| import Mathlib |
| import OAI.Analysis.CoulombIonization.Variational.InsertionSwap |
|
|
| namespace OAI |
|
|
| 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 |
|
|
| def FormZeroAt {N : ℕ} (ψ : FormVector N) (x : Configuration N) : Prop := |
| ∀ s, ψ.value s x = 0 ∧ ∀ i a, ψ.gradient s i a x = 0 |
|
|
| lemma insertionTerm_zero {N : ℕ} {T : FormVector (N+1)} {x : Configuration (N+1)} |
| (i : Fin (N+1)) (hx : FormZeroAt T (x ∘ insertionSwap i)) : |
| FormZeroAt (insertionTerm T i) x := by |
| intro s |
| constructor |
| · rw [insertionTerm_value,(hx _).1,mul_zero] |
| · intro j a |
| change _ * T.gradient (s ∘ (insertionSwap i).symm) (insertionSwap i j) a |
| (x ∘ (insertionSwap i).symm) = 0 |
| rw [insertionSwap_symm,(hx _).2,mul_zero] |
|
|
| lemma insertionTerms_disjoint {N : ℕ} {T : FormVector (N+1)} (R : ℝ) |
| (hc : ∀ x (j : Fin N), R ≤ ‖x j.castSucc‖ → FormZeroAt T x) |
| (he : ∀ x, ‖x (Fin.last N)‖ < R → FormZeroAt T x) : |
| FormsDisjoint (insertionTerm T) := by |
| have hh (k l : Fin (N+1)) (hkl : k ≠ l) (x : Configuration (N+1)) : |
| FormZeroAt (insertionTerm T k) x ∨ FormZeroAt (insertionTerm T l) x := by |
| by_cases hk : ‖x k‖ < R |
| · left |
| exact insertionTerm_zero k (he _ (by simpa only [Function.comp_apply,insertionSwap_last] using hk)) |
| · right |
| have hne : insertionSwap l k ≠ Fin.last N := by |
| intro h |
| have hh := congrArg (insertionSwap l) h |
| exact hkl (by simpa only [insertionSwap,Equiv.swap_apply_self,Equiv.swap_apply_right] using hh) |
| obtain ⟨j,hj⟩ := Fin.exists_castSucc_eq.mpr hne |
| apply insertionTerm_zero l |
| apply hc _ j |
| simpa only [Function.comp_apply,hj,insertionSwap,Equiv.swap_apply_self] using le_of_not_gt hk |
| constructor |
| · intro k l hkl s x |
| rcases hh k l hkl x with hk | hl |
| · exact Or.inl (hk s).1 |
| · exact Or.inr (hl s).1 |
| · intro k l hkl s i a x |
| rcases hh k l hkl x with hk | hl |
| · exact Or.inl ((hk s).2 i a) |
| · exact Or.inr ((hl s).2 i a) |
|
|
| lemma formMass_wedge {N : ℕ} {T : FormVector (N+1)} (hT : SobolevVector T) |
| (hd : FormsDisjoint (insertionTerm T)) : |
| formMass (wedgeForm T) = (N+1 : ℝ) * formMass T := by |
| rw [wedgeForm,formMass_sum_disjoint (fun i => hT.insertionTerm i) hd] |
| simp only [formMass_insertionTerm,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,Nat.cast_add,Nat.cast_one] |
|
|
| lemma formEnergy_wedge {N : ℕ} {T : FormVector (N+1)} (hT : SobolevVector T) |
| (hd : FormsDisjoint (insertionTerm T)) (Z : ℝ) : |
| formEnergy Z (wedgeForm T) = (N+1 : ℝ) * formEnergy Z T := by |
| rw [wedgeForm,formEnergy_sum_disjoint (fun i => hT.insertionTerm i) hd] |
