vibe-qc/Documentation

Finite H₂ operator comparison (#1003)

On the frozen H₂ fixture, the terminal BIPOLE RHF operator agrees with the bounded four-index source and an independent Gaussian evaluation at Gamma and on the full unshifted three-point mesh. Issue #1003 found that using a rounded third in exchange failed even for one reciprocal vector. Issue #1009 repaired the public RHF operands and retained that calculation as a negative control. The agreement is specific to the basis, supports and mesh below.

Run the executable experiment with a vibe-qc Python runtime. It prints the native and diagnostic digests, Gamma control and three-point result. The absolute gates were set before the run: 1e-10 Ha for normalized integrals and actions, 1e-12 Ha for forbidden blocks. Existing tighter component checks remain in force.

Frozen calculation

The orthorhombic cell lengths are (4.4, 6, 6.5) bohr. Neutral singlet H₂ has nuclei at (0,0,0) and (1.4,0,0) bohr, two electrons, no ECP, and one normalized primitive \(s\) shell per atom, exponent \(\alpha=0.8\) bohr⁻² and coefficient \((2\alpha/\pi)^{3/4}\) multiplying the bare Gaussian. The full mesh is \(k_x=0,+1/3,-1/3\) in reciprocal lattice units (the last stored as 2/3), with weights 1/3. The Gamma control has weight one. Both use \(\omega=0.45\) bohr⁻¹, Ewald precision 1e-6 and \(K_{\max}=4.28796887156131\) bohr⁻¹. Reciprocal support is \(0<|G+q|\le K_{\max}\); the three channels contain 226, 218 and 218 vectors. The private kinetic cutoff is \(K_{\max}^2/2\). The RHF energy and gradient convergence thresholds are 1e-11 and 1e-9. Both terminal rebuilds use the full density. Incremental, Schwarz, quartet permutation, point-group, reciprocal mirror and multipole shortcuts are off.

Component ledger

Use ordered AO indices \((a,b,c,d)\) at cells \((0,g,p,s)\). Cell labels in phases denote lattice vectors; \(k\) and \(q\) in dot products are Cartesian. Anchor the first product at zero and write \(h=s-p\) for the second product’s displacement. Both pair distances must be at most 3.15 bohr. For (00), (11) only the home cell qualifies; (01) admits \(g_x=-1,0\); (10) admits \(0,1\). All transverse labels are zero. For SR also require the two geometric pair midpoints to lie within 3.8 bohr of one another. The oracle independently enumerates these triples and compares the actual per-quartet lists, including multiplicity.

Contribution	Ordered integral \(I(q,l,r)_{ab,cd}\)	Direct versus crossed action
SR	Sum the exact translated erfc integrals with phase \(\exp i[l\cdot g+(q+r)\cdot p-r\cdot s]\).	For \(J_{ab}(t)\) use \(I(0,t,k)_{ab,cd}D_{cd}(k)\). For \(K_{ab}(t)\) use \(I(t-k,k,k)_{ac,bd}D_{cd}(k)\). Sum each over \(k,c,d\) with weight \(1/N_k\).
LR	Sum \(B_{ab}(l,P)^* W(P) B_{cd}(r,P)\) over the stated sphere, \(P=G+q\), \(W=4\pi e^{-P^2/(4\omega^2)}/(V P^2)\).	Same index maps and weights. Public J uses the unshifted sphere. For a certified full mesh, both public K and the private source center the integer target-minus-source transfer before division.
Finite zero subtraction	Only at \(q=0\), subtract \(c B_{ab}(l,0)^* B_{cd}(r,0)\), \(c=\pi/(V\omega^2)\).	\(J_0=-c S(t)N_e\) on fixed-charge densities; \(K_0=-(c/N_k) S(t)D(t)S(t)\). The public background uses the declared electron count.
Probe charge	Absent from the quartet.	Public K also adds \(\xi S(t)D(t)S(t)\), without another mesh weight. Include it in \(F=H+J-K/2\) and the physical orbital energies.

The SR phase follows by translating both products to the first cell. The four leg characters are \((-(l+q),l,q+r,-r)\): a common translation cancels, while moving just AO \(j\) to cell \(u_j\) changes the reanchored integral by \(\exp i[(l+q)u_a-lu_b-(q+r)u_c+ru_d]\) (dot products understood). In public real space the output fold contributes \(+t\) and the density fold contributes \(-k\) at displacement \(s-p\) for J. For K, crossing the second and third AO puts the output at \(p\) and the density at \(s-g\), giving \(t p-k(s-g)\). These give exactly the two selections in the ledger. A four-leg request with unequal transfers has zero projection under the common BvK translation sum. The private panel takes one transfer and therefore cannot itself be called with a forbidden block.

