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We compute the ground-state energy of atoms and quantum dots with a large number N of electrons. Both systems are described by a nonrelativistic Hamiltonian of electrons in a d-dimensional space. The electrons interact via the Coulomb potential. In the case of atoms (d = 3), the electrons are attracted by the nucleus via the Coulomb potential. In the case of quantum dots (d = 2), the electrons are confined by an external potential, whose shape can be varied. We show that the dominant terms of the ground-state energy are those given by a semiclassical Hartree-exchange energy, whose N -> infinity limit corresponds to Thomas-Fermi theory. This semiclassical Hartree-exchange theory creates oscillations in the ground-state energy as a function of N. These oscillations reflect the dynamics of a classical particle moving in the presence of the Thomas-Fermi potential. The dynamics is regular for atoms and some dots, but in general in the case of dots, the motion contains a chaotic component. We compute the correlation effects. They appear at the order N ln N for atoms, in agreement with available data. For dots, they appear at the order N.
Giuseppe Carleo, Gabriel Maria Pescia
Fabrizio Carbone, Giovanni Maria Vanacore, Ivan Madan, Ido Kaminer, Simone Gargiulo, Ebrahim Karimi