Connection to RPA / TDHF / CIS Equations¶
The stability matrix is built from the same \(A\) and \(B\) blocks that define the Random Phase Approximation (RPA), Time-Dependent Hartree–Fock (TDHF), and — in the simpler Tamm–Dancoff limit where \(B=0\) — Configuration Interaction Singles (CIS). This shared machinery is not a coincidence: both problems ask "what happens to the energy when the reference determinant is perturbed by occupied–virtual orbital mixing?" — an excited-state calculation asks it for the true, time-dependent excitation, while a stability analysis asks it for the static, time-independent second derivative.
This connection gives the stability eigenvalues a direct physical reading:
- The triplet stability eigenvalues, built from \( (A-B) \), track closely with the lowest true triplet excitation energy of the system. When this eigenvalue turns negative, it means the reference singlet determinant has become higher in energy than an accessible triplet-like rearrangement — the hallmark of a biradicaloid system that should really be treated with a spin-unrestricted or multireference method.
- The singlet stability eigenvalues, built from \( (A+B) \), signal a lower-energy solution within the same closed-shell, real-orbital manifold — often traceable to an over-constrained spatial symmetry.
Because a stability run only needs the sign of the lowest eigenvalue rather than the full excitation spectrum, it is run far more cheaply than an actual TD-DFT or CIS calculation on the same system.