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.