The spin of the ground state of a large atom: sharp bounds and extensions to molecules

For a large atom with nuclear charge $Z$ described by the non-relativistic Coulomb Hamiltonian, Fefferman and Seco proved that the total spin $|\langle S\Psi,\Psi\rangle|$ of a ground state is bounded by $CZ^{\gamma}$ with some $\gamma<1$ and a constant $C$ independent of $Z$, for neutral atoms. The physically expected scale, from Hund's rule, is only $Z^{1/3}$, recorded as a conjecture since their paper. The optimal exponent — the true order of magnitude of the ground-state spin — remains open, as does the molecular case, which belongs to the original question posed by M. and T. Hoffmann-Ostenhof for an atom or molecule.

Background

The setting is the exact non-relativistic quantum mechanics of atoms. An atom with nuclear charge $Z$ and $N$ electrons is described by the Coulomb Hamiltonian $H_{Z,N} = \sum_{i=1}^{N}\left(-\tfrac{1}{2}\Delta_i - \tfrac{Z}{|x_i|}\right) + \sum_{i<j}\tfrac{1}{|x_i-x_j|}$ acting on antisymmetric wave functions $\Psi(x_1,\sigma_1;\dots;x_N,\sigma_N)$, where each electron carries spin $\pm 1/2$ (antisymmetry encodes the Pauli exclusion principle). The total spin operator is $S = \sum_{i=1}^{N} s_i$, the sum of the individual electron spins, and $\langle S\Psi,\Psi\rangle$ measures the net spin of a state $\Psi$. A ground state is a minimizer of the energy. Physically, Hund's rule predicts that ground states prefer the maximal spin compatible with the Pauli principle within a shell; for a large atom the outermost shell has orbital quantum number $l \approx n \approx Z^{1/3}$, so its degeneracy $2l+1$ is of order $Z^{1/3}$ and the expected maximal ground-state spin is of order $Z^{1/3}$. Prior to their work the question — posed by M. and T. Hoffmann-Ostenhof — of whether the ground-state spin of a large atom is even $o(Z)$, i.e. sublinear in $Z$, was open. Fefferman and Seco (1996) proved, for neutral atoms, that $|\langle S\Psi,\Psi\rangle| \leq CZ^{\gamma}$ with $\gamma < 1$ and a constant $C$ independent of $Z$. The proof combines Thomas–Fermi theory (the $Z\to\infty$ limit of atomic structure, in which the leading energy is of order $Z^{7/3}$), Lieb's inequality (a lower bound on the many-electron energy by a sum of one-electron energies in a fixed effective potential, plus explicit remainder terms), and semiclassical spectral asymptotics with an explicit error term (a bound on the sum of negative eigenvalues of a one-particle Schrödinger operator), applied through a spin cutoff near the nucleus and a convexity/second-derivative argument for a perturbed Hamiltonian. Related rigorous progress on Hund's rule exists for itinerant (spatially delocalized, hopping rather than nucleus-bound) electrons in two-band lattice (Hubbard-type) models — Fröhlich and Ueltschi showed Hund's rule and metallic ferromagnetism for such systems when the on-site exchange anisotropy is large enough — but not for the three-dimensional Coulomb atom or molecule.

Problem Statement

Let $H_{Z,N}$ be the non-relativistic Coulomb Hamiltonian of an atom with nuclear charge $Z$ and $N$ electrons, and let $S = \sum_{i=1}^{N} s_i$ be the total spin operator, the sum of the individual electron spins (each $\pm 1/2$). For a ground state $\Psi$ of $H_{Z,N}$, determine the true order of magnitude of the total spin $|\langle S\Psi,\Psi\rangle|$ as $Z \to \infty$.

