Interpreting the antibonding nature between Cu and O$_2$ in two copper enzyme models from computational X-ray absorption spectra at the Cu L$_{2,3}$-edge

Cu-O$_2$ structures play important roles in bioinorganic chemistry and enzyme catalysis, where the bonding between Cu and O$_2$ parts serves as a fundamental research concern. Here we performed a multiconfigurational study on the copper L$_{2,3}$-edge X-ray absorption spectra (XAS) of two copper enzyme model complexes to gain a better understanding of the antibonding nature from clearly interpreted structure-spectroscopy relation. We obtained spectra in good agreement with the experiments by using the restricted active-space second-order perturbation theory (RASPT2) method, which facilitated reliable chemical analysis. Spectral feature interpretations were supported by computing the spin-orbit natural transition orbitals. All major features were assigned to be mainly from Cu $2p$ to antibonding orbitals between Cu $3d$ and O$_2$ $\pi^{*}$, Cu$3d-\pi^*_\text{O-O}$ (Type A), and a few also to mixed antibonding/bonding orbitals between Cu $3d$ and O$_2$ $\pi$, Cu$3d\pm\pi_\text{O-O}$ (Type M). Our calculations provided a clear illustration of the interactions between Cu $3d$ and O$_2$ $\pi^{*}$/$\pi$ orbitals that are carried in the metal L-edge XAS.