**1. Introduction**

Boron is the smallest and lightest semi-metal atom [1, 2] and a neighbor of carbon in the periodic table. Moreover, it has high ionization energy of 344.2 kJ/mol [3], and an affinity for oxygen atoms, which is the basis of borates [3, 4]. In recent years, the pure boron clusters, the metal, and non-metal doped boron clusters, have attracted considerable attention [1, 5–13] due to their unpredictable chemistry [14, 15] and high potential to form novel structures [16]. The potential of boron atoms to form stable molecular networks [17] lies in the fact that they have three valence electrons and four available orbitals, which implies they are electron-deficient. Boron electron deficiency gives origin to a vast number of allotropic forms and uncommon geometries [6, 16] such as nanotubes [13, 18], borospherenes [19], borophene [16], cages [13, 20], planar [21], quasi planar [22], rings [23, 24], chiral [22, 25–28], boron-based helix clusters [25, 29], and fluxional boron clusters [10, 29–32] that have recently attracted the interest of experimental and theoretical researchers. Aromaticity, antiaromaticity, and conflicting aromaticity dominate the chemical bonding in boron-based clusters [25, 33–35]. The two most-used indices for quantifying aromaticity are the harmonic oscillator model of aromaticity, based on the geometric structure, and the nucleus-independent chemical shift, based on the magnetic response. Aromaticity is not observable, cannot be directly measured [36], and correlates with electronic delocalization [37]. The fluxionality in boron and boron-doped-based molecular systems is highly relevant in terms of its catalytic activity [38] and is due to electronic delocalization [25]. Moreover, in boron-based nanoscale rotors, electronic localization o delocalization contributes significantly to stability, magnetic properties, and chemical reactivity [36], and it is a function of the atomic structure, size, bonding, charge, and temperature [39]. So far, doping a boron cluster with non-metals [40] dramatically affects its structure, stability, and reactivity, like shut-down the fluxionality of the boron-doped anion B19. In contrast, doping a boron cluster with metals [7, 9, 24, 41–43] like beryllium-doped boron clusters, exhibit remarkable properties such as fluxionality [16, 29, 32, 44–46], aromaticity [29, 47], and characteristics similar to borophene [1]. Furthermore, previous theoretical studies showed that the boron fullerenes B60 and B80 can be stabilized by surrounding the boron clusters with beryllium atoms [48, 49], which effectively compensates for boron electronic deficiency [49]. These effects make beryllium-doped boron clusters interesting, joined with the fact, nowadays, dynamic structural fluxionality in boron nanoclusters is a topic of interest in nanotechnology [19, 50]. Particularly attractive are the chiral helices Be6B11 , reported by Guo [29], and Buelna-Garcia et al. [39] as one of the low-lying and fluxional isomers. Later, a chemical bonding and mechanism of formation study of the beryllium-doped boron chiral cluster Be6B10<sup>2</sup> and coaxial triple-layered anionic Be6B11 sandwich structures were reported [25, 46]. In these structures, the chirality arises due to the formation of a boron helix. Particularly, the chirality of nanoclusters has attracted attention due to their chiroptical properties, potential application in efficient chiral discrimination [51, 52], nonlinear optics [53] and chiral materials with interesting properties [22, 54], and of course, not to mention that chiral structures play a decisive role in biological activity [55]. Previous theoretical studies joint with experimental photoelectron spectroscopy reported the first pure boron chiral B30 structure as the putative global minimum [22] at T = 0. In these pair of planar enantiomers, the chirality arises due to the hexagonal hole and its position. In the past years, the lowest energy structures of the B39 borospherene were reported as chiral due to their hexagonal and pentagonal holes [26]. Similarly, the B44 cluster was reported as a chiral structure due to its nonagonal holes [28]. That is, in these clusters, holes in the structure cause chirality. So far, the chirality depends on the geometry; In contrast, fluxionality strongly depends on temperature. A boron molecular Wankel motor [56–58] and sub nanoscale tank treads have been reported [59, 60]; however, the temperature have not been considered. Nevertheless, most theoretical density functional studies assume that the temperature is zero and neglect temperature-dependent and entropic contributions; consequently, their finite

