| import Mathlib |
| import OAI.Analysis.CoulombIonization.Fermionic.SlaterPairTrace |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace CoulombAtom |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| section Work_SlaterTwoBody_scope |
|
|
| open MeasureTheory Filter |
| open scoped BigOperators ComplexConjugate |
|
|
| variable {α : Type*} [MeasurableSpace α] {μ : Measure α} [SigmaFinite μ] {n : ℕ} |
|
|
| def slaterPairMarginal (φ : Fin n → α → ℂ) (j k : Fin n) (z : α × α) : ℝ := |
| if j ≠ k then (n.factorial : ℝ)⁻¹ * |
| ∫ y, ‖slaterPairSliceRaw φ j k z.1 z.2 y‖^2 |
| ∂Measure.pi (fun _ : SlaterPairRest j k => μ) else 0 |
|
|
| lemma slaterPairMarginal_sum {φ : Fin n → α → ℂ} |
| (hφ : ∀ i, MemLp (φ i) 2 μ) |
| (ho : ∀ i l, (∫ t, conj (φ i t) * φ l t ∂μ) = if i=l then 1 else 0) |
| (z : α × α) : |
| (∑ j : Fin n, ∑ k : Fin n, slaterPairMarginal (μ := μ) φ j k z) = |
| (∑ i : Fin n, ‖φ i z.1‖^2) * (∑ i : Fin n, ‖φ i z.2‖^2) - |
| ‖∑ i : Fin n, φ i z.1 * conj (φ i z.2)‖^2 := by |
| have he (j k : Fin n) : slaterPairMarginal (μ := μ) φ j k z = |
| (n.factorial : ℝ)⁻¹ * (if j ≠ k then |
| ∫ y, ‖slaterPairSliceRaw φ j k z.1 z.2 y‖^2 |
| ∂Measure.pi (fun _ : SlaterPairRest j k => μ) else 0) := by |
| unfold slaterPairMarginal |
| split_ifs <;> simp only [mul_zero] |
| simp_rw [he, ← Finset.mul_sum] |
| rw [slaterPairSliceRaw_sum_integral hφ ho, ← mul_assoc, |
| inv_mul_cancel₀ (show (n.factorial:ℝ) ≠ 0 by exact_mod_cast n.factorial_ne_zero), one_mul] |
|
|
| omit [SigmaFinite μ] in |
| lemma slaterPairMarginal_nonneg (φ : Fin n → α → ℂ) (j k : Fin n) (z : α × α) : |
| 0 ≤ slaterPairMarginal (μ := μ) φ j k z := by |
| unfold slaterPairMarginal |
| split_ifs |
| · exact mul_nonneg (by positivity) (integral_nonneg (fun _ => sq_nonneg _)) |
| · rfl |
|
|
| lemma slaterPairMarginal_observable_integrable {φ : Fin n → α → ℂ} |
| {j k : Fin n} (hjk : j ≠ k) (w : α × α → ℝ) |
| (hi : Integrable (fun x => w (x j,x k) * ‖slater φ x‖^2) |
| (Measure.pi fun _ : Fin n => μ)) : |
| Integrable (fun z => w z * slaterPairMarginal (μ := μ) φ j k z) (μ.prod μ) := by |
| have mp := (slaterPairSplit_preserving (μ := μ) hjk).symm |
| have h := (mp.integrable_comp_of_integrable hi).integral_prod_left |
| apply h.congr |
| filter_upwards [] with z |
| rcases z with ⟨u,v⟩ |
| simp only [Function.comp_def, slaterPairSplit_symm_apply_left, |
| slaterPairSplit_symm_apply_right, slater_pair_split_norm_sq, integral_const_mul, |
| slaterPairMarginal, ite_eq_left hjk] |
|
|
| theorem slater_two_body {φ : Fin n → α → ℂ} |
| (hφ : ∀ i, MemLp (φ i) 2 μ) |
| (ho : ∀ i l, (∫ t, conj (φ i t) * φ l t ∂μ) = if i=l then 1 else 0) |
| (w : α × α → ℝ) |
| (hi : ∀ j k : Fin n, j ≠ k → |
| Integrable (fun x => w (x j,x k) * ‖slater φ x‖^2) |
| (Measure.pi fun _ : Fin n => μ)) : |
| (∑ j : Fin n, ∑ k : Fin n, if j ≠ k then |
| (∫ x, w (x j,x k) * ‖slater φ x‖^2 ∂Measure.pi (fun _ : Fin n => μ)) else 0) = |
| ∫ z, w z * ((∑ i : Fin n, ‖φ i z.1‖^2) * (∑ i : Fin n, ‖φ i z.2‖^2) - |
| ‖∑ i : Fin n, φ i z.1 * conj (φ i z.2)‖^2) ∂μ.prod μ := by |
| have hm (j k : Fin n) : |