| simp only [formEnergy_insertionTerm,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,Nat.cast_add,Nat.cast_one] |
|
|
| lemma tensor_zero_left {N M : ℕ} (ψ : FormVector N) (φ : FormVector M) |
| {x : Configuration (N+M)} (hx : FormZeroAt ψ (leftList x)) : |
| FormZeroAt (tensorForm ψ φ) x := by |
| intro s |
| constructor |
| · change ψ.value _ _ * _ = 0 |
| rw [(hx _).1,zero_mul] |
| · intro i a |
| obtain ⟨i,rfl⟩ := finSumFinEquiv.surjective i |
| cases i with |
| | inl i => |
| simp only [tensorForm,Equiv.symm_apply_apply,Sum.elim_inl,(hx _).2,zero_mul] |
| | inr i => |
| simp only [tensorForm,Equiv.symm_apply_apply,Sum.elim_inr,(hx _).1,zero_mul] |
|
|
| lemma tensor_zero_right {N M : ℕ} (ψ : FormVector N) (φ : FormVector M) |
| {x : Configuration (N+M)} (hx : FormZeroAt φ (rightList x)) : |
| FormZeroAt (tensorForm ψ φ) x := by |
| intro s |
| constructor |
| · change _ * φ.value _ _ = 0 |
| rw [(hx _).1,mul_zero] |
| · intro i a |
| obtain ⟨i,rfl⟩ := finSumFinEquiv.surjective i |
| cases i with |
| | inl i => |
| simp only [tensorForm,Equiv.symm_apply_apply,Sum.elim_inl,(hx _).1,mul_zero] |
| | inr i => |
| simp only [tensorForm,Equiv.symm_apply_apply,Sum.elim_inr,(hx _).2,mul_zero] |
|
|
| lemma tensor_insertion_disjoint {N : ℕ} (ψ : FormVector N) (φ : FormVector 1) (R : ℝ) |
| (hc : ∀ x i, R ≤ ‖x i‖ → FormZeroAt ψ x) |
| (he : ∀ x, ‖x 0‖ < R → FormZeroAt φ x) : |
| FormsDisjoint (insertionTerm (tensorForm ψ φ)) := by |
| apply insertionTerms_disjoint R |
| · intro x i hx |
| apply tensor_zero_left |
| exact hc _ i hx |
| · intro x hx |
| apply tensor_zero_right |
| exact he _ hx |
|
|
| theorem normalized_wedge_trial {N : ℕ} {ψ : FormVector N} {φ : FormVector 1} |
| (hψ : FormAdmissible ψ) (hφ : FormAdmissible φ) (R : ℝ) |
| (hc : ∀ x i, R ≤ ‖x i‖ → FormZeroAt ψ x) |
| (he : ∀ x, ‖x 0‖ < R → FormZeroAt φ x) (Z : ℝ) : |
| ∃ F : FormVector (N+1), FormAdmissible F ∧ formEnergy Z F = formEnergy Z (tensorForm ψ φ) := by |
| let T := tensorForm ψ φ |
| have hT := hψ.sobolevFermion.sobolevVector.tensor hφ.sobolevFermion.sobolevVector |
| have hTa := tensor_coreAntisymmetric hψ.sobolevFermion φ |
| have hd := tensor_insertion_disjoint ψ φ R hc he |
| have hm : formMass T = 1 := by |
| rw [formMass_tensor,formMass,formMass,hψ.2.2.2.2.1,hφ.2.2.2.2.1,one_mul] |
| have hw := formMass_wedge hT hd |
| rw [hm,mul_one] at hw |
| have hp : 0 < formMass (wedgeForm T) := by rw [hw]; positivity |
| refine ⟨scaleForm (Real.sqrt (formMass (wedgeForm T)))⁻¹ (wedgeForm T), |
| (wedgeForm_fermion hT hTa).normalize hp,?_⟩ |
| rw [formEnergy_scale,formEnergy_wedge hT hd,hw,inv_pow, |
| Real.sq_sqrt (le_of_lt (show (0:ℝ) < N+1 by positivity))] |
| field_simp |
|
|
|
|
| open MeasureTheory Filter |
