Ab Initio Molecular Dynamics of Dimerization and Clustering in Alkali

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Ab Initio Molecular Dynamics of Dimerization and Clustering in Alkali Metal Vapors Vitaly V. Chaban, and Oleg V. Prezhdo J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b04609 • Publication Date (Web): 13 Jun 2016 Downloaded from http://pubs.acs.org on June 15, 2016

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The Journal of Physical Chemistry

Ab Initio Molecular Dynamics of Dimerization and Clustering in Alkali Metal Vapors Vitaly V. Chaban(1) and Oleg V. Prezhdo(2),* (1) Instituto de Ciência e Tecnologia, Universidade Federal de São Paulo, 12231-280, São José dos Campos, SP, Brazil (2) Departments of Chemistry, Physics and Astronomy, University of Southern California, Los Angeles, CA 90089, USA

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 distributions. 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.

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Introduction Clusters of metals represent a vibrant field of research, in which both experimental and theoretical methods are successfully employed.1-5 All alkali metals are known to form diatomic molecules through the formation of weak metal-metal bonds.6-11 The weak bond originates from the overlap between diffuse valence orbitals of two atoms, considered within the one-electron approximation.12 Various hetero-nuclear alkali metal dimers, such as NaLi, KNa, RbK, etc, are also possible. The metal-metal bonding leads to a partial association of the alkali metal vapors. The resulting structures constitute a significant research interest.13-18 According to spectroscopic and vapor pressure studies, a significant fraction of dimers is expected in alkali metal vapors. The association increases with the atomic mass of the element, Li < Na < K < Rb < Cs. Optical absorbance spectra of small clusters of alkali metals – dimers, trimers, and tetramers – point to existence of distinct vibrational energy levels. The spectra are qualitatively different for larger clusters. For instance, the clusters containing more than 8 atoms exhibit collective effects in their spectra.19,20 The heat of dimerization decreases as the size of the element increases. Ewing and coworkers21 employed the van’t Hoff equation to compute the standard enthalpies of dimerization, 2 Me ↔ Me2. All enthalpies derived this way are negative: ∆H (Li2) = 76.5 kJ mol-1;

∆H (K2) = -56.4 kJ mol-1;

∆H (Cs2) = -48.5 kJ mol-1.

Smaller

dimerization

enthalpies can be understood as generally weaker Me-Me bonds due to larger bond lengths. The same authors21 performed an analysis of the PVT data of the lithium, potassium, and cesium vapors. They concluded that the degree of dimerization is significant at moderate temperatures and pressures, suggesting that higher imperfections result primarily from the formation of stable tetramer molecules Li4, K4, and Cs4. The standard enthalpy for the formation of K4 from four potassium atoms in the vapor phase amounts to -146.0 kJ mol-1. In particular, it exceeds ∆H (K2) more than twofold.


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Urban and Sadlej applied a number of high-level ab initio methods to investigate the basic electric properties of the alkali atom dimers, including dipole moment and dipole polarizability.12 The authors criticized density functional theory (DFT) exchange-correlation models for underestimation of the parallel component of the dipole polarizability, as compared to more accurate ab initio methods and experimental observations. Sarkisyan and coworkers22 devised an interesting method to experimentally determine the metal dimer dissociation energy by observing the dependence of the relative dimer density on temperature in the superheated vapor of cesium. The proposed method is also sensitive to the amount of the investigated substance. Recently, an important simulation effort was described by Donoso and coworkers.6 Trimers Li3, Na3, K3, Rb3, Cs3 were studied using Born-Oppenheimer molecular dynamics as clusters in vacuum; Na7 was also simulated for comparison. Using fairly short simulations (8.0 ps) with using 10 replicas per system, the authors derived reliable structural and electric properties. Pseudorotation and crossover were considered as two coupled phenomena determining movement of the atoms. Dimerization of alkali metals constitutes an important fundamental phenomenon in chemistry, understanding of which is valuable for developing new reactions and adjusting yields of known chemistries. As exemplified above, the field develops gradually. At this point, we are not aware of efforts to simulate the vapor phase of the alkali metals at the atomic level of precision. For the first time, we report ab initio molecular dynamics (AIMD) simulations of the Li, Na, K, Rb, and Cs vapors above and below the critical points of these simple substances. The cluster analyses were performed to identify the structure and percentages of the Men clusters, where n > 1, in the vapor phase.



