Implementation of analytical excited state gradients for open shell systems in time-dependent density functional theory plus tight binding method

To investigate excited state electronic transitions, reaction dynamics, and photochemical and photophysical properties of molecules, it is essential to understand their excited state potential energy surfaces (PESs). These insights can be gained through their excited state structures, with the use of gradients of the excited state energies. Time-dependent density functional theory (TD-DFT) is a useful technique that can provide this information. As the size of the molecule gets bigger, the computational cost will increase exponentially. Therefore, to overcome this challenge, novel methods with similar accuracy as TD-DFT that are less computationally expensive are needed. The time-dependent density functional theory plus tight binding (TD-DFT+TB) method is one of the novel methods that can perform with an accuracy as TD-DFT but faster. In 2023, Havenridge et al. implemented analytical TD-DFT+TB excited state gradients for closed-shell molecules within the Amsterdam Modeling Suite (AMS). This was obtained through a Lagrangian-based method by differentiating the excitation energies. Therefore, in this study, we aim to advance this approach to make it available for open-shell molecules. The unrestricted TD-DFT+TB analytical gradient implementation for open-shell molecules is similar to the unrestricted TD-DFT based analytical gradient code within the Amsterdam Density Functional (ADF) engine within the AMS package. As the analytical gradients for the closed-shell molecules are already implemented, our focus is to incorporate an extra dimension for the spin component, which is necessary in terms of dealing with open-shell molecules. Compared to the numerical and analytical gradients computed in TD-DFT and the newly implemented TD-DFT+TB unrestricted method, closed-shell molecules reflect their gradients align closely, while open-shell cases have similar errors to the underlying unrestricted TD-DFT method.