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Uniform lattice-shell counts and explicit Gaussian tail controllers

Statement

Assume Countable Choice. For an invertible real $3 \times 3$ matrix $B$ , put $Γ = B Z^{3}$ and $C_{B} = (2 ∥ B^{- 1} ∥_{2} (1 + \sqrt{3} ∥ B ∥_{2}) + 1)^{3} .$ Here $∥ \cdot ∥_{2}$ is the Euclidean operator norm. Uniformly over every translate $y + Γ$ and every integer shell $m \geq 0$ , $# {z \in y + Γ : m \leq ∣ z ∣ < m + 1} \leq C_{B} (m + 1)^{3} .$

For $β > 0$ , any integer $p \geq 0$ , and integer $M \geq 0$ , set $T_{p} (β, M) = \sum_{m = M}^{\infty} (m + 1)^{p} e^{- β m^{2}} .$ For any integer $J \geq M$ satisfying $2 β J (J + 1) \geq p$ , define $I_{0} (J, β) = \frac{\sqrt{π}}{2 \sqrt{β}} erfc (\sqrt{β} J), I_{1} (J, β) = \frac{e^{- β J^{2}}}{2 β},$ and recursively for $2 \leq s \leq p$ , $I_{s} (J, β) = \frac{J^{s - 1} e^{- β J^{2}}}{2 β} + \frac{s - 1}{2 β} I_{s - 2} (J, β) .$ Then the finite controller $U_{p} (β, M, J) = \sum_{m = M}^{J} (m + 1)^{p} e^{- β m^{2}} + \sum_{s = 0}^{p} (\binom{p}{s}) I_{s} (J, β)$ satisfies $T_{p} (β, M) \leq U_{p} (β, M, J)$ . Such a $J$ exists for every $β > 0$ , $M \geq 0$ , and integer $p \geq 0$ .

Facts & Assumptions

Given: Countable Choice, the full-rank cell matrix $A$ and angular reciprocal lattice from Neutral point sources and Fourier data on a three-dimensional torus, an arbitrary invertible $B \in G L_{3} (R)$ , a translate $y + B Z^{3}$ , $β > 0$ , integer $p \geq 0$ , and $M \in N$ .

[F1]

Each coset of a full-rank lattice has a representative in its closed fundamental parallelotope, whose coordinates lie in $[0, 1]^{3}$ (Fundamental parallelotopes of a lattice tile Euclidean space with covolume volume).

[F2]

$erfc (s)$ is the nonnegative Gaussian tail $\frac{2}{\sqrt{π}} \int_{s}^{\infty} e^{- t^{2}} d t$ for $s \geq 0$ (Gaussian screened Coulomb kernel and its self limit).

[F3]

For every real $X$ there is an integer greater than $X$ (Every complete ordered field is Archimedean).

[F4]

Integration by parts holds on a finite interval when the factors and derivatives satisfy the stated integrability hypotheses (If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ ).

[A1]

Countable Choice is assumed by the Gaussian-tail and lattice-tiling interfaces used here (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1givenA1F1algebra

Under Countable Choice [A1], use [F1] to replace $y$ modulo $B Z^{3}$ by a vector $y_{0} = B v$ with $v \in [0, 1]^{3}$ . Thus $∣ y_{0} ∣ \leq \sqrt{3} ∥ B ∥_{2}$ . If $z = y_{0} + B n$ lies in the shell $m \leq ∣ z ∣ < m + 1$ , then $∣ n ∣ \leq ∥ B^{- 1} ∥_{2} (m + 1 + \sqrt{3} ∥ B ∥_{2})$ . The number of integer triples with this bound is at most $(2 ∥ B^{- 1} ∥_{2} (m + 1 + \sqrt{3} ∥ B ∥_{2}) + 1)^{3} \leq C_{B} (m + 1)^{3}$ , proving the uniform shell count.

1.2givenF3algebra

Choose $J \geq M$ with $2 β J (J + 1) \geq p$ , which exists by [F3]. For $x \geq J$ , the derivative of $f (x) = (x + 1)^{p} e^{- β x^{2}}$ has the sign of $p / (x + 1) - 2 β x \leq 0$ , so $f$ is nonincreasing. Therefore $\sum_{m = J + 1}^{\infty} f (m) \leq \int_{J}^{\infty} f (x) d x$ . Expanding $(x + 1)^{p}$ by the binomial theorem gives $\int_{J}^{\infty} f (x) d x = \sum_{s = 0}^{p} (\binom{p}{s}) I_{s} (J, β)$ .

2.1A1F2F4step 1.2algebra∎

The formulas for $I_{0}$ and $I_{1}$ follow by the substitutions $t = \sqrt{β} x$ and $u = β x^{2}$ . For $s \geq 2$ , choose an integer $L > (s + 1) / 2$ ; $e^{β x^{2}} \geq (β x^{2})^{L} / L!$ shows $x^{s} e^{- β x^{2}} = O (x^{s - 2 L})$ is integrable at infinity. Apply [F4] on $[J, R]$ and let $R \to \infty$ ; the same inequality makes the boundary term $R^{s - 1} e^{- β R^{2}}$ vanish. This gives $I_{s} = J^{s - 1} e^{- β J^{2}} / (2 β) + (s - 1) I_{s - 2} / (2 β)$ . Hence the finite sum through $J$ plus the integral tail in step 1.2 equals $U_{p}$ and bounds $T_{p}$ .

Remarks

The shell estimate is intentionally coarse. Its cubic factor is enough because every later term has Gaussian decay; no asymptotic lattice-point theorem is used.
For real-space estimates take $B = A$ . For reciprocal estimates take $B = 2 π A^{- T}$ , whose lattice points are exactly the angular physical frequencies $κ_{k}$ .

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Sources

Hark Lee and Wei Cai, Ewald Summation for Coulomb Interactions in a Periodic Supercell (2009 lecture notes) (standard reference, not scraped)
Cameron F. Abrams, Modern Molecular Simulations (CHE T580, 2021 lecture notes) (standard reference, not scraped)