| open scoped BigOperators |
|
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|
|
| def formKinetic {N : ℕ} (ψ : FormVector N) : ℝ := |
| (1/2:ℝ) * ∑ s, ∑ i, ∑ a, ∫ x, ‖ψ.gradient s i a x‖^2 |
|
|
| def formNuclear {N : ℕ} (ψ : FormVector N) : ℝ := |
| ∑ s, ∑ i, ∫ x, ‖ψ.value s x‖^2 / ‖x i‖ |
|
|
| def formRepulsion {N : ℕ} (ψ : FormVector N) : ℝ := |
| ∑ s, ∑ i, ∑ j, if i < j then ∫ x, ‖ψ.value s x‖^2 / ‖x i-x j‖ else 0 |
|
|
| lemma formEnergy_parts {N : ℕ} (ψ : FormVector N) (Z : ℝ) : |
| formEnergy Z ψ = formKinetic ψ - Z * formNuclear ψ + formRepulsion ψ := rfl |
|
|
| lemma formKinetic_nonneg {N : ℕ} (ψ : FormVector N) : 0 ≤ formKinetic ψ := by |
| unfold formKinetic |
| positivity |
|
|
| lemma formNuclear_nonneg {N : ℕ} (ψ : FormVector N) : 0 ≤ formNuclear ψ := by |
| apply Finset.sum_nonneg; intro s _ |
| apply Finset.sum_nonneg; intro i _ |
| exact integral_nonneg fun x => div_nonneg (sq_nonneg _) (norm_nonneg _) |
|
|
| lemma integral_product_join {N M : ℕ} (f : Configuration N → ℝ) (g : Configuration M → ℝ) : |
| (∫ z : Configuration (N+M), f (leftList z)*g (rightList z)) = (∫ x, f x)*(∫ y, g y) := by |
| rw [← (configurationJoin_preserving N M).integral_comp |
| (configurationJoin N M).toHomeomorph.measurableEmbedding] |
| simp only [configurationJoin_apply_prod,leftList_join,rightList_join] |
| rw [Measure.volume_eq_prod] |
| exact integral_prod_mul f g |
|
|
| lemma tensor_kinetic_left {N M : ℕ} (ψ : FormVector N) (φ : FormVector M) |
| (s : Spins N) (t : Spins M) (i : Fin N) (a : Fin 3) : |
| (∫ z, ‖(tensorForm ψ φ).gradient (joinLists s t) (finSumFinEquiv (Sum.inl i)) a z‖^2) = |
| (∫ x, ‖ψ.gradient s i a x‖^2)*(∫ y, ‖φ.value t y‖^2) := by |
| simp only [tensorForm,leftList_join,rightList_join,Equiv.symm_apply_apply,Sum.elim_inl,norm_mul,mul_pow] |
| exact integral_product_join (N := N) (M := M) (fun x => ‖ψ.gradient s i a x‖^2) (fun y => ‖φ.value t y‖^2) |
|
|
| lemma tensor_kinetic_right {N M : ℕ} (ψ : FormVector N) (φ : FormVector M) |
| (s : Spins N) (t : Spins M) (i : Fin M) (a : Fin 3) : |
| (∫ z, ‖(tensorForm ψ φ).gradient (joinLists s t) (finSumFinEquiv (Sum.inr i)) a z‖^2) = |
| (∫ x, ‖ψ.value s x‖^2)*(∫ y, ‖φ.gradient t i a y‖^2) := by |
| simp only [tensorForm,leftList_join,rightList_join,Equiv.symm_apply_apply,Sum.elim_inr,norm_mul,mul_pow] |
| exact integral_product_join (N := N) (M := M) (fun x => ‖ψ.value s x‖^2) (fun y => ‖φ.gradient t i a y‖^2) |
|
|
| lemma formKinetic_tensor {N M : ℕ} (ψ : FormVector N) (φ : FormVector M) : |
| formKinetic (tensorForm ψ φ) = formKinetic ψ * formMass φ + formMass ψ * formKinetic φ := by |
| unfold formKinetic formMass |
| rw [← sum_spin_join] |