### *Boltzmann Populations of the Fluxional Be6B11 and Chiral Be4B8 Clusters at Finite… DOI: http://dx.doi.org/10.5772/intechopen.100771*

temperature properties remain unexplored [61, 62]. Experimental studies are carried out in non-zero temperatures, then it is necessary to understand the effect of the temperature on the cluster properties and the lowest energy structure's determination [61–63]. Herein, we investigate the effect of temperature-entropy term on the Boltzmann population, which needs the elucidation of the putative global minimum and its low-energy isomers [39, 64–68]. The properties observed in a molecule are statistical averages over the ensemble of geometrical conformations that are ruled by the Boltzmann distributions of isomers. So we need an efficiently sampling of the free energy surface to know the distribution of isomers at different temperatures [39, 68– 71]. A considerable change in the isomer distribution and the energetic separation among them is the first notable effect of temperature [39]. Useful materials work at finite temperatures; in that conditions, Gibbs free energy is minimized whereas, the entropy of the atomic cluster is maximized [39, 72]. and determines the putative global minimum at a finite temperature [39]. Although in the mid 1960's, Mermin et al. [73] studied the thermal properties of the inhomogeneous electron gas, most DFT calculations are typically performed at zero temperature. Recently, over again, DFT was extended to finite temperature [74–76], but nowadays, as far as we know, it is not implemented in any public software and practical calculations are not possible. Taking temperature into account requires dealing with small systems' thermodynamics; The Gibbs free energy of classical thermodynamics also applies for small systems, known as thermodynamics of small systems [77–79]. The thermodynamics of clusters have been studied by various theoretical and simulation tools [61, 68, 77, 80–86] like molecular-dynamics simulations. Previous reports investigated the behavior of Al12C cluster at finite temperature employing Car-Pirinello molecular dynamics [61], and dynamical behavior of Borospherene in the framework of Born-Oppenheimer Molecular Dynamics [5, 10]. Under the harmonic superposition approximation, the temperature-entropy term can be computed with the vibrational frequencies on hand. The entropy and thermal effects have been considered for gold, copper, water, and sodium clusters [71, 87–95]. Franco-Perez et al. [96] reported the thermal corrections to the chemical reactivity at finite temperature, their piece of work validates the usage of reactivity indexes calculated at zero temperature to infer chemical behavior at room temperature. Gazquez et al. [97] presented a unified view of the temperature-dependent approach to the DFT of chemical reactivity. Recently, the effect of temperature was considered by Castillo-Quevedo et al. reported the reaction rate and the lowest energy structure of copper Cu13 clusters at finite temperature [98, 99]. Dzib et al. reported Eyringpy; A Python code able to compute the rate constants for reactions in the gas phase and in solution [100], Vargas-Caamal et al. computed the temperature-dependent dipole moments for the HCl(H2O)n clusters [101], Shkrebtii et al. computed the temperature-dependent linear optical properties of the Si(100) surface [102], several authors take into account the temperature in gold clusters [87–89], and thermochemical behavior study of the sorghum molecule [103], and more recently, Buelna-Garcia et al. [104] employing density functional theory and nanothermodynamics reported the lowest energy structure of neutral chiral Be4B8 at a finite temperature. and reported that the fluxionality of the anionic Be6B11 clusters depends strongly on temperature [39]. In this work, we employed density functional theory, statistical thermodynamics, and CCSD(T) to compute the Gibbs free energy and the Boltzmann population at absolute temperature T for each neutral chiral Be4B8 and anionic Be6B11 isomers. We think that this provides useful information about which isomers will be dominant at hot temperatures. No work has previously been attempted to investigate entropy-driven isomers in the fluxional Be6B11 and chiral Be4B8 cluster at CCSDT level of theory to the best of our knowledge. The remainder of the manuscript is organized as follows: Section 2 gives the computational details and a brief overview of the theory and algorithms

used. The results and discussion are presented in Section 3. We discuss the effect of the symmetry in the energetic ordering and clarify the origin of the 0.41 kcal/mol difference in energy between two structures with symmetries C2 and C1 appear when we compute the Gibbs free energy. A comparison among the energies computed at a single point CCSD(T) against the DFT levels of theory and the T1 diagnostic is presented. Conclusions are given in Section 4.
