| import Mathlib |
| import OAI.Analysis.CoulombIonization.Variational.SpinSpaceIntegrable |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace CoulombAtom |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
|
|
| open MeasureTheory Filter |
| open scoped BigOperators ComplexConjugate ContDiff |
|
|
| abbrev slaterDirectDensity {n : ℕ} (φ : Fin n → SlaterParticle → ℂ) |
| (z : SlaterParticle) : ℝ := ∑ i : Fin n, ‖φ i z‖^2 |
|
|
| abbrev slaterExchangeDensity {n : ℕ} (φ : Fin n → SlaterParticle → ℂ) |
| (z : SlaterParticle × SlaterParticle) : ℝ := |
| ‖∑ i : Fin n, φ i z.1 * conj (φ i z.2)‖^2 |
|
|
| lemma slater_repulsion_spinSpace {n : ℕ} {φ : Fin n → SlaterParticle → ℂ} |
| (hφ : ∀ i s, ContDiff ℝ ∞ (fun x : Space => φ i (s,x))) |
| (hc : ∀ i s, HasCompactSupport (fun x : Space => φ i (s,x))) : |
| formRepulsion (slaterForm φ) = (1/2:ℝ) * ∑ j : Fin n, ∑ k : Fin n, |
| if j ≠ k then ∫ z, ‖slater φ z‖^2 / ‖(z j).2-(z k).2‖ |
| ∂Measure.pi (fun _ : Fin n => slaterParticleMeasure) else 0 := by |
| rw [formRepulsion_ordered] |
| congr 1 |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro j _ |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro k _ |
| by_cases hjk : j ≠ k |
| · simp only [ite_eq_left hjk] |
| exact (spinSpace_integral _ (slater_coulomb_integrable hφ hc j k hjk)).symm |
| · simp only [ite_eq_right hjk,Finset.sum_const_zero] |
|
|
| theorem slaterForm_repulsion {n : ℕ} {φ : Fin n → SlaterParticle → ℂ} |
| (hφ : ∀ i s, ContDiff ℝ ∞ (fun x : Space => φ i (s,x))) |
| (hc : ∀ i s, HasCompactSupport (fun x : Space => φ i (s,x))) |
| (ho : ∀ i k, (∫ z, conj (φ i z) * φ k z ∂slaterParticleMeasure) = if i=k then 1 else 0) : |
| formRepulsion (slaterForm φ) = (1/2:ℝ) * ∫ z : SlaterParticle × SlaterParticle, |
| (slaterDirectDensity φ z.1 * slaterDirectDensity φ z.2 - slaterExchangeDensity φ z) / |
| ‖z.1.2-z.2.2‖ ∂slaterParticleMeasure.prod slaterParticleMeasure := by |
| have hl (i : Fin n) := memLp_spinSpace (fun s => (hφ i s).continuous) (hc i) |
| have hi (j k : Fin n) (hjk : j ≠ k) : |
| Integrable (fun z : Fin n → SlaterParticle => ‖(z j).2-(z k).2‖⁻¹ * ‖slater φ z‖^2) |
| (Measure.pi fun _ : Fin n => slaterParticleMeasure) := by |
| simpa only [div_eq_mul_inv,mul_comm] using slater_coulomb_integrable hφ hc j k hjk |
| have ht := slater_two_body hl ho (fun z : SlaterParticle × SlaterParticle => ‖z.1.2-z.2.2‖⁻¹) hi |
| rw [slater_repulsion_spinSpace hφ hc] |
| simpa only [div_eq_mul_inv,mul_comm] using congrArg (fun x : ℝ => (1/2:ℝ)*x) ht |
|
|
| open MeasureTheory Filter |
| open scoped BigOperators ComplexConjugate ContDiff |
|
|
| lemma smooth_pair_integrable {n : ℕ} {f : Configuration n → ℂ} |
| (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) (i j : Fin n) (hij : i ≠ j) : |
