Atomistic Scale

Many macroscopic phenomena originate from elementary processes that take place in the atoms. The atomistic scale considers the internal dynamics of atoms, i.e., the behavior of matter at a subatomic scale. Table 3.1 shows the properties of the most relevant subatomic particles, considered in polymerization processes. The knowledge of the forces between subatomic particles should, in principle, be sufficient to determine the behavior of matter at all scales. The corresponding space and time scales considered are in the order of 10"¹⁵ m to 10"¹⁰ m (corresponding to the range of sizes from the nucleus to the electronic cloud around a single atom) and 10"¹⁸ s to 10"¹⁵ s (from the relaxation time of an electron to the vibration period of a molecular bond). At this scale, the governing equation describing the interaction between subatomic particles is the Schrodinger equation (Eq. 3.1). It is worth noticing that the Schrodinger equation is a fundamental equation, which does not involve any empirical parameter but only fundamental constants such as the mass and charge of the electron, and Planck constant. The more general time-dependent Schrodinger equation is expressed as follows:

where 4'(t) is the wavefunction of the system, (|^VP²| describes the probability of finding a particle in space), H is the Hamiltonian operator (representing the total energy of the system, as a function of its wavefunction), b is the reduced Planck constant (1.055 x 10"³⁴ J-s), i is the unit imaginary number (V-l), t is time, and d/dt is the partial derivative with respect to time.

Table 3.1 Relevant subatomic particles and their properties

At stationary states of the system, the Schrodinger equation can be simplified into its time-independent form:

where E is the energy of the wavefunction at the stationary state.

Even though Eq. 3.1 and Eq. 3.2 are relatively simple expressions, due to the analytical complexity of the wavefunction (ЧК), exact analytical solutions to the Schrodinger equation are only possible for just a few cases, the most notable one being the Hydrogen atom [6]. On the other hand, the numerical solution to the Schrodinger equation for a particular system becomes difficult as the size of the system increases, because one equation is needed for each electron or nucleus (conformed by protons and neutrons) present in the system. In addition, for larger systems, the range of characteristic energies considered is wider, leading to larger numerical errors. Thus, the numerical solution to the Schrodinger equation is usually restricted by computational capability to just individual atoms or very small molecules. For these reasons, some strategies have been proposed for efficiently solving the Schrodinger equation in larger systems, including the Born-Oppenheimer (BO) [7] and the Hartree- Fock (HF) [8] approximations.

The BO approximation is based on the fact that the mass of nuclei is three orders of magnitude larger than the mass of electrons, and since their momenta is in the same order of magnitude, the velocities of the nuclei are negligible compared to the velocities of the electrons. Thus, the BO approximation assumes that all nuclei are stationary (classical point-like particles with zero kinetic energy) and calculates the energy of the system for this electronic ground-state configuration. The BO approximation is the basis of the concept of potential surface energy from which the geometry of chemical structures is extracted [9].

Electronic ground state: State of lowest energy of the electrons in the

system.

The solution to the problem after the BO approximation is still difficult because the electrons are correlated as a result of inter- electronic interactions. In the HF approximation, the wavefunction of the system is assumed to be the sum of separable basis functions, representing each electron. Since the exact function representing each electron is not known, a set of basis functions is assumed, and the total energy of the system is determined. Then, by means of an iterative process, the set of basis functions is modified until a minimum value for the ground-state energy is found, which would be the best approximation to the exact ground-state energy of the system. The basis functions are usually based either on Slater-type orbitals (STO), which are radial functions containing a long-range exponential decay contribution term (Eq. 3.3), or on Gaussian-type orbitals (GTO), which contains a Gaussian distribution term (Eq. 3.4).

where R[r) describes the radial orbital function, r is the distance of the electron to the nucleus, A is a normalization constant, n is a natural number, and a is a constant related to the effective charge of the nucleus.

Methods using the direct solution to the Schrodinger equation or any of the previous approximations (BO or HF) are usually denoted as ab initio methods, as they do not require any empirical parameter, but only the first principles of quantum mechanics and fundamental physical constants. Due to computational limitations, ab initio methods are suitable only for systems of up to thousands of atoms during up to just a few picoseconds (10^-12 s) of simulation.

Some additional approximations allow the simulation of larger systems, but at the price of reduced accuracy in the calculations. These methods are usually known as parameterized methods since they incorporate experimental parameters, such as atom sizes, bond lengths, bond angles, and torsion angles. The parameterized methods can be either semi-empirical or empirical. The semi- empirical methods include the free-electron molecular orbital (MO) method, the Pariser-Parr-Pople (PPP) MO method, the Hiickel MO method, and the extended Hiickel method [10]. Empirical methods only consider the nuclei, completely neglecting the electrons in the atoms, and thus, they belong to the next scale: the molecular scale.

An additional method at the atomistic scale is the dens/ty/unct/ona/ theory (DFT) method, where the wavefunction of the system is not calculated directly, but instead the electronic probability density (p) for the system is determined and then the electronic energy of the system is calculated as a function of p, according to the Hohenberg- Kohn theorems [11] and the Kohn-Sham (KS) approximation [12].

Atomistic modeling is useful for determining fundamental properties of molecular species such as their electronic structures, which are useful for determining the kinetics and thermodynamics of chemical reactions, i.e., rate of reaction and equilibrium constants [13]. It is also used for determining the effect of the solvents on chemical reactions. It is particularly useful for investigating chemical reactions involving radicals, such as the free-radical polymerization reactions and reversible addition-fragmentation chain transfer (RAFT) polymerization [14-19]; although other types of polymerization reactions have also been considered [20-22]. For polymerization reactions in general, and for radical polymerization in particular, atomistic modeling allows the determination of copolymerization ratios and individual reaction rates for each molecular or macromolecular species [23-26].