| import Mathlib |
| import OAI.Analysis.CoulombIonization.Localization.PacketDensity |
|
|
| namespace OAI |
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| noncomputable section |
|
|
| namespace CoulombAtom |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace Convolution |
| open Filter |
| open scoped Topology InnerProductSpace |
| open MeasureTheory Complex Filter |
| open scoped Topology InnerProductSpace |
| open MeasureTheory Complex Filter |
| open scoped Topology InnerProductSpace ContDiff |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace Convolution |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace |
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff InnerProductSpace ENNReal |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open Set Filter Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter |
| open scoped Topology |
| open MeasureTheory Filter |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter |
| open scoped Topology BigOperators InnerProductSpace |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter |
| open scoped Topology BigOperators Pointwise |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open scoped Convolution |
| open scoped RealInnerProductSpace |
| open MeasureTheory Filter Set Metric |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Set Filter Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Set Filter Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology InnerProductSpace Laplacian |
| open Filter Set Metric Topology InnerProductSpace Laplacian |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open Filter |
| open scoped Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Filter Set Metric Topology |
| open MeasureTheory Complex Filter |
| open scoped Topology InnerProductSpace ContDiff BigOperators |
| open MeasureTheory Filter Set |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators InnerProductSpace |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open scoped BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
|
|
| def formOneBody {N : ℕ} (ψ : FormVector N) (w : Space → ℝ) : ℝ := |
| ∑ s : Spins N, ∑ i : Fin N, ∫ x, ‖ψ.value s x‖^2 * w (x i) |
|
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| def OneBodyIntegrable {N : ℕ} (ψ : FormVector N) (w : Space → ℝ) : Prop := |
| ∀ s i, Integrable (fun x => ‖ψ.value s x‖^2 * w (x i)) |
|
|
| lemma formOneBody_scale {N : ℕ} (ψ : FormVector N) (w : Space → ℝ) (c : ℝ) : |
| formOneBody (scaleForm c ψ) w = c^2 * formOneBody ψ w := by |
| unfold formOneBody |
| simp only [scaleForm,norm_real_mul_sq,mul_assoc,integral_const_mul,Finset.mul_sum] |
|
|
| lemma OneBodyIntegrable.scale {N : ℕ} {ψ : FormVector N} {w : Space → ℝ} |
| (h : OneBodyIntegrable ψ w) (c : ℝ) : OneBodyIntegrable (scaleForm c ψ) w := by |
| intro s i |
| simpa only [scaleForm,norm_real_mul_sq,mul_assoc] using (h s i).const_mul (c^2) |
|
|
| lemma OneBodyIntegrable.reindex {N M : ℕ} {ψ : FormVector N} {w : Space → ℝ} |