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Methodology Adiabatic AIMD simulations were conducted within the DFT framework employing a pure DFT functional and a converged plane-wave basis set. Plane waves constitute an efficient means to optimize wave functions in the three-dimensional periodic systems. Periodic boundary conditions allow simulating virtually infinite system, while eliminating undesirable boundary effects. The exchange-correlation functional proposed by Perdew, Burke, and Ernzerhof in the generalized gradient approximation was employed.23 All core electrons were simulated by the projector-augmented wave method,24 providing high computational efficiency. The selfconsistent field (SCF) convergence threshold was set to 10-5 Hartree. A minimum of three SCF iterations were conducted per time-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 timesteps. 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, since 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 generate initial configurations for the AIMD simulations. VMD (Visual Molecular Dynamics, version 1.9.1)28 and Gabedit (version 2)29 were the tools to manipulate particles and visualize atomic trajectories. Table 1 enumerates the simulated systems and the basic simulation parameters, such as the energy cut-off for plane waves, simulated temperature, density, and sampling duration. Each simulated system contains 20 alkali metal atoms (Li, Na, K, Rb, Cs), which were distributed randomly in space at time zero. The generated configurations were placed in cubic boxes of volumes varying between 8.0×103 Å and 2.7×104 Å. The considered cube sides were 20, 25, 28, and 30 Å. Note that although the simulated systems of the same volume (8.0×103 Å) have the ACS Paragon Plus Environment

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same number density, their mass densities differ by large factors. The simulated temperatures were selected to cover the regions both above and below the corresponding critical points (Figure 1). All temperatures are significantly larger than the corresponding normal boiling points.

Table 1. Simulated systems, their key parameters, and sampling durations. Each system contains 20 alkali atoms. #

Metal

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Li Li Li Li Li Na Na Na Na K K K K Rb Rb Rb Rb Cs Cs Cs Cs Li Li Li

Temperature, K 2100 2400 2700 3000 3300 1800 2100 2400 2700 1400 1700 2000 2300 1600 1800 2000 2200 1400 1600 1800 2000 3300 3300 3300

Density, kg m-3 28.8 28.8 28.8 28.8 28.8 95.4 95.4 95.4 95.4 162.3 162.3 162.3 162.3 354.7 354.7 354.7 354.7 551.6 551.6 551.6 551.6 14.7 10.5 8.5

Mass, a.m.u. 138.82 138.82 138.82 138.82 138.82 459.80 459.80 459.80 459.80 781.97 781.97 781.97 781.97 1709.36 1709.36 1709.36 1709.36 2658.11 2658.11 2658.11 2658.11 138.82 138.82 138.82


Plane wave cut-off, eV 100 100 100 100 100 76.5 76.5 76.5 76.5 87.5 87.5 87.5 87.5 91.4 91.4 91.4 91.4 165.2 165.2 165.2 165.2 100 100 100

Sampling duration, ps 100 90 70 70 70 100 80 80 70 100 90 90 80 100 100 90 90 100 100 90 90 100 100 100


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Figure 1. Experimental normal boiling temperatures (red solid line) and critical temperatures (green dashed line) of the simple substances composed of alkali metal atoms.30 Simple substance is a homogeneous form of a chemical element existing in a free state.

The atom-atom radial distribution functions (RDFs) were computed following their standard definition, in which RDF=1 coincides with the average density of atoms in the system. Therefore, the maximum height of an RDF shows by what factor the probability of finding an atom at a given distance from another atom exceeds the probability to find two atoms at an (infinitely) large distance in the periodic system. The cluster analysis was performed based on a geometric criterion. If the distance between any two atoms is equal to or is below the defined cut-off, they are assigned to the same cluster. The cut-offs were defined using covalent radii of the alkali metals, as discussed below.

