# ATOM⁴ — literature notes

The prior work that ATOM⁴'s first paper (the 4D periodic table) has to cite and stand against, gathered in a first
literature pass (2026-10-25). The standard is arXiv's: every reference verified against its source before it is
cited, every claim of novelty checked against what's known. This file is the working record, not the paper.

How far each reference has been verified:

- **read**: the source itself was read.
- **index**: the bibliographic details were confirmed from a publisher's or index's listing, but the text wasn't read.
- **cited**: seen only as a citation in another work. Must be checked before it goes in a paper.

## The question, and where it stands

**Paper 1's claim to defend:** with the Coulomb interaction kept at 1/r in four spatial dimensions, the neutral
atoms' ground states, solved self-consistently, form a periodic table with rows of 2, 10, 10, 28, 28 and noble gases
at Z = 2, 12, 22, 50, 78. Inner electrons screen only partly (there is no shell theorem), so outer electrons are held
much harder than shell-theorem screening would allow.

What the pass found:

1. **The setup is established.** Keeping V = −Ze²/r while varying the number of dimensions D is the convention of the
   D-dimensional hydrogen literature and of Herschbach's dimensional-scaling programme. The alternative, the Poisson
   potential r^(2−D), is noted there and set aside, as we did for the same reason (no ground state in 4D). We cite
   the convention. Our physical argument for it (the lens vacuum) is ours, and belongs to a later paper.
2. **The hydrogenic part is known.** E = −Z²/2[N + (D − 1)/2]², with N = n − 1, and the degeneracy g(l, D), which is
   (l + 1)² in 4D. These are standard (Nieto 1979; Négadi & Kibler 1996, eqs. 10 and 14a).
3. **The period lengths were predicted, not computed.** Négadi & Kibler (1996) state a generalized Madelung rule for
   any D: fill (N + 1 + l) increasing, then N + 1, each subshell holding 2g(l, D). They work it out only for D = 2.
   Applied to D = 4, it gives rows of 2, 10, 10, 28, 28 and noble gases at 2, 12, 22, 50, 78. That is exactly what
   our self-consistent solver finds. Their rule is combinatorics plus a fitted q-deformation, with no
   electron–electron interaction.
4. **A self-consistent table exists only for D = 2.** Pyykkö & Zhao (1991) solved Hartree–Fock ground states for
   Z = 1–24 in two dimensions, with 1/r interactions. The Herschbach group computed Hartree–Fock and exact energies of
   two-electron atoms as a function of D (Loeser & Herschbach 1986, and later). No self-consistent many-electron
   table for D = 4 turned up in four searches, two of them extended.
5. **Our electron-gas constants are known.** The 4D kinetic and exchange coefficients, (4π/3)ρ^(3/2) and
   −16/(15√π)ρ^(5/4), reproduce the general-D formulas in Loos & Gill's review (checked numerically:
   εₓ = −2Dα/π(D² − 1) and εₜ = Dα²/2(D + 2), with α = 2^((D−1)/D) Γ(D/2 + 1)^(2/D), give 0.60180 and 4.18879 at D = 4).
   The forces chapter now credits them.

So paper 1's contribution, as far as the pass can tell, is:

- **The first self-consistent periodic table for four dimensions,** to Z = 78. Its period structure confirms the
  generalized Madelung rule's prediction.
- **What the rule cannot give:**
  - the ground configurations, including where 4D's transition metals depart from Madelung;
  - ionization energies and radii;
  - the strength of partial screening, measured against the shell-theorem counterfactual (noble-gas ionization
    about twice as high);
  - the far pull of a neutral atom (−Σ⟨r²⟩/8s³).
- **Validation by the same solver:** our own table in 3D (32 of 36 ground configurations, ionization energies within
  a few percent), helium against Hartree–Fock, and hydrogen exact in both worlds.

## Remaining risk to the novelty claim

The pass did not reach:

- The full text of *Dimensional Scaling in Chemical Physics* (1992/1993). Its Pyykkö–Zhao chapter, "Low D Regime"
  (with Goodson and López-Cabrera, pp. 115–138), is on low D.
- Loeser & Herschbach (1986).
- Citation-forward searches from Pyykkö & Zhao and from Négadi & Kibler, for later work at D = 4.

A first citation-forward search (Pyykkö's publication list, works citing Négadi & Kibler, D-dimensional Madelung
work since 2000) found nothing at D = 4. It was a web search, not a citation index, so the claim in the paper will read
"to our knowledge", and a Web of Science or Scopus forward search on both papers should be run before it's posted.

