Helium Atom & Two-Electron Correlation

Electron-Electron Coulomb Repulsion $r_{12}$, Correlated Wavefunctions & Isoelectronic Scaling

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Foundations Root · Two-Electron Systems, Correlation Energy & Isoelectronic Scaling

Public Status Boundary. The Helium atom cannot be solved analytically due to the non-separable electron-electron Coulomb repulsion term $1/r_{12}$. Benchmark values are anchored to high-precision Hylleraas variational calculations and NIST ASD experimental ionization energies ($I_1=24.587387\text{ eV}$).

$$\boxed{{\displaystyle \mathbf{\text{Independent-particle model}}\ne \mathbf{\text{correlated two-electron model}}}}$$$$\boxed{{\displaystyle \mathbf{\text{Correlation energy }}{E}_{\text{corr}}=E_{\text{exact,nonrel}}-E_{\text{HF}}<0}}$$$$\boxed{{\displaystyle \mathbf{\text{NIST ASD / Drake Benchmark}}=\mathbf{\text{VERSIONED EXTERNAL REFERENCE ANCHOR}}}}$$

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1. The Two-Electron Hamiltonian

For a nucleus of charge $Z$ with two electrons, the non-relativistic Hamiltonian in atomic units is:

$$H=-\frac{1}{2}{\mathrm{\nabla }}_1^2-\frac{1}{2}{\mathrm{\nabla }}_2^2-\frac{Z}{r_1}-\frac{Z}{r_2}+\frac{1}{r_{12}}$$

where $r_{12}=|{\mathbf{r}}_1-{\mathbf{r}}_2|=\sqrt{r_1^2+r_2^2-2r_1{r}_2\cos {\theta }_{12}}$.

The singular inter-electronic term $\frac{1}{r_{12}}$ prevents separation of variables in spherical coordinates, establishing electron correlation as the fundamental physical problem of multi-body atomic physics.

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2. Independent-Particle vs Correlated Models

Model Level / Taxonomy	Mathematical Ansatz	Ground-State Energy ($Z=2$)	Correlation Captured
Unshielded Hydrogenic	$\psi ({\mathbf{r}}_1,{\mathbf{r}}_2)={\varphi }_{1s}({\mathbf{r}}_1){\varphi }_{1s}({\mathbf{r}}_2)$	$-4.000000{\text{ E}}_h$ ($-108.84\text{ eV}$)	$0\mathrm{\%}$ (Unphysical baseline)
Variational Screening ($Z_{\text{eff}}=27/16$)	${\varphi }_{1s}(Z_{\text{eff}},{\mathbf{r}}_1){\varphi }_{1s}(Z_{\text{eff}},{\mathbf{r}}_2)$	$-2.847656{\text{ E}}_h$ ($-77.489\text{ eV}$)	Mean-field screening
Hartree-Fock Limit (HF_LIMIT)	Self-consistent central field orbit product	$-2.861680{\text{ E}}_h$ ($-77.8703\text{ eV}$)	$0\mathrm{\%}$ ($E_{\text{HF}}$ baseline)
Correlated Nonrelativistic Benchmark	Hylleraas explicitly correlated expansion ($r_{12}$)	$-2.903724{\text{ E}}_h$ ($-79.0152\text{ eV}$)	$100\mathrm{\%}$ ($E_{\text{nonrel}}$)

2.1 Definitive Correlation Energy

The correlation energy is formally defined as the exact nonrelativistic ground-state energy minus the Hartree-Fock limit:

$$E_{\text{corr}}=E_{\text{nonrel}}-E_{\text{HF}}=-2.903724377034{\text{ E}}_h-(-2.861679995612{\text{ E}}_h)=-0.042044381422{\text{ E}}_h(\approx -1.14409\text{ eV})$$

2.2 Versioned External Reference Anchor: Sequential Two-Electron Removal

In physical spectroscopy, the total energy required to sequentially strip both electrons from Helium is measured via laser and synchrotron spectroscopy (NIST ASD v5.11, DOI: 10.18434/T4W30F):

• First Ionization Potential ($\text{He}\to {\text{He}}^{+}+e^{-}$): $I_1=24.587389011(10)\text{ eV}$
• Second Ionization Potential (${\text{He}}^{+}\to {\text{He}}^{2+}+e^{-}$): $I_2=54.4177655282(12)\text{ eV}$
• Total Sequential Removal Energy: $E_{\text{remove}}=I_1+I_2=79.0051545392(101)\text{ eV}$ (incorporates relativistic mass shift, Darwin term, spin-orbit, and QED Lamb shifts).

