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arxiv: 2605.06634 · v1 · submitted 2026-05-07 · ⚛️ physics.comp-ph

Recognition: unknown

libwignernj: a reusable C/C++/Fortran/Python library for exact Wigner symbols and related coefficients

Susi Lehtola
Authors on Pith no claims yet

Pith reviewed 2026-05-08 03:18 UTC · model grok-4.3

classification ⚛️ physics.comp-ph
keywords Wigner symbolsangular momentum coefficientsprime factorizationRacah sumClebsch-Gordan coefficientsexact arithmeticcomputational physicsGaunt coefficients
0
0 comments X

The pith

A library computes Wigner 3j, 6j, and 9j symbols with all intermediates kept as exact rationals.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces libwignernj, a standards-compliant C99 library that evaluates Wigner 3j, 6j, 9j symbols along with Clebsch-Gordan, Racah W, Fano X, and Gaunt coefficients for both real and complex spherical harmonics. It encodes factorials as vectors of signed prime exponents and performs the Racah sum using multiword integers. Every intermediate quantity therefore remains an exact rational, with rounding restricted to the final conversion into floating-point numbers. The implementation delivers single-, double-, and long-double-precision results that match the last representable bit and provides optional paths to higher precision. Bindings for C++, Fortran 90, and CPython together with package files allow drop-in use across atomic, molecular, nuclear, and scattering codes without runtime dependencies or initialization steps.

Core claim

Representing factorials by their signed prime-exponent decomposition and combining that representation with the multiword-integer Racah sum keeps every intermediate quantity an exact rational. All rounding is confined to the final floating-point conversion, so that single-, double-, and long-double-precision results are correct to the last representable bit.

What carries the argument

The prime-factorization representation of factorials combined with the multiword-integer Racah sum, under which intermediates stay exact rationals.

If this is right

  • Atomic and molecular structure codes can obtain Wigner symbols and Gaunt coefficients without accumulated rounding error inside the angular-momentum algebra.
  • Half-integer angular momenta are handled exactly through integer 2j arguments, removing the need for separate half-integer logic in calling programs.
  • Optional exposure of binary128 and MPFR evaluation paths lets the same code path scale from double precision to arbitrary precision without changing the calling interface.
  • CMake and pkg-config integration files together with the absence of mandatory dependencies allow the library to be embedded directly into existing C, C++, Fortran, and Python projects.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same prime-exponent and multiword-integer technique could be applied to other families of combinatorial coefficients that involve large factorials.
  • Embedding the library inside symbolic-algebra systems would allow exact symbolic manipulation of angular-momentum expressions before any numerical evaluation.
  • The approach demonstrates that exact rational arithmetic can be made practical inside performance-critical scientific libraries when the final output is floating point.

Load-bearing premise

The prime-factorization vectors and multiword integers can represent the necessary factorials and sums without overflow, excessive memory use, or loss of exactness for all angular momenta that arise in practical atomic, molecular, nuclear, and scattering calculations.

What would settle it

A computed Wigner symbol for any concrete set of angular momenta whose floating-point value differs from the result obtained by arbitrary-precision arithmetic on the same exact rational expression.

Figures

Figures reproduced from arXiv: 2605.06634 by Susi Lehtola.

Figure 1
Figure 1. Figure 1: Relative error of libwignernj, WIGXJPF 1.13, and GSL 2.8 against an mpmath quadruple-precision reference for the four benchmark inputs at j = 1 . . . 200. The dashed black line marks the double-precision unit roundoff ϵM. GSL points are shown only for j at which GSL returned a finite value without raising its internal error handler. Values where GSL trapped or returned a non-finite result are omitted from … view at source ↗
Figure 2
Figure 2. Figure 2: Per-call wall time of libwignernj (in-tree multiword bigint kernel, solid. With the optional FLINT back-end, dash-dotted), WIGXJPF 1.13 (dashed), and GSL 2.8 (dotted) on the four benchmark inputs at j = 1 . . . 200. The two 3j panels share the y-axis. The 6j and 9j panels are scaled independently. GSL timing samples are dropped where GSL did not return a meaningful value (same mask as fig. 1). Timing view at source ↗
read the original abstract

