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A neutral dipole cell and its reciprocal modes

Example

Take the unit cubic torus $R^{3} / Z^{3}$ , with charges $q_{1} = 1$ at $x_{1} = (0, 0, 0)$ and $q_{2} = - 1$ at $x_{2} = (1 / 2, 0, 0)$ . The dipole is neutral and its Fourier structure factor vanishes on every even first-coordinate mode and is equal to $2$ on every odd first-coordinate mode.

Facts & Assumptions

Given: Countable Choice, $α > 0$ , the unit cell, these two charges and positions, and the Fourier coefficient and reciprocal-energy conventions for neutral periodic data.

[A1]

The Gaussian Ewald formula uses Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

$ρ_{k} = \sum_{i} q_{i} e^{- 2 π i k \cdot x_{i}}$ and neutral data have zero coefficient at $k = 0$ (Neutral point sources and Fourier data on a three-dimensional torus).

[F2]

The reciprocal Ewald energy is $\frac{1}{2} \sum_{k \neq 0} c_{k} ∣ ρ_{k} ∣^{2}$ with $c_{k} = 4 π e^{- ∣ κ_{k} ∣^{2} / (4 α^{2})} / (V ∣ κ_{k} ∣^{2})$ , and the full renormalized energy adds the real-space and self terms (The alpha-independent Ewald potential and renormalized energy).

Verification

technique · direct

1.1givenF1algebra

The two positions are distinct in $R^{3} / Z^{3}$ and $q_{1} + q_{2} = 0$ . For $k \in Z^{3}$ , [F1] gives $ρ_{k} = 1 - e^{- 2 π i k_{1} / 2} = 1 - (- 1)^{k_{1}}$ ; hence $ρ_{0} = 0$ , $ρ_{k} = 0$ when $k_{1}$ is even, and $ρ_{k} = 2$ when $k_{1}$ is odd.

2.1givenA1F2step 1.1algebra∎

For any finite symmetric reciprocal mode set $K \subset Z^{3} ∖ {0}$ , the dipole contributes $\frac{1}{2} \sum_{k \in K} c_{k} ∣ ρ_{k} ∣^{2} = 2 \sum_{k \in K, k_{1} odd} c_{k}$ to the retained reciprocal energy. The remaining real-space and self contributions are exactly the terms in [F2], under its neutrality, positive-parameter and Countable Choice hypotheses [A1]. Thus the zero mode is absent and the reciprocal selection is precisely the odd- $k_{1}$ subset, not an assumed continuum average.

Depends on

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