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3. Isoelectronic Sequence Scaling ($Z$-Progression)

The balance between nuclear attraction ($\propto -Z^2$) and electron repulsion ($\propto Z$) governs the isoelectronic sequence:

$$E(Z)=-Z^2(1-\frac{5}{8Z}+\frac{0.1576}{Z^2}+\mathcal{O}(Z^{-3}))\text{ a.u.}$$

Ion	Nuclear Charge $Z$	Experimental Ground State ($E_0$)	Ionization Potential ($I_1$)	Status
${\text{H}}^{-}$	$Z=1$	$-14.35\text{ eV}$ ($-0.5278\text{ a.u.}$)	$0.754\text{ eV}$	Weakly bound ion (correlated)
$\text{He}$	$Z=2$	$-79.005\text{ eV}$ ($-2.9037\text{ a.u.}$)	$24.587\text{ eV}$	Standard atomic benchmark
${\text{Li}}^{+}$	$Z=3$	$-198.09\text{ eV}$ ($-7.2799\text{ a.u.}$)	$75.640\text{ eV}$	Bound core ion
${\text{Be}}^{2+}$	$Z=4$	$-371.60\text{ eV}$ ($-13.6556\text{ a.u.}$)	$153.896\text{ eV}$	High-$Z$ isoelectronic test

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4. Authorial Relational Model & Geometric Boundaries

In the Science of Fabric Reality, the Helium three-body system $(e_1,e_2,\text{nucleus})$ is modeled as a 3-simplex relational constraint network:

• Relational Coordinate Frame: Parameterized by internal distances $(r_1,r_2,r_{12})$ rather than absolute coordinate vectors $({\mathbf{r}}_1,{\mathbf{r}}_2)$.
• Cusp Condition Preservation: Exact adherence to Kato's electron-electron cusp condition:$${\frac{\partial \overline{\psi }}{\partial r_{12}}|}_{r_{12}=0}=\frac{1}{2}\psi (r_{12}=0)$$
• Non-Perturbative Boundary: Treating correlation not as a perturbation, but as an irreducible topological constraint on admissible 2-electron state space.

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5. Canonical Descent & Navigation

Explore adjacent atomic and experimental sectors:

$$\boxed{{\displaystyle \mathbf{\text{Continue → }}\text{LPFR Laboratory Interface: Laser-Plasma Protocols}}}$$

Related foundations: Hydrogen Dynamics $\parallel$ Atomic Physics Root $\parallel$ Simulation Atlas.

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SOURCE AUTHORITY & BOUNDARY LOCK

This route enforces strict cryptographic and epistemic boundaries between consensus reference data, comparator theoretical literature, and authorial candidate predictions.

ESTABLISHED BASELINE

• NIST ASD v5.12 (2024-11-07)
  Helium neutral I1 = 24.587389011(25) eV, ionic I2 = 54.4177655282(10) eV, sequential removal sum = 79.0051545392 eV.
• CODATA 2022 (2025)
  Hartree energy conversion E_h = 27.211386245981(30) eV.

COMPARATOR LITERATURE

• Drake (2006) (2006)
  Hylleraas-type non-relativistic benchmark (-2.903724377 Eh) vs restricted Hartree-Fock limit (-2.861680 Eh).

METROLOGY / DATA STANDARDS

• JCGM 100:2008 (GUM)
  Evaluation of sequential ionization energy sums with unevaluated cross-covariance.

AUTHORIAL EXTENSION BOUNDARY

VERSIONED_EXTERNAL_REFERENCE_PLUS_THEORETICAL_BENCHMARK

Helium ground state and correlation energy are established external benchmarks. Geometric correlation boundary functionals are authorial research models.

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