Fefferman and Seco proved that, for neutral atoms, every ground state satisfies $|\langle S\Psi,\Psi\rangle| \leq C Z^{\gamma}$ with some $\gamma < 1$ and a constant $C$ independent of $Z$. The open problem is the optimal exponent $\gamma$ — indeed the true order of magnitude — and in particular whether the conjectured Hund's-rule scale is attained. The two sides carry different quantifiers, reflecting the spin degeneracy of ground-state eigenspaces: the upper bound is asserted for every ground state $\Psi$, while the conjecture, as recorded since the original paper, is existential — that for large $Z$ there exist atoms whose ground states have total spin of order $Z^{1/3}$ (the outermost shell has $l \approx n \approx Z^{1/3}$, so its degeneracy $2l+1$ is of order $Z^{1/3}$). The original question was posed for an atom or molecule, and the molecular case is part of the target. For every molecule with a fixed number of nuclei at fixed positions and nuclear charges scaled together by a common factor $Z$ (charges $a_j Z$ with $a_j > 0$ fixed, and electron number $N$ equal to the total nuclear charge, so that the molecule is neutral), the same determination is required in the limit $Z \to \infty$: the true order of magnitude of the ground-state total spin, with the sublinear bound $|\langle S\Psi,\Psi\rangle| \le C Z^{\gamma}$ ($\gamma < 1$, $C$ independent of $Z$) asserted for every ground state and the attainability of the $Z^{1/3}$ scale asked existentially.

References

1. Charles L. Fefferman and Luis A. Seco, The spin of the ground state of an atom, Revista Matemática Iberoamericana 12 (1996), no. 1, 19–36. DOI: 10.4171/rmi/192. URL: https://doi.org/10.4171/rmi/192. (Source paper: it proves the theorem that the total spin is $\le CZ^{\gamma}$ with $\gamma < 1$ for a neutral atom with a $Z$-independent constant, and records the conjecture that the order of magnitude of the spin can be as large as $Z^{1/3}$.)
2. Charles L. Fefferman and Luis A. Seco, On the Dirac and Schwinger corrections to the ground-state energy of an atom, Advances in Mathematics 107 (1994), no. 1, 1–185. DOI: 10.1006/aima.1994.1060. URL: https://doi.org/10.1006/aima.1994.1060. (Part of the semiclassical-asymptotics machinery underlying the spin bound.)
3. Charles L. Fefferman and Luis A. Seco, The eigenvalue sum for a three-dimensional radial potential, Advances in Mathematics 119 (1996), no. 1, 26–116. DOI: 10.1006/aima.1996.0027. URL: https://doi.org/10.1006/aima.1996.0027. (Refined spectral asymptotics of the type underlying the semiclassical machinery.)
4. Jürg Fröhlich and Daniel Ueltschi, Hund's rule and metallic ferromagnetism, Journal of Statistical Physics 118 (2005), no. 5-6, 973–978. DOI: 10.1007/s10955-004-2174-y. URL: https://doi.org/10.1007/s10955-004-2174-y. Preprint: arXiv:cond-mat/0404483, https://arxiv.org/abs/cond-mat/0404483. (Neighboring result: rigorous Hund's rule for itinerant lattice electrons, not the 3D Coulomb atom.)
5. Ioannis Anapolitanos and Israel Michael Sigal, Long-Range Behavior of the van der Waals Force, Communications on Pure and Applied Mathematics 70 (2017), no. 9, 1633–1671. DOI: 10.1002/cpa.21695. URL: https://doi.org/10.1002/cpa.21695. (It uses large-atom asymptotics.)
6. Ioannis Anapolitanos and Israel Michael Sigal, On long range behaviour of van der Waals force, in: Mathematical Results in Quantum Mechanics (Proceedings of the QMath12 conference, Berlin, September 10–13, 2013), World Scientific, pp. 47–75. DOI: 10.1142/9789814618144_0003. URL: https://doi.org/10.1142/9789814618144_0003.
7. Charles L. Fefferman and Luis A. Seco, On the energy of a large atom, Bulletin of the American Mathematical Society, New Series 23 (1990), no. 2, 525–530. DOI: 10.1090/S0273-0979-1990-15969-5. URL: https://doi.org/10.1090/S0273-0979-1990-15969-5. (Expository account of the large-atom energy program in which the spin question sits.)