**3. Results and discussion**

#### **3.1 The lowest energy structures and energetics**

**Figure 1** shows the lowest energy structure of Be6B11 clusters and seven lowenergy competing isomers computed at the PBE0-D3/def2-TZVP basis set. For the putative global minimum at the PBE0-D3/def2-TZVP, the optimized average B-B bond length is 1.64 Å. In contrast, the optimized B-Be bond length is 2.01 Å. At the PBE0-D3/def2-TZVP and temperature of 298.15 K, the putative global minimum with 54% of the relative population has C1 symmetry with a singlet electronic state <sup>1</sup> A. It is a distorted, oblate spheroid with three berylliums in one face and two in the other face. Nine-boron and one-beryllium atoms are forming a ring located around the spheroid's principal axes and the remaining two boron atoms are located close to the boron ring in one of its faces. The second higher energy structure, at 298.15 K, lies only 0.61 kcal/mol Gibbs free energy above the putative global minima, and it has C1 symmetry with a singlet electronic state <sup>1</sup> A. It is a prolate spheroid with 19% of the relative population at a temperature of 298.15 K. The next two higher energy isomers, at 298.15 K, lies at 0.85 and 1.23 kcal/mol Gibbs energy above the putative global minimum. They are prolate, coaxial Triple-Layered structures with Cs, and C2v symmetries with singlet electronic states, <sup>1</sup> A, respectively. This clearly, shows that the low-symmetry structure C1 become more energetically preferred than the C2v symmetry by Gibbs free energy difference of 0.38 kcal/mol at 298.15 K, due to entropic effects and in agreement with a similar result found in Au32 [105]. Indeed,

#### **Figure 1.**

*The optimized geometries of Be6B11 cluster. The most important energy isomers show in two orientations, front, and rotated 90 degrees up to plane paper. Relative Gibbs free energies in kcal/mol (in round parenthesis) and the relative population [in square parenthesis], at PBE0-D3/Def2-TZVP level of theory. The criterium to plot them is until the probability occupation is zero. The pink- and yellow-colored spheres represent the boron and beryllium atoms, respectively.*

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

and according to our computations, those structures are strongly dominating at temperatures higher than 377 K. The next structure is shown in **Figure 1**(**5**), is located at 1.48 kcal/mol above the global minimum; it is close to a spherical shape and correspond to a prolate structure with C1 symmetry, and a singlet electronic state <sup>1</sup> A; this structure only has 4.4% of the relative population at 298.15 K. The next two structures, located at 2.37 kcal/mol Gibbs free energy above the global minimum, are the chiral helix-type structures, reported by Guo [29] as minimum global. They are prolate structures with C2v symmetries, and their relative population is around only 1%. We must point out that those chiral-helix structures never become the lowest energy structures in all ranges of temperature. The relative population is zero for structures located at higher relative Gibbs free energy than 5.1 kcal/mol, and at 298.15 K, there is no contribution of these isomers to any total molecular property. A full understanding of the molecular properties requires the search of global minimum and all its closest low-energy structures [64]. The separation among isomers by energy-difference is an important and critical characteristic that influences the relative population and, consequently, the total molecular properties. We computed the global minima and the first seven low-energy to gain insight into how the energy-gap among isomers change and how the energy-ordering of the low-energy structures is affected at a single point CCSD(T)/def2-TZVP level of theory corrected with the zero-point energy computed at the PBE0-D3/def2-TZVP level of theory. At the CCSD(T) level of theory, the global minima, the seven lowest energy isomers, and the energy order agree with previous work [39], as seen in the first row of **Table 3**. The second row of **Table 3** shows the corrected CCSDT+EZPE energy. Interestingly, the energetic ordering does not change when we take into account the ZPE energy. Nevertheless, the energy difference among isomers was reduced drastically. we can deduce that the ZPE energy inclusion is essential in the isomers' energy ordering and molecular properties. The third row of **Table 3** shows the energy-order considering the Gibbs free energy computed at 298.15 K; at this temperature, the isomers energy-ordering is changed, the second isomers take the putative global minima place, and the first isomers take the fifth place. Interestingly, this energy-ordering is at 298.15 K. This energy-ordering is a complete function of the temperature that we will discuss later in the relative population section. The fourth row in **Table 3** shows the electronic energy taking into account the ZPE energy. It follows the same trend in energy-ordering when considering the Gibbs free energy, and it is the same putative global minima. The fifth row in **Table 3** is just electronic energy. It almost follows the CCSD(T) energies trend, except the


#### **Table 3.**

*The relative energies in kcal*�*mol*�*<sup>1</sup> , coupled-cluster single-double and perturbative triple, CCSD T*ð Þ*, CCSD T*ð Þ *with zero-point energy (εZPE), (CCSD T*ð Þþ *εZPE), Gibbs free energy (ΔG) at 298.15 K, electronic energy with εZPE (*%*mcalEε0), electronic energy (ε*0*), point group symmetry, electronic ground state, and the lowest frequency in cm*�*<sup>1</sup> for eight low-energy isomers.*

isomers number eight that take the second place located at 0.52 kcal/mol above the putative global minima. The sixth, seventh and eighth rows on **Table 3** show the point group symmetry, electronic ground state, and the lowest vibrational frequency of each isomer. When we take the Gibbs free energy to energy-ordering structures, the second isomers interchange to the first place, becoming the lowest energy structure; The energy ordering change drastically, whereas the electronic energy almost follows the same trend CCSD(T) energy-ordering. This shows us that the level of theory and the inclusion of entropy and temperature change the energyordering; therefore, the total molecular properties.
