The Multi-Electron Unified Wave Equation in Quantum Chemistry: Symmetrized Riesz Exchange Kernels, Sobolev Domain Classification, and O(N4) Quasiparticle Electronic Structure across Bond Dissociation
Abstract
A central challenge in ab initio electronic structure theory is capturing both short-range dynamic electron correlation (the Kato inter-electronic cusp) and long-range static multireference correlation across covalent bond dissociation without incurring the exponential O(exp N) cost of Full Configuration Interaction (FCI), the O(N⁶) cost and single-reference breakdown of Coupled Cluster Singles and Doubles (CCSD), or the one-electron self-interaction error (SIE) of Density Functional Theory (DFT). In this work (the first of two companion papers on the Multi-Electron Unified Wave Equation, MEUWE), we formulate and rigorously analyze the closed-system stationary sector of MEUWE as an ab initio two-electron correlation theory. First, we establish a foundational No-Go result against nonlinear state powers Ψᵖ (p ≠ 1) and unscaled nonlocal kernels (Proposition II.1), and establish operator uniqueness of the Jordan-symmetrized Riesz exchange kernel K̂sym = ½ ∑ᵢ₍ⱼ₎ {Ŵᵢ, r₀ᵅ|rᵢ − rⱼ|⁻ᵅ P̂ᵢⱼ} within the bilinear screening-permutation pencil P_W,K, showing that unsymmetrized operators ŴᵢV̂ᵅP̂ᵢⱼ violate Hermiticity (ΔHerm ∈ [1.68, 3.76]) and leak fermion states outside the Pauli sector ∧ᴺL²(R³) (Theorem III.1). Second, we establish a four-regime functional-analytic self-adjointness classification of Ĥ₀ + ΛK̂sym on ∧ᴺL²(R³): operator Kato–Rellich self-adjointness on H²(R³ᴺ) for α ∈ (0, 3/2), quadratic-form KLMN self-adjointness on H¹(R³ᴺ) via Hardy’s inequality for α ∈ [3/2, 2) (including our canonical α = 3/2), the critical Hardy threshold at α = 2, and bounded-density L¹ ∩ L∞ expectation finiteness for α ∈ (2, 3) with ultraviolet divergence at α ≥ 3. Third, for closed bound states below the excitation threshold (Eₙ < Eₜₕ), we prove via Feshbach–Schur resolvent projection that integrating out the virtual excitation continuum yields a strictly real, Hermitian, energy-dispersive quasiparticle self-energy Σₘₑₘ(E) = Λγₘₑₘ(E)K̂sym whose self-consistent fixed point converges geometrically via Banach contraction (qₘₐₓ = 0.091226 < 1, error 4.44 × 10⁻¹⁵ Ha in ≤ 13 iterations, matching exact Stinespring resolvent poles to 5.82 × 10⁻⁷ Ha, Theorem III.5). Fourth, using exact 6D Mellin–Laplace Gaussian integrals implemented in PySCF/libcint, we verify that MEUWE is 100% self-interaction-free on all one-electron ions Z = 1 … 6 (|ΔE| ≤ 2.66 × 10⁻¹² mHa) and achieves an overall correlation RMSE of 0.3824 mHa across the Z = 2 … 6 two-electron isoelectronic series (7.31× more accurate than MP2, Section III C 1). Finally, freezing all parameters strictly at the two-electron Helium calibration (α = 1.50, μ = 0.606503, Λ₀ = 0.01150) with zero molecule-fitted parameters and zero FCI calls, our standalone O(N⁴) Dyson pair solver reproduces exact FCI potential energy surfaces across H₂, HeH⁺, and LiH (R ∈ [0.5, 3.0] Å) with mean absolute errors of 0.1382, 0.1618, and 0.1302 mHa, captures the avoided-crossing ionic-to-neutral dipole turnover in LiH (μ(3.0 Å) = 5.0398 D vs. FCI 5.0265 D where RHF diverges to 9.1647 D, Section III C 2), and resolves atomic K/L shell structures and Coulson–Neilson Coulomb holes across H–Ne (Section III C 3).
