**Table 1.**

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


#### **Table 2.**

*Contributions to internal energy and entropy.*

$$P(T) = \frac{e^{-\beta \Delta G^{\circ}}}{\sum e^{-\beta \Delta G^{\circ}}} \tag{5}$$

where *β* ¼ 1*=kBT*, and *kB* is the Boltzmann constant, T is the temperature in Kelvin, *ΔG* is the Gibbs free energy of the kth isomer. Eq. (5) establishes that the distribution of molecules will be among energy levels as a function of the energy and temperature. Eq. (5) is restricted so that the sum of all probabilities of occurrence, at fixed temperature T, Pi (T) is equal to 1 and given by Eq. (6)

$$\sum\_{i=1}^{n} P(T) = \mathbf{1} \tag{6}$$

It is worth mentioning that the energy difference among isomers is determinant in the computation of the solid–solid transition, Tss point. Tss occurs when two competing structures are energetically equaled, and there is simultaneous coexistence of isomers at T. in other words, the Tss point is a function of the energy difference between two isomers and the relative energy *ΔG* that the cluster possesses. Boltzmann distribution finds a lot of applications as like simulated method annealing applied to the search of structures of minimum energy, rate of chemical reaction [100], among others. For the calculation of the Boltzmann populations, we used a homemade Python/Fortran code called BOFA (Boltzmann-Optics-Full-Ader).

#### **2.4 Computational details**

The global exploration of the potential and free energy surfaces of the Be6B11� and Be4 B8 clusters were done with a hybrid Cuckoo-genetic algorithm written in Python. All local geometry optimization and vibrational frequencies were carried out employing the density functional theory (DFT) as implemented in the Gaussian 09 [136] suite of programs, and no restrictions in the optimizations were imposed. Final equilibrium geometries and relative energies are reported at PBE0 [141] /def2- TZVP [142] level of theory, taking into account the D3 version of Grimme's dispersion corrections [143] and including the zero-point (ZPE) energy corrections. As Pan et al. [144] reported, the computed relative energies with PBE0 functional are very close to the CCSD(T) values in B9 � boron cluster. The def2-TZVP basis from the Ahlrichs can improve computations accuracy and describe the Be-B clusters [29]. To gain insight into its energetics, we evaluated the single point energy CCSD (T)/def2TZVP//PBEO-D3/def2-TZVP level of theory for the putative global

minima and the low-energy Be6B11 isomers, and employing Orca code at the DLPNO-CCSD(T) theoretical level for the low-energy isomers of Be4B8 cluster. Boltzmann-Optics-Full-Ader, (BOFA) is employed in the computation of the Boltzmann populations. The code is available with the corresponding author.