Results and Discussion The simulated systems equilibrate quickly at high temperatures, within a few thousands of steps. This was revealed by the evolution of their thermodynamic properties. The equilibration parts of the trajectories were deleted, whereas the remaining trajectory parts were used to sample the structure properties. Figure 2 exemplifies configuration of the equilibrated sodium system. Note that we simulate equal mass densities of the same substance at different temperatures, but different mass densities of different substances. The simulated vapor densities are high, since ACS Paragon Plus Environment

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they roughly correspond to the critical points, while the critical pressures much exceed the atmospheric pressure, e.g. 670, 450, 160 bar for Li, Na, and K, respectively.30

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.

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, since the bond energy decreases uniformly in the above series. The bond energy decrease correlates with the bond length increase, which, in turn, is due to larger covalent radii of the heavier 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.



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Figure 3. Atom-atom radial distribution functions for Li, Na, K, Rb, Cs at the simulated temperatures above and below the corresponding critical points.

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 cut-off distance for the cluster analysis is challenging and the final choice cannot be unique. Figure 4 compares a number of bond related distances. The van der Waals diameters (VDWDs) are significantly larger than the other distances. The VDWDs can be applied for identifying all metallic clusters, including those, which do not form covalent bonds. The van der Waals diameters are 33% (Li), 43% (Na), 36% (K) higher than the covalent bond lengths. The covalent diameters and the MeMe bond lengths computed by DFT in Me2 at 0 K almost coincide, with a certain discrepancy in the case of Cs. The positions of the RDF maxima are systematically larger. We derived the cutoff distances from the covalent diameters by increasing those by 12%. The reason of the increase ACS Paragon Plus Environment

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is thermal expansion, which is not included in the tabulated covalent diameters. It accounts for bond length fluctuations upon molecular dynamics at a finite temperature. A number of prospective percentages were tested with a step of 2%. The value of 12% resulted in the smallest change of the cluster distribution diagrams, as compared to the distributions at the next value of 14%.

Figure 4. Various definitions of the Me-Me distance: covalent diameter (red solid line),30 simulated RDF maximum (green dashed line), computed bond length in the Me2 dimers (blue dash-dotted line), VDW diameter (pink dash-dot-dotted line).30 Reliable van der Waals radii are not yet available for Rb and Cs.30

Cluster analysis performed at the highest simulated temperature, i.e. in the vicinity of the critical point of the corresponding simple substance, reveals that most atoms exist as single particles. This result does not significantly depend on the atomic mass of the alkali metal. Compare 70% of single atoms in the case of Na to 67% in the case of Rb, and to 64% in the case of Cs. The percentage of dimers Me2 slightly increases from Li to Cs, from 15 to 21%. This result is in accordance with the previous knowledge. Temperature increase reduces the fraction of Me2 only slightly. For instance, Na forms 23% of Na2 at 1800 K, but 21% at 2700 K. A very similar trend is observed for Rb2. In comparison, the fraction of Li2 does not change at all in the range from 2100 K to 3300 K, within the error bars, ca. 15-16%. This result is quite interesting, since the vapor pressure is much higher at 3300 K than at 2100 K, considering the constant volume of the simulated systems. Larger molecules are also observed. Their probability ACS Paragon Plus Environment


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decreases sharply as size of the Men molecule increases. The fraction of trimers Me3 ranges between 4 and 8%, being systematically higher for Rb3 and Cs3. The fraction of tetramers Me4 is 1-4%, with the heavier metals maintaining larger fractions. While we also detected larger molecules, their probabilities are below 1%, which is smaller than the standard error of the present calculations. Using larger systems and longer times may help to provide more accurate insights into these fascinating vaporized structures. Note that comparison of the investigated alkali metals is not direct, since their critical temperatures are significantly different, and thus, the kinetic energies at the simulated conditions differ as well.

