Scaling Approaches to Quantum Many-Body Problems

In the present thesis, we will focus on a less studied aspect of Thomas-Fermi theory: the highly non-trivial scaling relations following from it. The main objective of this thesis is to introduce this scaling approach, not as a method to solve the many-body problem, but as an efficient way of organizing the information contained in its solution in order to extract yet more – sometimes non-trivial – information. To this goal we apply the scaling approach to a wide range of system, from nanostructures (quantum dots) to atoms and atomic ions.Our main findings can be summarized as follows: (i) the obtainment of scaling relations for the correlation energy of quantum dots and atomic ions, respectively. This allows us to extend our scaling approach to complex quantities that are beyond mean-field methods; (ii) the obtainment of scaling relations for the chemical potentials and addition energies of two-dimensional quantum dots, which allows us to compare our results to experimental data; and (iii) the obtainment of scaling relations for the ground-state energy, chemical potentials, and addition energies of three-dimensional quantum dots, which allows us to explore the dimensionality effects on the scaling relations.In all cases, we not only showed the functional form of the scaling relations, but we also provided explicit analytical expressions for the scaled quantities. Such expressions are not simple by-products of the approach, but approximations that can be used for estimating relevant quantities with practically no computational cost. Furthermore, the obtained scaling relation may serve as a starting point for the improvement of more elaborated theories, for example, in the optimization of density functionals within density functional theory.The above results are reported in four publications which constitute the basis of the thesis

Scaling in the correlation energies of two-dimensional artificial atoms

We find an unexpected scaling in the correlation energy of artificial atoms, i.e., harmonically confined two-dimensional quantum dots. The scaling relation is found through extensive numerical examinations including Hartree-Fock, variational quantum Monte Carlo, density-functional, and full configuration-interaction calculations. We show that the correlation energy, i.e., the true ground-state total energy subtracted by the Hartree-Fock total energy, follows a simple function of the Coulomb energy, confimenent strength and, the number of electrons. We find an analytic expression for this function, as well as for the correlation energy per particle and for the ratio between the correlation and total energies. Our tests for independent diffusion Monte Carlo and coupled-cluster results for quantum dots -- including open-shell data -- confirm the generality of the obtained scaling. As the scaling is also well applicable to $\gtrsim$ 100 electrons, our results give interesting prospects for the development of correlation functionals within density-functional theory.Comment: Accepted to Journal of Physics: Condensed Matte