**2. Theoretical methods and computational details**

### **2.1 Global minimum search**

Despite advances in computing power, the minimum global search in molecular and atomic clusters remains a complicated task due to several factors. The exploration should be systematic and unbiased [68, 105]; a molecule's degrees of freedom increase with the number of atoms [68, 106–109]; a molecule composed of N number of atoms possesses 3 N degrees of freedom (i.e., a linear molecule has three degrees of translation, two of rotation, and [3 N-6] of vibrational modes); and, as a consequence, the potential/free energy surface depends on a large number of variables. The number of local minima increases exponentially as a function of the number of atoms in the molecule. Moreover, the total energy computation requires a quantum mechanical methodology to produce a realistic value for energy. In addition to that, there should be many initial structures. It is essential to sample a large region of the configuration space to ensure that we are not missing structures, making an incomplete sampling of the configurational space and introducing a significant problem to calculating the thermodynamic properties [64]. A complete sampling of the potential/free energy surface is impossible, but a systematic exploration of the potential energy surface is extremely useful. Although searching for a global minimum in molecular systems is challenging, the design and use of algorithms dedicated to the search for global minima, such as simulated annealing, [110–115] kick method [116, 117], genetic algorithms [118–120], Gradient Embedded Genetic Algorithm [121–123] and basin hopping [124, 125], has been accomplished over the years. In the past few years, one of us designed and employed genetic algorithms [12, 13, 29, 39, 98, 99, 104, 126, 127] and kick methodology [101, 127–133] coupled with density functional theory to explore atomic and molecular clusters' potential energy surfaces. They have led us to solve the minimum global search in a targeted way. In this chapter, our computational procedure to elucidate the low-energy structures employs a recently developed natureinspired hybrid strategy that combines a *Cuckoo* search [134] and genetic algorithms coupled to density functional theory that has been implemented in the GALGOSON code v1.0. Nature-inspired metaheuristic algorithms have been applied in almost all areas of science, engineering, and industry, work remarkably efficiently, and have many advantages over deterministic methods [135]. GALGOSON systematically and efficiently explores potential/free energy surfaces (PES/FES) of the atomic clusters to find the minimum energy structure. The methodology consists of a three-step search strategy where, in the first and second steps, we explore the PES, and in the third step, we explore the FES. First, the code builds a generation of random initial structures with an initial population of two hundred individuals per atom in the Be-B clusters using a kick methodology. The process to make 1D, 2D, and 3D structures is similar to that used in previous work [12] and are restricted by two conditions [12] that can be summarized as follows: (a) All the atoms are confined inside a sphere with a radius determined by adding all atoms' covalent radii and multiplied by a factor established by the user,

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

typically 0.9. (b) The bond length between any two atoms is the sum of their covalent radii, modulated by a scale factor established by the user, typically close to 1.0; this allows us to compress/expand the bond length. These conditions avoid the high-energy local minima generated by poorly connected structures (too compact/ loose). Then, structures are optimized at the PBE0/3-21G level of theory employing Gaussian 09 code. As the second step, all energy structures lying in the energy range of 20 kcal/mol were re-optimized at the PBE0-GD3/LANL2DZ level of theory and joints with previously reported global minimum structures. Those structures comprised the initial population for the genetic algorithm. The optimization in this stage was at the PBE0-D3/LANL2DZ level of theory. The criterion to stop the generation is if the lowest energy structure persists for 10 generations. In the third step, structures lying in 10 kcal/mol found in the previous step comprised the initial population for the genetic algorithm that uses Gibbs free energy extracted from the local optimizations at the PBE0-D3/def2-TZVP, taking into account the zero-point energy (ZPE) corrections. The criterion to stop is similar to that used in the previous stage. In the final step, the lowest energy structures are evaluated at a single point energy at the CCSD(T)/def2-TZVP// PBE0-D3/def2-TZVP level of theory. All the calculations were done employing the Gaussian 09 code [136].

