TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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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.

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Gaussian product theorem and finite polynomial shifts

Statement

Let $a, b > 0$ , $A, B \in R^{3}$ , $p = a + b$ , $μ = a b / p$ , and $P = (a A + b B) / p$ . Then for every $x \in R^{3}$ , $\exp (- a ∥ x - A ∥_{2}^{2}) \exp (- b ∥ x - B ∥_{2}^{2}) = \exp (- μ ∥ A - B ∥_{2}^{2}) \exp (- p ∥ x - P ∥_{2}^{2}) .$ For any multi-indices $α, β \in N_{0}^{3}$ , the polynomial factor $(x - A)^{α} (x - B)^{β}$ is a finite polynomial in $x - P$ of total degree at most $∣ α ∣ + ∣ β ∣$ . Moreover, $P = (a / p) A + (b / p) B$ lies on the closed segment from $A$ to $B$ .

Facts & Assumptions

Given: $a, b > 0$ , centers $A, B \in R^{3}$ , and multi-indices $α, β \in N_{0}^{3}$ as in Cartesian Gaussian primitives, contractions, and integral conventions.

[F1]

Cartesian Gaussian primitives, contractions, and integral conventions: a primitive has exponential factor $\exp (- a ∥ x - A ∥_{2}^{2})$ and polynomial factor $(x - A)^{α}$ .

Proof

1.1givenF1algebra

Since $p = a + b > 0$ , expand the left exponent using $∥ x - A ∥^{2} = ∥ x ∥^{2} - 2 A \cdot x + ∥ A ∥^{2}$ . The quadratic and linear terms agree with $p ∥ x - P ∥^{2}$ because $p P = a A + b B$ . The remaining constant satisfies $a ∥ A ∥^{2} + b ∥ B ∥^{2} - p ∥ P ∥^{2} = \frac{a b}{p} ∥ A - B ∥^{2} = μ ∥ A - B ∥^{2},$ by substituting $P = (a A + b B) / p$ and expanding the inner products. Thus $a ∥ x - A ∥^{2} + b ∥ x - B ∥^{2} = p ∥ x - P ∥^{2} + μ ∥ A - B ∥^{2}$ , which gives the exponential identity.

1.2givenF1algebra

Each coordinate obeys $x_{j} - A_{j} = (x_{j} - P_{j}) + (P_{j} - A_{j})$ and $x_{j} - B_{j} = (x_{j} - P_{j}) + (P_{j} - B_{j})$ . The finite binomial theorem expands each integer power as a polynomial in $x_{j} - P_{j}$ of no larger degree; multiplying the six coordinate polynomials gives a finite polynomial of total degree at most $∣ α ∣ + ∣ β ∣$ .

2.1givenalgebra∎

The coefficients $a / p$ and $b / p$ are positive and sum to one. Hence $P$ is a convex combination of $A$ and $B$ , so it lies on their closed line segment, including the case $A = B$ .

Depends on

Cartesian Gaussian primitives, contractions, and integral conventions

Used by

Dependency tree · two levels

5 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)