**3.3 The lowest energy structures of Be4B8 clusters**

**Figure 3** shows the low-energy configurations of Be4B8 clusters optimized at PBE0-D3/def2-TZVP level of theory taking into account ZPE energy correction. The optimized average B-B bond length of the putative chiral global minimum is 1.5867 Å, in good agreement with an experimental bond length of 1.57–1.59 Å [145, 146], and also within agreement with others previous DFT calculations [39]. The most recurring motif within the lower energy isomers of B8Be4 is a sandwich structure, (SSh) in which the boron atoms form a hollow distorted ellipsoid ring

#### **Figure 3.**

*Optimized geometries of a neutral Be4B8 cluster at the PBE0-D3/def2TZVP level of theory with zero-point correction energy. These are shown in front and side views. The first letter is the isomer label, the relative Gibbs free energies in kcalmol<sup>1</sup> (in round parenthesis) at 298.15 K, the relative population (in square brackets), and the group symmetry point (in red round parenthesis). The structures with labels (a and b), (c and d), (e and f), (i and j), (k and l) and (h) are chiral. The purple- and yellow-colored spheres represent the boron and beryllium atoms, respectively.*

with each of the Be-Be dimers capping the top and bottom with C1 point group symmetry. Isomers 1 and 2 are also listed as i1 and i2 in **Table 4**, are enantiomers differing in the orientation of the Be-Be dimers with respect to the boron skeleton. The Be-Be bond length for the six lowest energy enantiomers is 1.9874, 1.9876, and 1.9881 Å for symmetries C1, C2, and D2, respectively, in good agreement with the bond length of the Be-Be in Be2B8 cluster 1.910 Å [44]. To gain insight into the energy hierarchy of isomers and validate our DFT calculations, relative energies were computed at different levels of theory, and differences between them are shown in **Table 4**. Energy computed at different methods yield different energies due mainly to the functional and basis-set employed, [39, 147], so the energetic ordering change; consequently, the probability of occurrence and the molecular properties will change. The first line of **Table 4** shows the relative Gibbs free energy computed at PBE0-D3/def2-TZVP and room temperature. The small relative Gibbs free energies (0.41, and 0.81 kcal/mol) differences among the six enantiomer structures i1 to i6 in **Table 4** are caused by the rotational entropy being a function of the symmetry number that in turn depends on the point group symmetry. An increase/decrease in the value of rotational entropy changes the Gibbs free energy. The Gibbs free energy computed with and without symmetry will differ by a factor RTln(σ). Here, R is the universal gas constant, T, the temperature, and σ is the symmetry number. The computed factor at room temperature with σ = 2 is RTln (σ) = 0.41 kcal/mol, and it is RTln(σ) = 0.81 kcal/mol with σ = 4, in agreement with the values shown in the first line of **Table 4**. As the temperature increases, the energy differences between the factors RTln(σ) become larger. These small relative Gibbs free energies are responsible for different values of probability of occurrence at low temperatures for the similar isomers with different point group symmetry. This strongly suggests that there must be atomic clusters with low and high symmetries in the Boltzmann ensemble to compute the molecular properties correctly. The second line in **Table 4** shows single point (SP) relative energies computed at the CCSD(T) [148], the energetic ordering of isomers listed in the first line of **Table 4** follows almost the trend of energetic ordering at SP CCSD(T) level, notice that just the achiral isomers label i7 to i8 in **Table 4** are interchanged in energetic ordering. The third line **Table 4** shows single point relative energies computed at


#### **Table 4.**

*Single-point relative energy calculations of the low-energy structures from* i*<sup>1</sup> to* i*<sup>10</sup> at different levels of theory: coupled cluster single-double and perturbative triple (CCSD(T)), CCSD(T) with zero-point energy (*CCSDð Þþ *T* εZPE*, CCSD(T)) employing the domain-based local pair natural orbital coupled-cluster theory (DLPNO-CCSD(T)), with TightPNO setting, and with <sup>ε</sup>*ZPE *(*DLPNO‐CCSDð Þþ *<sup>T</sup>* <sup>ε</sup>ZPE*), Gibbs free energy (*ΔG*) at 298.15 K, electronic energy with ε*ZPE *(ε*<sup>0</sup> þ εZPE*), electronic energy (ε*0*), point group symmetry, and T1 diagnostic. All relative energies are given in kcal*�*mol*�*<sup>1</sup> .*

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

the CCSD(T) [148]/def2-TZVP//PBE0-D3/def2-TZVP; the energetic ordering is similar to pure CCSD(T) energy. DLPNO-CCSD(T) relative energies, with and without ZPE correction, are shown in lines four and five of **Table 4**, the first follows the trend of pure CCSD(T) energy, and the second, the ZPE value, interchange the isomers, label i7 in **Table 4**, to be the putative global minimum. Here we can say that the ZPE energy inclusion is essential in distributing isomers and molecular properties. The sixth and seventh lines of **Table 4** show the electronic energy with and without ZPE correction, and both of them follow the trend of the Gibbs free energy given in line number one. Line number 8 in **Table 4** shows the point group symmetry for each isomer. The T1 diagnostic for each isomer is shown in line nine of **Table 4**, all of them are lower than the recommended value 0.02 [148] so the systems are appropriately characterized.
