Mapping DFT to effective Hamiltonians

This thesis proposes a new effective Hamiltonian to represent the interactions taking place in the atomic nucleus. This estimated Hamiltonian is analogous to the Pairing-Plus-Quadrupole model and this work is focused exclusively on the quadrupole part, with the aim of describing nuclear structure in a computationally efficient way. We have compared the accuracy of our proposed expression with the one proposed by K. Kumar and B. Sørensen [Nucl. Phys. A146 (1970) 1] studying the root-mean-squared error between those estimates and the exact formula. We also compare the spectra yielded by the different expressions. After this study, we conclude that the derived approximations appear to be more precise than the one proposed by Kumar and Sørensen.Matter is made of atoms that are constituted by a massive core, called nucleus, with charges, called electrons, orbiting around it. The nucleus itself is made up of two types of particles : protons and neutrons. Neutrons are not electrically charged, unlike protons which are positively charged. The atomic nucleus can be imagined like a bag full of marbles, some neutrals, some are charged. The difficulty is then to describe the behavior of such an object. On the one hand, all those positive charges repel each other (in the same fashion as the '+' side of two magnets next to each other), but on the other hand, the bag keeps them together with the rest of the marbles. Protons and neutrons even go beyond this analogy, for they also have the strange habit of pairing with each other. In addition to all these effects, the bag is not rigid and can be deformed into non-spherical shapes : squeezed like a rugby ball, or flattened like a pancake. Describing the atomic nucleus properly requires taking into account all of these features. This leads to a problem : solving exactly the equations that rule these interactions involves mathematical techniques that take a lot of computing power. As a result we usually approximate these equations, which raises a new question : how does one do that, and how can we judge the validity of such an approximation ? One answer (among many) is given in this thesis where we consider one estimate, called the Pairing-Plus-Quadrupole model. We try to enhance it using new formulas in combination with the work of two scientists, K. Kumar and B. Sørensen. The aim here is to propose an approximation that will bring results with a satisfactory level of precision and which will be easy and fast to run on a computer. The atomic nucleus is the heart of matter and is widely used for many applications such as energy production, medical imaging with MRI or cancer treatment. As a consequence, an efficient description of nuclei is very useful and can help unveiling their full potential