This book introduces a novel method to extend the generator coordinate method (GCM) framework by simultaneously determining the weights and the Slater determinants based on the variational principle. Collective motions—a characteristic feature of quantum many-body systems—represent coherent motion of the constituent particles, arising from many-body correlations. In atomic nuclei, various types of collective motion exist, including rotational motion and surface vibrations, as well as phenomena such as giant resonances, shape coexistence, and nuclear fission. Developing theories that describe these collective motions based on the dynamics of individual nucleons is one of the major goals in theoretical nuclear physics, and GCM is a standard approach for this purpose. By incorporating the present extension, the space spanned by the Slater determinants (the collective subspace) becomes optimal for the ground states of the systems.

This new approach, referred to as the optimized GCM (OptGCM), is applied to nuclear systems in this book. Analyses of the ground states of ¹⁶O and ²⁸Si, as well as low-lying excited states in ²⁰Ne, ²⁴Mg, and ²⁸Si, are presented. Notably, this study represents the first application of basis optimization within density functional theory using a Skyrme-type effective interaction. Using OptGCM, we extract the optimal collective subspace and demonstrate that the method improves the collective subspace used in conventional GCM calculations. Furthermore, significant effects of basis optimization on physical observables are observed. This work therefore represents an important step in the development of nuclear many-body theory, and further development of the method is expected to improve the description of a wide range of collective phenomena in nuclear systems.