| (h : OneBodyIntegrable ψ w) (e : Fin M ≃ Fin N) : OneBodyIntegrable (reindexForm e ψ) w := by |
| intro s i |
| simpa only [Function.comp_def,configurationReindex_apply,reindexForm,Equiv.symm_apply_apply] |
| using (configurationReindex_preserving e).integrable_comp_of_integrable (h (s ∘ e.symm) (e i)) |
|
|
| lemma formOneBody_reindex {N M : ℕ} (ψ : FormVector N) (w : Space → ℝ) (e : Fin M ≃ Fin N) : |
| formOneBody (reindexForm e ψ) w = formOneBody ψ w := by |
| have hh (s : Spins M) (i : Fin M) : |
| (∫ x, ‖(reindexForm e ψ).value s x‖^2 * w (x i)) = |
| ∫ x, ‖ψ.value (s ∘ e.symm) x‖^2 * w (x (e i)) := by |
| simpa only [Function.comp_apply,Equiv.symm_apply_apply,reindexForm] using |
| integral_reindex e (fun x => ‖ψ.value (s ∘ e.symm) x‖^2 * w (x (e i))) |
| have hi (s : Spins M) : |
| (∑ i : Fin M, ∫ x, ‖ψ.value (s ∘ e.symm) x‖^2 * w (x (e i))) = |
| ∑ i : Fin N, ∫ x, ‖ψ.value (s ∘ e.symm) x‖^2 * w (x i) := |
| Equiv.sum_comp e (fun i : Fin N => ∫ x, ‖ψ.value (s ∘ e.symm) x‖^2 * w (x i)) |
| unfold formOneBody |
| simp_rw [hh,hi] |
| exact sum_spin_reindex e (fun s => ∑ i, ∫ x, ‖ψ.value s x‖^2 * w (x i)) |
|
|
| lemma formOneBody_sum_disjoint {ι : Type*} [Fintype ι] {N : ℕ} |
| {ψ : ι → FormVector N} {w : Space → ℝ} |
| (h : ∀ k, OneBodyIntegrable (ψ k) w) (hd : FormsDisjoint ψ) : |
| formOneBody (sumForm ψ) w = ∑ k, formOneBody (ψ k) w := by |
| have hh (s : Spins N) (i : Fin N) : |
| (∫ x, ‖(sumForm ψ).value s x‖^2 * w (x i)) = |
| ∑ k, ∫ x, ‖(ψ k).value s x‖^2 * w (x i) := by |
| simp only [sumForm,norm_sum_sq_disjoint _ (fun k l hkl => hd.1 k l hkl s _),Finset.sum_mul] |
| exact integral_finsetSum _ (fun k _ => h k s i) |
| unfold formOneBody |
| simp_rw [hh] |
| calc |
| _ = ∑ s, ∑ k, ∑ i, ∫ x, ‖(ψ k).value s x‖^2 * w (x i) := |
| Finset.sum_congr rfl (fun _ _ => Finset.sum_comm) |
| _ = _ := Finset.sum_comm |
|
|
| lemma formOneBody_insertionTerm {N : ℕ} (T : FormVector (N+1)) (w : Space → ℝ) (i : Fin (N+1)) : |
| formOneBody (insertionTerm T i) w = formOneBody T w := by |
| rw [insertionTerm,formOneBody_scale,formOneBody_reindex,real_sign_sq,one_mul] |
|
|
| lemma formOneBody_wedge {N : ℕ} {T : FormVector (N+1)} {w : Space → ℝ} |
| (h : OneBodyIntegrable T w) (hd : FormsDisjoint (insertionTerm T)) : |
| formOneBody (wedgeForm T) w = (N+1 : ℝ) * formOneBody T w := by |
| have hk (i : Fin (N+1)) : OneBodyIntegrable (insertionTerm T i) w := |
| (h.reindex (insertionSwap i)).scale _ |
| rw [wedgeForm,formOneBody_sum_disjoint hk hd] |
| simp only [formOneBody_insertionTerm,Finset.sum_const,Finset.card_univ,Fintype.card_fin, |
| nsmul_eq_mul,Nat.cast_add,Nat.cast_one] |
|
|
| lemma formOneBody_tensor {N M : ℕ} (ψ : FormVector N) (φ : FormVector M) (w : Space → ℝ) : |
| formOneBody (tensorForm ψ φ) w = formOneBody ψ w * formMass φ + |
| formMass ψ * formOneBody φ w := by |
| have hl (s : Spins N) (t : Spins M) (i : Fin N) : |
| (∫ z, ‖(tensorForm ψ φ).value (joinLists s t) z‖^2 * w (z (finSumFinEquiv (Sum.inl i)))) = |
