Extensions to Extended Tight-Binding Methods for Transition-Metal Containing Systems

Semi-empirical quantum-chemical methods such as extended tight-binding (xTB) models are widely used for large-scale simulations. Despite their popularity, their accuracy for transition-metal containing systems is lower than, for example, closed-shell organic molecules. In this work, we extend the Q-Chem-xTB framework with a geometric direct minimization (GDM) scheme for robust self-consistent convergence and Hubbard correction (+ $U$ ) to improve the description of local interactions and reduce self-interaction errors similar to those characteristic of density-functional theory calculations for transition-metal complexes. The Hubbard correction term is integrated self-consistently within the xTB Hamiltonian, allowing shell-specific $U$ values for each atom. The performance of Q-Chem-xTB+ $U$ is assessed for four benchmark sets of iron complexes, focusing on their spin-state energetics. Sensitivity and optimization analyses of the spin parameters show that parameter tuning alone cannot systematically reduce the error or consistently recover correct spin ground-state predictions across different datasets. In contrast, introducing the + $U$ correction yields significant error reduction and improved electronic linearity with respect to fractional occupation, demonstrating that the correction fulfills its intended role of reducing self-interaction error. However, the optimized $U$ values remain system-dependent, and the resulting improvements are only partially transferable. As a side effect, the + $U$ correction stabilizes the self-consistent field optimization by widening the HOMO-LUMO gap, thereby overcoming convergence instabilities of the conventional direct inversion of the iterative subspace (DIIS) scheme at low electronic temperatures.