Excited-state spin-contamination in time-dependent density-functional theory for molecules with open-shell ground states

While most applications of the linear response formulation of time-dependent density-functional theory (TDDFT) have been to the calculation of the excited states of molecules with closed-shell ground states, Casida's formulation of TDDFT opened the way to TDDFT calculations on molecules with open-shell ground states by allowing for different-orbitals-for-different-spin, Although a number of publications have now appeared applying TDDFT to molecules with open-shell ground states and give surprisingly good results for simple excitations, it is relatively easy to show that some excited states of open-shell molecules will have unphysically large amounts of spin contamination. There is thus a clear need for computational tools which can separate physical from unphysical excited spin states in TDDFT. We address this need by using analytic derivative techniques to develop formulae for the 1- and 2-electron reduced density difference matrices, in essential agreement with those obtained by Rowe in the equation-of-motion superoperator approach to Green's functions in nuclear physics. The corresponding formula for excited-state spin contamination appears to be generally good enough for assigning excited-state spin symmetries, but does lead to a small overestimation of S^2 in the cases considered here. This (apparently small) problem is eliminated when the Tamm-Dancoff approximation is used in TDDFT.