Completing the large-$Z$ expansion of the neutral atom's total binding energy: the oscillatory fourth term and the definitive relativistic supplement

For a neutral atom of nuclear charge $Z$, the total binding energy expands for large $Z$ as $-E=(0.768745\,Z^{7/3}-\tfrac{1}{2}Z^2+0.269900\,Z^{5/3}+\cdots)e^2/a_0$: the Thomas-Fermi (TF) value, the Scott correction for the strongly bound electrons, and the Dirac-Schwinger exchange term, all established (the last rigorously). Schwinger (1980) derived the Scott correction by grafting the hydrogenic treatment of the inner electrons onto the TF model and explicitly acknowledged unresolved issues, two of which survive in narrowed form. First, the grafting splits the electrons at a cutoff principal quantum number and the strongly-bound electron count is a staircase (step) function, leaving an ambiguity of relative order $Z^{-1}$; Englert and Schwinger (1985) later showed that these contributions are oscillatory atomic-shell terms of order $Z^{4/3}$, in good agreement with Hartree-Fock data, but no mathematical proof of the conjectured fourth term exists (Englert, 2019), and rigorous work has so far produced only bounds. Second, the relativistic supplement for physical atoms (fixed fine-structure constant) beyond Schwinger's leading term $-0.854\,Z^2(Z\alpha)^2(1-2.43\,Z^{-1/3})\,e^2/a_0$ (the printed coefficient $2.33$ was corrected to $2.43$ in the notes added in proof) still lacks a definitive treatment that joins the relativistic TF and hydrogenic domains smoothly and fixes all coefficients. The task is to complete both pieces of the expansion - the full oscillatory and higher-order structure at relative order $Z^{-1}$ and beyond, and the definitive relativistic supplement.

Background

An atom with nuclear charge $Z$ and $Z$ electrons has, in nonrelativistic quantum mechanics, a ground-state energy $E(Z)$ whose large-$Z$ asymptotics are organized as an expansion in powers of $Z^{-1/3}$. The leading term is given by the Thomas-Fermi (TF) model, the simplest density-functional description of an atom: a continuous electron density $n(\mathbf{r})$ minimizes an energy functional that combines a local Fermi-gas approximation to the electrons' quantum kinetic energy with the classical electrostatic electron-electron repulsion and nuclear attraction. Lieb and Simon (1977) proved that this approximation becomes exact in the limit of large nuclear charge: the binding energy $-E$ (positive; $a_0$ denotes the Bohr radius and $e^2/a_0$ the Hartree energy unit) satisfies $-E\sim 0.768745\,Z^{7/3}\,e^2/a_0$.

The first correction, of relative order $Z^{-1/3}$, arises because TF treats even the innermost electrons semiclassically, whereas the electrons near the nucleus (the 'strongly bound' electrons) occupy hydrogenic orbitals of the unscreened nucleus with discrete, widely spaced binding energies $\varepsilon\sim-Z^2/(2n^2)$ in the principal quantum number $n$. Scott (1952) estimated the resulting correction to the binding energy as $\tfrac{1}{2}Z^2\,e^2/a_0$ - the Scott correction. Schwinger (1980) rederived it by a 'grafting' construction: electrons are split at a cutoff principal quantum number $\bar{n}'$ with $1\ll\bar{n}'\ll Z^{1/3}$, those with $n\ge\bar{n}'$ treated by TF and those with $n<\bar{n}'$ hydrogenically, with the total independent of $\bar{n}'$. In this construction the number of strongly bound electrons as a function of the separating energy is a staircase (integer-valued step) function, and Schwinger acknowledged that the choice of continuous interpolant through it is ambiguous: 'The proper procedure cannot be clear in the absence of an improved treatment that permits the modification to be performed smoothly, at the $Z^{-1}$ level of accuracy.'

The second correction, of relative order $Z^{-2/3}$, combines the Dirac exchange energy (the leading correction to the electrostatic electron-electron interaction arising from the Pauli exclusion principle) with the next quantum correction to the kinetic energy. The 1980 paper adopted the coefficient $0.266$ from March's review (Ref. 3 there); Schwinger's 1981 sequel derived it within the same grafting framework as $0.269900$ multiplying $Z^{5/3}$, and Fefferman and Seco (1994) proved this Dirac-Schwinger term rigorously. Thus $-E=(0.768745\,Z^{7/3}-\tfrac{1}{2}Z^2+0.269900\,Z^{5/3}+\cdots)e^2/a_0$, and the fourth term, of relative order $Z^{-1}$, is the first unsettled one. The Englert-Schwinger trilogy (1985) supplied the missing smooth treatment at physics level: applying Poisson summation (the identity converting a sum over equally spaced quantum numbers into a sum over Fourier components) to the discrete quantum-number sums, they showed that the $Z^{-1}$-level contributions are oscillatory functions of $Z^{1/3}$ - atomic shell structure - of order $Z^{4/3}$, estimated the leading oscillatory term, approximately $-0.4805\,Z^{4/3}\,e^2/a_0$ times the sine series $\sum_{k\ge1}(-1)^k\sin(2\pi k\lambda_0)/(\pi k)^3$ in $\lambda_0=0.928\,Z^{1/3}$ (a factor that is a cubic function of the distance of $\lambda_0$ from the nearest integer and vanishes when $\lambda_0$ is an integer or half-integer), and verified agreement with Hartree-Fock data (Hartree-Fock being the standard self-consistent-field approximation). But per Englert's 2019 retrospective, 'As of today (2019), there is no mathematical proof of its correctness', and the rigorous work of Córdoba, Fefferman and Seco on the relevant Weyl sums (exponential sums of the form $\sum e^{i\,f(n)}$ over quantum numbers) has so far yielded two-sided bounds rather than the asymptotics.

