. Simulated Systems, Their Key Parameters, and a of the atoms. Sampling Durations Dimerization of alkali metals constitutes an important density, mass, plane wave sampling fundamental phenomenon in chemistry, the understanding of no. metal temp, K kg m−3 amu cutoff,eV duration, ps which is valuable for developing new reactions and adjusting 1 Li 2100 28.8 138.82 100 100 yields of known chemistries. As exemplified above, the field 2 Li 2400 28.8 138.82 100 90 develops gradually. At this point, we are not aware of efforts to 3 Li 2700 28.8 138.82 100 70 simulate the vapor phase of the alkali metals at the atomic level 4 Li 3000 28.8 138.82 100 70 of precision. For the first time, we report ab initio molecular 5 Li 3300 28.8 138.82 100 70 dynamics (AIMD) simulations of the Li, Na, K, Rb, and Cs 6 Na 1800 95.4 459.80 76.5 100 vapors above and below the critical points of these simple 7 Na 2100 95.4 459.80 76.5 80 substances. The cluster analyses were performed to identify the 8 Na 2400 95.4 459.80 76.5 80 structure and percentages of the Men clusters, where n >1,in 9 Na 2700 95.4 459.80 76.5 70 the vapor phase. 10 K 1400 162.3 781.97 87.5 100 ■ METHODOLOGY 11 K 1700 162.3 781.97 87.5 90 12 K 2000 162.3 781.97 87.5 90 Adiabatic AIMD simulations were conducted within the DFT 13 K 2300 162.3 781.97 87.5 80 framework employing a pure DFT functional and a converged 14 Rb 1600 354.7 1709.36 91.4 100 plane-wave basis set. Plane waves constitute an efficient means 15 Rb 1800 354.7 1709.36 91.4 100 to optimize wave functions in the three-dimensional periodic 16 Rb 2000 354.7 1709.36 91.4 90 systems. Periodic boundary conditions allow simulating a 17 Rb 2200 354.7 1709.36 91.4 90 virtually infinite system, while eliminating undesirable boundary 18 Cs 1400 551.6 2658.11 165.2 100 effects. The exchange−correlation functional proposed by 19 Cs 1600 551.6 2658.11 165.2 100 Perdew, Burke, and Ernzerhof in the generalized gradient 20 Cs 1800 551.6 2658.11 165.2 90 23 approximation was employed. All core were 21 Cs 2000 551.6 2658.11 165.2 90 24 simulated by the projector-augmented wave method, 22 Li 3300 14.7 138.82 100 100 providing high computational efficiency. The self-consistent 23 Li 3300 10.5 138.82 100 100 −5 field (SCF) convergence threshold was set to 10 Hartree. A 24 Li 3300 8.5 138.82 100 100 a minimum of three SCF iterations were conducted per time Each system contains 20 alkali atoms. step. The initial geometries were optimized prior to performing finite-temperature AIMD. The nuclear equations-of-motion were propagated with a 1.0 fs integration time step. The Cartesian coordinates of all atoms were saved every 100 time steps. The simulations were carried out in the constant temperature constant volume ensemble. The Nose thermostat was used to maintain constant temperature25 with the relaxation time constant equaling to 100 time steps. Empirical dispersion corrections were not employed, because the contribution of the van der Waals interaction to the overall interaction energy is much smaller than the metal−metal covalent bond energies. The Vienna Ab initio Simulation Package (VASP)26 was employed. Packmol27 was used to Figure 1. Experimental normal boiling temperatures (red solid line) generate initial configurations for the AIMD simulations. VMD 28 and critical temperatures (green dashed line) of the simple substances (Visual Molecular Dynamics, version 1.9.1) and Gabedit 30 29 composed of alkali metal atoms. Simple substance is a homogeneous (version 2) were the tools to manipulate particles and form of a chemical element existing in a free state. visualize atomic trajectories. Table 1 enumerates the simulated systems and the basic simulation parameters, such as the energy cutoff for plane Therefore, the maximum height of an RDF shows by what waves, simulated temperature, density, and sampling duration. factor the probability of finding an atom at a given distance Each simulated system contains 20 alkali metal atoms (Li, Na, from another atom exceeds the probability to find two atoms at K, Rb, Cs), which were distributed randomly in space at time an (infinitely) large distance in the periodic system. The cluster zero. The generated configurations were placed in cubic boxes analysis was performed on the basis of a geometric criterion. If of volumes varying between 8.0 × 103 and 2.7 × 104 Å. The the distance between any two atoms is equal to or is below the considered cube sides were 20, 25, 28, and 30 Å. Note that defined cutoff, they are assigned to the same cluster. The although the simulated systems of the same volume (8.0 × 103 cutoffs were defined using covalent radii of the alkali metals, as Å) have the same number density, their mass densities differ by discussed below. large factors. The simulated temperatures were selected to cover the regions both above and below the corresponding ■ RESULTS AND DISCUSSION critical points (Figure 1). All temperatures are significantly The simulated systems equilibrate quickly at high temperatures, larger than the corresponding normal boiling points. within a few thousands of steps. This was revealed by the The atom−atom radial distribution functions (RDFs) were evolution of their thermodynamic properties. The equilibration computed following their standard definition, in which RDF = parts of the trajectories were deleted, whereas the remaining 1 coincides with the average density of atoms in the system. trajectory parts were used to sample the structure properties.

