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

Define $h_{pq}=⟨p|\hat{h}|q⟩$, the one-electron kinetic plus nuclear-attraction integral, and physicists' two-electron integralsin atomic units. With $E_{\mathrm{NN}}$ the zero-electron nuclear-repulsion term, the Slater determinant expectation isThe occupied sums include spin-orbitals, so the exchange integral vanishes for incompatible spins without needing a separate closed-shell factor.

A Hartree-Fock wavefunction is stationary in energy with respect to all infinitesimal occupied-virtual orbital rotations, subject to orbital orthonormality. The determinant then satisfies the self-consistent Hartree-Fock equations. Stationarity alone does not guarantee that it is the lowest-energy or even a locally stable determinant.

${\Phi }_i^a$ is the normalized determinant obtained by replacing occupied spin-orbital $i$ in ${\Phi }_0$ with previously unoccupied orbital $a$, using a consistent antisymmetric determinant ordering. Thus $i=1,\dots ,N$ labels occupied orbitals and $a=N+1,\dots ,2M$ labels virtual orbitals. Occupied-to-occupied rotations only change the representation of the same occupied subspace; the occupied-virtual rotations describe distinct nearby determinants.

Orthogonality of the determinants gives $⟨{\Phi }^{(1)}|{\Phi }^{(1)}⟩=1+O({\lambda }^2)$. To first order the normalized energy is thereforeSlater-Condon rules give the matrix element as the occupied-virtual Fock elementStationarity for independently variable real ${\lambda }_i^a$ requires every $F_{ia}=0$. This is Brillouin's theorem. Because $F$ is Hermitian, occupied and virtual blocks can separately be diagonalized; canonical orbitals then satisfy $\hat{F}{\varphi }_p={\gamma }_p{\varphi }_p$.

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.