## References

### Dimension and the inverse-square law
- P. Ehrenfest, Proc. Amsterdam Acad. **20**, 200 (1917); Ann. Phys. **61**, 440 (1920). Orbits and atoms are
  unstable for D > 3 under the Poisson potential. — *cited* (via Tangherlini and later reviews)
- F. R. Tangherlini, "Schwarzschild field in n dimensions and the dimensionality of space problem", Nuovo Cimento
  **27**, 636–651 (1963). Extends the argument to general relativity and to the Schrödinger hydrogen atom. — *index*
  (INSPIRE)
- C. Callender, "Answers in search of a question: 'proofs' of the tri-dimensionality of space" (2005), and F. Caruso &
  R. Moreira Xavier, on the stability argument's weakness. — *cited*; the exact references are to be fixed.

### D-dimensional hydrogen and periodic tables
- M. M. Nieto, "Hydrogen atom and relativistic pi-mesic atom in N-space dimensions", Am. J. Phys. **47**, 1067
  (1979). — *cited* (via hep-th/9410161)
- S. M. Al-Jaber, "Hydrogen atom in N dimensions", Int. J. Theor. Phys. (1998). — *cited*; volume and pages to be fixed.
- G. F. Kventsel & J. Katriel, Phys. Rev. A **24**, 2299 (1981). The Poisson-equation potential in D dimensions.
  — *cited* (via Négadi & Kibler)
- T. Négadi & M. Kibler, "The periodic table in Flatland", Int. J. Quantum Chem. **57**, 53 (1996);
  arXiv:atom-ph/9512001. — *read*
- P. Pyykkö & Y. Zhao, "The elements of Flatland: Hartree–Fock atomic ground states in two dimensions for Z = 1–24",
  Int. J. Quantum Chem. **40**, 527–544 (1991), doi:10.1002/qua.560400408. — *index* (Wiley)
- E. Madelung, *Die mathematischen Hilfsmittel des Physikers* (Springer, 1936); V. M. Klechkovskii (1955, 1961).
  The Madelung rule. — *cited* (via Négadi & Kibler)

### Dimensional scaling
- D. R. Herschbach, J. Avery & O. Goscinski (eds.), *Dimensional Scaling in Chemical Physics* (Kluwer, Dordrecht,
  1992; softcover 1993). — *index*
- P. Pyykkö & Y. Zhao, "Flatland: the periodic system for D = 2", in the above. — *index* (Pyykkö's publication list).
  A Springer listing found by search gives a "Low D Regime" chapter, pp. 115–138, by D. Z. Goodson, M. López-Cabrera,
  P. Pyykkö & Y. Zhao. Whether these are the same chapter is to be settled from the book.
- J. G. Loeser & D. R. Herschbach, J. Chem. Phys. **84**, 3882–3892 (1986). Hartree–Fock and exact energies of He-like
  ions against D. — *cited* (via Ghosh, Kais & Herschbach 2020; Bernard, Loos & Gill 2013)
- A. Ghosh, S. Kais & D. R. Herschbach, "Unorthodox dimensional interpolations for He, Li, Be atoms and hydrogen
  molecule", Front. Phys. (2020), doi:10.3389/fphy.2020.00331; arXiv:2004.11489. — *index*

### The uniform electron gas in D dimensions
- P.-F. Loos & P. M. W. Gill, "The uniform electron gas", arXiv:1601.03544 (2016); published in WIREs Comput. Mol.
  Sci. (2016), volume and pages to be confirmed. — *read* (the general-D kinetic and exchange formulas)

### Methods the solver uses (to be verified before citing)
- Numerov integration; Sturm bisection for eigenvalues.
- Local exchange (Dirac/Slater); J. P. Perdew & A. Zunger, Phys. Rev. B **23**, 5048 (1981), self-interaction
  correction. — *from memory; verify*
- Hund's rules; the shell theorem (Newton).

### Molecules: Hartree–Fock in any dimension (chapters 9 and 10)
- S. F. Boys, "Electronic wave functions. I. A general method of calculation for the stationary states of any molecular
  system", Proc. R. Soc. A **200**, 542–554 (1950), doi:10.1098/rspa.1950.0036. Gaussians, and the Boys function our
  G_n(T) generalizes. — *index*
- L. E. McMurchie & E. R. Davidson, "One- and two-electron integrals over Cartesian Gaussian functions", J. Comput.
  Phys. **26**, 218–231 (1978), doi:10.1016/0021-9991(78)90092-X. The integral scheme `js/gauss.js` follows. — *index*
- S. F. Boys & F. Bernardi, Mol. Phys. **19**, 553 (1970). The counterpoise correction. — *from memory; verify*
- P. Pulay, Chem. Phys. Lett. **73**, 393 (1980). DIIS. — *from memory; verify*
- R. S. Mulliken, J. Chem. Phys. **23**, 1833 (1955). Population analysis. — *from memory; verify*
- Water at the Hartree–Fock limit (r ≈ 1.776 bohr, θ ≈ 106.3°, μ ≈ 0.78 au; dimer ≈ 3.6 kcal/mol) and measured
  (1.809 bohr, 104.5°, 0.730 au; dimer De ≈ 5.0 kcal/mol, O–O 2.91 Å): the values the chapters quote. — *from memory;
  each needs a primary source*