4303 DOI: 10.1021/acs.jpca.6b04609 J. Phys. Chem. A 2016, 120, 4302−4306 The Journal of Physical Chemistry A Article

Figure 2 exemplifies configuration of the equilibrated system. Note that we simulate equal mass densities of the same

Figure 2. Molecular snapshot obtained after equilibration of system 9, Table 1 (20 Na atoms at 2700 K). The depicted box sides visualize periodic boundary conditions. When the atom crosses the boundary, it immediately appears on the opposite side with the same translational velocity vector. substance at different temperatures, but different mass densities of different substances. The simulated vapor densities are high, because they roughly correspond to the critical points, whereas the critical pressures much exceed the atmospheric pressure, e.g., 670, 450, 160 bar for Li, Na, and K, respectively.30 RDFs (Figure 3) are scattered significantly as should be expected for high-temperature simulations. Only the first peak is present at each RDF. The height of the peaks decreases in the row Li > Na > K > Rb > Cs. This trend is expected, because the bond energy decreases uniformly in the above series. The bond Figure 3. Atom−atom radial distribution functions for Li, Na, K, Rb, energy decrease correlates with the bond length increase, andCsatthesimulatedtemperaturesaboveandbelowthe which, in turn, is due to larger covalent radii of the heavier corresponding critical points. metals. The computed positions of the first RDF peaks are 2.94, 3.36, 4.36, 4.60, and 5.14 Å for Li, Na, K, Rb, and Cs, respectively. The RDF peaks include both the covalently bound atoms and the atoms interacting via the weaker van der Waals forces. The RDF analysis by itself does not allow differentiating these contributions. A more sophisticated approach should be employed. To identify alkali metal atoms that are bound covalently, the following cluster analysis was conducted. The bond energy profiles contain no barrier, which could be used to define the internuclear distance at which the bonds break. Thus, identification of the cutoff distance for the cluster analysis is challenging and the final choice cannot be unique. Figure 4 Figure 4. Various definitions of the Me−Me distance: covalent compares a number of bond related distances. The van der diameter (red solid line),30 simulated RDF maximum (green dashed Waals diameters (VDWDs) are significantly larger than the line), computed bond length in the Me2 dimers (blue dash-dotted line), and VDW diameter (pink dash-dotted line).30 Reliable van der other distances. The VDWDs can be applied for identifying all 30 metallic clusters, including those that do not form covalent Waals radii are not yet available for Rb and Cs. bonds. The van der Waals diameters are 33% (Li), 43% (Na), and 36% (K) higher than the covalent bond lengths. The cluster distribution diagrams, as compared to the distributions covalent diameters and the Me−Me bond lengths computed by at the next value of 14%. DFT in Me2 at 0 K almost coincide, with a certain discrepancy Cluster analysis performed at the highest simulated temper- in the case of Cs. The positions of the RDF maxima are ature, i.e., in the vicinity of the critical point of the systematically larger. We derived the cutoff distances from the corresponding simple substance, reveals that most atoms exist covalent diameters by increasing those by 12%. The reason for as single particles. This result does not significantly depend on the increase is thermal expansion, which is not included in the the atomic mass of the alkali metal. Compare 70% of single tabulated covalent diameters. It accounts for bond length atoms in the case of Na to 67% in the case of Rb, and to 64% in fl fi uctuations upon molecular dynamics at a nite temperature. A the case of Cs. The percentage of dimers Me2 slightly increases number of prospective percentages were tested with a step of from Li to Cs, from 15 to 21%. This result is in accordance with 2%. The value of 12% resulted in the smallest change of the the previous knowledge. Temperature increase reduces the

