Skip to content

The Electronic Hessian: A and B Matrices

The orbital Hessian splits naturally into two blocks, conventionally labeled \( A \) and \( B \), built from orbital energies and two-electron repulsion integrals in the molecular-orbital basis:

\[ A_{ai,bj} = \delta_{ab}\delta_{ij}(\varepsilon_a - \varepsilon_i) + (ai|jb) - (ab|ij) \]
\[ B_{ai,bj} = (ai|bj) - (aj|bi) \]

Here \( i,j \) index occupied orbitals, \( a,b \) index virtual orbitals, \( \varepsilon \) are orbital energies, and \( (pq|rs) \) are two-electron repulsion integrals in chemists' notation. These are the identical matrix elements that build the particle–hole response matrix in RPA/TDHF theory and the CIS Hamiltonian.

For a real, closed-shell reference, Seeger and Pople showed that the full orbital Hessian block-diagonalizes into two independent, real, symmetric stability conditions:

\[ \text{Real (RHF} \rightarrow \text{RHF) stability:} \qquad (A + B) \succeq 0 \]
\[ \text{Triplet (RHF} \rightarrow \text{UHF) stability:} \qquad (A - B) \succeq 0 \]

where \( \succeq 0 \) means "positive semi-definite" (all eigenvalues non-negative). Gaussian's default stable test diagonalizes the combined alpha-alpha/beta-beta rotation space directly — the <AA,BB:AA,BB> block seen in the output — and classifies each resulting eigenvector by its computed spin expectation value \( \langle S^2 \rangle \) as belonging to the singlet (\(A+B\)-like) or triplet (\(A-B\)-like) channel, rather than diagonalizing the two combinations separately.