Obstructions to four-body perimetric coordinates
Abstract
Pekeris’ perimetric coordinates make the three-body Coulomb problem tractable. They are linear in the interparticle distances, range independently over [0,∞), and turn the Schrödinger equation into a sparse recursion. Whether a four-body analogue exists has been debated since 1966, with claims in both directions and no agreed definition. We formulate a hierarchy of definitions and settle each level of it. (i) No linear coordinates exist, because the configuration domain is not convex. (ii) No coordinates smooth up to the boundary, in the sense of a diffeomorphism of manifolds with corners, exist even locally, because the domain has a round tangent cone at collinear configurations. (iii) Independent coordinates on the closed orthant whose squared distances are polynomial do exist. They cover the whole domain and are bijective on its interior. Degree 4 is attained. (iv) No independent coordinates exist in which the six distances themselves are rational functions, for instance polynomials. The proof of (iv) shows that the variety of coplanar configurations, written in the distances, is birational to a quotient (X ×X)/G of the square of a Kummer K3 surface. It therefore carries a nonzero holomorphic 4-form and cannot be parametrized rationally. Finally, we identify an exact Galois symmetry in a broad class of Pekeris-type sparse schemes, the coefficientwise graded schemes. If one distance enters only through its square, such a scheme computes identical spectra for repulsion and for attraction in that pair, so it cannot converge for both. If it is consistent for the attractive problem, it has spurious eigenvalues below the true ground state of the repulsive one. A helium calculation exhibits this behaviour. Within this class, uniform convergence for four-body Coulomb systems therefore requires rational distances, which (iv) excludes. Every computational step is accompanied by a runnable check.
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BibTeX
@misc{baskerville2026obstructions,
title = {{Obstructions to four-body perimetric coordinates}},
author = {Adam L. Baskerville},
howpublished = {ChemRxiv},
year = {2026},
month = oct,
doi = {10.26434/chemrxiv.15010676/v1},
url = {https://chemrxiv.org/doi/full/10.26434/chemrxiv.15010676/v1},
note = {CC BY},
}
RIS
TY - UNPB
TI - Obstructions to four-body perimetric coordinates
AU - Adam L. Baskerville
PY - 2026
DA - 2026/10/19
PB - ChemRxiv
DO - 10.26434/chemrxiv.15010676/v1
UR - https://chemrxiv.org/doi/full/10.26434/chemrxiv.15010676/v1
AB - Pekeris’ perimetric coordinates make the three-body Coulomb problem tractable. They are linear in the interparticle distances, range independently over [0,∞), and turn the Schrödinger equation into a sparse recursion. Whether a four-body analogue exists has been debated since 1966, with claims in both directions and no agreed definition. We formulate a hierarchy of definitions and settle each level of it. (i) No linear coordinates exist, because the configuration domain is not convex. (ii) No coordinates smooth up to the boundary, in the sense of a diffeomorphism of manifolds with corners, exist even locally, because the domain has a round tangent cone at collinear configurations. (iii) Independent coordinates on the closed orthant whose squared distances are polynomial do exist. They cover the whole domain and are bijective on its interior. Degree 4 is attained. (iv) No independent coordinates exist in which the six distances themselves are rational functions, for instance polynomials. The proof of (iv) shows that the variety of coplanar configurations, written in the distances, is birational to a quotient (X ×X)/G of the square of a Kummer K3 surface. It therefore carries a nonzero holomorphic 4-form and cannot be parametrized rationally. Finally, we identify an exact Galois symmetry in a broad class of Pekeris-type sparse schemes, the coefficientwise graded schemes. If one distance enters only through its square, such a scheme computes identical spectra for repulsion and for attraction in that pair, so it cannot converge for both. If it is consistent for the attractive problem, it has spurious eigenvalues below the true ground state of the repulsive one. A helium calculation exhibits this behaviour. Within this class, uniform convergence for four-body Coulomb systems therefore requires rational distances, which (iv) excludes. Every computational step is accompanied by a runnable check.
ER -