Soundness and accumulated error bounds for the finite Gaussian integral evaluator

For overlap and kinetic entries, the procedure's binomial polynomial expansion and shifted moments are exact by [F3]; the pointwise derivative recurrence gives the kinetic identity. For attraction and two-electron entries, [F2] identifies the finite Hermite sums with the exact integrals. Every Boys input interval used in those sums is enclosed by [F4]. For attraction the cutoff is the sum of the two primitive angular degrees; for Coulomb it is the sum of all four. These dominate every Hermite derivative degree in [F2]. At derivative degree $d$, an entry with $j\le L-d$ uses predecessors of smaller derivative degree and index $j+1\le L-(d-1)$, so induction proves every required $R$ entry is available and enclosed. Finite contraction then uses $\overline{d_{ai}}{d}_{bj}$ for the one-body integrals and $d_{ai}\overline{d_{bj}}\overline{d_{ck}}{d}_{dl}$ for this definition's sesquilinear pair-density Coulomb convention; these are precisely the coefficients obtained by expanding the defining integrands.

The unscreened contracted diagonal intervals contain $G_{ab,ab}$ including all primitive contraction coefficients. Positivity in [F6] therefore gives $0\le G_{ab,ab}\le U_{ab}$ uniformly on the parameter boxes. Each skipped entry is assigned the interval $[-Q_{abcd},Q_{abcd}]$ with $Q_{abcd}\ge \sqrt{U_{ab}{U}_{cd}}$, which contains it by [F6]. The density interval gives $\mathrm{\mid }{W}_{ab}\mathrm{\mid }\le A_{ab}$ uniformly, so its weighted absolute energy contribution is at most $w_{abcd}\ge \frac{1}{2}{A}_{ab}{A}_{cd}{Q}_{abcd}$. Adding these bounds over the actual ordered skipped quartets, as done before each screen decision, bounds their entire omitted contribution. The reported outward sum dominates that sum; entrywise $Q$ is not confused with weighted $w$. No statement sets an omitted integral equal to zero.

The recurrence arrays and contractions are finite expression graphs. The exponential and $\pi$ constants used in those graphs are enclosed by [F8], and every remaining finite interval node encloses the exact node value by [F5]. Combining this induction with step 1.1 proves that every returned integral interval contains its exact range over the input boxes.

The midpoint tensors and radii from step 2.1 satisfy the entrywise error hypotheses of [F7], including the screened entries. For every fixed parameter choice, [F7] bounds the integral-tensor contribution to energy and gradient error uniformly using $B,M_D$. The finite polynomial evaluations over the complete $C,D$ boxes contain the midpoint-tensor energy and every component of its gradient. Enlarging the former by ${\delta }_E$ contains the exact energy. The gradient component intervals lie within Frobenius distance $\sqrt{\sum {\rho }_{a\mu }^2}$ of their returned midpoint center; the triangle inequality adds ${\delta }_{\mathrm{\nabla }}$ and gives the reported gradient bound, including input-matrix uncertainty. All final enlargements and norm upper bounds are rounded outward by [F5].

All primitive, pair, quartet, Boys-order, and auxiliary-table loops range over finite input lists or the finite integer bounds stated in [F1]. The shared work counter is checked before every primitive operation and loop update, including operations inside Boys, Gamma and elementary subcalls. A child receives finite algorithm parameters and no more than the remaining parent work, and is interrupted when that counter is exhausted by [F1]. Thus a large child construction cannot escape the finite parent cap. Therefore every invocation returns a finite status after finitely many operations.