Quantifying the impact of the Tamm-Dancoff approximation on the computed spectra of transition-metal systems

Abstract: The Tamm-Dancoff Approximation (TDA) offers a computationally efficient alternative to full linear-response Time-Dependent Density Functional Theory (TDDFT) for calculating electronic excited states, particularly in large molecular systems. By neglecting the coupling between excitation and de-excitation channels, TDA simplifies the TDDFT response equations into a Hermitian form. This not only reduces computational cost but also eliminates numerical instabilities that can arise in the full non-Hermitian formalism. While TDA has been widely explored for valence excitations, its reliability for transition metal complexes and core-level spectroscopies remains largely untested. In this work, we address this gap by systematically comparing TDA and full TDDFT results for a series of transition metal species, focusing on absorption spectra across the UV-Vis, metal K-edges, and L-edges. Our results show that, for core-level excitations, TDA yields excitation energies and oscillator strengths nearly indistinguishable from those obtained with full TDDFT. This agreement is attributed to the negligible contribution of de-excitation amplitudes at high excitation energies, indicating that the omitted coupling terms play an insignificant role in these spectral regimes. These findings validate the accuracy and robustness of TDA in core-level spectra simulations and support its broader application in studies involving transition metal systems.