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, 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.

The shapes of the larger clusters Men are in good agreement with the cluster geometries reported in the recent work of Donoso and co-workers.6 At the same time, thermal geometry fluctuations are much more significant in our simulations, which have been conducted at


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substantially higher temperatures compared to the earlier simulations carried out at 300 K. Further, we did not observe Me7 molecules. Temperature increase leads to destruction of the Me-Me covalent bonds (Figure 6), irrespective of the alkali metal. Apart from Lin, the fractions of Men are very similar ranging within 29-37%. These results are important for understanding of the PVT data and deviations from ideality in the alkali metal vapors. Noteworthy, significant percentages of Me2 and Men (n > 2) exist even at the critical temperatures. Because of the experimental hurdles, the critical parameters of the alkali metals, including those used as a reference in the present work, are determined based on extrapolation.

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), Csn (cyan diamonds).

The size of Men was investigated as a function of density (simulation volume). The results are shown in Figure 7. While lowering the density has a marginal effect on the percentages of lone atoms and metallic dimers, the fractions of Me3 and Me4 decay very significantly. Therefore, pressure plays an important role in the formation of larger structures. For instance, the percentage of Li4 becomes barely detectable when the box side increases to 25 Å and above.



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Figure 7. Percentage of the alkali atom molecules (Me2, Me3, Me4) and lone atoms vs. the simulated box side length. The standard error of the computation was estimated to be ca. 1-2% based on the statistical processing of several trajectory parts.

A fairly small number of alkali metal atoms (20) was used in the present work to derive the fractions of different size molecules in the vapor phase. This number is sufficient to observe all possible compositions, since it is several times larger than the number of atoms in the biggest vapor phase structure. The cluster distribution was investigated as a function of the simulated system size in the recent work31 focusing on alkali metal chlorides. No undesirable deviations for the smaller systems were found.

Conclusions AIMD simulations of the alkali metal vapors were reported. While formation of alkali metal dimers in the gas phase is known from experiment, to the best of our knowledge, no theoretical simulation of this process has been attempted previously. Earlier modeling of small alkali metal clusters focused on low and ambient temperatures, which do not generate the vapor phase. The distribution of the cluster size in the vapor phase at high temperatures was reported here theoretically for the first time. ACS Paragon Plus Environment

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The analyses of the structure properties, such as RDFs and cluster size distributions, confirmed that the alkali metal atoms form molecules Men. The existence of n=2-4 was confirmed with good confidence. Larger molecules were also observed in the course of the AIMD simulations, but their percentages were smaller than the error bars. High pressure strongly favors larger structures, whereas high temperature decreases the fraction of larger clusters in the vapors. Heavier alkali metals maintain larger fractions of dimers, trimers, and tetramers. A single atom is the most frequently observed structure, irrespective of the element and temperature. The reported results are of fundamental importance, since understanding of the chemical behavior of Li, Na, K, Rb, and Cs can lead to development of new chemical reactions involving these metals. Because high temperatures and pressures are difficult to access in current experimental setups, AIMD presents a particularly valuable and promising tool for studies of such systems.

Acknowledgments O.V.P. acknowledges financial support of the US Department of Energy, grant No. DE-SC0014429.

Author Information E-mail for correspondence: [email protected] (O.V.P.); tel.:+1 (213) 821-3116.

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(29) Allouche, A. R. Gabedit-A Graphical User Interface for Computational Chemistry Softwares. J. Comput. Chem. 2011, 32, 174-182. (30) Periodic Table. http://periodictable.com. (accessed May 5, 2016). (31) Chaban, V. V.; Prezhdo, O. V. Ionic Vapor Composition in Critical and Supercritical States of Strongly Interacting Ionic Compounds. J. Phys. Chem. B 2016, 120, 4302–4309.



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Ab Initio Molecular Dynamics of Dimerization and Clustering in Alkali

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