Geometrization for Energy Levels of Isotropic Hyperfine Hamiltonian Block and Related Central Spin Problems for an Arbitrarily Complex Set of Spin-1/2 Nuclei

Description of interacting spin systems relies on understanding the spectral properties of the corresponding spin Hamiltonians. However, the eigenvalue problems arising here lead to algebraic problems too complex to be analytically tractable. This is already the case for the simplest nontrivial [Formula: see text] block for an isotropic hyperfine Hamiltonian for a radical with spin- [Formula: see text] nuclei, where [Formula: see text] nuclei produce an n-th order algebraic equation with n independent parameters. Systems described by such blocks are now physically realizable, e.g., as radicals or radical pairs with polarized nuclear spins, appear as closed subensembles in more general radical settings, and have numerous counterparts in related central spin problems. We provide a simple geometrization of energy levels in this case: given [Formula: see text] spin- [Formula: see text] nuclei with arbitrary positive couplings [Formula: see text] take an n-dimensional hyper-ellipsoid with semiaxes [Formula: see text] stretch it by a factor of [Formula: see text] along the spatial diagonal [Formula: see text] read off the semiaxes of thus produced new hyper-ellipsoid [Formula: see text] augment the set [Formula: see text] with [Formula: see text] and obtain the sought [Formula: see text] energies as [Formula: see text] This procedure provides a way of seeing things that can only be solved numerically, giving a useful tool to gain insights that complement the numeric simulations usually inevitable here, and shows an intriguing connection to discrete Fourier transform and spectral properties of standard graphs.