Algebraic coefficients of fractional parentage for three particles in the oscillator Jacobi basis
Abstract
We present a closed-form construction of the orbital coefficients of fractional parentage (CFP) of three identical particles in the translationally invariant harmonic oscillator (HO) Jacobi basis. These are usually obtained by numerically diagonalizing a permutation operator in every energy-angular-momentum block. We show this diagonalization is unnecessary. Dressing the class operator P13 + P23 with the single unitary W = H1 Φ -- a fixed complexification followed by the HO bracket with mass parameter d=1 -- turns it into the label swap times an integer. Every CFP is then the single ket |Ki,s⟩ = 2Qs W†|i⟩ -- a 2-scaled, sign-decorated column of the d=1 bracket restricted to one e2-parity of the labels -- whose norm, 1 + s⟨ i | J | i ⟩, is itself a single matrix element. The same W carries label parity into swap symmetry, W Qs W† = Rs, making the bookkeeping collapse. The eigenvalues are exact integers ± 2 and ± 1 labeling the S(3) symmetry classes. The construction is uniform in E, L and symmetry class, reducing CFP-specific work from O(n3) to O(n2) in block dimension. An eigensolver remains only inside the bracket engine supplying H1, where all residual numerical error originates. We implement this on an SU(3)-basis bracket engine, checking all 1301 blocks up to E=50 (912 366 CFPs, largest block 2907 × 2907): the S(3) class assignment is exact everywhere. The residual numerical error (entirely the bracket engine's) stays below 10−8 for E ≤ 35, restoring to ≤ 10−22 via a quadruple-precision build beyond that. We further validate against independently generated antisymmetrization data of a production three-nucleon code, including the 3He ground-state energy.
Versions
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BibTeX
@misc{stepsys2026algebraic,
title = {{Algebraic coefficients of fractional parentage for three particles in the oscillator Jacobi basis}},
author = {A. Stepšys and S. Mickevičius and D. Germanas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21385},
eprint = {2610.21385},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21385},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Algebraic coefficients of fractional parentage for three particles in the oscillator Jacobi basis
AU - A. Stepšys
AU - S. Mickevičius
AU - D. Germanas
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21385
UR - https://arxiv.org/abs/2610.21385
AB - We present a closed-form construction of the orbital coefficients of fractional parentage (CFP) of three identical particles in the translationally invariant harmonic oscillator (HO) Jacobi basis. These are usually obtained by numerically diagonalizing a permutation operator in every energy-angular-momentum block. We show this diagonalization is unnecessary. Dressing the class operator P_13 + P_23 with the single unitary W = H^1 Φ -- a fixed complexification followed by the HO bracket with mass parameter d=1 -- turns it into the label swap times an integer. Every CFP is then the single ket |K_i,s⟩ = 2Q^s W^†|i⟩ -- a 2-scaled, sign-decorated column of the d=1 bracket restricted to one e_2-parity of the labels -- whose norm, 1 + s⟨ i | J | i ⟩, is itself a single matrix element. The same W carries label parity into swap symmetry, W Q^s W^† = R^s, making the bookkeeping collapse. The eigenvalues are exact integers ± 2 and ± 1 labeling the S(3) symmetry classes. The construction is uniform in E, L and symmetry class, reducing CFP-specific work from O(n^3) to O(n^2) in block dimension. An eigensolver remains only inside the bracket engine supplying H^1, where all residual numerical error originates. We implement this on an SU(3)-basis bracket engine, checking all 1301 blocks up to E=50 (912 366 CFPs, largest block 2907 × 2907): the S(3) class assignment is exact everywhere. The residual numerical error (entirely the bracket engine's) stays below 10^−8 for E ≤ 35, restoring to ≤ 10^−22 via a quadruple-precision build beyond that. We further validate against independently generated antisymmetrization data of a production three-nucleon code, including the ^3He ground-state energy.
ER -