#### **2.2 Thermochemistry properties**

All the information about a quantum system is contained in the wave function; similarly, the partition function provides all the information need to compute the thermodynamic properties and it indicates the states accessible to the system at temperature T. Previous theoretical studies used the partition function to compute temperature-dependent entropic contributions [137] on [Fe(pmea)(NCS)2] complex, infrared spectroscopy on anionic Be6B11 cluster [39], and rate constant [100]. In this work, the thermodynamic functions are calculated using the temperaturedependent partition function Q shown in Eq. (1).

$$Q(T) = \sum\_{i} \mathbf{g}\_{i} \, e^{-\Delta E\_{i}/k\_{B}T} \tag{1}$$

In Eq. (1), the *gi* is the degeneracy or multiplicity, using degeneracy numbers is equivalent to take into account all degenerate states and the sum runs overall energy levels, and *kB* is the Boltzmann constant, T is the temperature and *ΔEi* is the total energy of a molecule [100, 138]. At high temperatures, all thermal states are accessible due to the term �*ΔEi=kBT* tends to zero, and the partition tends to infinity. An exact calculation of Q could be complicated due to the coupling of the internal modes, a way to decouple the electronic and nuclei modes is through the use of Born-Oppenheimer approximation. (BOA) This approach says that the electron movement is faster than the nuclei and assumes that the molecular wave function is the electronic and nuclear wavefunction product *ψ* ¼ *ψeψn*. The vibrations change the momentum of inertia as a consequence, affect the rotations; this fact tightly couple the vibrational and rotational degrees of freedom; The separation of rotational and vibrational modes is called the rigid rotor, harmonic oscillator (RRHO) approximation, under this approximation, the molecule is treated rigidly, this is generally good when vibrations are of small amplitude. Here the vibration will be modeled in terms of harmonic oscillator and rotations in terms of the rigid rotor. Within BOA and RRHO approximations, the partition function is factorized into electronic, translational, vibrational, and rotational energies. Consequently, the partition function, Q, can be given in Eq. (2) as a product of the corresponding

contributions [100, 139], and under the rigid rotor, harmonic oscillator, Born-Oppenheimer, ideal gas, and a particle-in-a-box approximations.

$$\mathbb{Q} = q\_{trans} q\_{rot} q\_{vib} q\_{elec.} \tag{2}$$

**Table 1** shows the contributions of electronic, translational, vibrational, and rotational to the partition function.

We computed all partition functions at temperature T and a standard pressure of 1 atm. The equations are equivalent to those given in the Ref. [100], and any standard text of thermodynamics [138, 139] and they apply for an ideal gas. The implemented translational partition function in the Gaussian code [136] is the partition function, *q* ¼ *qtrans*, given in **Table 1**. In this study, the *q* ¼ *qtrans* is computed as a function of T and is used to calculate the translational entropy. In addition to using vibrational modes to identify true lowest energy structures from transition states, we also used them to compute the vibrational partition function. In this study is considered vibrational modes, *ν*, under the harmonic oscillator approximation, and total vibrational energy consists of the sum of the energies of each vibrational mode. In computing the electronic partition, we considered that the energy gap between the first and higher excited states is more considerable than *kBT*, as a consequence electronic partition function, *q* ¼ *qelect*, is given by *qelect* ¼ *ω0*, *qrot*, *qnl rot*, *q* ¼ *qtrans* are used to compute the entropy contributions given in **Table 2**.

The vibrational frequencies are calculated employing the Gaussian code, and all the information needed to compute the total partition function is collected from the output. The Gibbs free energy and the enthalpy are computed employing the Eqs. (3) and (4).

$$H = U + nRT,\tag{3}$$

$$\mathbf{G} = \mathbf{H} - \mathbf{T} \mathbf{S}.\tag{4}$$

#### **2.3 Boltzmann population**

The properties observed in a molecule are statistical averages over the ensemble of geometrical conformations or isomers accessible to the cluster [140]. So, the molecular properties are ruled by the Boltzmann distributions of isomers that can change due to temperature-entropic term [23, 71, 101], and the soft vibrational modes that clusters possess make primary importance contributions to the entropy [93]. The Boltzmann populations of the low-energy isomers of the cluster Be6B11 � and Be4B8 are computed through the probabilities defined in Eq. (5)

