ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Exact same-center s-orbital overlap, kinetic, attraction, and Coulomb integrals

Example

Let $g_{a} (x) = e^{- a ∥ x ∥^{2}}$ and $g_{b} (x) = e^{- b ∥ x ∥^{2}}$ for $a, b > 0$ , both centered at the origin. Put $p = a + b$ . Then $S (g_{a}, g_{b}) = \frac{π^{3 / 2}}{p^{3 / 2}}, T (g_{a}, g_{b}) = \frac{3 a b π^{3 / 2}}{p^{5 / 2}}, V_{0} (g_{a}, g_{b}) = \frac{2 π}{p} .$ For $c, d > 0$ , put $q = c + d$ . The same-center two-electron integral is $G (g_{a}, g_{b}; g_{c}, g_{d}) = \frac{2 π^{5 / 2}}{p q \sqrt{p + q}} .$

Facts & Assumptions

Given: Positive exponents $a, b, c, d$ ; the integral conventions in Cartesian Gaussian primitives, contractions, and integral conventions; and all centers equal to zero.

[F1]

Cartesian Gaussian moments and exponent-parameter derivatives: $\int_{R^{3}} e^{- p ∥ x ∥^{2}} d x = (π / p)^{3 / 2}$ and $\int_{R^{3}} ∥ x ∥^{2} e^{- p ∥ x ∥^{2}} d x = \frac{3}{2 p} (π / p)^{3 / 2}$ .

[F2]

Cartesian Gaussian raising, lowering, and kinetic-energy recurrences: $- \frac{1}{2} Δ g_{b} = (3 b - 2 b^{2} ∥ x ∥^{2}) g_{b}$ for an s primitive.

[F3]

Boys integral, power series, recurrence, and incomplete-gamma representation: $F_{0} (0) = 1$ .

[F4]

Hermite recurrence formulas for Gaussian nuclear-attraction and two-electron Coulomb integrals: the s-pair Coulomb formulas reduce to $2 π / p F_{0} (p ∥ P - C ∥^{2})$ and $2 π^{5 / 2} / (p q \sqrt{p + q}) F_{0} (η ∥ P - Q ∥^{2})$ ; an s-pair has only its $E^{000} = 1$ coefficient.

[F5]

Gaussian product theorem and finite polynomial shifts: multiplying two primitives gives the product exponent $p = a + b$ and product center $P = (a A + b B) / p$ , with its stated Gaussian prefactor.

Verification

1.1givenF1F2F5algebra

For $A = B = 0$ , [F5] gives $g_{a} g_{b} = e^{- p ∥ x ∥^{2}}$ , so [F1] evaluates the overlap as $(π / p)^{3 / 2}$ . For kinetic energy, substitute [F2] and use both moments in [F1]: $3 b (π / p)^{3 / 2} - 2 b^{2} \frac{3}{2 p} (π / p)^{3 / 2} = \frac{3 a b π^{3 / 2}}{p^{5 / 2}} .$

2.1givenF3F4algebra∎

In [F4], set $C = P = 0$ ; then the Boys argument is zero and [F3] gives $V_{0} = 2 π / p$ . For the two-electron integral, both pair centers are zero and their only Hermite coefficient is $1$ , so [F4] with $F_{0} (0) = 1$ gives $2 π^{5 / 2} / (p q \sqrt{p + q})$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

23 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Trygve Helgaker, Molecular Integral Evaluation, 10th Sostrup Summer School lecture notes (2008) (standard reference, not scraped)
L. E. McMurchie and E. R. Davidson, One and Two Electron Integrals over Cartesian Gaussian Functions, UCRL-13791 (1976) (standard reference, not scraped)