We describe libwignernj, a freely available, BSD-licensed library that evaluates Wigner 3j, 6j, and 9j symbols, Clebsch--Gordan, Racah $W$, and Fano $X$ coefficients, and Gaunt coefficients over both complex and real spherical harmonics in standards-compliant C99. libwignernj represents factorials by the vector of their signed prime-exponent decomposition - a prime-factorization technique introduced for the angular-momentum coefficients by Dodds and Wiechers (Comput. Phys. Commun. 4, 268 (1972)) and refined in a long line of subsequent work - and combines that representation with the multiword-integer Racah sum of Johansson and Forss\'en (SIAM J. Sci. Comput. 38, A376 (2016)), under which every intermediate quantity is an exact rational and all rounding is confined to the final floating-point conversion. Single-, double-, and long-double-precision results are correct to the last representable bit, and IEEE 754 binary128 evaluation through libquadmath and arbitrary-precision evaluation through the GNU Multiple-Precision Floating-Point Reliable (MPFR) library are optionally exposed. libwignernj has no mandatory runtime dependencies and no caller-side initialization step, making it easy to embed across the atomic, molecular, nuclear, and electromagnetic-scattering applications in which these coefficients arise. C++, CPython, and Fortran 90 bindings ship alongside the C library. Half-integer angular momenta are encoded exactly via integer $2j$ arguments throughout the application programming interface (API). CMake-package and pkg-config files ship for drop-in integration into downstream projects, and a continuous-integration (CI) pipeline runs the full test suite on Linux (shared and static), macOS, and Windows on every push.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript describes libwignernj, a BSD-licensed C99 library (with C++, Fortran 90, and CPython bindings) for computing Wigner 3j, 6j, 9j symbols, Clebsch-Gordan, Racah W, Fano X, and Gaunt coefficients. It uses prime-factorization representations of factorials (following Dodds-Wiechers) combined with multiword-integer Racah sums (Johansson-Forssén) so that every intermediate is an exact rational and all rounding is confined to the final floating-point conversion. The library claims bit-exact results in single, double, and long-double precision, supports optional quad and MPFR arbitrary precision, encodes half-integers via 2j integers, has no mandatory dependencies, and ships with CMake/pkg-config integration plus a CI test suite.

Significance. If the implementation realizes the exact-rational guarantee, the library supplies a reusable, dependency-free tool for high-accuracy angular-momentum coefficients that are ubiquitous in atomic, molecular, nuclear, and scattering computations. The packaging of established exact-arithmetic techniques into a standards-compliant, multi-language library with straightforward embedding features represents a practical contribution that can reduce floating-point artifacts in downstream applications.

major comments (1)
  1. Abstract: the central claim that single-, double-, and long-double-precision results are 'correct to the last representable bit' is load-bearing, yet the manuscript supplies no tabulated numerical comparisons, error metrics, or explicit test cases against reference values or other implementations for any range of angular momenta. While the CI suite is mentioned, concrete verification data inside the paper is required to substantiate the exact-arithmetic assertion.
minor comments (2)
  1. The optional IEEE 754 binary128 and MPFR interfaces are described only at a high level; a short paragraph or table clarifying the corresponding API calls and build flags would improve usability.
  2. The bibliography entries for Dodds-Wiechers (1972) and Johansson-Forssén (2016) should be checked for complete and consistent formatting.

Simulated Author's Rebuttal

1 responses · 0 unresolved

Thank you for the positive assessment of the manuscript and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: Abstract: the central claim that single-, double-, and long-double-precision results are 'correct to the last representable bit' is load-bearing, yet the manuscript supplies no tabulated numerical comparisons, error metrics, or explicit test cases against reference values or other implementations for any range of angular momenta. While the CI suite is mentioned, concrete verification data inside the paper is required to substantiate the exact-arithmetic assertion.

