The role of symmetry in solving large-scale eigenvalue problems: Applications to quantum mechanics and vibrational analysis

Abstract

This work explores the significance of symmetry in addressing eigenvalue problems of a large scale, namely in the fields of quantum mechanics and vibrational analysis. Through the utilization of group theory and block-diagonalization techniques, symmetry-based approaches effectively decrease both computing complexity and time. Comparative analysis of symmetry-adapted methodologies reveals superior performance compared to non-symmetry methods, particularly in systems characterized by significant levels of symmetry, such as multi-electron molecules and crystalline lattices. The work serves to illustrate the efficacy of these techniques in diminishing processing time, therefore confirming their utility in the fields of scientific computing and engineering. Proposed avenues for future investigation encompass the extension of these methodologies to more intricate systems and the integration of machine learning and quantum computing.