**2. Introducing the relevant method for modeling and simulations**

The optimum structure of the materials and their corresponding applications can be predicted by modeling and simulation methods. They require analogous levels of

#### *Boron-Based Cluster Modeling and Simulations: Application Point of View DOI: http://dx.doi.org/10.5772/intechopen.105828*

precision and control that can also accurately describe the pertinent processes and conditions. As shown in **Figure 1**, across length and timescales, these methods can equip a wide range of opportunities to shed light on properties and phenomena that are unattainable through experimental effort. Among these methods, *ab initio* methods (such as density functional theory (DFT) calculations), standard molecular dynamics (MDs) simulations, and *ab initio* molecular dynamics simulations (AIMD) have been employed, mainly to study the properties of materials including boron clusters at the nanoscale.

*Ab initio* methods refer to those methods derived directly from theoretical principles, and their equations have not contained any empirical or semi-empirical parameters and the inclusion of experimental ones. The Hartree-Fock (HF) method is the simplest type of *ab initio* electronic structure calculations in which the correlated electron-electron repulsion is not explicitly included, and only its average effect is taken into account in the calculations [27, 28].

In DFT calculations, the ground-state energy is obtained as a function of a set of *n* one-electron Schrodinger-like equations, which are known as Kohn-Sham orbitals. This equation expresses the ground-state energy as a function of the interactions between the electrons, the nuclei, and themselves, the kinetic energy, and the exchange-correlation energy (see Eq. (1)). In these calculations, functionals (functions of another function) are employed to determine the properties of a manyelectron system. There is an approximation in the hamiltonian and the expression for

**Figure 1.** *Typical length and timescales in the simulation of the materials.*

the total electron density in the DFT calculations. However, these type of calculations can be very accurate for little computational cost [29–32]:

$$E[n] = E\_{Kin}[n] + E\_{Coul}[n] + E\_{xc}[n] + E\_{ext}[n],\tag{1}$$

Determining the exact functionals for exchange and correlation is the main problem in DFT. Accordingly, a bunch of functionals for DFT calculations is developed which can be classified from the simplest to the most accurate functionals. The exchange-correlation energy term in the functionals is constructed based on some approximation, i.e. local density approximation (LDA, see Eq. (2)), generalized gradient approximations (GGAs, see Eq. (3)), meta-GGA (see Eq. (4)), and hybrid functionals. For example, hybrid functionals are termed based on the density functional exchange functional in combination with the Hartree-Fock exchange term:

$$E\_{\rm xc}^{LDA}[\rho] = \int \rho(r) \varepsilon\_{\rm xc}(\rho(r)) dr,\tag{2}$$

$$E\_{\mathbf{x}\varepsilon}^{GGA}[\rho] = \int \rho(r) \varepsilon\_{\mathbf{x}\varepsilon}(\rho(r), \nabla \rho(r)) dr,\tag{3}$$

$$E\_{\rm xc}^{\rm MGGA}[\rho] = \int \rho(r) \varepsilon\_{\rm xc} \left( \rho(r), \nabla \rho(r), \nabla^2 \rho(r) \right) dr,\tag{4}$$

where *ρ* and *εxc* refer to the electronic density and the exchange-correlation energy per particle of a homogeneous electron gas with the charge density of *ρ*, respectively.

In the standard molecular dynamics (MDs) simulations, by considering classical treatment, Newton's second law is applied to the atomic coordinates. Then, force fields (FFs) which are a gradient of a prescribed interatomic potential functions are employed to calculate instantaneous force on each atom. FFs are the heart of MDs which are a function of the atomic coordinates and containing parameter sets (see Eq. (5)):

$$\overrightarrow{F}\left(\overrightarrow{R}\right) = \mathbf{U}\_{bonds} + \mathbf{U}\_{angles} + \mathbf{U}\_{dihedral} + \mathbf{U}\_{impprper} + \mathbf{U}\_{nonbond},\tag{5}$$

$$\mathbf{U}\_{bonds} = \sum\_{bonds} k\_i^{bonds} (r\_i - r\_0)^2,$$

$$\mathbf{U}\_{angles} = \sum\_{angles} k\_i^{angles} (\theta\_i - \theta\_0)^2,$$

$$\mathbf{U}\_{dihedral} = \sum\_{dihedrals} k\_i^{dihedral} (1 + \cos\left(n\_i \theta\_i - \delta\_i\right)),$$

$$\mathbf{U}\_{impmer} = \mathbf{V}\_{imp}$$

$$\mathbf{U}\_{nonbond} = \sum\_{ij} 4k\varepsilon\_{ij} \left(\frac{\sigma\_{ij}^2}{r\_{ij}^2} - \frac{\sigma\_{ij}^6}{r\_{ij}^6}\right) + \sum\_{dec} \frac{q\_i q\_j}{r\_{ij}}$$

where the local contributions to the total energy are included in the first four terms, i.e. bond stretching, angle bending, dihedral, and improper torsions. In this case, when considering a 12-6 Lennard-Jones potential, the repulsive, van der Waals, and coulombic interactions are described in the last two terms. The parameters are derived from experiments and quantum mechanics. After that, the position and velocity of the particles can be calculated by numerical integration [33–35].

#### *Boron-Based Cluster Modeling and Simulations: Application Point of View DOI: http://dx.doi.org/10.5772/intechopen.105828*

In the *ab initio* MD simulations (AIMDs), at first, the interatomic forces are found at a given time instant. Then, from a quantum-mechanical perspective, the system is parameterized as a function of nuclei and electrons coordinates at a fixed time. Using the Born-Oppenheimer (BO) approximation, the nuclei are considered to be fixed at the instantaneous positions of the atoms. Consequently, the time-independent Schrodinger equation can be invoked to calculate the many-body electron wave function and the energy. In fact, the obtained energy is a function of the nuclei coordinates which were previously considered fixed. This energy can be considered as an interatomic potential to obtain the forces in Newton's equation of motion. In the other words, the gradients of the DFT energy at this fixed point can be calculated to obtain forces in which the nuclei are moved by this force to reach the next time step. This whole mentioned process is then repeated for these new atomic positions [36, 37].

The calculation way of the interatomic forces and the computational costs are the manifest and the origin of the difference between the standard MDs, AIMDs, and DFT calculations. AIMDs can be applied only for small system sizes, due to its huge computational cost. Also, AIMDs allow determining the dynamics of the systems that have no FFs. Intrinsically, AIMDs can deal with some effects such as polarization, bonding, many-body effects, and charge transfer, whereas in standard MDs these effects are artificially imposed from the data. Moreover, DFT as a quantum mechanical method for calculating energy as well as other properties of the material is a timeconsuming technique. However, empirical potentials (FFs) are much faster but less accurate than the *ab initio* method like DFT. Finaly, the method selection for a specific case should be made based on these factors.