Independently, relativity modifies every term of the expansion. For physical atoms the fine-structure constant $\alpha$ is fixed, so relativistic kinematics enters through powers of $Z\alpha$, and the leading relativistic correction is of order $Z^2(Z\alpha)^2$ - a Scott-level term with a $Z$-dependent coefficient. Schwinger (1980) computed it from a TF treatment of the relativistic electron density, obtaining $\Delta E=-0.854\,(Z^2e^2/a_0)(Z\alpha)^2(1-2.43\,Z^{-1/3})$ (the printed coefficient $2.33$ was replaced by $2.43$ in the notes added in proof; the leading constant equals $5\pi^2/24-\zeta(3)\approx 0.854$ with $\zeta$ the Riemann zeta function, as identified in the later rigorous literature), but flagged that the treatment was not definitive: the strongly-bound electrons were handled nonrelativistically, spin-orbit coupling was only qualitatively dismissed, the higher-order relativistic terms were omitted, and Schwinger's estimates of the relativistic correction showed discrepancies with numerical evidence that he left unresolved. Dmitrieva and Plindov (1982) subsequently took more corrections into account within Schwinger's method and reported total binding energies within 1% of accurate data for $10\le N\le Z\le 100$ at physics level - an accuracy that, as Solovej, Sørensen and Spitzer (2010) note, goes beyond a rigorous mathematical treatment. Density-based relativistic treatments followed (Hill, Grout and March, 1984 and 1987, including the Darwin term (the leading relativistic correction of order $(Z\alpha)^2$ to the hydrogenic levels) and the Breit interaction (the leading magnetic electron-electron correction)), but Englert and Schwinger (1985) judged the existing attempts at the relativistic supplement 'none of which has been really satisfactory so far'. Rigorous progress covers idealized relativistic models: the Chandrasekhar (pseudo-relativistic) operator, and the no-pair Brown-Ravenhall and Furry operators in which the negative-energy Dirac sea states are projected out; in all of them the nuclear coupling $Z\alpha=\gamma$ is held fixed as $Z\to\infty$. The Scott-type correction was proved for the Chandrasekhar model by Solovej, Sørensen and Spitzer (2010) and, by a different technique, Frank, Siedentop and Warzel (2008); for Brown-Ravenhall by Frank, Siedentop and Warzel (2009); and for the Furry model by Handrek and Siedentop (2015), the last recovering Schwinger's coefficient as the weak-coupling limit of the relativistic Scott function and comparing with reference energies from the NIST database (NIST, the U.S. National Institute of Standards and Technology) and from Dirac-Fock calculations (Dirac-Fock being the relativistic self-consistent-field counterpart of Hartree-Fock). These theorems do not supply the definitive supplement for physical atoms at fixed $\alpha$, nor the higher-order spin-dependent terms; the Frank-Merz-Siedentop review (2023) still lists the corresponding questions among its open problems.

Problem Statement

Complete the asymptotic expansion of the total binding energy of neutral atoms beyond the three established terms,

$$-E=\left(0.768745\,Z^{7/3}-\tfrac{1}{2}Z^2+0.269900\,Z^{5/3}\right)\frac{e^2}{a_0}+R(Z),$$

by settling, along the two lines that Schwinger (1980) explicitly left open, the remainder $R(Z)$ of the nonrelativistic expansion displayed above (part a), and the relativistic supplement for physical atoms at fixed fine-structure constant $\alpha$ (part b), whose relativistic and spin-dependent contributions are additional terms of the physical binding energy rather than terms of $R(Z)$. The two parts are the two channels through which the same overarching objective - a complete, unambiguous large-$Z$ expansion of the neutral-atom binding energy with fixed numerical coefficients - remains incomplete.

(a) The oscillatory discrete-sum terms (relative order $Z^{-1}$). Determine the complete structure of the contributions generated by the discreteness of the one-electron quantum numbers (the staircase structure of the strongly-bound electron count): the leading oscillatory term of order $Z^{4/3}$ conjectured by Englert-Schwinger (1985) and Fefferman-Seco (1994), including its dependence on the fractional part of $\lambda_0=0.928\,Z^{1/3}$, together with the subsequent oscillatory and non-oscillatory terms at orders $Z$, $Z^{2/3}$, and so on; and establish (or refute) the mathematical correctness of the conjectured fourth term, going beyond the currently available two-sided bounds.

(b) The relativistic supplement for physical atoms (fixed fine-structure constant $\alpha$). Compute all relativistic and spin-dependent contributions to the binding energy beyond the leading term $-0.854\,Z^2(Z\alpha)^2(1-2.43\,Z^{-1/3})\,e^2/a_0$ of Schwinger (1980) (printed as $2.33$ and corrected to $2.43$ in the notes added in proof) - the higher powers of $Z\alpha$ and the relativistic corrections to the lower-order terms, including spin-orbit, Darwin, and Breit (magnetic) interactions - in a treatment that joins the relativistic TF and hydrogenic domains smoothly at relative order $Z^{-1}$ (no cutoff or interpolation ambiguity) and fixes all numerical coefficients.

Rigorous theorems for idealized models at fixed $\gamma=Z\alpha$ (Chandrasekhar, Brown-Ravenhall, or Furry operators) concern the fixed-$\gamma$ idealization, whereas part (b) requires the physical scaling at fixed fine-structure constant $\alpha$.

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