<i>Ab Initio</i> Methodology To Describe the Static Mechanism of Electrodipolar Luminescence in Lanthanides

Quantification of the electron-vibrational couplings in lanthanide complexes and identification of the vibrations that are strongly coupled to electronic transitions are important for applications of optical lanthanide spectroscopy. While information about the electron-vibrational couplings can be extracted from the emission spectra, this is not common because of the challenges associated with interpreting the complex vibronic structures of the spectra. To overcome this challenge, we develop a fully <i>ab initio</i> methodology for predicting the vibronic peaks in the emission spectra of lanthanide complexes by calculating the electron-vibrational couplings associated with individual vibrational modes. We show that the energy gradients of the emitting and ground spin-orbit states, which are the key quantities required for calculating the electron-vibrational couplings, can be obtained analytically from the energy gradients of the spin-diabatic states and the corresponding nonadiabatic coupling matrix elements. To illustrate this methodology, we calculate the ⁴F_9/2→⁴I_15/2 emission spectrum of the erbium trensal complex, provide the full decomposition of the vibronic structure of the spectrum, and investigate the effects of spin-orbit interaction and metal-ligand hybridization on the electron-vibrational couplings. In addition, to validate our methodology, we calculate and compare with experiment the vibronic structure of the ⁴S_3/2→⁴I_15/2 green emission band.