  Density functional calculations of Rydberg excited states up to high energy
are carried out for several molecules using an approach where the orbitals are
variationally optimized by converging on saddle points on the electronic energy
surface within a real space grid representation. Remarkably good agreement with
experimental estimates of the excitation energy is obtained using the
generalized gradient approximation (GGA) functional of Perdew, Burke and
Ernzerhof (PBE) when Perdew-Zunger self-interaction correction is applied in
combination with complex-valued orbitals. Even without the correction, the PBE
functional gives quite good results despite the fact that corresponding Rydberg
virtual orbitals have positive energy in the ground state calculation. Results
obtained using the TPSS and r2SCAN meta-GGA functionals are also presented, but
they do not provide a systematic improvement over the results from the
uncorrected PBE functional. The grid representation combined with the projector
augmented-wave approach gives a simpler and better representation of the
diffuse Rydberg orbitals than a linear combination of atomic orbitals with
commonly used basis sets, the latter leading to an overestimation of the
excitation energy due to confinement of the excited states.