| Integrable (fun z => w z * slaterPairMarginal (μ := μ) φ j k z) (μ.prod μ) := by |
| by_cases hjk : j ≠ k |
| · exact slaterPairMarginal_observable_integrable hjk w (hi j k hjk) |
| · simp [slaterPairMarginal, hjk] |
| have he (j k : Fin n) : |
| (if j ≠ k then (∫ x, w (x j,x k) * ‖slater φ x‖^2 |
| ∂Measure.pi (fun _ : Fin n => μ)) else 0) = |
| ∫ z, w z * slaterPairMarginal (μ := μ) φ j k z ∂μ.prod μ := by |
| by_cases hjk : j ≠ k |
| · simp only [ite_eq_left hjk, slaterPairMarginal] |
| exact slater_pair_observable hjk w (hi j k hjk) |
| · simp only [ite_eq_right hjk, slaterPairMarginal, mul_zero, integral_zero] |
| simp_rw [he] |
| simp_rw [← integral_finsetSum _ (fun k _ => hm _ k)] |
| rw [← integral_finsetSum _ (fun j _ => integrable_finsetSum _ (fun k _ => hm j k))] |
| apply integral_congr_ae |
| filter_upwards [] with z |
| simp_rw [← Finset.mul_sum] |
| rw [slaterPairMarginal_sum hφ ho] |
|
|
| theorem slater_two_body_le_direct {φ : Fin n → α → ℂ} |
| (hφ : ∀ i, MemLp (φ i) 2 μ) |
| (ho : ∀ i l, (∫ t, conj (φ i t) * φ l t ∂μ) = if i=l then 1 else 0) |
| (w : α × α → ℝ) (hw : ∀ z, 0 ≤ w z) |
| (hi : ∀ j k : Fin n, j ≠ k → |
| Integrable (fun x => w (x j,x k) * ‖slater φ x‖^2) |
| (Measure.pi fun _ : Fin n => μ)) |
| (hd : Integrable (fun z => w z * |
| ((∑ i : Fin n, ‖φ i z.1‖^2) * (∑ i : Fin n, ‖φ i z.2‖^2))) (μ.prod μ)) : |
| (∑ j : Fin n, ∑ k : Fin n, if j ≠ k then |
| (∫ x, w (x j,x k) * ‖slater φ x‖^2 ∂Measure.pi (fun _ : Fin n => μ)) else 0) ≤ |
| ∫ z, w z * ((∑ i : Fin n, ‖φ i z.1‖^2) * (∑ i : Fin n, ‖φ i z.2‖^2)) ∂μ.prod μ := by |
| rw [slater_two_body hφ ho w hi] |
| have hn : ∀ z, 0 ≤ w z * ((∑ i : Fin n, ‖φ i z.1‖^2) * (∑ i : Fin n, ‖φ i z.2‖^2) - |
| ‖∑ i : Fin n, φ i z.1 * conj (φ i z.2)‖^2) := by |
| intro z |
| rw [← slaterPairMarginal_sum hφ ho] |
| exact mul_nonneg (hw z) (Finset.sum_nonneg (fun j _ => |
| Finset.sum_nonneg (fun k _ => slaterPairMarginal_nonneg (μ := μ) φ j k z))) |
| exact integral_mono_of_nonneg (Filter.Eventually.of_forall hn) hd |
| (Filter.Eventually.of_forall (fun z => |
| mul_le_mul_of_nonneg_left (sub_le_self _ (sq_nonneg _)) (hw z))) |
|
|
| end Work_SlaterTwoBody_scope |
|
|
| open MeasureTheory Filter |
| open scoped BigOperators ComplexConjugate |
|
|
| variable {α : Type*} [MeasurableSpace α] {μ : Measure α} [SigmaFinite μ] {n : ℕ} |
|
|
| def slaterCoordinate (φ ψ : Fin n → α → ℂ) (j : Fin n) (x : Fin n → α) : ℂ := |
| (Real.sqrt (n.factorial : ℝ) : ℂ)⁻¹ * slaterCoordinateRaw φ ψ j x |
|
|
| lemma memLp_slaterCoordinate {φ ψ : Fin n → α → ℂ} |
| (hφ : ∀ i, MemLp (φ i) 2 μ) (hψ : ∀ i, MemLp (ψ i) 2 μ) (j : Fin n) : |
| MemLp (slaterCoordinate φ ψ j) 2 (Measure.pi fun _ : Fin n => μ) := |
| (memLp_slaterCoordinateRaw hφ hψ j).const_mul _ |
|
|
| lemma slaterCoordinate_sum_norm_sq_integral {φ ψ : Fin n → α → ℂ} |
| (hφ : ∀ i, MemLp (φ i) 2 μ) (hψ : ∀ i, MemLp (ψ i) 2 μ) |
| (ho : ∀ i k, (∫ t, conj (φ i t) * φ k t ∂μ) = if i=k then 1 else 0) : |
| (∑ j : Fin n, ∫ x, ‖slaterCoordinate φ ψ j x‖^2 |
| ∂Measure.pi (fun _ : Fin n => μ)) = ∑ i, ∫ t, ‖ψ i t‖^2 ∂μ := by |