| (∫ x, ‖ψ.value s x‖^2 * w (x i)) * (∫ y, ‖φ.value t y‖^2) := by |
| change (∫ z, ‖ψ.value (leftList (joinLists s t)) (leftList z) * |
| φ.value (rightList (joinLists s t)) (rightList z)‖^2 * w (leftList z i)) = _ |
| simp only [leftList_join,rightList_join,norm_mul,mul_pow] |
| simp_rw [mul_right_comm _ _ (w _)] |
| exact integral_product_join (N := N) (M := M) (fun x => ‖ψ.value s x‖^2 * w (x i)) |
| (fun y => ‖φ.value t y‖^2) |
| have hr (s : Spins N) (t : Spins M) (i : Fin M) : |
| (∫ z, ‖(tensorForm ψ φ).value (joinLists s t) z‖^2 * w (z (finSumFinEquiv (Sum.inr i)))) = |
| (∫ x, ‖ψ.value s x‖^2) * (∫ y, ‖φ.value t y‖^2 * w (y i)) := by |
| change (∫ z, ‖ψ.value (leftList (joinLists s t)) (leftList z) * |
| φ.value (rightList (joinLists s t)) (rightList z)‖^2 * w (rightList z i)) = _ |
| simp only [leftList_join,rightList_join,norm_mul,mul_pow,mul_assoc] |
| exact integral_product_join (N := N) (M := M) (fun x => ‖ψ.value s x‖^2) |
| (fun y => ‖φ.value t y‖^2 * w (y i)) |
| have hk (s : Spins N) (t : Spins M) : |
| (∑ i, ∫ z, ‖(tensorForm ψ φ).value (joinLists s t) z‖^2 * w (z i)) = |
| (∑ i, ∫ x, ‖ψ.value s x‖^2 * w (x i))*(∫ y, ‖φ.value t y‖^2) + |
| (∫ x, ‖ψ.value s x‖^2)*(∑ i, ∫ y, ‖φ.value t y‖^2 * w (y i)) := by |
| rw [← Equiv.sum_comp finSumFinEquiv,Fintype.sum_sum_type] |
| simp only [hl,hr,Finset.sum_mul,Finset.mul_sum] |
| unfold formOneBody formMass |
| rw [← sum_spin_join] |
| simp_rw [hk,Finset.sum_add_distrib,← Finset.sum_mul,← Finset.mul_sum] |
|
|
| def insertOrbital {N : ℕ} (ψ : FormVector N) (φ : FormVector 1) : FormVector (N+1) := |
| scaleForm (Real.sqrt (N+1 : ℝ))⁻¹ (wedgeForm (tensorForm ψ φ)) |
|
|
| lemma insertOrbital_sobolev {N : ℕ} {ψ : FormVector N} {φ : FormVector 1} |
| (hψ : SobolevFermion ψ) (hφ : SobolevFermion φ) : SobolevFermion (insertOrbital ψ φ) := |
| (wedgeForm_fermion (hψ.sobolevVector.tensor hφ.sobolevVector) |
| (tensor_coreAntisymmetric hψ φ)).scale _ |
|
|
| lemma insertion_normalizer (N : ℕ) : (Real.sqrt (N+1 : ℝ))⁻¹^2 * (N+1 : ℝ) = 1 := by |
| rw [inv_pow,Real.sq_sqrt (by positivity)] |
| exact inv_mul_cancel₀ (by positivity) |
|
|
| lemma formMass_insertOrbital {N : ℕ} {ψ : FormVector N} {φ : FormVector 1} |
| (hψ : SobolevVector ψ) (hφ : SobolevVector φ) |
| (hd : FormsDisjoint (insertionTerm (tensorForm ψ φ))) : |
| formMass (insertOrbital ψ φ) = formMass ψ * formMass φ := by |
| rw [insertOrbital,formMass_scale,formMass_wedge (hψ.tensor hφ) hd,← mul_assoc, |
| insertion_normalizer,one_mul,formMass_tensor] |
|
|
| lemma formEnergy_insertOrbital {N : ℕ} {ψ : FormVector N} {φ : FormVector 1} |
| (hψ : SobolevVector ψ) (hφ : SobolevVector φ) |
| (hd : FormsDisjoint (insertionTerm (tensorForm ψ φ))) (Z : ℝ) : |
| formEnergy Z (insertOrbital ψ φ) = formEnergy Z (tensorForm ψ φ) := by |
| rw [insertOrbital,formEnergy_scale,formEnergy_wedge (hψ.tensor hφ) hd,← mul_assoc, |
| insertion_normalizer,one_mul] |
|
|
| lemma formOneBody_insertOrbital {N : ℕ} {ψ : FormVector N} {φ : FormVector 1} {w : Space → ℝ} |
