# Genetic Algorithms in Application to the Geometry Optimization of Nanoparticles

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Genetic algorithms

#### 2.1. Representation

**Figure 1.**Mutation and crossover operations are shown on binary strings of 8 bits length. Red bits show mutation operation of alternating bits and last line shows the child individual formed by crossover operation of merging parts of parent individuals

#### 2.2. Variation generation

#### 2.3. Selection

#### 2.4. Reproduction

#### 2.5. Lamarckian GAs

#### 2.6. GA parameters

## 3. GAs for geometry optimization of nanoparticles

#### 3.1. Floating point array representation

_{n}H

_{2}clusters [34]. Some other studies facilitating this continuous representation can be seen in the references [19,35,36,37,38]. As stated in Section 2.1., floating point array representation allows applications of phenotype genetic operations (see next section) considering the nanoparticle geometry in a more convenient way and it avoids the necessity for decoding and encoding the atomic coordinates.

#### 3.2. Phenotype genetic operations for geometry optimization problem

**Particle permutation :**Positions of two or more particles are exchanged. This mutation technique is used for heteronuclear nanoparticles.**Particle displacement :**Positions of one or more particles are modified slightly in a randomized manner.**Piece rotation :**One part of the nanoparticle is rotated a certain amount around a chosen axis. Axis and rotation amount are usually determined randomly. This mutation technique is preferred for atomic or molecular clusters having spherical shapes.**Piece reflection :**One part of the nanoparticle is exchanged with a reflection of the same part or another part of the nanoparticle.**Shrinkage :**Size of the nanoparticle is shrank by multiplying atomic coordinates with a factor less than unity.

**Figure 2.**Illustration of phenotype cut and splice crossover operation developed by Deaven and Ho [19].

#### 3.3. Local relaxation

## 4. Single parent parallel Lamarckian GA implementation

#### 4.1. Method overview

^{10}, V being the potential energy of the molecular system, is used in the roulette wheel selection mechanism since potential energies of different isomers are usually very close to each other. The fittest five individuals are selected certainly for elitism and the fittest individual obtained so far during the optimization process is not allowed to mutate in a way that reduces its fitness value. Classical Monte Carlo method is used as the local optimizer in each GA step for all individuals, causing the method to be a Lamarckian GA.

#### 4.2. Algorithm layout

**C**object oriented programming language and

_{++}**MPI**parallelization library, may be summarized as follows:

**Population initialization :**Each computing node initializes a number of**(populationSize / numberOfNodes)**individual atomic clusters by generating random positions for the atoms in a cubical box of a certain size which is smaller than the expected size of the cluster.**GA loop**- -
**Local optimization :**Each individual cluster is relaxed by the MC local optimizer at the computing node initializing the cluster.- -
**Gathering at master node :**Atomic coordinates and potential energy values of locally optimized clusters are gathered at the master node by**MPI**communication.- -
**Selection :**Half of the individuals are selected by**roulette wheel selection**with elitism using the fitness function V^{10}. The number of GA steps with no progress in the potential energy of the fittest individual is counted and when this number reaches a certain amount, an alternative selection mechanism is applied in which all the individuals are transformed to the fittest individual.- -
**Termination check :**Number of alternative selection mechanism loops with no progress in best fitness value is counted. When this number reaches a certain amount GA loop terminates.- -
**Variation generation :**Phenotype mutation operations of**piece rotation**and**shrinkage**are applied on the selected individuals.- -
**Distribution of individuals :**Individual clusters are distributed to available computing nodes by**MPI**communication for local optimization.

**Output :**Atomic coordinates and potential energy value of the fittest individual are given as the output of GA implementation.

_{20}cluster in which three GA steps of the evolution of the fittest individual winning the competition is given ignoring some intermediate GA steps. This illustration is for the evolution of a single individual and therefore natural selection and reproduction steps are not shown.

#### 4.3. Application on carbon clusters

**Figure 3.**Illustration of the single parent GA implementation showing evolution of the fittest individual in C

_{20}optimization. Three GA step samples are shown in separate lines ignoring some intermediate GA steps. 1. Column shows the cluster just after the shrinkage mutation operation (initial geometry in top line), 2. Column shows the result of local optimization and 3. Column shows the cluster after the piece rotation mutation step together with the cutting plane.

_{20}cluster with the other GA method mentioned above which also has the Lamarckian property (13 in authors work and 800 in the comparison work) suggests that MC optimization used in the current GA may have some advantages compared to deterministic local optimization routines even though single MC local optimization is more time consuming.

**Figure 4.**Optimized structures of carbon clusters. Top line: 11 atoms, 20 atoms (I

_{h}symmetry); Bottom line: 22 atoms, 32 atoms (D

_{3}symmetry)

#### 4.4. Application on silicon germanium core-shell structures

_{25}Ge

_{75}, Si

_{50}Ge

_{50}and Si

_{75}Ge

_{25}clusters have been studied in 100 atom case and Si

_{50}Ge

_{100}, Si

_{75}Ge

_{75}and Si

_{100}Ge

_{50}clusters have been studied for 150 atom case. Si-core and Ge-core initial geometries for Si

_{50}Ge

_{50}cluster is shown in Figure 5 for exemplification. Population size has been chosen to be 16 and 24 individuals for 100 atom and 150 atom clusters respectively. Value of the shrinkage factor is not crucial for SiGe clusters and thus it has been fixed to the value of 0.7 for all separate run instances.

_{25}Ge

_{75}(initial Si-core), Si

_{50}Ge

_{50}(initial mixed) and Si

_{75}Ge

_{25}(initial Ge-core) clusters. Tendency of Ge atoms for going up to surface can be seen clearly in these images.

**Figure 6.**Optimized structures of Si-core initial geometry Si

_{25}Ge

_{75}(left), mixed initial geometry Si

_{50}Ge

_{50}(middle) and Ge-core initial geometry Si

_{75}Ge

_{25}(right) clusters .

## 5. Discussion and guidelines

## Acknowledgements

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Dugan, N.; Erkoç, Ş. Genetic Algorithms in Application to the Geometry Optimization of Nanoparticles. *Algorithms* **2009**, *2*, 410-428.
https://doi.org/10.3390/a2010410

**AMA Style**

Dugan N, Erkoç Ş. Genetic Algorithms in Application to the Geometry Optimization of Nanoparticles. *Algorithms*. 2009; 2(1):410-428.
https://doi.org/10.3390/a2010410

**Chicago/Turabian Style**

Dugan, Nazım, and Şakir Erkoç. 2009. "Genetic Algorithms in Application to the Geometry Optimization of Nanoparticles" *Algorithms* 2, no. 1: 410-428.
https://doi.org/10.3390/a2010410