Spin-Flip TDDFT within the Sternheimer Formulation: A Gaussian and Plane Wave Implementation

We report the first implementation of spin-flip time-dependent density functional theory (SF-TDDFT) within the Tamm-Dancoff approximation in the Sternheimer formulation including the use of the noncollinear kernel. The noncollinear kernel was stabilized by introducing a screening method for the numerical integration, realizing a robust scheme of excited energy and gradient calculations of SF-TDDFT using generalized gradient approximation functionals. The implementation is evaluated by benchmark calculations of vertical excitation energies and optimized molecular geometries. The benchmark for vertical excitations consists of 19 excitations with high level of theory reference data from the QUESTDB. An underestimation of vertical excitation energies was observed for the PBE and PBE0 functionals, as seen by their average deviations of -0.3 eV. The benchmark for optimized geometries consists of 25 optimized structures with high level of theory, comprising CCSD, CISD, and FCI data, and 10 reference structures optimized with other implementations of collinear and noncollinear SF-TDDFT. The optimized structures using PBE and PBE0, with a noncollinear kernel, were found to be close to the high-level reference structures, with mean deviations of 0.010 and -0.004 Å, respectively. The extension to the auxiliary density matrix method (ADMM) is also presented. We found an average deviation of 0.003 Å in the calculated bond lengths when employing the ADMM for the PBE0 functional.