Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 4 / 60 / c

The stationary determinant should be stable against orbital rotations: its orbital-energy Hessian must have no negative direction. A suitable local minimum, preferably the lowest accessible determinant of the appropriate symmetry and spin, can approximate the ground state; a stationary excited determinant or saddle generally cannot. Stability is still not a guarantee of small electron-correlation error.

An example is the doubly occupied antibonding configuration $({\sigma }_u{)}^2$, rather than the bonding $({\sigma }_g{)}^2$ configuration dominant near equilibrium. It can be a self-consistent stationary determinant because symmetry makes the corresponding occupied-virtual Fock mixing vanish, while its bonding energy and nodal structure are inappropriate for the ground state. Solving the Hartree-Fock stationarity equations does not impose ground-state occupancy automatically.

The reference, singly excited and doubly excited determinants are mutually orthogonal. Singles have amplitude $O(\lambda )$ and doubles $O({\lambda }^2)$, so only the reference norm and squares of single-excitation amplitudes contribute through second order:There is no reference-double cross term. The stated expansion has no cubic contribution because its single and double sectors are orthogonal.

At a Hartree-Fock stationary point the reference-single terms vanish. Expanding the normalized expectation through second order givesThe reference-double term arises twice in the numerator and cancels the factor $1/2$ in the wavefunction. The normalization subtracts $E_0{\sum }_{ia}({\lambda }_i^a{)}^2$.

Using Slater-Condon rules in canonical orbitals,and $⟨{\Phi }_0|\hat{H}|{\Phi }_{ij}^{ab}⟩=⟨ab|ij⟩-⟨ab|ji⟩$. Combining them gives the requested $X_{ij}^{ab}$. The orbital energies arefor either occupied or virtual $p$. Diagonalizing the occupied and virtual Fock blocks is the suitable orbital choice that makes the delta form valid for coincident as well as distinct indices.

The real symmetric quadratic form is positive for every nonzero vector of occupied-virtual rotation parameters precisely when its matrix $X$ is positive definite. Then the Hartree-Fock determinant is a strict local minimum within the considered real orbital-rotation space. A negative eigenvalue gives an unstable lowering direction; a zero eigenvalue requires higher-order analysis. Stability in this space does not by itself prove global minimality or stability under more general spin/complex-orbital variations.