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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
end OAI
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