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

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Nearly duplicate Gaussian primitives produce an ill-conditioned overlap matrix

Example

For $a, b > 0$ , define normalized s primitives $h_{a} (x) = (2 a / π)^{3 / 4} e^{- a ∥ x ∥^{2}}$ . Their overlap is $S (h_{a}, h_{b}) = {(\frac{2 \sqrt{a b}}{a + b})}^{3 / 2} .$ For $a = b$ the two-function overlap matrix has eigenvalues $2$ and $0$ . For distinct exponents $a, b$ it has eigenvalues $1 \pm S (h_{a}, h_{b})$ , and as $b \to a$ its condition number tends to infinity.

Facts & Assumptions

Given: Positive exponents $a, b$ , normalized Gaussian functions as defined in the Example, and the overlap integral in Cartesian Gaussian primitives, contractions, and integral conventions.

[F1]

Gaussian product theorem and finite polynomial shifts: $e^{- a ∥ x ∥^{2}} e^{- b ∥ x ∥^{2}} = e^{- (a + b) ∥ x ∥^{2}}$ .

[F2]

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

Verification

1.1givenF1F2algebra

By [F1]–[F2], the unnormalized overlap is $(π / (a + b))^{3 / 2}$ and each norm squared is $(π / (2 a))^{3 / 2}$ or $(π / (2 b))^{3 / 2}$ . Multiplying by the normalization constants gives $S (h_{a}, h_{b}) = (2 \sqrt{a b} / (a + b))^{3 / 2}$ .

2.1givenalgebra∎

The Gram matrix is $(\begin{array}{cc} 1 & S \\ S & 1 \end{array})$ , whose characteristic polynomial is $(1 - λ)^{2} - S^{2}$ , so its eigenvalues are $1 \pm S$ . At $a = b$ , $S = 1$ and the lower eigenvalue is zero. For $a \neq b$ , $2 \sqrt{a b} < a + b$ , hence $0 < S < 1$ ; continuity of the displayed formula gives $S \to 1$ as $b \to a$ , and therefore $(1 + S) / (1 - S) \to \infty$ .

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)