Variational Forward-Backward Charge Transfer Analysis Based on Absolutely Localized Molecular Orbitals: Energetics...

To facilitate the understanding of charge-transfer (CT) effects in dative complexes, we propose a variational forward-backward (VFB) approach to decompose the overall CT stabilization energy into contributions from forward and backward donation in the framework of energy decomposition analysis based on absolutely localized molecular orbitals (ALMO-EDA). Such a decomposition is achieved by introducing two additional constrained intermediate states in which only one direction of CT is permitted. These two "one-way" CT states are variationally relaxed such that the associated nuclear forces can be readily obtained. This allows for a facile integration into the previously developed adiabatic EDA scheme, so that the molecular property changes arising from forward and back donation can be separately assigned. Using ALMO-EDA augmented by this VFB model, we investigate the energetic, geometric, and vibrational features of complexes composed of CO and main group Lewis acids (BH₃, BeO/BeCO₃) and complexes of the N₂, CO, and BF isoelectronic series with [Ru(II)(NH₃)₅]²⁺. We identify that the shift in the stretching frequency of a diatomic π-acidic ligand (XY), such as CO, results from a superposition of the shifts induced by permanent electrostatics and backward CT: permanent electrostatics can cause an either red or blue shift depending on the alignment of the XY dipole in the dative complex, and this effect becomes more pronounced with a more polar XY ligand; the back-donation to the antibonding π orbital of XY always lowers the X-Y bond order and thus red-shifts its stretching frequency, and the strength of this interaction decays rapidly with the intermolecular distance. We also reveal that while σ forward donation contributes significantly to energetic stabilization, it affects the vibrational feature of XY mainly by shortening the intermolecular distance, which enhances both the electrostatic interaction and backward CT but in different rates. The synergistic effect of the forward and backward donations appears to be more significant in the transition-metal complexes, where the forward CT plays an essential role in overcoming the strong Pauli repulsion. These findings highlight that the shift in the XY stretching frequency is not a reliable metric for the strength of π back-donation. Overall, the VFB-augmented EDA scheme that we propose and apply in this work provides a useful tool to characterize the role played by each physical component that all together lead to the frequency shift observed.