### Molecular dynamics (js/md.js)
- P. P. Ewald, Ann. Phys. **64**, 253 (1921). The lattice sum, which `md.js` generalizes to any dimension. — *from
  memory; verify*
- H. M. Evjen, Phys. Rev. **39**, 675 (1932). The weighted direct sum used as the independent check on the 4D Madelung
  constant. — *from memory; verify*
- The rock-salt Madelung constant 1.747565. — *standard value; cite a table*
- J.-P. Ryckaert, G. Ciccotti & H. J. C. Berendsen, J. Comput. Phys. **23**, 327 (1977), SHAKE; H. C. Andersen,
  J. Comput. Phys. **52**, 24 (1983), RATTLE. — *from memory; verify*
- G. Bussi, D. Donadio & M. Parrinello, "Canonical sampling through velocity rescaling", J. Chem. Phys. **126**,
  014101 (2007), doi:10.1063/1.2408420; arXiv:0803.4060. The thermostat. — *index*
- H. J. C. Berendsen, J. R. Grigera & T. P. Straatsma, "The missing term in effective pair potentials", J. Phys.
  Chem. **91**, 6269–6271 (1987). SPC/E, the published water model used to check the engine. — *index*
- NIST Standard Reference Simulation Website, "SPC/E water reference calculations — 10 Å cutoff":
  https://www.nist.gov/mml/csd/chemical-informatics-group/spce-water-reference-calculations-10a-cutoff, and the
  configurations (https://www.nist.gov/document/spcesampleconfigurations-targz). The Ewald parameters (α = 5.6/L,
  kmax 5, k² < 27, 10 Å) and the reference energies, which `md.js` matches part by part. Configuration 1 is in
  `test/fixtures/`. — *read*
- NIST, "LAMMPS MD equation of state: pressure vs density, SPC/E water":
  https://www.nist.gov/mml/csd-mml/informatics/lammps-md-equation-state-pressure-vs-density-spce-water. SPC/E's
  parameters, with ε/k = 78.2 K and its 5.213 kJ/mol induction term; the 300 K rows couldn't be read reliably. —
  *read (in part)*
- C. Vega & J. L. F. Abascal, "Simulating water with rigid non-polarizable models: a general perspective", Phys. Chem.
  Chem. Phys. (2011). The water models' liquid properties, needed to quote SPC/E's U at 298 K exactly (ours gives
  −46.9 kJ/mol). — *cited (via arXiv:0905.4009); volume and pages to be fixed*

### Energy landscapes and searching them (chapter 11)
- F. H. Stillinger & T. A. Weber, "Hidden structure in liquids", Phys. Rev. A **25**, 978 (1982). Inherent
  structures: quench each configuration to its minimum. The idea behind settling every start in the live search. —
  *index* (cited with these details in a 2003 PRL abstract and in a Springer reference list; full text not read)
- F. H. Stillinger, "Exponential multiplicity of inherent structures", Phys. Rev. E **59**, 48 (1999),
  doi:10.1103/PhysRevE.59.48. At fixed density, the number of distinct minima grows exponentially with the number of
  molecules. — *index* (APS record and abstract)
- D. J. Wales, *Energy Landscapes* (Cambridge University Press, 2003), ISBN 978-0-521-81415-7. The standard text on
  minima, funnels and searching them. — *index* (publisher's copyright page; not read)
- B. J. Smith, D. J. Swanton, J. A. Pople, H. F. Schaefer III & L. Radom, "Transition structures for the interchange
  of hydrogen atoms within the water dimer", J. Chem. Phys. **92**, 1240–1247 (1990). Ten stationary points on the
  water pair's surface; only the trans-linear one is a minimum. This is the known answer our 3D search must reproduce. —
  *index* for the ten stationary points (via the supplementary data of van Duijneveldt-van de Rijdt et al., PCCP 2003);
  that just one is a minimum is *from memory; verify*