| Integrable (fun x => ‖f x‖^2 / ‖x i-x j‖) := |
| pair_integrable i j hij (hf.continuous.memLp_of_hasCompactSupport hc) |
| (fun _a => (smooth_derivative_continuous hf _).memLp_of_hasCompactSupport (compact_derivative hc _)) |
| (fun a _g hg hgc => smooth_weak_gradient hf hc hg hgc (direction i a)) |
|
|
| lemma smooth_nuclear_integrable {f : Space → ℂ} |
| (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) : |
| Integrable (fun x => ‖f x‖^2 / ‖x‖) := |
| linear_coulomb_integrable (ContinuousLinearMap.id ℝ Space) |
| (fun a => EuclideanSpace.single a 1) (fun _ => rfl) |
| (hf.continuous.memLp_of_hasCompactSupport hc) |
| (fun _a => (smooth_derivative_continuous hf _).memLp_of_hasCompactSupport (compact_derivative hc _)) |
| (fun a _g hg hgc => smooth_weak_gradient hf hc hg hgc (EuclideanSpace.single a 1)) |
|
|
| lemma orbital_coulomb_product_integrable {f g : SlaterParticle → ℂ} |
| (hf : ∀ s, ContDiff ℝ ∞ (fun x : Space => f (s,x))) |
| (hg : ∀ s, ContDiff ℝ ∞ (fun x : Space => g (s,x))) |
| (hfc : ∀ s, HasCompactSupport (fun x : Space => f (s,x))) |
| (hgc : ∀ s, HasCompactSupport (fun x : Space => g (s,x))) : |
| Integrable (fun z : SlaterParticle × SlaterParticle => |
| (‖f z.1‖^2 * ‖g z.2‖^2) / ‖z.1.2-z.2.2‖) |
| (slaterParticleMeasure.prod slaterParticleMeasure) := by |
| apply ((measurePreserving_finTwoArrow slaterParticleMeasure).integrable_comp_emb |
| MeasurableEquiv.finTwoArrow.measurableEmbedding).mp |
| change Integrable (fun z : Fin 2 → SlaterParticle => |
| (‖f (z 0)‖^2 * ‖g (z 1)‖^2) / ‖(z 0).2-(z 1).2‖) |
| (Measure.pi fun _ : Fin 2 => slaterParticleMeasure) |
| apply spinSpace_integrable |
| · have hfm := (continuous_prod_of_discrete_left.mpr (fun s => (hf s).continuous)).measurable |
| have hgm := (continuous_prod_of_discrete_left.mpr (fun s => (hg s).continuous)).measurable |
| exact (((hfm.comp (measurable_pi_apply 0)).norm.pow_const 2).mul |
| ((hgm.comp (measurable_pi_apply 1)).norm.pow_const 2)).div |
| (((measurable_pi_apply 0).snd.sub (measurable_pi_apply 1).snd).norm) |
| · intro s |
| let φ : Fin 2 → SlaterParticle → ℂ := ![f,g] |
| have hφ (i : Fin 2) (t : Fin 2) : ContDiff ℝ ∞ (fun x : Space => φ i (t,x)) := by |
| fin_cases i |
| · exact hf t |
| · exact hg t |
| have hc (i : Fin 2) (t : Fin 2) : HasCompactSupport (fun x : Space => φ i (t,x)) := by |
| fin_cases i |
| · exact hfc t |
| · exact hgc t |
| have hi := smooth_pair_integrable (slater_spatial_tensor_smooth hφ s 1) |
| (slater_spatial_tensor_compact hc s 1) (0 : Fin 2) 1 (by decide) |
| simpa only [slaterTensor,Fin.prod_univ_two,Equiv.Perm.one_apply,φ, |
| Matrix.cons_val_zero,Matrix.cons_val_one,Matrix.head_cons,norm_mul,mul_pow, |
