Excitation energies for open-shell systems

Excitation energies can be obtained for open-shell systems in a spin-unrestricted TDDFT calculation [154]. This can not be used in case of spin-orbit coupling. To perform an open-shell TDDFT calculation one just needs to do an unrestricted SCF calculation and use the EXCITATION keyword. Presently the excitation energies can only be found with Davidson's procedure.

The printed symmetry in the output in TDDFT calculations is actually the symmetry of transition density. For closed-shell systems, the symmetry of the excited state is the same as the symmetry of the transition density, while for open-shell systems, the symmetry of the excited states is the direct product between the symmetry of the transition density and the ground state symmetry. Note that the ground state symmetry of an open shell molecule is not necessarily A1.

For degenerate representations such as the 2-dimensional E-representations or the 3-dimensional T-representations, the occupation should be either fully occupied or zero. For example, for an orbital in an E-representation the α and β occupation number should be either 2 or 0. The α occupation number can of course be different from the β occupation number.

As for the spin-state, the general rule is that if the excited state mainly results from transitions from the singly occupied orbitals to virtual orbitals or from fully occupied orbitals to the singly occupied orbitals, the spin state of the excited state should roughly be the same as that of the ground state. However, if the excited state mainly comes from transitions from fully occupied orbitals to virtual orbitals, the spin state of the excited state are usually a mixture since TDDFT can only deal with single excitations within adiabatic approximation for the XC kernel [155]. Sometimes we just suppose the spin state of this kind of excited states to be the same as that of ground state [154]. In the MO → MO transitions part for the excitations of the output file, the spin of each molecular orbitals are also specified to help assign the spin state of the excited states. The transitions are always from α spin-orbital to α spin-orbital or from β spin-orbital to β spin-orbital.