How Do VirtualOrbitals Influence Charge-TransferExcitations in ΔSCF Calculations? Insights from ConstrictedVariational DFT

A comprehensive evaluation of constricted variational density functional theory (CV-DFT) is presented for intra- and intermolecular singlet-singlet excitations with varying degrees of charge transfer (CT) to understand the role of virtual orbitals in variational calculations of the excited state. The effect of initial guess (TDDFT, sTDDFT, or TDDFT+TB) is investigated to assess the robustness of CV-DFT for these excitations. Mild states (dCT ≤ 1.50 Å) do not require variational calculations, whereas strong states (dCT ≥ 1.50 Å) benefit from CV-DFT calculations. Strong CT states demonstrating charge transfer over intermediate distances (1.50 Å ≤ dCT ≤ 2.50 Å) require a self-consistent optimization of the natural transition orbitals (NTOs) for the excitation. In such cases, the SCF-CV(∞)-DFT scheme (and additional orbital relaxation) efficiently captures missing NTOs compared to the linear response regime, as well as yielding improved excitation energies. In calculations employing LDA and PBE, the SCF-CV(∞)-DFT scheme yields excitation energies with root mean squared error (RMSE) of 0.63 eV, whereas, with hybrid functionals, the RMSE is reduced to ≈ 0.33 eV against ab initio data. The RMSE with PBE0 is 0.23 eV and is comparable to double-hybrid TDDFT results in the literature. However, for fully charge-separated states (dCT ≥ 2.50 Å), the SCF-CV(∞)-DFT scheme collapses to a charge-localized state as it is deemed variationally unstable. For these states, a variational calculation wherein the NTOs are frozen at their initial guess in the single-orbital replacement (SOR)-R-CV(∞)-DFT scheme removes the variational instability, yielding excellent agreement with theoretical best estimates for both intramolecular and intermolecular CT excitations. These findings show that variational ΔSCF calculations of excited states can be achieved with CV-DFT by optimizing and relaxing higher-lying virtual orbitals.