4304 DOI: 10.1021/acs.jpca.6b04609 J. Phys. Chem. A 2016, 120, 4302−4306 The Journal of Physical Chemistry A Article fraction of Me2 only slightly. For instance, Na forms 23% of Na2 at 1800 K, but 21% at 2700 K (Figure 5). A very similar

Figure 6. Influence of temperature on formation of Men particles, n > 1, in vapors of the alkali metals: Lin (red circles), Nan (green squares), Kn (blue triangles up), Rbn (pink triangles down), and Csn (cyan diamonds).

The size of Men was investigated as a function of density (simulation volume). The results are shown in Figure 7.

Figure 5. Distribution of the alkali metal cluster sizes in the vapor phase: Li at 3300 K, Na at 2700 K, K at 2300 K, Rb at 2200 K, and Cs at 2000 K. See Figure 2 for the experimental critical temperatures. The standard errors of the computations were estimated to range 1−10%, with higher errors corresponding to larger structures. These errors were obtained from statistical processing of several trajectory parts.

trend is observed for Rb2. In comparison, the fraction of Li2 does not change at all in the range from 2100 to 3300 K, within the error bars, ca. 15−16%. This result is quite interesting, because the vapor pressure is much higher at 3300 K than at

2100 K, considering the constant volume of the simulated Figure 7. Percentage of the alkali atom molecules (Me2,Me3,Me4) systems. Larger molecules are also observed. Their probability and lone atoms vs the simulated box side length. The standard error of decreases sharply as size of the Men molecule increases. The the computation was estimated to be ca. 1−2% on the basis of the fraction of trimers Me3 ranges between 4 and 8%, being statistical processing of several trajectory parts. systematically higher for Rb3 and Cs3. The fraction of tetramers − ff Me4 is 1 4%, with the heavier metals maintaining larger Though lowering the density has a marginal e ect on the fractions. Although we also detected larger molecules, their percentages of lone atoms and metallic dimers, the fractions of fi probabilities are below 1%, which is smaller than the standard Me3 and Me4 decay very signi cantly. Therefore, pressure plays error of the present calculations. Using larger systems and an important role in the formation of larger structures. For longer times may help to provide more accurate insights into instance, the percentage of Li4 becomes barely detectable when these fascinating vaporized structures. Note that comparison of the box side increases to 25 Å and above. the investigated alkali metals is not direct, because their critical A fairly small number of alkali metal atoms (20) was used in temperatures are significantly different, and thus, the kinetic the present work to derive the fractions of different size energies at the simulated conditions differ as well. molecules in the vapor phase. This number is sufficient to The shapes of the larger clusters Men are in good agreement observe all possible compositions, because it is several times with the cluster geometries reported in the recent work of larger than the number of atoms in the biggest vapor phase Donoso and co-workers.6 At the same time, thermal geometry structure. The cluster distribution was investigated as a function 31 fluctuations are much more significant in our simulations, of the simulated system size in the recent work focusing on which have been conducted at substantially higher temperatures alkali metal . No undesirable deviations for the smaller compared to the earlier simulations carried out at 300 K. systems were found. Further, we did not observe Me7 molecules. Temperature increase leads to destruction of the Me−Me ■ CONCLUSIONS covalent bonds (Figure 6), irrespective of the alkali metal. Apart AIMD simulations of the alkali metal vapors were reported. from Lin, the fractions of Men are very similar ranging within Although formation of alkali metal dimers in the gas phase is 29−37%. These results are important for understanding of the known from experiment, to the best of our knowledge, no PVT data and deviations from ideality in the alkali metal vapors. theoretical simulation of this process has been attempted fi n Noteworthy, signi cant percentages of Me2 and Men ( >2) previously. Earlier modeling of small alkali metal clusters exist even at the critical temperatures. Because of the focused on low and ambient temperatures, which do not experimental hurdles, the critical parameters of the alkali generate the vapor phase. The distribution of the cluster size in metals, including those used as a reference in the present work, the vapor phase at high temperatures was reported here are determined on the basis of extrapolation. theoretically for the first time.

