vibe-qc/Documentation

Molecular ROHF shell projectors and effective Fock blocks

In integer-occupation, high-spin restricted open-shell Hartree-Fock, the alpha and beta electrons share spatial orbitals. Both spins occupy the closed shell, only alpha occupies the open shell, and the remaining orbitals are virtual. Roothaan describes the mutually orthogonal closed and open spaces in section 2, equation (16). His treatment also allows more general open-shell occupations. Here the block coefficients specify the particular effective operator used by vibeqc.rohf.roothaan_effective_fock. Roothaan (1960)

From densities to shell projectors

Let \(S\) be the real symmetric AO overlap and let the columns of \(C\) be metric-orthonormal, \(C^T S C=I\). For a diagonal occupation mask \(M\) with entries zero or one, the AO density is \(D_M=C M C^T\). The action of its metric projector on orbital coefficients follows by associativity,

\[ (D_M S)C = C M(C^T S C)=CM. \]

rohf_mask_action proves this identity, even for a rectangular orbital frame. Let \(A=D_\alpha S\) and \(B=D_\beta S\). In this occupation model the beta mask selects a subset of the alpha mask. Thus \(A^2=A, B^2=B, AB=BA=B\). Production uses three action maps,

\[ P_c=B,\qquad P_o=A-B,\qquad P_v=I-A. \]

Their sum is the identity by cancellation. For example, \(P_o^2=A^2-AB-BA+B^2=A-B\) and \(P_o P_v=A-B-A^2+BA=0\). rohf_shell_resolution proves the sum, and rohf_shell_projectors proves all three idempotences and all six ordered cross products from the four nesting identities. In AO coordinates these maps can be oblique under the ordinary Euclidean inner product. Their placement on the bra side therefore requires a transpose.

Which spin Fock enters each block

Set \(F_m=(F_\alpha+F_\beta)/2\). The entry in row \(i\), column \(j\) of this table defines the AO operator \(B_{ij}\) whose matrix element is used.

Bra / ket	Closed	Open	Virtual
Closed	\(F_m\)	\(F_\beta\)	\(F_m\)
Open	\(F_\beta\)	\(F_m\)	\(F_\alpha\)
Virtual	\(F_m\)	\(F_\alpha\)	\(F_m\)

The code forms \(F_{\mathrm{eff}}=\sum_{ij}P_i^T B_{ij}P_j\). For orbital columns \(x,y\) in shells \(s,t\), respectively, \(P_i x=\delta_{is}x\) and \(P_j y=\delta_{jt}y\). Substituting gives

\[ x^T F_{\mathrm{eff}}y = \sum_{ij}(P_i x)^T B_{ij}(P_j y) = x^T B_{st} y . \]

rohf_effective_block proves this selection rule for each of the nine ordered shell pairs. With real symmetric spin Focks, \(B_{ij}^T=B_{ji}\). Transposing the sum and exchanging its two indices recovers the same matrix, as proved by rohf_effective_symmetric. The implementation averages the computed matrix with its transpose to remove numerical asymmetry.

Implementation and energy checks

The fixture supplies a nonorthogonal AO basis with \(S=C^{-T}C^{-1}\) and two distinct symmetric spin Focks. It transforms the production result to MO coordinates and compares every entry with an independent reference assembled by slicing the two MO spin Focks directly. The maximum absolute discrepancy over all four cases was 1.34e-15 or smaller at a tolerance of 1e-12. The cases include all three populated subspaces, an empty open shell, an empty closed shell and an empty virtual shell. Swapping the spin inputs and substituting the identity for the AO metric gave maximum errors of 4.12 and 2.04, respectively.

A previously accepted full molecular calculation used neutral doublet OH, an O-H separation of 1.81 bohr, STO-3G and direct all-electron ROHF. Its energy was -74.36009225705583 Hartree. The matched ORCA 6.1.1 ROHF reference gave -74.360092257600 Hartree, an absolute difference of 5.44e-10 Hartree. The effective-Fock function text matched in the source under review, the installed fixture and the completed calculation. The energy comparison exercises the complete driver. The synthetic check isolates the spin-block algebra that a single final energy cannot distinguish.

Run python formal/check_rohf.py with a vibe-qc installation and run python formal/check_proofs.py with the pinned Lean toolchain. The five ROHF claims participate in the standard method audit and its transitive axiom check. The proof models exact real matrices and integer shell masks. Fractional occupations, complex periodic orbitals, convergence, canonical orbital energies, gradients and floating-point error bounds remain open for this route.