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BibTeX
@misc{singh2026multielectron,
title = {{The Multi-Electron Unified Wave Equation in Quantum Chemistry: Symmetrized Riesz Exchange Kernels, Sobolev Domain Classification, and O(N4) Quasiparticle Electronic Structure across Bond Dissociation}},
author = {Mehardeep Singh},
howpublished = {ChemRxiv},
year = {2026},
month = oct,
doi = {10.26434/chemrxiv.15010852/v1},
url = {https://chemrxiv.org/doi/full/10.26434/chemrxiv.15010852/v1},
note = {CC BY-NC-ND},
}
RIS
TY - UNPB
TI - The Multi-Electron Unified Wave Equation in Quantum Chemistry: Symmetrized Riesz Exchange Kernels, Sobolev Domain Classification, and O(N4) Quasiparticle Electronic Structure across Bond Dissociation
AU - Mehardeep Singh
PY - 2026
DA - 2026/10/22
PB - ChemRxiv
DO - 10.26434/chemrxiv.15010852/v1
UR - https://chemrxiv.org/doi/full/10.26434/chemrxiv.15010852/v1
AB - A central challenge in ab initio electronic structure theory is capturing both short-range dynamic electron correlation (the Kato inter-electronic cusp) and long-range static multireference correlation across covalent bond dissociation without incurring the exponential O(exp N) cost of Full Configuration Interaction (FCI), the O(N⁶) cost and single-reference breakdown of Coupled Cluster Singles and Doubles (CCSD), or the one-electron self-interaction error (SIE) of Density Functional Theory (DFT). In this work (the first of two companion papers on the Multi-Electron Unified Wave Equation, MEUWE), we formulate and rigorously analyze the closed-system stationary sector of MEUWE as an ab initio two-electron correlation theory. First, we establish a foundational No-Go result against nonlinear state powers Ψᵖ (p ≠ 1) and unscaled nonlocal kernels (Proposition II.1), and establish operator uniqueness of the Jordan-symmetrized Riesz exchange kernel K̂sym = ½ ∑ᵢ₍ⱼ₎ {Ŵᵢ, r₀ᵅ|rᵢ − rⱼ|⁻ᵅ P̂ᵢⱼ} within the bilinear screening-permutation pencil P_W,K, showing that unsymmetrized operators ŴᵢV̂ᵅP̂ᵢⱼ violate Hermiticity (ΔHerm ∈ [1.68, 3.76]) and leak fermion states outside the Pauli sector ∧ᴺL²(R³) (Theorem III.1). Second, we establish a four-regime functional-analytic self-adjointness classification of Ĥ₀ + ΛK̂sym on ∧ᴺL²(R³): operator Kato–Rellich self-adjointness on H²(R³ᴺ) for α ∈ (0, 3/2), quadratic-form KLMN self-adjointness on H¹(R³ᴺ) via Hardy’s inequality for α ∈ [3/2, 2) (including our canonical α = 3/2), the critical Hardy threshold at α = 2, and bounded-density L¹ ∩ L∞ expectation finiteness for α ∈ (2, 3) with ultraviolet divergence at α ≥ 3. Third, for closed bound states below the excitation threshold (Eₙ < Eₜₕ), we prove via Feshbach–Schur resolvent projection that integrating out the virtual excitation continuum yields a strictly real, Hermitian, energy-dispersive quasiparticle self-energy Σₘₑₘ(E) = Λγₘₑₘ(E)K̂sym whose self-consistent fixed point converges geometrically via Banach contraction (qₘₐₓ = 0.091226 < 1, error 4.44 × 10⁻¹⁵ Ha in ≤ 13 iterations, matching exact Stinespring resolvent poles to 5.82 × 10⁻⁷ Ha, Theorem III.5). Fourth, using exact 6D Mellin–Laplace Gaussian integrals implemented in PySCF/libcint, we verify that MEUWE is 100% self-interaction-free on all one-electron ions Z = 1 … 6 (|ΔE| ≤ 2.66 × 10⁻¹² mHa) and achieves an overall correlation RMSE of 0.3824 mHa across the Z = 2 … 6 two-electron isoelectronic series (7.31× more accurate than MP2, Section III C 1). Finally, freezing all parameters strictly at the two-electron Helium calibration (α = 1.50, μ = 0.606503, Λ₀ = 0.01150) with zero molecule-fitted parameters and zero FCI calls, our standalone O(N⁴) Dyson pair solver reproduces exact FCI potential energy surfaces across H₂, HeH⁺, and LiH (R ∈ [0.5, 3.0] Å) with mean absolute errors of 0.1382, 0.1618, and 0.1302 mHa, captures the avoided-crossing ionic-to-neutral dipole turnover in LiH (μ(3.0 Å) = 5.0398 D vs. FCI 5.0265 D where RHF diverges to 9.1647 D, Section III C 2), and resolves atomic K/L shell structures and Coulson–Neilson Coulomb holes across H–Ne (Section III C 3).
ER -