For a translated product with overlap \(S_g\) and midpoint \(X_g\), \(B_{ab}(k,P)=\sum_g S_g e^{-ikg-iPX_g}e^{-P^2/(8\alpha)}\). For SR, the independent Gaussian calculation uses \(S_g S_h[\mathrm{erf}(\sqrt{\alpha}R)-\mathrm{erf}(vR)]/R\), \(v^{-2}=\alpha^{-1}+\omega^{-2}\), with the analytic coincident limit. This is Izmaylov et al., Eqs. 41-42; neither the screening approximation in Eq. 40 nor the asymptote in Eq. 43 supplies an integral here. Sun, Eqs. 18-22 gives the finite erfc transform at zero when full and LR Coulomb omit zero. With the same overlap on both sides, a same-k occupied-virtual overlap vanishes; an occupied-occupied overlap need not. A nonzero transfer has no zero term. The separate exchange probe shifts occupied RHF eigenvalues by \(-\xi\) and virtual ones by zero at this integer-occupation stationary point. Use those physical eigenvalues in any denominator; promoting the probe to a universal transition integral would also change J. See McClain et al., Sec. 4 for the momentum-conserving ERIs and the normal-ordered Fock term.

Original discriminator (#1003)

For source \(k=0\), target \(t=+1/3\) or \(-1/3\), use crossed quartet \((00|00)\), \(g=h=(0,0,0)\). Here \(B(0,P)=e^{-P^2/(8\alpha)}\). At \(G=(0,0,0)\) the shifted vector is nonzero. Its exact fractional x-coordinate is \(\pm1/3\); the original public K evaluated \(\pm0.333333333\). All differences below are public minus bounded, in Ha.

Operand	Public	Bounded	Signed difference
LR term at this one \(G\)	0.22764014041062708	0.22764013979575884	+6.1486824e-10
LR quartet, all 218 terms	0.29598306826853865	0.29598306767632576	+5.9221289e-10
Accepted density, \(K_{01}^{LR}(0)\)			+6.1194688e-10
Accepted density, physical \(F_{01}(0)\)			−3.0597347e-10

The public Fourier evaluator with its existing exact-address diagnostic agrees with the independent quartet to 5e-14 Ha. At Gamma alone all public components agree within 5.6e-16 Ha. On the three-point mesh the other five action components agree within 6.7e-16 Ha. The physical RHF gap is 1.245202439739 Ha; the probe contribution to the occupied energy is −0.269452406436 Ha at each k. These are finite-fixture results, with no claim of basis, support, mesh or external HF accuracy.

The independent sum checked all 16 AO quartets at every \((q,l,r)\), the adjoint and opposite-transfer relations, reanchoring, and a mixed complex Coulomb bilinear. Omitting the conjugate changed an LR element by 0.3570 Ha; the worst forbidden projection was 9.6e-17 Ha. The native all-k action on the 12 Hermitian matrix units (including imaginary off-diagonal probes) agreed with the independent sum within 8.4e-17 Ha. For public RHF, paired time reversal and fixed charge leave a six-dimensional real tangent space. Positive perturbations of \(S^{-1}\) spanned all six directions, and the accepted density reproduced the returned physical Fock to 1e-12 Ha.

Exact mesh replay (#1009)

Write \(B=2\pi A^{-T}\) for lattice columns \(A\). For mesh axis length \(N_i\) and half-step flag \(s_i\), the native address is \(2m_i+s_i\) modulo \(2N_i\). Subtract the source address from the target. The common shift cancels. Reduce modulo \(2N_i\), then subtract \(2N_i\) if the remainder is at least \(N_i\). The resulting numerator \(d_i\) gives \(f_i=d_i/(2N_i)\), with a half-face represented by \(-1/2\). Use \(P=B(G+f)\) for both the kernel and the pair FT; the fold still uses the source character \(e^{-ik'R}\). Only \(d=0\) selects the diagonal J tables. For \(d\ne0\), \(G=0\) is a regular, nonzero term. Neither the finite short-range subtraction nor the separate occupied-level probe correction changes. An arbitrary Cartesian list keeps its original convention; no inferred rational grid is used.

