**3.2 Boltzmann population of Be6B11 cluster**

**Figure 2a** shows the most important and strongly dominating Tss1-g point that is located at 377 K temperature scale with a relative population of 33%. For temperatures ranging from 10 to 377 K, the relative population is strongly dominated by the putative global minima isomer distorted oblate spheroid with C1 symmetry and this relative population is similar to -T<sup>3</sup> function with one point of inflection located at 180 K. After decreases monotonically up to 377 K. At the Tss1-g point, the distorted oblate spheroid with C1 symmetry co-exist and compete with the coaxial Triple-Layered structures with Cs symmetry; This implies that the distorted oblate spheroid will be replaced with the coaxial Triple-Layered structures. Above temperature 377 K, the relative population is strongly dominated by the coaxial Triple-Layered structures with Cs symmetry, located at 0.85 kcal/mol above the global minima at temperature 298.15 K. This relative population depicted in blue-solid line in panel (a) has behavior as a sigmoid function, from temperatures ranging from 377 to 600 K, it grows rapidly and from temperatures ranging from 600 to 1500 K, it almost keeps constant with 60%. The second Tss2-g point is located at temperature 424 K with a relative population of 22.9%, and this point the global minima distorted oblate spheroid with C1 symmetry co-exist, and compete with the coaxial Triple-Layered structures with C2v symmetry, located at 1.23 kcal/mol above the global minima at 298.15 K. The relative population of the coaxial Triple-Layered C2v symmetry depicted in green-solid line in panel (a) also has a behavior of a sigmoid function and up to 600 K it keeps constant with 32% of relative population. The Tss3-g, and Tss4-g points are located at 316.7 K, and 349 K axis temperature with

#### **Figure 2.**

*Panel (a) shows the Boltzmann population of the Be6B11 (ensemble at thermal equilibrium) for the temperatures ranging from 10 to 1500 K computed at the PBE0-D3/def2-TZVP level of theory. Panel (b) shows the Boltzmann population for the temperatures ranging from 10 to 1500 K computed at the CCSDT/ def2-TZVP//PBE0-GD3/def2-TZVP level of theory. At the temperature of 350 K, four structures co-exist with 20% of probability.*

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

relative populations 14% and 17%, respectively. These relative populations correspond to the second isomer located just 0.61 kcal/mol at 298.15 K above the global minima, and co-existing at the temperatures 316.7 K and 349 K with the coaxial Triple-Layered structures with Cs, and C2v symmetries, respectively. At low temperatures range, this isomer's relative population depicted in red-solid line of **Figure 2a** is around only 20%, and up to room temperature, it decreases exponentially to zero. At temperatures up to 600 K, the relative population is zero; hence, at high temperatures these isomers do not contribute to the molecular properties. The relative population lower than 10%, depicted in violet-solid line shows in **Figure 2a**, correspond to the isomers located at 1.48 kcal/mol above global minima at 298.15 K. Interesting, this structure is the putative minimum global when the CCSD(T) energy is employed in the ordering energetic, Despite that, this structure's relative population clearly shows that this structure does not contribute to molecular properties in all ranges of temperatures.
