Development and Implementation of Excited-State Gradients for Local Hybrid Functionals

Local hybrid functionals are a relatively recent class of exchange-correlation functionals that use a real-space dependent admixture of exact exchange. Here, we present the first implementation of time-dependent density functional theory excited-state gradients for these functionals. Based on the ansatz of a fully variational auxiliary Lagrangian of the excitation energy, the working equations for the case of a local hybrid functional are deduced. For the implementation, we derive the third-order functional derivatives used in the hyper-kernel and kernel-gradients following a seminumerical integration scheme. The first assessment for a test set of small molecules reveals competitive performance for excited-state structural parameters with typical mean absolute errors (MAEs) of 1.2 pm (PBE0: 1.4 pm) for bond lengths as well as for vibrational frequencies with typical MAEs of 81 cm–1 (PBE0: 76 cm–1). Excellent performance was found for adiabatic triplet excitation energies with typical MAEs of 0.08 eV (PBE0: 0.32 eV). In a detailed case analysis of the first singlet and triplet excited states of formaldehyde the conceptional (dis-)­advantages of the local hybrid scheme for excited-state gradients are exposed.