Algebraic Varieties in Quantum Chemistry

Electronic structure theory is a powerful quantum mechanical framework for investigating the intricate behavior of electrons within molecules and crystals. In coupled cluster theory (CC), the high-dimensional eigenvalue problems that encode the electronic Schrödinger equation are approximated by a hierarchy of polynomial systems at various levels of truncation. We develop algebraic geometry for CC theory of quantum many-body systems. The quantum states of systems are approximated by the exponential parametrisation, a nonlinear map constructed by elements in the Fermi-Dirac algebra, a Clifford algebra over the exterior algebra. This reveals well-known varieties, such as the Grassmannian. We offer a detailed study of the truncation varieties and their CC degrees, a complexity measure for solving the CC equations. We explore the full solution spectrum of the CC equations for small molecules such as LiH.