| have hk (s : Spins N) (t : Spins M) : |
| (∑ i, ∑ a, ∫ z, ‖(tensorForm ψ φ).gradient (joinLists s t) i a z‖^2) = |
| (∑ i, ∑ a, ∫ x, ‖ψ.gradient s i a x‖^2)*(∫ y, ‖φ.value t y‖^2) + |
| (∫ x, ‖ψ.value s x‖^2)*(∑ i, ∑ a, ∫ y, ‖φ.gradient t i a y‖^2) := by |
| rw [← Equiv.sum_comp finSumFinEquiv, Fintype.sum_sum_type] |
| simp only [tensor_kinetic_left,tensor_kinetic_right,Finset.sum_mul,Finset.mul_sum] |
| simp_rw [hk,Finset.sum_add_distrib,← Finset.sum_mul,← Finset.mul_sum] |
| ring |
|
|
| lemma tensor_nuclear_left {N M : ℕ} (ψ : FormVector N) (φ : FormVector M) |
| (s : Spins N) (t : Spins M) (i : Fin N) : |
| (∫ z, ‖(tensorForm ψ φ).value (joinLists s t) z‖^2 / ‖z (finSumFinEquiv (Sum.inl i))‖) = |
| (∫ x, ‖ψ.value s x‖^2 / ‖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 / ‖leftList z i‖) = _ |
| simp only [leftList_join,rightList_join,norm_mul,mul_pow] |
| simp_rw [mul_div_right_comm] |
| exact integral_product_join (N := N) (M := M) (fun x => ‖ψ.value s x‖^2 / ‖x i‖) (fun y => ‖φ.value t y‖^2) |
|
|
| lemma tensor_nuclear_right {N M : ℕ} (ψ : FormVector N) (φ : FormVector M) |
| (s : Spins N) (t : Spins M) (i : Fin M) : |
| (∫ z, ‖(tensorForm ψ φ).value (joinLists s t) z‖^2 / ‖z (finSumFinEquiv (Sum.inr i))‖) = |
| (∫ x, ‖ψ.value s x‖^2)*(∫ y, ‖φ.value t y‖^2 / ‖y i‖) := by |
| change (∫ z, ‖ψ.value (leftList (joinLists s t)) (leftList z) * |
| φ.value (rightList (joinLists s t)) (rightList z)‖^2 / ‖rightList z i‖) = _ |
| simp only [leftList_join,rightList_join,norm_mul,mul_pow,mul_div_assoc] |
| exact integral_product_join (N := N) (M := M) (fun x => ‖ψ.value s x‖^2) (fun y => ‖φ.value t y‖^2 / ‖y i‖) |
|
|
| lemma formNuclear_tensor {N M : ℕ} (ψ : FormVector N) (φ : FormVector M) : |
| formNuclear (tensorForm ψ φ) = formNuclear ψ * formMass φ + formMass ψ * formNuclear φ := by |
| unfold formNuclear formMass |
| rw [← sum_spin_join] |
| have hk (s : Spins N) (t : Spins M) : |
| (∑ i, ∫ z, ‖(tensorForm ψ φ).value (joinLists s t) z‖^2 / ‖z i‖) = |
| (∑ i, ∫ x, ‖ψ.value s x‖^2 / ‖x i‖)*(∫ y, ‖φ.value t y‖^2) + |
| (∫ x, ‖ψ.value s x‖^2)*(∑ i, ∫ y, ‖φ.value t y‖^2 / ‖y i‖) := by |
| rw [← Equiv.sum_comp finSumFinEquiv,Fintype.sum_sum_type] |
| simp only [tensor_nuclear_left,tensor_nuclear_right,Finset.sum_mul,Finset.mul_sum] |
| simp_rw [hk,Finset.sum_add_distrib,← Finset.sum_mul,← Finset.mul_sum] |
|
|
| lemma tensor_pair_left {N M : ℕ} (ψ : FormVector N) (φ : FormVector M) |
| (s : Spins N) (t : Spins M) (i j : Fin N) : |
| (∫ z, ‖(tensorForm ψ φ).value (joinLists s t) z‖^2 / |
| ‖z (finSumFinEquiv (Sum.inl i))-z (finSumFinEquiv (Sum.inl j))‖) = |
| (∫ x, ‖ψ.value s x‖^2 / ‖x i-x j‖)*(∫ 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 / ‖leftList z i-leftList z j‖) = _ |
