From Metal to Atom: Ionisation Energies of 74 Elements from the Photoelectric Threshold

The first ionisation energy IE of a free atom follows from the work function W of its metal by one formula with no adjustable constant. In atomic units IE = W + (3/8) W^(2/3); in electronvolts IE = W + 1.128 W^(2/3). The first term is ħ times the Kepler orbital frequency of one electron at the edge of a sphere whose radius R is fixed by the work function through the threshold law of the companion paper, W = E_h (a0/R)^(3/2). The second term is the classical image-charge cost, (3/8) e^2/R, of leaving that same sphere charged. Metals with a closed outer s2 pair shift R by −a0/2, and metals with a lone p electron shift it by +a0/2. In hydrogen terms, W equals the correspondence frequency of the Bohr levels at an effective quantum number n* = (R/a0)^(1/2), and IE/W = 1 + (3/8) n*. Results: on 57 metals (Kawano 2022 work functions, NIST ionisation energies) the median error is 3.6% (51 of 57 within 10%). On 17 further elements predicted with the law frozen (a frozen-model holdout), the median error is 3.7%, and 2.4% on the six with measured work functions. On the same 17, the rule IE = 2W gives 14%, the Wong–Vongehr–Kresin ratio IE ≈ 1.8 W gives 7.1%, and the best straight line fitted to the 57 gives 4.6%. Measurement uncertainty in W accounts for only about 1% of the error, so the remaining misses (rare earths, Sc and Y, mid-d metals, mercury) point to physics the law does not yet capture. The law fails at the hydrogen limit. The charged-sphere term is due to Perdew (1988) and cluster physics; to our knowledge the closed link from the work function alone is new. Companion to: Kapil, R. (2026), A Halfway Bridge Between Classical and Quantum Physics, Zenodo, DOI 10.5281/zenodo.24124173. Files: the paper (PDF); kapil2026_metal_to_atom_data.csv, all 74 elements with work function, uncertainty, basis, step and NIST ionisation energy; reproduce.py, which recomputes every statistic in the paper. Version history: Version 2 corrected how the half-step enters the law, gave a full account of what was known before the 17-element test, and added fitted-line and Wong–Vongehr–Kresin baselines, error propagation, the electronvolt form, the data file and the reproduction script. Version 3 adds the data file and script, which were missing from the version 2 record; the paper is unchanged from version 2. Results are unchanged throughout.