4305 DOI: 10.1021/acs.jpca.6b04609 J. Phys. Chem. A 2016, 120, 4302−4306 The Journal of Physical Chemistry A Article

The analyses of the structure properties, such as RDFs and (12) Urban, M.; Sadlej, A. J. Electronic-Structure and Electric cluster size distributions, confirmed that the alkali metal atoms Properties of the Alkali-Metal Dimers. J. Chem. Phys. 1995, 103, 9692− form molecules Men. The existence of n =2−4 was confirmed 9704. with good confidence. Larger molecules were also observed in (13) Musial, M.; Kowalska-Szojda, K.; Lyakh, D. I.; Bartlett, R. J. the course of the AIMD simulations, but their percentages were Potential energy curves via double electron-attachment calculations: Dissociation of alkali metal dimers. J. Chem. Phys. 2013, 138, 194103. smaller than the error bars. High pressure strongly favors larger (14) Zuchowski, P. S.; Hutson, J. M. Reactions of ultracold alkali- structures, whereas high temperature decreases the fraction of metal dimers. Phys. Rev. A: At., Mol., Opt. Phys. 2010, 81, 060703. larger clusters in the vapors. Heavier alkali metals maintain (15) Porsev, S. G.; Derevianko, A. Accurate relativistic many-body larger fractions of dimers, trimers, and tetramers. A single atom calculations of van der Waals coefficients C-8 and C-10 for alkali-metal is the most frequently observed structure, irrespective of the dimers. J. Chem. Phys. 2003, 119, 844−850. element and temperature. (16) Huang, R. H.; Ward, D. L.; Dye, J. L. Alkali-Metal-Anion Dimers The reported results are of fundamental importance, because and Chains in Alkalide Structures. J. Am. Chem. Soc. 1989, 111, 5707− understanding of the chemical behavior of Li, Na, K, Rb, and 5708. Cs can lead to development of new chemical reactions (17) Banerjee, A.; Autschbach, J.; Chakrabarti, A. Time-dependent involving these metals. Because high temperatures and density-functional-theory calculation of the van der Waals coefficient ffi C(6) of alkali-metal atoms Li, Na, K; alkali-metal dimers Li(2), Na(2), pressures are di cult to access in current experimental setups, K(2); sodium clusters Na(n); and fullerene C(60). Phys. Rev. A: At., AIMD presents a particularly valuable and promising tool for Mol., Opt. Phys. 2008, 78, 032704. studies of such systems. (18) Bahadur, R.; McClurg, R. B. Nucleation rates for the condensation of monovalent metals. J. Chem. Phys. 2004, 121, ■ AUTHOR INFORMATION 12499−12510. Corresponding Author (19) Ashby, R. A. Absorption-Spectra of Alkali-Metal Vapors. J. *O.V.P. E-mail: [email protected]. Tel: +1 (213) 821-3116. Chem. Educ. 1978, 55, 500−501. (20) Wang, Y.; Lewenkopf, C.; Tomanek, D.; Bertsch, G.; Saito, S. Notes fi Collective Electronic Excitations and Their Damping in Small Alkali The authors declare no competing nancial interest. Clusters. Chem. Phys. Lett. 1993, 205, 521−528. (21) Ewing, C. T.; Stone, J. P.; Spann, J. R.; Miller, R. R. Molecular ■ ACKNOWLEDGMENTS Association in Sodium Potassium and Cesium Vapors at High O.V.P. acknowledges financial support of the US Department Temperatures. J. Phys. Chem. 1967, 71, 473−477. of Energy, grant No. DE-SC0014429. (22) Sarkisyan, D. H.; Sarkisyan, A. S.; Yalanusyan, A. K. Thermal dissociation of cesium dimers. Appl. Phys. B: Lasers Opt. 1998, 66, ■ REFERENCES 241−244. (23) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient (1) Cruz-Olvera, D.; Vasquez, A. D.; Geudtner, G.; Vasquez-Perez, J. approximation made simple. Phys. Rev. Lett. 1996, 77, 3865−3868. M.; Calaminici, P.; Koster, A. M. Transition-State Searches in Metal (24) Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the Clusters by First-Principle Methods. J. Phys. Chem. A 2015, 119, Phys. Rev. B: Condens. Matter − projector augmented-wave method. 