We describe the Kondo resonance in quantum dots employing the atomic model. We calculate approximate Green's functions of the impurity Anderson model employing the exact solution of the system with a conduction band with zero width, and we use the completeness condition to choose the position of that band. At low temperatures, there are two solutions close to the chemical potential µ, satisfying this condition, and we choose the one with minimum Helmholtz free energy, considering that this corresponds to the Kondo solution. At high temperatures, this solution no longer exist, corresponding to the disappearance of the Kondo peak. We present curves of density of states that characterize the Kondo peak structure problem. As a simple application we calculate the conductance of a side-coupled quantum dot and we obtain good agreement with recent experimental results.

We consider the ESR of Co2+ in different environments: in an regular octahedron (Co2+ in a MgO crystal), in a deformed octahedron (Co2+ in single crystals and powder samples of NH4NiPO4.6H2O) or in a trigonal bipyramid (Co2+ in powders of Co2(OH)PO4 and Co2(OH)AsO4). We study the effect of the non-cubic crystal fields in the ESR of Co2+ in the deformed octahedron, by employing the normal modes of this structure to simplify the systematic study of the effect of these fields. A similar study was done for the deformed trigonal bipiramid, and it was necessary to derive the normal modes of this complex that are relevant to our problem.

It is sometimes convenient to emphasize the local aspects of a part of a crystalline system, and use the corresponding localized states to build a basis of the states of the whole system. In many cases it is only a subspace of these local states that is relevant, and the Hubbard operators provide a fairly simple way to write the corresponding projected Hamiltonian. Two examples of this type of treatment are presented int this work. The first is a Co2+ impurity in a MgO crystal interacting through a Jahn-Teller term with the crystal phonons, and it is shown how this interaction affects the electronic Raman scattering. The second is the Anderson lattice when the local electrons have an infinite Coulomb repulsion, and a diagrammatic expansion with cumulants is discussed. We propose a method to obtain approximate Green's functions for the Anderson lattice that employs the exact solutions of an atomic problem, and the corresponding spectral density of the local electrons is calculated.