Small-Z certificates — Simon ionization

Compute notes for the two- and three-electron trials in this folder. This is a replay of published facts about hydrogen, not a new bound.

Result

The Hylleraas trial

$\psi=\exp\bigl(-\tfrac56(r_1+r_2)\bigr)\bigl(1+\tfrac12 r_{12}\bigr)$

has exact variational energy $-815/1602$ on $H(2,1)$ . That is strictly below the hydrogen threshold $-1/2$ , with gap $7/801$ . Binding of the second electron follows, so the maximal binding number at $Z=1$ is at least 2.

Lieb's published theorem is $N_c<2Z+1$ . At unit nuclear charge that is strictly less than 3, hence at most 2 electrons. Combined:

N_0(1)=2.

Lieb, Phys. Rev. A 29, 3018 (1984), already concludes that hydrogen binds exactly two electrons and that $\mathrm{H}^{--}$ is not stable. Nam, arXiv:2206.15393, restates that this settles the ionization conjecture for the hydrogen atom. The certificate is a replay with an explicit rational energy. It is not a new bound.

A stronger Chandrasekhar trial with $a=26/25$ , $b=7/25$ gives exact energy $-1076189452297/2096620401250\approx -0.513297$ , gap $13939625836/1048310200625\approx 0.013297$ below $-1/2$ . Same conclusion.

Helium at nuclear charge 2 with two electrons has a Hylleraas energy $-54353/18800\approx -2.891117$ , about $0.0126$ above the published non-relativistic value $-2.9037243770341195$ (Nakashima–Nakatsuji 2007). That is a variational object, not a uniqueness theorem for helium.

He $^-$ does not bind in the trials below. That search is residue, not a lower bound. We do not claim a unique maximal $N$ for any nuclear charge larger than 1.

Hamiltonian and threshold

Infinite nuclear mass, Hartree atomic units:

H=-\tfrac12\Delta_1-\tfrac12\Delta_2-\frac{Z}{r_1}-\frac{Z}{r_2}+\frac1{r_{12}}.

Hydrogen is exact: $E(1,1)=-1/2$ . Binding of H $^-$ is $E(2,1)<-1/2$ .

The uncorrelated product $\exp(-\alpha(r_1+r_2))$ has energy $\alpha^2-2(Z-5/16)\alpha$ , minimum $-(Z-5/16)^2$ . At $Z=1$ this is $-121/256>-1/2$ , so a product of 1s orbitals does not bind.

The correlation factor $(1+c r_{12})$ , or Chandrasekhar's split-exponent symmetrised product, is the classical repair (Hylleraas 1929; Bethe 1929; Chandrasekhar 1944). Closed forms are written out in hylleraas.py. Høgaasen–Richard–Sorba, arXiv:0907.2614, record the Chandrasekhar algebra we re-derived from hydrogenic factorisation plus the Coulomb integral

\int\!\!\int\frac{e^{-\lambda r_1-\mu r_2}}{r_{12}}\,d^3r_1\,d^3r_2 =\frac{32\pi^2(\lambda^2+3\lambda\mu+\mu^2)}{\lambda^2\mu^2(\lambda+\mu)^3}.

Lieb's triangle $N_c<2Z+1$ is invoked, not re-proved.

Files and replay

From problems/simon-ionization-excess/compute/q1/:

python3 hylleraas.py
python3 hylleraas.py --write-certs
python3 he_hylleraas.py
python3 three_electron_try.py

file	role
`hylleraas.py`	exact Hylleraas $(α,c)$ and Chandrasekhar $(a,b)$ at general $Z$
`he_hylleraas.py`	helium $Z=2$ replay vs Nakashima–Nakatsuji
`three_electron_try.py`	three-electron Slater search; residue for He $^-$
`certs/hminus.json`	parameters, exact energy, enclosure, gap, script SHA
`certs/n0_z1.json`	uniqueness at hydrogen, both legs

The primary energy comparison is integer arithmetic on a rational. The decimal enclosure in the cert is floor/ceil of that rational times $10^{40}$ , not a floating-point evaluation of the Hamiltonian.

Helium object

Uncorrelated $α=27/16$ gives the textbook $-729/256=-2.84765625$ . Hylleraas $α=37/20$ , $c=37/100$ gives $-54353/18800$ . Chandrasekhar $a=109/50$ , $b=119/100$ gives about $-2.875658$ , matching the Chandrasekhar column in arXiv:0907.2614 Table 1. Zhislin already binds two electrons at helium. We do not extract a unique $N$ from this.

Three-electron search (residue)

Hydrogenic $J$ and $K$ used in the $N=3$ Slater energies were replayed from the same Coulomb generating function: $J(1s,2s)=17ζ/81$ , $K(1s,2s)=16ζ/729$ , and the $1s$ – $2p$ and $2s$ – $2p$ values listed in three_electron_try.py.

A single-charge hydrogenic $1s^2 2s$ or $1s2s2p\,{}^4P$ trial, and a screened Slater $1s^2 2s$ with a nodeless $2s$ orthogonalised against $1s$ , were minimised at $Z=2$ and $Z=3$ .

He $^-$ ( $Z=2$ , $N=3$ ). Best screened $1s^2 2s$ in the scan is about $-2.8465$ , above the published helium energy $-2.9037$ and above our own helium Hylleraas $-2.8911$ . The hydrogenic ${}^4P$ is worse (about $-2.01$ ). No trial here has $E_{\mathrm{var}}(3,2)$ below a lower bound on $E(2,2)$ . He $^-$ is not shown to bind. Residue, not a lower bound.

Lithium ( $Z=3$ , $N=3$ ). Screened $1s^2 2s$ reaches about $-7.418$ , and an optimised hydrogenic $1s^2 2s$ reaches $-6642153923171/911250000000\approx-7.28906$ . Both sit below the published Li $^+$ energy $\approx-7.27991$ (Nakashima–Nakatsuji, $Z=3$ ). That comparison is not a certificate: we do not own a lower bound on $E(2,3)$ . Our own Li $^+$ Hylleraas/Chandrasekhar numbers are upper bounds and cannot be used as a threshold. Zhislin already binds the neutral lithium atom. We do not claim a unique $N$ at lithium.

A three-electron Hylleraas–CI fragment was not built. The Slater search is the attempt.

What this is not

Not a new bound on the maximal electron number, or on the excess charge as $Z\to\infty$ .
Not a proof that He $^-$ binds.
Not a uniqueness theorem for any $Z>1$ .
Lieb's functional-analytic proof is cited, not reproduced.