| (h : OneBodyIntegrable (tensorForm ψ φ) w) |
| (hd : FormsDisjoint (insertionTerm (tensorForm ψ φ))) : |
| formOneBody (insertOrbital ψ φ) w = formOneBody ψ w * formMass φ + |
| formMass ψ * formOneBody φ w := by |
| rw [insertOrbital,formOneBody_scale,formOneBody_wedge h hd,← mul_assoc, |
| insertion_normalizer,one_mul,formOneBody_tensor] |
|
|
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
|
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|
|
| def FormAvoids {N : ℕ} (ψ : FormVector N) (S : Set Space) : Prop := |
| ∀ x i, x i ∈ S → FormZeroAt ψ x |
|
|
| lemma FormAvoids.scale {N : ℕ} {ψ : FormVector N} {S : Set Space} |
| (h : FormAvoids ψ S) (c : ℝ) : FormAvoids (scaleForm c ψ) S := by |
| intro x i hx s |
| constructor |
| · change _ * ψ.value s x = 0 |
| rw [(h x i hx s).1,mul_zero] |
| · intro j a |
| change _ * ψ.gradient s j a x = 0 |
| rw [(h x i hx s).2,mul_zero] |
|
|
| lemma FormAvoids.reindex {N M : ℕ} {ψ : FormVector N} {S : Set Space} |
| (h : FormAvoids ψ S) (e : Fin M ≃ Fin N) : FormAvoids (reindexForm e ψ) S := by |
| intro x i hx s |
| have hz := h (x ∘ e.symm) (e i) (by simpa only [Function.comp_apply,Equiv.symm_apply_apply] using hx) |
| exact ⟨(hz (s ∘ e.symm)).1, fun j a => (hz (s ∘ e.symm)).2 (e j) a⟩ |
|
|
| lemma FormAvoids.sum {ι : Type*} [Fintype ι] {N : ℕ} {ψ : ι → FormVector N} {S : Set Space} |
| (h : ∀ k, FormAvoids (ψ k) S) : FormAvoids (sumForm ψ) S := by |
| intro x i hx s |
| constructor |
| · change (∑ k, (ψ k).value s x) = 0 |
| exact Finset.sum_eq_zero (fun k _ => (h k x i hx s).1) |
| · intro j a |
| change (∑ k, (ψ k).gradient s j a x) = 0 |
| exact Finset.sum_eq_zero (fun k _ => (h k x i hx s).2 j a) |
|
|
| lemma FormAvoids.tensor {N M : ℕ} {ψ : FormVector N} {φ : FormVector M} {S : Set Space} |
| (hψ : FormAvoids ψ S) (hφ : FormAvoids φ S) : FormAvoids (tensorForm ψ φ) S := by |
| intro x i hx |
| obtain ⟨i,rfl⟩ := finSumFinEquiv.surjective i |
| cases i with |
| | inl i => exact tensor_zero_left ψ φ (hψ (leftList x) i hx) |
| | inr i => exact tensor_zero_right ψ φ (hφ (rightList x) i hx) |
|
|
| lemma FormAvoids.wedge {N : ℕ} {T : FormVector (N+1)} {S : Set Space} |
| (h : FormAvoids T S) : FormAvoids (wedgeForm T) S := by |
| apply FormAvoids.sum |
| intro i |
| exact (h.reindex (insertionSwap i)).scale _ |
|
|
| lemma FormAvoids.insertOrbital {N : ℕ} {ψ : FormVector N} {φ : FormVector 1} {S : Set Space} |
| (hψ : FormAvoids ψ S) (hφ : FormAvoids φ S) : FormAvoids (insertOrbital ψ φ) S := |
| (hψ.tensor hφ).wedge.scale _ |
|
|
| lemma FormAvoids.mono {N : ℕ} {ψ : FormVector N} {S T : Set Space} |
| (h : FormAvoids ψ T) (hst : S ⊆ T) : FormAvoids ψ S := |
| fun x i hx => h x i (hst hx) |
|
|
| lemma radialLocalOrbital_avoids (y z : Space) {R B : ℝ} (hR : 0 < R) |
| (hsep : R+B ≤ ‖y-z‖) : FormAvoids (radialLocalOrbital y R) (Metric.ball z B) := by |
| intro x i hx |
| have hi : i = 0 := Subsingleton.elim _ _ |
| subst i |
