Implementation of a Particle Swarm Optimization Algorithm with a Hooke’s Potential, to Obtain Cluster Structures of Carbon Atoms, and of Tungsten and Oxygen in the Ground State

Particle Swarm Optimization (PSO) is a metaheuristic optimization technique based on population behavior, inspired by the movement of a flock of birds or a school of fish. In this method, particles move in a search space to find the global minimum of an objective function. In this work, a modified PSO algorithm written in Fortran 90 is proposed. The optimized structures obtained with this algorithm are compared with those obtained using the basin-hopping (BH) method written in Python (3.10), and complemented with density functional theory (DFT) calculations using the Gaussian 09 software. Additionally, the results are compared with the structural parameters reported from single crystal X-ray diffraction data for carbon clusters Cn(n = 3–5), and tungsten–oxygen clusters, WOnm−(n = 4–6, m=2,4,6). The PSO algorithm performs the search for the minimum energy of a harmonic potential function in a hyperdimensional space ∈R3N (where N is the number of atoms in the system), updating the global best position ( gbest) and local best position ( pbest), as well as the velocity and position vectors for each swarm cluster. A good approximation of the optimized structures and energies of these clusters was obtained, compared to the geometric optimization and single-point electronic energies calculated with the BH and DFT methods in the Gaussian 09 software. These results suggest that the PSO method, due to its low computational cost, could be useful for approximating a molecular structure associated with the global minimum of potential energy, accelerating the prediction of the most stable configuration or conformation, prior to ab initio electronic structure calculation.