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Brazilian Journal of Physics

Ground state energy of charged particles clusters by quantum Monte Carlo method

N. L. Moreira, J. N. Teixeira Rabelo, Ladir Cândido

Published Sep 1, 2006• 0 citations
Publisher

Abstract

Using variational and diffusion Monte Carlo methods, we have calculated the ground state energy of spinless charged particles (for N < 10) interacting through a repulsive Coulomb potential, moving in two-dimensions and kept together by an external parabolic potential. Using a very simple trial wave function, we obtain results comparable to those of a sophisticated model of a quantum dot.

DOI
10.1590/s0103-97332006000500024

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Using variational and diffusion Monte Carlo methods, we have calculated the ground state energy of spinless charged particles (for N < 10) interacting through a repulsive Coulomb potential, moving in two-dimensions and kept together by an external parabolic potential. Using a very simple trial wave function, we obtain results comparable to those of a sophisticated model of a quantum dot.
The Pad\'e approximant technique and the variational Monte Carlo method are applied to determine the ground-state energy of a finite number of charged bosons in two dimensions confined by a parabolic trap. The particles interact repulsively through a Coulombic 1/r1/r1/r potential. Analytic expressions for the ground-state energy are obtained. The convergence of the Pad\'e sequence and comparison with the Monte Carlo results show that the error of the Pad\'e estimate is less than 4% at any boson density and is exact in the extreme situations of very dilute and high density.
By variational Monte-Carlo method developed Ceperley et al. for the simulation of fermi systems in macroscopic confining potential well we simulate various spin ground states of the coulomb clusters with 2,3 and 4 particles in a broad two-dimensional (2D) parabolic well. In this method quantum state numbers determining the variational wave function are not the numbers of well quantum states but numbers of the equilibrium spatial positions of particles that give a minimum of the system potential energy. The ground states with parallel, antiparallel spins and as well, as bose state are simulated. For the cluster with three particles it is also simulated the state when two particles have one direction of spin and third opposite. The simulation shows that clusters with parallel spins have lower ground state energy than clusters with other spin configurations and bose state. That reminds a Hund's rule in atomic physics when in not full filled atomic shells electrons prefer to have a state with parallel spins.
The diffusion Monte Carlo method is applied to study the ground-state properties of charged bosons in one dimension confined in a harmonic double-well trap. The particles interact repulsively through a Coulombic 1/ r potential. Numerical results show that the well separation has significant influence on the ground-state properties of the system. When the interaction of the system is weak, ground-state energy decreases with the increasing well separation and has a minimal value. If the well separation increases continually, the ground-state energy increases and approaches to a constant gradually. This effect will be abatable in the strong interacting system. In addition, by calculating the density of the systems for different interaction strengths with various well separations, we find that the density increases abnormally when the well separation is large at the centre of the system.
The ground state of a two-dimensional charged quantum system is studied using the method of correlated-basis functions. For particles obeying the Bose-Einstein statistics we compute the ground-state energy variationally for all particle densities using the Bijl-Dingle-Jastrow trial wave function. The Bose ground-state energy and wave function so determined are used next to study the Fermi system. We compute the Fermi ground-state energy using a cluster expansion approach, which is again valid for all particle densities, and find that the ground state is paramagnetic. While these calculations provide variational upper bounds to the ground-state energies, we also establish independent lower bounds to the ground-state energies.
This thesis aims to give an understanding of quantum dots trapped in different potentials by computationally finding the ground state energies of the system.The potentials studied are the single and double harmonic oscillator wells and the finite square well.To achieve this, we have implemented a robust Variational Monte Carlo framework that allows for different single-particle wavefunctions.This allowed us to study the use of a linear expansion of the single harmonic oscillator basisfunctions as an approximation to the single-particle wavefunctions of the other potential wells.We investigated how many of these basisfunctions needed to achieve sufficiently good single-particle wavefunctions.The framework has several optimizations that drastically reduce the computational time of the Monte Carlo method.The framework has a rigorous testing regime that ensures the authenticity of the results.To verify the results, we have compared them to the results of other Variatonal Monte Carlo computations, as well as Diffusion Monte Carlo, the Coupled Cluster method, the Similarity Renormalization Group method and the Full Configuration Interaction method.In addition, we have verified the results of the finite square well by comparing with a general series solution.