| simp only [slaterCoordinate, norm_mul, mul_pow, integral_const_mul, norm_inv, |
| Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg (Real.sqrt_nonneg _), inv_pow, |
| Real.sq_sqrt (show (0:ℝ) ≤ n.factorial by positivity)] |
| rw [← Finset.mul_sum, slaterCoordinateRaw_sum_norm_sq_integral hφ hψ ho, |
| ← mul_assoc, inv_mul_cancel₀ (show (n.factorial:ℝ) ≠ 0 by exact_mod_cast n.factorial_ne_zero), |
| one_mul] |
|
|
| omit [MeasurableSpace α] in |
| lemma slaterCoordinateTensor_mul (φ : Fin n → α → ℂ) (b : α → ℂ) |
| (j : Fin n) (σ : Equiv.Perm (Fin n)) (x : Fin n → α) : |
| slaterCoordinateTensor φ (fun i t => b t * φ i t) j σ x = |
| b (x j) * slaterTensor φ σ x := by |
| unfold slaterCoordinateTensor slaterTensor |
| have he (i : Fin n) : |
| (if i=j then (fun k t => b t * φ k t) else φ) (σ i) (x i) = |
| (if i=j then b (x j) else 1) * φ (σ i) (x i) := by |
| by_cases hij : i=j |
| · subst i; simp |
| · simp only [ite_eq_right hij, one_mul] |
| simp_rw [he] |
| rw [Finset.prod_mul_distrib] |
| congr 1 |
| simp |
|
|
| omit [MeasurableSpace α] in |
| lemma slaterCoordinateRaw_mul (φ : Fin n → α → ℂ) (b : α → ℂ) |
| (j : Fin n) (x : Fin n → α) : |
| slaterCoordinateRaw φ (fun i t => b t * φ i t) j x = b (x j) * slaterRaw φ x := by |
| simp only [slaterCoordinateRaw, slaterCoordinateTensor_mul, slaterRaw, Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro σ _ |
| ring |
|
|
| omit [MeasurableSpace α] in |
| lemma slaterCoordinate_mul (φ : Fin n → α → ℂ) (b : α → ℂ) |
| (j : Fin n) (x : Fin n → α) : |
| slaterCoordinate φ (fun i t => b t * φ i t) j x = b (x j) * slater φ x := by |
| rw [slaterCoordinate, slaterCoordinateRaw_mul, slater] |
| ring |
|
|
| lemma slater_local_squares {φ : Fin n → α → ℂ} (hφ : ∀ i, MemLp (φ i) 2 μ) |
| (ho : ∀ i k, (∫ t, conj (φ i t) * φ k t ∂μ) = if i=k then 1 else 0) |
| (b : α → ℂ) (hb : ∀ i, MemLp (fun t => b t * φ i t) 2 μ) : |
| (∫ x, (∑ j : Fin n, ‖b (x j)‖^2) * ‖slater φ x‖^2 |
| ∂Measure.pi (fun _ : Fin n => μ)) = |
| ∑ i, ∫ t, ‖b t‖^2 * ‖φ i t‖^2 ∂μ := by |
| have hi (j : Fin n) : Integrable (fun x => ‖b (x j)‖^2 * ‖slater φ x‖^2) |
| (Measure.pi fun _ : Fin n => μ) := by |
| simpa only [slaterCoordinate_mul, norm_mul, mul_pow] using |
| (memLp_slaterCoordinate hφ hb j).norm.integrable_sq |
| simp_rw [Finset.sum_mul] |
| rw [integral_finsetSum _ (fun j _ => hi j)] |
| simpa only [slaterCoordinate_mul, norm_mul, mul_pow] using |
| slaterCoordinate_sum_norm_sq_integral hφ hb ho |
|
|
| lemma slater_nonnegative_potential {φ : Fin n → α → ℂ} |
| (hφ : ∀ i, MemLp (φ i) 2 μ) |
| (ho : ∀ i k, (∫ t, conj (φ i t) * φ k t ∂μ) = if i=k then 1 else 0) |
| (V : α → ℝ) (hV : ∀ t, 0 ≤ V t) |
| (hv : ∀ i, MemLp (fun t => (Real.sqrt (V t) : ℂ) * φ i t) 2 μ) : |
| (∫ x, (∑ j : Fin n, V (x j)) * ‖slater φ x‖^2 |
| ∂Measure.pi (fun _ : Fin n => μ)) = |
| ∑ i, ∫ t, V t * ‖φ i t‖^2 ∂μ := by |
| simpa only [Complex.norm_real, Real.norm_eq_abs, |
| abs_of_nonneg (Real.sqrt_nonneg _), Real.sq_sqrt (hV _)] using |
| slater_local_squares hφ ho (fun t => (Real.sqrt (V t) : ℂ)) hv |
|
|
| end CoulombAtom |
|
|
| end |
|
|
| end OAI |
|
|