| slaterParticles] using hi |
|
|
| lemma slater_direct_coulomb_integrable {n : ℕ} {φ : Fin n → SlaterParticle → ℂ} |
| (hφ : ∀ i s, ContDiff ℝ ∞ (fun x : Space => φ i (s,x))) |
| (hc : ∀ i s, HasCompactSupport (fun x : Space => φ i (s,x))) : |
| Integrable (fun z : SlaterParticle × SlaterParticle => |
| ((∑ i : Fin n, ‖φ i z.1‖^2) * (∑ i : Fin n, ‖φ i z.2‖^2)) / ‖z.1.2-z.2.2‖) |
| (slaterParticleMeasure.prod slaterParticleMeasure) := by |
| simp only [Finset.sum_mul,Finset.mul_sum,Finset.sum_div] |
| apply integrable_finsetSum |
| intro i _ |
| apply integrable_finsetSum |
| intro j _ |
| exact orbital_coulomb_product_integrable (hφ j) (hφ i) (hc j) (hc i) |
|
|
| lemma orbital_nuclear_integrable {φ : SlaterParticle → ℂ} |
| (hφ : ∀ s, ContDiff ℝ ∞ (fun x : Space => φ (s,x))) |
| (hc : ∀ s, HasCompactSupport (fun x : Space => φ (s,x))) : |
| Integrable (fun z => ‖φ z‖^2 / ‖z.2‖) slaterParticleMeasure := by |
| have hm : Measurable (fun z : SlaterParticle => ‖φ z‖^2 / ‖z.2‖) := |
| (((continuous_prod_of_discrete_left.mpr (fun s => (hφ s).continuous)).measurable.norm).pow_const 2).div |
| measurable_snd.norm |
| exact (integrable_prod_iff hm.aestronglyMeasurable).mpr |
| ⟨Eventually.of_forall (fun s => smooth_nuclear_integrable (hφ s) (hc s)),Integrable.of_finite⟩ |
|
|
| open MeasureTheory Filter |
| open scoped BigOperators ComplexConjugate ContDiff |
|
|
| lemma memLp_potential_mul {α : Type*} [MeasurableSpace α] {μ : Measure α} |
| {f : α → ℂ} {V : α → ℝ} (hf : Measurable f) (hV : Measurable V) |
| (hVn : ∀ x, 0 ≤ V x) (hi : Integrable (fun x => V x * ‖f x‖^2) μ) : |
| MemLp (fun x => (Real.sqrt (V x) : ℂ)*f x) 2 μ := by |
| apply (memLp_two_iff_integrable_sq_norm |
| ((Complex.measurable_ofReal.comp (Real.continuous_sqrt.measurable.comp hV)).mul hf).aestronglyMeasurable).mpr |
| change Integrable (fun x => ‖(Real.sqrt (V x) : ℂ)*f x‖^2) μ |
| simpa only [norm_mul,mul_pow,Complex.norm_real,Real.norm_eq_abs, |
| abs_of_nonneg (Real.sqrt_nonneg _),Real.sq_sqrt (hVn _)] using hi |
|
|
| lemma slaterForm_nuclear {n : ℕ} {φ : Fin n → SlaterParticle → ℂ} |
| (hφ : ∀ i s, ContDiff ℝ ∞ (fun x : Space => φ i (s,x))) |
| (hc : ∀ i s, HasCompactSupport (fun x : Space => φ i (s,x))) |
| (ho : ∀ i k, (∫ z, conj (φ i z)*φ k z ∂slaterParticleMeasure) = if i=k then 1 else 0) : |
| formNuclear (slaterForm φ) = ∑ i : Fin n, ∫ z : SlaterParticle, |
| ‖φ i z‖^2 / ‖z.2‖ ∂slaterParticleMeasure := by |
| have hl (i : Fin n) := memLp_spinSpace (fun s => (hφ i s).continuous) (hc i) |
| have hi (i : Fin n) : Integrable (fun z => ‖z.2‖⁻¹*‖φ i z‖^2) slaterParticleMeasure := by |
| simpa only [div_eq_mul_inv,mul_comm] using orbital_nuclear_integrable (hφ i) (hc i) |
| have hv (i : Fin n) := memLp_potential_mul |