1494 1501. Mater. Phys. 1999, 59, 1758−1775. (2) Jamshidi, Z.; Far, M. F.; Maghari, A. Binding of Noble Metal (25) Nose, S. A molecular dynamics method for simulations in the Clusters with Rare Gas Atoms: Theoretical Investigation. J. Phys. Mol. Phys. 52 − Chem. A 116 − canonical ensemble. 1984, , 255 268. 2012, , 12510 12517. (26) Kresse, G.; Furthmuller, J. Efficiency of ab-initio total energy (3) Ben-Xia, Z.; Dong, D.; Ling, W.; Ji-Xian, Y. Density Functional calculations for metals and semiconductors using a plane-wave basis Study on the Structural, Electronic, and Magnetic Properties of 3d set. Comput. Mater. Sci. 1996, 6,15−50. Transition-Metal-Doped Au-5 Clusters. J. Phys. Chem. A 2014, 118, − (27) Martinez, L.; Andrade, R.; Birgin, E. G.; Martinez, J. M. 4005 4012. PACKMOL: a package for building initial configurations for molecular (4) Mandado, M.; Krishtal, A.; Van Alsenoy, C.; Bultinck, P.; dynamics simulations. J. Comput. Chem. 2009, 30, 2157−2164. Hermida-Ramon, J. M. Bonding study in all-metal clusters containing J. Phys. Chem. A 111 − (28) Humphrey, W.; Dalke, A.; Schulten, K. VMD: Visual molecular Al-4 units. 2007, , 11885 11893. dynamics. J. Mol. Graphics 1996, 14,33−38. (5) Lee, J. S. Accurate ab initio binding energies of alkaline earth J. Phys. Chem. A 109 − (29) Allouche, A. R. Gabedit-A Graphical User Interface for metal clusters. 2005, , 11927 11932. Computational Chemistry Softwares. J. Comput. Chem. 2011, 32, (6) Donoso, R.; Cardenas, C.; Fuentealba, P. Ab Initio Molecular 174−182. Dynamics Study of Small Alkali Metal Clusters. J. Phys. Chem. A 2014, 118 − (30) Periodic Table. http://periodictable.com. (accessed May 5, , 1077 1083. 2016). (7) Tomza, M.; Madison, K. W.; Moszynski, R.; Krems, R. V. (31) Chaban, V. V.; Prezhdo, O. V. Ionic Vapor Composition in Chemical reactions of ultracold alkali-metal dimers in the lowest- Phys. Rev. A: At., Mol., Opt. Phys. 88 Critical and Supercritical States of Strongly Interacting Ionic energy (3)Sigma state. 2013, , Compounds. J. Phys. Chem. B 2016, 120, 4302−4309. 050701. (8) Lima, N. A.; Caldas, M. J. Long range van der Waals density functional: Dimers involving alkali-metal, alkaline-earth-metal, and noble-gas atoms. Phys. Rev. B: Condens. Matter Mater. Phys. 2005, 72, 033109. (9) Gorshkov, A. V.; Manmana, S. R.; Chen, G.; Demler, E.; Lukin, M. D.; Rey, A. M. Quantum magnetism with polar alkali-metal dimers. Phys. Rev. A: At., Mol., Opt. Phys. 2011, 84, 033619. (10) Aldegunde, J.; Hutson, J. M. Hyperfine energy levels of alkali- metal dimers: Ground-state homonuclear molecules in magnetic fields. Phys. Rev. A: At., Mol., Opt. Phys. 2009, 79, 013401. (11) Liu, X.; Ito, H.; Torikai, E. Exchange-Correlation Interaction and AO-Hybridization of Alkali-Metal Atomic Clusters. J. Phys. Chem. A 2013, 117, 9099−9107.

4306 DOI: 10.1021/acs.jpca.6b04609 J. Phys. Chem. A 2016, 120, 4302−4306