    Authors: We agree that the manuscript would be strengthened by the inclusion of explicit numerical verification data. Although the library's CI pipeline executes an extensive test suite that checks bit-exact agreement across precisions, these results are not presented as tabulated comparisons or error metrics within the current manuscript text. In the revised version we will add a dedicated verification section (or appendix) containing tabulated comparisons for a representative range of angular momenta, relative-error metrics against reference values obtained from arbitrary-precision arithmetic, and direct confirmation that single-, double-, and long-double results match to the last representable bit. This addition will be concise and will directly substantiate the central claim without changing the paper's scope or conclusions. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript describes a software library that implements standard Wigner symbols and related coefficients using the prime-factorization representation of factorials (cited to Dodds & Wiechers 1972) combined with the multiword-integer Racah sum (cited to Johansson & Forssén 2016). Every load-bearing algorithmic step is taken from these external references; the paper supplies the C99 realization, bindings, and test suite but does not derive any new mathematical identity, fit parameters to data, or invoke a uniqueness theorem from the authors' own prior work. The exact-rational guarantee follows directly from the cited techniques once the multiword arithmetic is realized without overflow inside the documented range; no step reduces to a self-definition or to a fitted input renamed as a prediction. The result is therefore self-contained against the cited external literature.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The library rests on the standard algebraic definitions of Wigner symbols and the Racah sum; it introduces no new free parameters, axioms beyond conventional mathematics, or postulated entities.

axioms (1)
  • standard math Standard definitions and algebraic properties of Wigner 3j, 6j, 9j symbols, Clebsch-Gordan coefficients, Racah W, Fano X, and Gaunt coefficients hold as given in angular-momentum theory.
    The library computes these established quantities using the cited exact-arithmetic algorithms.

pith-pipeline@v0.9.0 · 5643 in / 1426 out tokens · 152078 ms · 2026-05-08T03:18:49.782054+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

101 extracted references · 80 canonical work pages

  1. [1]

    Fousse, G

    L. Fousse, G. Hanrot, V . Lefèvre, P. Pélissier, P. Zimmermann, MPFR: A multiple-precision binary floating- point library with correct rounding, ACM Trans. Math. Softw. 33 (2007) 13. doi:10.1145/1236463.1236468

    work page doi:10.1145/1236463.1236468 2007

  2. [2]

    W. B. Hart, Fast Library for Number Theory: An Introduction, Springer Berlin Heidelberg, 2010, pp. 88–91. doi:10.1007/978-3-642-15582-6_18

    work page doi:10.1007/978-3-642-15582-6_18 2010

  3. [3]

    H. T. Johansson, C. Forssén, Fast and accurate evaluation of Wigner 3j, 6j, and 9jsymbols using prime factorization and multiword integer arithmetic, SIAM J. Sci. Comput. 38 (2016) A376–A384. doi:10.1137/15m1021908

    work page doi:10.1137/15m1021908 2016

  4. [4]

    E. P. Wigner, On the Matrices Which Reduce the Kronecker Products of Representations of S. R. Groups, Springer Berlin Heidelberg, 1993, pp. 608–654. doi:10.1007/978-3-662-02781-3_42

    work page doi:10.1007/978-3-662-02781-3_42 1993

  5. [5]

    Racah, Theory of Complex Spectra

    G. Racah, Theory of Complex Spectra. I, Phys. Rev. 61 (1942) 186–197. doi:10.1103/PhysRev.61.186

    work page doi:10.1103/physrev.61.186 1942

  6. [6]

    Racah, Theory of Complex Spectra

    G. Racah, Theory of Complex Spectra. II, Phys. Rev. 62 (1942) 438–462. doi:10.1103/PhysRev.62.438

    work page doi:10.1103/physrev.62.438 1942

  7. [7]

    Racah, Theory of Complex Spectra

    G. Racah, Theory of Complex Spectra. III, Phys. Rev. 63 (1943) 367–382. doi:10.1103/PhysRev.63.367

    work page doi:10.1103/physrev.63.367 1943

  8. [8]

    Racah, Theory of Complex Spectra

    G. Racah, Theory of Complex Spectra. IV, Phys. Rev. 76 (1949) 1352–1365. doi:10.1103/PhysRev.76.1352

    work page doi:10.1103/physrev.76.1352 1949

  9. [9]