| simp only [leftList_join,rightList_join,norm_mul,mul_pow] |
| simp_rw [mul_div_right_comm] |
| exact integral_product_join (N := N) (M := M) (fun x => ‖ψ.value s x‖^2 / ‖x i-x j‖) (fun y => ‖φ.value t y‖^2) |
|
|
| def tensorCross {N : ℕ} (ψ : FormVector N) (φ : FormVector 1) : ℝ := |
| ∑ s : Spins N, ∑ t : Spins 1, ∑ i : Fin N, |
| ∫ z, ‖(tensorForm ψ φ).value (joinLists s t) z‖^2 / ‖z i.castSucc-z (Fin.last N)‖ |
|
|
| lemma formRepulsion_tensor_one {N : ℕ} (ψ : FormVector N) (φ : FormVector 1) : |
| formRepulsion (tensorForm ψ φ) = formRepulsion ψ * formMass φ + tensorCross ψ φ := by |
| classical |
| unfold formRepulsion tensorCross formMass |
| rw [← sum_spin_join] |
| have hk (s : Spins N) (t : Spins 1) : |
| (∑ i : Fin (N+1), ∑ j : Fin (N+1), if i < j then |
| ∫ z, ‖(tensorForm ψ φ).value (joinLists s t) z‖^2 / ‖z i-z j‖ else 0) = |
| (∑ i : Fin N, ∑ j : Fin N, if i < j then ∫ x, ‖ψ.value s x‖^2 / ‖x i-x j‖ else 0)* |
| (∫ y, ‖φ.value t y‖^2) + |
| ∑ i : Fin N, ∫ z, ‖(tensorForm ψ φ).value (joinLists s t) z‖^2 / |
| ‖z i.castSucc-z (Fin.last N)‖ := by |
| rw [Fin.sum_univ_castSucc] |
| have he (i : Fin N) : (∑ j : Fin (N+1), if i.castSucc < j then |
| ∫ z, ‖(tensorForm ψ φ).value (joinLists s t) z‖^2 / ‖z i.castSucc-z j‖ else 0) = |
| (∑ j : Fin N, if i < j then ∫ x, ‖ψ.value s x‖^2 / ‖x i-x j‖ else 0)* |
| (∫ y, ‖φ.value t y‖^2) + |
| ∫ z, ‖(tensorForm ψ φ).value (joinLists s t) z‖^2 / ‖z i.castSucc-z (Fin.last N)‖ := by |
| rw [Fin.sum_univ_castSucc] |
| simp only [Fin.castSucc_lt_castSucc_iff,Fin.castSucc_lt_last,ite_true] |
| congr 1 |
| rw [Finset.sum_mul] |
| apply Finset.sum_congr rfl; intro j _ |
| split_ifs |
| · exact tensor_pair_left ψ φ s t i j |
| · simp only [zero_mul] |
| simp_rw [he,Finset.sum_add_distrib,← Finset.sum_mul] |
| have he0 : (∑ j : Fin (N+1), if Fin.last N < j then |
| ∫ z, ‖(tensorForm ψ φ).value (joinLists s t) z‖^2 / ‖z (Fin.last N)-z j‖ else 0) = 0 := by |
| apply Finset.sum_eq_zero; intro j _ |
| exact ite_eq_right (not_lt_of_ge (Fin.le_last j)) |
| rw [he0,add_zero] |
| simp_rw [hk,Finset.sum_add_distrib,← Finset.sum_mul,← Finset.mul_sum] |
| congr 1 |
| exact Finset.sum_comm |
|
|
| lemma formEnergy_tensor_one {N : ℕ} (ψ : FormVector N) (φ : FormVector 1) (Z : ℝ) : |
| formEnergy Z (tensorForm ψ φ) = formEnergy Z ψ * formMass φ + |
| formMass ψ * (formKinetic φ - Z * formNuclear φ) + tensorCross ψ φ := by |
| rw [formEnergy_parts,formKinetic_tensor,formNuclear_tensor,formRepulsion_tensor_one,formEnergy_parts] |
| ring |
|
|
|
|
| open MeasureTheory Filter |
| open scoped Topology ContDiff BigOperators |
|
|
| end CoulombAtom |
|
|
| end |
|
|
| end OAI |
|
|
