Physical Meaning of Virtual Kohn–Sham Orbitals and Orbital Energies: An Ideal Basis for the Description of Molecular Excitations

In recent years, several benchmark studies on the performance of large sets of functionals in time-dependent density functional theory (TDDFT) calculations of excitation energies have been performed. The tested functionals do not approximate exact Kohn–Sham orbitals and orbital energies closely. We highlight the advantages of (close to) exact Kohn–Sham orbitals and orbital energies for a simple description, very often as just a single orbital-to-orbital transition, of molecular excitations. Benchmark calculations are performed for the statistical average of orbital potentials (SAOP) functional for the potential [<i>J. Chem. Phys.</i> <b>2000</b>, <i>112</i>, 1344; <b>2001</b>, <i>114</i>, 652], which approximates the true Kohn–Sham potential much better than LDA, GGA, mGGA, and hybrid potentials do. An accurate Kohn–Sham potential does not only perform satisfactorily for calculated vertical excitation energies of both valence and Rydberg transitions but also exhibits appealing properties of the KS orbitals including occupied orbital energies close to ionization energies, virtual-occupied orbital energy gaps very close to excitation energies, realistic shapes of virtual orbitals, leading to straightforward interpretation of most excitations as single orbital transitions. We stress that such advantages are completely lost in time-dependent Hartree–Fock and partly in hybrid approaches. Many excitations and excitation energies calculated with local density, generalized gradient, and hybrid functionals are spurious. There is, with an accurate KS, or even the LDA or GGA potentials, nothing problematic about the “band gap” in molecules: the HOMO–LUMO gap is close to the first excitation energy (the optical gap)