Types of Wavefunction Instabilities¶
Not every negative eigenvalue means the same thing. Gaussian classifies instabilities along two independent axes: whether the rotation stays within real orbitals or requires complex ones, and whether it preserves or breaks spin symmetry.
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RHF → RHF (real, singlet) — a lower-energy solution exists within the same real, closed-shell space. Usually a sign that an artificial spatial-symmetry constraint is holding the wavefunction at an unnecessarily high-energy solution.
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RHF → UHF (real, triplet) — a lower-energy spin-unrestricted solution exists. This is the most common instability in practice, appearing in biradicaloid molecules, stretched or partially broken bonds, and some transition-metal complexes where the restricted determinant over-delocalizes spin density.
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RHF → complex RHF — a lower-energy solution exists only when the molecular orbital coefficients are allowed to be complex, while remaining spin-restricted. This is uncommon and mostly relevant to specific symmetric or extended systems.
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RHF → complex UHF — the least common case: a lower-energy solution requires both spin-unrestriction and complex orbitals.
The Stable keyword with no arguments (Stable=Opt) tests only the real, internal channels — RHF→RHF and RHF→UHF together — which covers the great majority of practical cases, including the formic acid dimer example used throughout this section. The Complex option extends the test to the remaining two channels when needed.