Full three-point mesh, calculated minus reference (Ha)	Signed error
Crossed (00,00), \(G=0\), either third	+2.78e-17
Crossed (00,00), all 218 vectors, positive / negative third	0 / -5.55e-17
Largest full public J/K probe residual, J at Gamma (00)	+7.49e-16 - 1.04e-18 i
Largest private J/K probe residual, J at Gamma (00)	+8.33e-17 - 2.61e-19 i
Terminal \(K^{LR}_{01}(0)\)	-5.55e-17 - 1.39e-17 i
Terminal physical \(F_{01}(0)\)	-1.11e-16 + 6.96e-18 i

The forbidden projection is below 9.58e-17 Ha, and omitting the conjugate changes an LR integral by 0.357 Ha. The physical gap is 1.245202439362 Ha; the probe lowers occupied levels by 0.269452406436 Ha and leaves virtual levels unchanged within roundoff. Substituting the historical rounded third still gives +6.15e-10 Ha for the single vector and +5.92e-10 Ha for the full quartet.

Decision. #1004 can begin its bounded correlation experiment on this fixture. The public D93 refusal remains in force. This comparison does not qualify a post-HF method, another cell or a real-space truncation policy under #531.

Raw full-space MP2 on the qualified fixture (#1004)

The public calculation and independent replays use the geometry, basis and controls above. With those operands:

from vibeqc import run_pbc_bipole_rhf
from vibeqc.dlpno.translation_driver import run_translation_dlpno_mp2

limit = 64 << 20
rhf = run_pbc_bipole_rhf(system, basis, mesh, options,
                         terminal_transition_bytes=limit, **fixture_options)
mp2 = run_translation_dlpno_mp2(
    rhf, (3, 1, 1), factor_handle=rhf.terminal_transitions,
    max_factor_bytes=limit, n_frozen=0, tcut_mkn=0, tcut_pairs=0, tcut_pno=0,
    conv_tol_energy=1e-13, conv_tol_residual=1e-12)
raw = mp2.solution.e_pair_per_cell

This experimental library call returns the raw energy in Ha per primitive cell, the convergence trace and the source identity. The generic D93 refusal remains in force. The handle binds one confirmed RHF generation, its two physical bands at each k, metric projectors, weights, integer addresses, native module and finite support. A different result cannot lend its handle, even with identical orbital dimensions and energies. The supported settings digest pins the full executed domain, including screening, for this experiment.

rhf.terminal_transitions.transition(i,a,j,b,max_bytes=limit) reads a normalized torus integral for the single occupied and virtual band at the four specified k indices. The native selection has \(l=a\), \(q=i-a\), \(r=-b\), giving four characters \((-i,a,-j,b)\). It returns zero unless \(b=i-a+j\) modulo the mesh. One cell translation sum converts the normalized four-leg integral to the primitive ordered integral divided by three. The exchange integral interchanges the two virtual k labels. Both signs are required.

For the shared solver, a unitary Fourier transform maps each real occupied or virtual trial column back to those same Bloch AO axes. Each scalar quartet is contracted directly into the current representative pair matrix. No full AO ERI tensor or auxiliary factors are retained. The handle requires 42,991,616 bytes: 8 MiB for the owner and one physical domain, 32 MiB for one native panel including backend margin, and 1 MiB for the small transforms and snapshots. One numerical read at a time holds this reservation; cancellation or an exception releases it. This is a source budget, not a process RSS limit. Shared pair preparation and transport have their own bounds.

For comparison the ordered Bloch sum evaluates the analytic Gaussians with the second AO product reversed and uses \(\mathrm{Re}[J^*(2J-K)]/(\epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b)\). Divide the extensive sum once by three. The independent six-AO supercell instead places SR images modulo three, Fourier transforms translated pairs on a single 662-vector reciprocal lattice with the supercell volume, and applies the finite overlap subtraction. It diagonalizes the folded physical Fock in the independently assembled overlap metric. Rotating the two degenerate occupied and virtual subspaces preserves the projectors and energy. The exchange probe remains in the physical Fock and its denominators.

Excitation transfer \(k_a-k_i\)	Ordered contribution (Ha/cell)
0	-0.0028548609263644
+1/3	-0.0086921590587205
-1/3	-0.0086921590587205

Raw MP2 is -0.0202391790438056 Ha/cell. Omitting the negative third raises it by 0.0086921590587207 Ha/cell. This verifies a finite H₂ Hamiltonian; it does not qualify another material, locality truncation, or correlation beyond MP2. See Nejad et al., Sec. II for the BvK normalization and Pisani et al., Sec. II for the local periodic pair equations.