Implementation of analytical excited state gradients for open-shell systems in time-dependent density functional...

Excited state potential energy surfaces are essential for understanding molecular behavior in excited states, including electronic transitions, reaction dynamics, and other photochemical and photophysical properties. When excited state minima or reaction dynamics are of interest, the gradients of the total excited state energy are required. Time-dependent density functional theory (TDDFT) is a widely used method, as it provides reasonably accurate excited state energies compared to experiment; however, its computational cost scales unfavorably with molecule size. To address this limitation, methods that maintain TDDFT accuracy while reducing the computational cost are needed. The time-dependent density functional theory plus tight binding (TDDFT+TB) method offers comparable accuracy to TDDFT with significantly improved efficiency. Previously, Havenridge et al. implemented TDDFT+TB analytical excited state gradients for closed-shell molecules. This was achieved using a Lagrangian-based approach by taking analytical derivatives of the excitation energies with respect to nuclear coordinates. In this study, we extend this implementation to handle open-shell molecular systems. Our unrestricted TDDFT+TB analytical gradient implementation follows the framework of the unrestricted TDDFT analytical gradient code in the Amsterdam Density Functional engine within the Amsterdam Modeling Suite. Building on the existing closed-shell implementation, we focus on key modifications to the coupling matrix and the incorporation of spin indices in the gradient expressions required for open-shell systems. We validate the method through calculations on various molecular systems, comparing gradient accuracy, emission energies, and computational cost to demonstrate that it delivers reliable results for open-shell molecular excited states with improved efficiency compared to the standard TDDFT method.