| have hx' : ‖x 0-z‖ < B := Metric.mem_ball.mp hx |
| have ht : ‖y-z‖ ≤ ‖x 0-y‖+‖x 0-z‖ := by |
| calc |
| _ ≤ ‖y-x 0‖+‖x 0-z‖ := norm_sub_le_norm_sub_add_norm_sub _ _ _ |
| _ = _ := by rw [norm_sub_rev y (x 0)] |
| exact radialLocalOrbital_zero y hR (by linarith) |
| |
| lemma integrable_coreCoulombAt {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) |
| (y : Space) {B : ℝ} (hB : 0 < B) (hc : FormAvoids ψ (Metric.ball y B)) (s : Spins N) (i : Fin N) : |
| Integrable (fun x => ‖ψ.value s x‖^2/‖x i-y‖) := by |
| have hm1 : AEStronglyMeasurable (fun x : Configuration N => ‖ψ.value s x‖^2) volume := |
| (hψ.1 s).aestronglyMeasurable.norm.pow 2 |
| have hm2 : AEStronglyMeasurable (fun x : Configuration N => ‖x i-y‖⁻¹) volume := |
| (((continuous_apply i).sub continuous_const).norm.measurable.inv).aestronglyMeasurable |
| have hm : AEStronglyMeasurable (fun x : Configuration N => ‖ψ.value s x‖^2/‖x i-y‖) volume := by |
| simpa only [div_eq_mul_inv] using hm1.fun_mul hm2 |
| apply Integrable.mono' (((hψ.1 s).integrable_norm_pow (by norm_num)).div_const B) hm |
| apply Eventually.of_forall |
| intro x |
| rw [Real.norm_eq_abs,abs_of_nonneg (div_nonneg (sq_nonneg _) (norm_nonneg _))] |
| by_cases hz : ψ.value s x = 0 |
| · simp only [hz,norm_zero,zero_pow (by decide : (2:ℕ) ≠ 0),zero_div,le_refl] |
| · have hd : B ≤ ‖x i-y‖ := by |
| by_contra hh |
| exact hz (hc x i (Metric.mem_ball.mpr (not_le.mp hh)) s).1 |
| exact div_le_div_of_nonneg_left (sq_nonneg _) hB hd |
|
|
| lemma coreCoulombAt_eq_formOneBody {N : ℕ} (ψ : FormVector N) (y : Space) : |
| coreCoulombAt ψ y = formOneBody ψ (fun x => 1/‖x-y‖) := by |
| simp only [coreCoulombAt,formOneBody,mul_one_div] |
|
|
| lemma coreCoulombAt_insertOrbital {N : ℕ} {ψ : FormVector N} {φ : FormVector 1} |
| (hψ : SobolevVector ψ) (hφ : SobolevVector φ) |
| (hd : FormsDisjoint (insertionTerm (tensorForm ψ φ))) |
| (y : Space) {B : ℝ} (hB : 0 < B) |
| (hcψ : FormAvoids ψ (Metric.ball y B)) (hcφ : FormAvoids φ (Metric.ball y B)) : |
| coreCoulombAt (insertOrbital ψ φ) y = coreCoulombAt ψ y * formMass φ + |
| formMass ψ * coreCoulombAt φ y := by |
| simp only [coreCoulombAt_eq_formOneBody] |
| apply formOneBody_insertOrbital _ hd |
| intro s i |
| simpa only [mul_one_div] using integrable_coreCoulombAt (hψ.tensor hφ) y hB |
| (hcψ.tensor hcφ) s i |
|
|
| lemma coreCoulombAt_packet (y z : Space) {R : ℝ} (hR : 0 < R) |
| (hsep : R < ‖y-z‖) : coreCoulombAt (radialLocalOrbital y R) z = 1/‖y-z‖ := by |
| have hh (s : Spins 1) := radialLocalOrbital_potential y hR |
| (z := z) (by rwa [norm_sub_rev]) s |
| simp only [coreCoulombAt,Fin.sum_univ_one] |
| have he (s : Spins 1) : (∫ x : Configuration 1, |
| ‖(radialLocalOrbital y R).value s x‖^2/‖x 0-z‖) = (1/2:ℝ)/‖y-z‖ := by |
| simpa only [norm_sub_rev] using hh s |
| simp only [he,Finset.sum_const,Finset.card_univ,nsmul_eq_mul] |
| norm_num [Spins] |
| ring |
|
|
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators |
|
|
| end CoulombAtom |
|
|
| end |
|
|
| end OAI |
|
|