| (continuous_prod_of_discrete_left.mpr (fun s => (hφ i s).continuous)).measurable |
| measurable_snd.norm.inv (fun z : SlaterParticle => inv_nonneg.mpr (norm_nonneg z.2)) (hi i) |
| have ht := slater_nonnegative_potential hl ho (fun z : SlaterParticle => ‖z.2‖⁻¹) |
| (fun z => inv_nonneg.mpr (norm_nonneg z.2)) hv |
| have his (j : Fin n) := slater_nuclear_integrable hφ hc j |
| have hs : (∫ z : Fin n → SlaterParticle, (∑ j, ‖(z j).2‖⁻¹)*‖slater φ z‖^2 |
| ∂Measure.pi (fun _ : Fin n => slaterParticleMeasure)) = formNuclear (slaterForm φ) := by |
| simp_rw [Finset.sum_mul] |
| rw [integral_finsetSum _ (fun j _ => by |
| simpa only [div_eq_mul_inv,mul_comm] using his j)] |
| unfold formNuclear |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro j _ |
| calc |
| _ = ∫ z : Fin n → SlaterParticle, ‖slater φ z‖^2 / ‖(z j).2‖ |
| ∂Measure.pi (fun _ : Fin n => slaterParticleMeasure) := by |
| apply integral_congr_ae |
| filter_upwards [] with z |
| rw [div_eq_mul_inv,mul_comm] |
| _ = _ := spinSpace_integral _ (his j) |
| rw [hs] at ht |
| simpa only [div_eq_mul_inv,mul_comm] using ht |
|
|
| theorem slaterForm_repulsion_le_direct {n : ℕ} {φ : Fin n → SlaterParticle → ℂ} |
| (hφ : ∀ i s, ContDiff ℝ ∞ (fun x : Space => φ i (s,x))) |
| (hc : ∀ i s, HasCompactSupport (fun x : Space => φ i (s,x))) |
| (ho : ∀ i k, (∫ z, conj (φ i z)*φ k z ∂slaterParticleMeasure) = if i=k then 1 else 0) : |
| formRepulsion (slaterForm φ) ≤ (1/2:ℝ)*∫ z : SlaterParticle × SlaterParticle, |
| (slaterDirectDensity φ z.1 * slaterDirectDensity φ z.2) / ‖z.1.2-z.2.2‖ |
| ∂slaterParticleMeasure.prod slaterParticleMeasure := by |
| have hl (i : Fin n) := memLp_spinSpace (fun s => (hφ i s).continuous) (hc i) |
| have hi (j k : Fin n) (hjk : j ≠ k) : |
| Integrable (fun z : Fin n → SlaterParticle => ‖(z j).2-(z k).2‖⁻¹*‖slater φ z‖^2) |
| (Measure.pi fun _ : Fin n => slaterParticleMeasure) := by |
| simpa only [div_eq_mul_inv,mul_comm] using slater_coulomb_integrable hφ hc j k hjk |
| have hd : Integrable (fun z : SlaterParticle × SlaterParticle => |
| ‖z.1.2-z.2.2‖⁻¹ * ((∑ i, ‖φ i z.1‖^2)*(∑ i, ‖φ i z.2‖^2))) |
| (slaterParticleMeasure.prod slaterParticleMeasure) := by |
| simpa only [div_eq_mul_inv,mul_comm] using slater_direct_coulomb_integrable hφ hc |
| have ht := slater_two_body_le_direct hl ho (fun z : SlaterParticle × SlaterParticle => ‖z.1.2-z.2.2‖⁻¹) |
| (fun z => inv_nonneg.mpr (norm_nonneg _)) hi hd |
| rw [slater_repulsion_spinSpace hφ hc] |
| simpa only [div_eq_mul_inv,mul_comm] using mul_le_mul_of_nonneg_left ht (by norm_num : (0:ℝ) ≤ 1/2) |
|
|
| end CoulombAtom |
|
|
| end |
|
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| end OAI |
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|
