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The Stability Eigenvalue Problem

The general linear-response problem built from the same \(A\) and \(B\) blocks — as used to find excitation energies in RPA/TDHF — is a non-Hermitian eigenvalue equation:

\[ \begin{pmatrix} A & B \\ -B & -A \end{pmatrix} \begin{pmatrix} X \\ Y \end{pmatrix} = \omega \begin{pmatrix} X \\ Y \end{pmatrix} \]

Solving this in general requires care because the matrix on the left is not symmetric, and its eigenvalues \( \omega \) can in principle be complex.

The static stability test, however, does not need excitation energies — it only needs the sign of the curvature along each rotation direction. Because real orbital rotations are self-inverse (\( \kappa_{ai} = \kappa_{ai}^{*} \), unlike the independent forward/backward amplitudes \(X\) and \(Y\) of a true excitation), the stability condition collapses to two much simpler, Hermitian, real-symmetric problems:

\[ (A+B)\, v = \lambda_{\text{singlet}}\, v, \qquad (A-B)\, v = \lambda_{\text{triplet}}\, v \]

Both \( (A+B) \) and \( (A-B) \) are real symmetric matrices, so their eigenvalues are guaranteed real, and standard symmetric-matrix iterative solvers (Davidson-type methods) can be used directly. This is what makes a stability test computationally far cheaper than a full TDHF/RPA excited-state calculation covering the same rotation space.