Zero Excitation Energy Theorem and the Spin-Flip Kernel

This work establishes the zero excitation energy theorem, which ensures that the TDDFT equations constructed from an open-shell reference state must admit excited-state solutions with zero excitation energy. This theorem holds exactly in TDDFT but only approximately when the Tamm-Dancoff approximation is used. From this theorem, we derive an identity connecting the spin-conserving and spin-flip kernels. Based on this identity, a method to construct the spin-flip kernel solely from the spin-conserving kernel is proposed. This method is applicable to all types of collinear functional, is numerically stable, and preserves the expected energy degeneracy. Since this spin-flip kernel is merely a simple geometric average of the spin-conserving kernel, the spin-flip TDDFT based on it is easy to implement, especially in programs that already support spin-conserving TDDFT. Numerical tests show that the spin-flip TDDFT and its analytic gradient are as efficient as spin-conserving TDDFT, making them practical for routine use.