Mathematics for Nerds: Wavefunction Stability

This section derives the theory behind the stability test, starting from the orbital-rotation parameterization of the SCF energy and ending with the full iterative algorithm Gaussian uses to check it.

• Orbital Rotation Parameterization — how the wavefunction is expanded as a function of occupied–virtual orbital mixing.
• The Electronic Hessian (A and B Matrices) — the second-derivative matrix that defines stability, and its link to RPA/TDHF theory.
• Types of Wavefunction Instabilities — real vs. complex, singlet vs. triplet.
• The Stability Eigenvalue Problem — why the static stability test reduces to a real, symmetric eigenvalue problem.
• Connection to RPA/TDHF Equations — why stability analysis and excited-state theory share the same machinery.
• Davidson Iterative Diagonalization — how Gaussian finds the lowest eigenvalues without ever forming the full Hessian.
• Complete Wavefunction Stability Algorithm — the full procedure, start to finish.

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