ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Highly contracted Gaussian coefficients amplify primitive integral error

Example

Let $n = 2$ , $r = 1$ , $D = (1)$ , $C = (K, K)^{T}$ with $K > 0$ , and suppose every primitive four-index tensor error is $Δ G_{a b c d} = ε > 0$ while $Δ H = 0$ . Then the transformed four-index error is $16 K^{4} ε$ , the energy error is $8 K^{4} ε$ , and the gradient error has Frobenius norm $16 \sqrt{2} K^{3} ε$ . Both bounds in Explicit finite-dimensional contraction error bounds for energy and gradient are attained for this aligned-sign perturbation.

Facts & Assumptions

Given: The finite energy functional in Finite tensor contractions and their variational energy and gradient, with the two-dimensional coefficient vector and entrywise perturbation in the Example.

[F1]

Finite tensor contractions and their variational energy and gradient: $G_{1111}^{'} = \sum_{a, b, c, d = 1}^{2} C_{a 1} C_{b 1} C_{c 1} C_{d 1} G_{a b c d}$ and $E_{2} = \frac{1}{2} D_{11}^{2} G_{1111}^{'}$ .

[F2]

Explicit finite-dimensional contraction error bounds for energy and gradient: energy and gradient errors are bounded by the explicit $n, r, B, M$ formulas in its Statement.

Verification

1.1givenF1algebra

There are $2^{4} = 16$ primitive entries, each perturbed by $ε$ , and each product of four coefficients is $K^{4}$ . Thus $Δ G_{1111}^{'} = 16 K^{4} ε$ and $Δ E = \frac{1}{2} Δ G_{1111}^{'} = 8 K^{4} ε$ . The perturbed energy is $\frac{1}{2} ε (C_{1} + C_{2})^{4}$ , so each gradient component is $2 ε (C_{1} + C_{2})^{3} = 16 K^{3} ε$ ; its two-entry Frobenius norm is $16 \sqrt{2} K^{3} ε$ .

2.1givenF2step 1.1algebra∎

Here $B = ∥ C ∥_{F} = \sqrt{2} K$ and $M = ∥ D ∥_{F} = 1$ . Substitution into [F2] gives energy bound $\frac{1}{2} (1)^{2} (1)^{2} (2)^{2} (\sqrt{2} K)^{4} ε = 8 K^{4} ε$ and gradient bound $2 (2)^{2} (1)^{2} (\sqrt{2} K)^{3} (1)^{2} ε = 16 \sqrt{2} K^{3} ε$ , equal to the errors in step 1.1.

Depends on

Used by

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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)