### Gases (chapter 12)
- Critical points of the elements, Wikipedia data page, compiling the CRC Handbook and others: neon 44.4 K, argon
  150.87 K (other columns 150.85–151.15), krypton 209.41 K, helium 5.19 K. The measured critical points the rules are
  checked against. — *index* (the data page read; the CRC itself not)
- Argon's Joule–Thomson inversion temperature at low pressure, about 794 K: from a cryogenics lecture table (Seoul
  National University) listing N₂ at 621–623 K and He at 34–51 K beside it. Hydrogen's ≈ 202 K and helium's ≈ 51 K
  also appear in encyclopedia summaries of the effect. — *cited* (secondary tables only; a primary measurement is
  still to be found before it goes in a paper)
- Argon's Boyle temperature: 407 K in an Oxford thermal-physics handout (A. Schekochihin), 411 K and 392 K in other
  sources (the spread is noted in arXiv:1303.5307). The chapter quotes "about 407, sources give 392 to 411". — *cited*
- Argon's second virial coefficient at 298.15 K, about −15.8 cm³/mol (lecture tables); the measured data are compiled
  in J. H. Dymond & E. B. Smith, *The Virial Coefficients of Pure Gases and Mixtures* (Clarendon, Oxford, 1980), and
  argon's reference equation of state is Ch. Tegeler, R. Span & W. Wagner, J. Phys. Chem. Ref. Data **28**, 779
  (1999). — *cited* (the value from secondary tables; Dymond & Smith and Tegeler et al. not read)
- Lennard-Jones B₂: Boyle temperature 3.418 ε/k and low-pressure inversion 6.43 ε/k, which our integrator
  reproduces (3.4179 and 6.4308). — *from memory; verify* against a published table (e.g. Vega et al., or the
  standard texts) before citing
- Solid argon's cohesive energy, 0.080 eV per atom: C. Kittel, *Introduction to Solid State Physics*, the crystal
  binding chapter's table of inert-gas crystals; also −2941 μHa in arXiv:1908.00601. — *cited* (Kittel's table seen
  only as reproduced in lecture slides)
- The liquid–vapour critical point is in the Ising universality class, whose upper critical dimension is four: at
  d = 4 the exponents are mean field's, with multiplicative logarithmic corrections. K. G. Wilson & M. E. Fisher,
  "Critical exponents in 3.99 dimensions", Phys. Rev. Lett. **28**, 240 (1972); the logarithms as in R. Kenna's work
  on the 4D Ising and O(N) models. — *cited* (standard; the specific papers not read for this)

- J. A. van Meel, B. Charbonneau, A. Fortini & P. Charbonneau, "Hard-sphere crystallization gets rarer with
  increasing dimension", Phys. Rev. E **80**, 061110 (2009), arXiv:0907.3100; building on van Meel, Frenkel &
  Charbonneau's 4D study, which found a larger crystallization barrier in 4D than in 3D. — *index* (APS and AMOLF
  listings and abstract)
- The liquid's range shrinks, and the liquid finally disappears, as the attraction's range shortens (colloids;
  M. H. J. Hagen & D. Frenkel, J. Chem. Phys. **101**, 4093 (1994)); fluid–fluid coexistence needs a range above
  about 30% of the core. — *cited* (from later abstracts; Hagen & Frenkel not read)
- Aluminium's ionization energies 5.986, 18.829, 28.448 eV and oxygen's electron affinity 1.461 eV (NIST). — *from
  memory; verify* against the NIST tables before citing

- Sodium fluoride: nearest-neighbour distance 2.31 Å (4.38 bohr) and lattice energy about 923 kJ/mol (9.57 eV),
  sodium's ionization 5.14 eV and fluorine's affinity 3.40 eV. — *from memory; verify* (CRC / NIST) before citing.
  The electron-gas model applied to ionic crystals: Y. S. Kim & R. G. Gordon, Phys. Rev. B **9**, 3548 (1974). —
  *cited* (not read; details from memory)

- The hydrogen fluoride dimer is bent (a near-linear F–H···F bond, the acceptor tilted well off the line), as measured;
  an effective potential reproducing its F–F distance and both angles: arXiv:cond-mat/9903051. Ab initio (HF)₂ and
  (HCl)₂ at double-ζ: S. Scheiner & Z. Latajka (1988). — *index* (abstracts; the structures' numbers not read)
- The ammonia dimer is very floppy; microwave work (Nelson, Fraser & Klemperer) measured a dipole of 0.74 D against
  ~2 D for a linear hydrogen bond; a 2022 ab initio surface (PMC8933541) finds it fluxional but hydrogen-bonded. —
  *index* (summaries of these, not the papers)

## Next

1. A citation-index forward search on Pyykkö & Zhao (1991) and Négadi & Kibler (1996). The web search is done.
2. Turn every *cited* entry into *read* or *index*.
3. An outline for paper 1.