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18 CHAPTER2. COMPUTATIONALMETHODS mum in a certain search space. There are ways to check whether the solution has converged close to the optimum, e.g. by repeating a certain search for more gener- ations, or by repeating it with another random seed to see if the best chromosome is similar to the one previously found. The second problem is that there are no good rules for how to choose the values for the parameters in the GA, e.g. what mutation rate (how many percent of the individual chromosomes in a certain gen- eration should be mutated) or what crossover rate to use (i.e. how many of the best individuals should be used for crossover to generate new chromosomes). Which parameters to use is problem-dependent. Studies exist of which GA parameters are optimal for a certain class of problems[126, 127], but when being applied to a new problem, several attempts with different GA parameter values are necessary. In fact, there are even methods which attempt to optimize the GA parameters while the GA is being used to solve a main problem, which indicates the difficulty in deciding parameters[128]. Finally, as in any global optimization method, for the solution to be found quickly the evaluation of the fitness function must require a short time. This is the case when applying GAs for optimization of FF parameters, as one structural relaxation using a force field based on Coulomb interactions and two-body potentials such as those of the Buckingham type usually requires just a few seconds on a single modern processor core. Despite these problems, GAs have been used with success for many optimization problems, including the determination of low energy crystal structures[129] and fitting of parameters for force fields[130]. It is used in the present thesis to attempt to find a globally optimal force field, to describe both structural and energetic properties of DSA, based on Buckingham potentials and the AOM model. 2.1.2 Quantum mechanics After the discussion of semi-classical modeling of molecules and matter in the last section, we will now continue the discussion by focusing on quantum mechanics. This theory, which was developed to describe the behavior of small particles, such as atoms, electrons and photons, had to also account for the wave-like characteris- tics of matter. This was achieved through the description of a particle using a wave function, Ψ. The wave function contains all information that can be known about a system, such as that of electrons around atoms. For example, for the one-electron wave function, the square (or complex conjugate |Ψ|2 = Ψ∗Ψ, for complex wave functions) of the wave function at a certain point is the probability of finding the electron around that point, and the integral of the square of the wave function over all space is unity (the probability of finding the electron somewhere in space is 1). If matter is described as a wave, then there should be a similar kind of wave equation for matter as for waves in classical physics. The equation that was found to describe the properties of matter is the Schrödinger equation (SE),PDF Image | Studies of Electrode Processes in Industrial Electrosynthesis
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