| import Mathlib |
| import OAI.Analysis.CoulombIonization.Variational.OrbitalPole |
| |
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
|
|
| open scoped BigOperators |
| |
| namespace CoulombAtom |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
|
|
| lemma occupation_pair_sum_complex (p : ι → ℝ) (f : ι → ι → ℂ) (hf : ∀ i, f i i = 0) : |
| (∑ m : ι → Bool, (occupationWeight p m : ℂ) * |
| ∑ i, ∑ j, if m i && m j then f i j else 0) = |
| ∑ i, ∑ j, (p i : ℂ)*(p j : ℂ)*f i j := by |
| have hre (b : Bool) (z : ℂ) : (if b then z else 0).re = if b then z.re else 0 := by |
| cases b < |
| have him (b : Bool) (z : ℂ) : (if b then z else 0).im = if b then z.im else 0 := by |
| cases b < |
| apply Complex.ext |
| · simpa only [Complex.re_sum,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im, |
| mul_zero,zero_mul,sub_zero,hre,Complex.mul_im,add_zero] using |
| occupation_pair_sum p (fun i j => (f i j).re) (fun i => by rw [hf i] |
| · simpa only [Complex.im_sum,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im, |
| mul_zero,zero_mul,zero_add,add_zero,him,Complex.mul_re,sub_zero] using |
| occupation_pair_sum p (fun i j => (f i j).im) (fun i => by rw [hf i] |
|
|
| end CoulombAtom |
|
|
| end |
|
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| end OAI |
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|
