Why Variational Density Functional Theory Is More Accurate Than Time-Dependent Density Functional Theory for Certain “Difficult” Excited States?

Despite its remarkable success for a wide variety of excited states, time-dependent density functional theory (TDDFT) can fail severely for specific types of systems─for example, charge-transfer excited states, inverted singlet-triplet gap (STG) molecules, core excitations, Rydberg states, and others. Variational excited-state density functional theory (eDFT) has been demonstrated to provide accuracy similar to that of TDDFT for "well-behaved" systems. However, eDFT consistently performs better for difficult excited states, where TDDFT yields significant errors or incorrect qualitative results. In this article, the energy equations of eDFT and two-state Tamm-Dancoff approximation (TDA) TDDFT are compared to investigate the origins of both the successes and failures of these two methods for three types of excitations: core excitations, inverted STG molecules, and intermolecular charge-transfer excitations. A general decomposition method is developed for singlet and triplet excitation energies computed by using eDFT and TDDFT. Using this decomposition approach, the effects of different functionals on various components of excitation energies are analyzed. The sensitivity of eDFT excitation energies to the underlying densities is also evaluated. Our analysis reveals that eDFT excitation energies include higher contributions from the Hartree and exchange-correlation kernels compared with TDDFT. However, greater kernel contributions in eDFT by themselves do not guarantee improved accuracy. The inclusion of orbital relaxation, together with greater kernel contributions, is responsible for the superior performance of eDFT over conventional TDDFT for difficult excited states.