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A scaled incomplete-gamma tail bound for large Boys arguments

Statement

Let $m \in N_{0}$ , $s = m + \frac{1}{2}$ , and $T > \max {s - 1, 0}$ . Then $F_{m} (T) = \frac{Γ (s) - R_{s} (T)}{2 T^{s}}, R_{s} (T) = \int_{T}^{\infty} u^{s - 1} e^{- u} d u,$ and $0 \leq R_{s} (T) \leq \frac{T^{s - 1} e^{- T}}{1 - \max {s - 1, 0} / T} .$ Consequently a certified enclosure of $Γ (s)$ and $e^{- T}$ gives the interval $[\frac{Γ (s)}{2 T^{s}} - \frac{T^{s - 1} e^{- T}}{2 T^{s} (1 - \max {s - 1, 0} / T)}, \frac{Γ (s)}{2 T^{s}}]$ after intersecting its lower endpoint with zero. In the evaluator's large branch $T > 2 m + 1$ , the denominator is at least $1 / 2$ .

Facts & Assumptions

Given: $m \geq 0$ , $s = m + 1 / 2$ , $T > \max {s - 1, 0}$ , and the definitions in Boys functions and lower incomplete-gamma notation and Euler's Gamma function on the right half-plane.

[F1]

Boys integral, power series, recurrence, and incomplete-gamma representation: $F_{m} (T) = γ (s, T) / (2 T^{s})$ .

[F2]

Euler's Gamma function on the right half-plane: $Γ (s) = \int_{0}^{\infty} u^{s - 1} e^{- u} d u$ for $s > 0$ .

[F3]

Additivity of the nonnegative Lebesgue integral: splitting a nonnegative integral over adjacent intervals adds the two integrals.

Proof

1.1givenF1F2F3algebra

By [F2]–[F3], $Γ (s) = γ (s, T) + R_{s} (T)$ , so [F1] gives the scaled identity. The tail is nonnegative.

1.2givenalgebra

Substitute $u = T + v$ into $R_{s} (T)$ . If $s \geq 1$ , then $(T + v)^{s - 1} = T^{s - 1} (1 + v / T)^{s - 1} \leq T^{s - 1} e^{(s - 1) v / T}$ because $1 + w \leq e^{w}$ for $w \geq 0$ . Hence $R_{s} (T) \leq T^{s - 1} e^{- T} \int_{0}^{\infty} e^{- (1 - (s - 1) / T) v} d v$ , whose integral is $1 / (1 - (s - 1) / T)$ . If $0 < s < 1$ , then $(T + v)^{s - 1} \leq T^{s - 1}$ , yielding $R_{s} (T) \leq T^{s - 1} e^{- T}$ . These are the combined bound.

2.1givenstep 1.1step 1.2algebra∎

For $s = m + 1 / 2$ , $\max {s - 1, 0} \leq m$ ; if $T > 2 m + 1$ , then $\max {s - 1, 0} / T < 1 / 2$ . Substitution into step 1.2 gives the evaluator's large-branch denominator bound and the interval enclosure.

Depends on

Used by

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19 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)