    H. A. Jahn, J. Hope, Symmetry properties of the Wigner 9jsymbol, Phys. Rev. 93 (1954) 318–321. doi:10.1103/PhysRev.93.318

    work page doi:10.1103/physrev.93.318 1954

  10. [10]

    J. A. Gaunt, The Triplets of Helium, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 228 (1929) 151–196. doi:10.1098/rsta.1929.0004

    work page doi:10.1098/rsta.1929.0004 1929

  11. [11]

    W. R. Johnson, Atomic Structure Theory, Springer Berlin Heidelberg, 2007. doi:10.1007/978-3-540-68013-0

    work page doi:10.1007/978-3-540-68013-0 2007

  12. [12]

    Itô-Wentzell-Lions Formula for Mea- sure Dependent Random Fields under Full and Conditional Measure Flows

    H. Friedrich, Theoretical Atomic Physics, Springer International Publishing, 2017. doi:10.1007/978-3-319- 47769-5. 25

    work page doi:10.1007/978-3-319- 2017

  13. [13]

    The shell model as a unified view of nuclear structure

    E. Caurier, G. Martínez-Pinedo, F. Nowacki, A. Poves, A. P. Zuker, The shell model as a unified view of nuclear structure, Rev. Mod. Phys. 77 (2005) 427–488. doi:10.1103/RevModPhys.77.427

    work page doi:10.1103/revmodphys.77.427 2005

  14. [14]

    Helgaker, P

    T. Helgaker, P. Jørgensen, J. Olsen, Molecular electronic-structure theory, John Wiley & Sons, Ltd., 2000

    2000

  15. [15]

    doi:10.1090/S0025- 5718-1970-0274029-X

    Y .-l. Xu, Fast evaluation of the Gaunt coefficients, Math. Comput. 65 (1996) 1601–1612. doi:10.1090/s0025- 5718-96-00774-0

    work page doi:10.1090/s0025- 1996

  16. [16]

    R. S. Caswell, L. C. Maximon, Fortran programs for the calculation of Wigner 3j, 6j, and 9j coefficients for angular momenta greater than or equal to 80, Technical Report, National Institute of Standards, 1966. doi:10.6028/NBS.TN.409

    work page doi:10.6028/nbs.tn.409 1966

  17. [17]

    Tamura, Angular momentum coupling coefficients, Comput

    T. Tamura, Angular momentum coupling coefficients, Comput. Phys. Comm. 1 (1970) 337–342. doi:10.1016/0010-4655(70)90034-2

    work page doi:10.1016/0010-4655(70)90034-2 1970

  18. [18]

    J. G. Wills, On the evaluation of angular momentum coupling coefficients, Comput. Phys. Commun. 2 (1971) 381–382. doi:10.1016/0010-4655(71)90030-0

    work page doi:10.1016/0010-4655(71)90030-0 1971

  19. [19]

    Bretz, An improved method for calculation of angular momentum coupling coefficients, Acta Phys

    V . Bretz, An improved method for calculation of angular momentum coupling coefficients, Acta Phys. Hung. 40 (1976) 255–259. doi:10.1007/bf03157502

    work page doi:10.1007/bf03157502 1976

  20. [20]

    Srinivasa Rao, K

    K. Srinivasa Rao, K. Venkatesh, New Fortran programs for angular momentum coefficients, Comput. Phys. Commun. 15 (1978) 227–235. doi:10.1016/0010-4655(78)90093-0

    work page doi:10.1016/0010-4655(78)90093-0 1978

  21. [21]

    Srinivasa Rao, Computation of angular momentum coefficients using sets of generalised hypergeometric functions, Comput

    K. Srinivasa Rao, Computation of angular momentum coefficients using sets of generalised hypergeometric functions, Comput. Phys. Commun. 22 (1981) 297–302. doi:10.1016/0010-4655(81)90063-1

    work page doi:10.1016/0010-4655(81)90063-1 1981

  22. [22]

    I. I. Guseinov, A. Özmen, Ü. Atav, H. Yüksel, Computation of Clebsch–Gordan and Gaunt coefficients using binomial coefficients, J. Comput. Phys. 122 (1995) 343–347. doi:10.1006/jcph.1995.1220

    work page doi:10.1006/jcph.1995.1220 1995

  23. [23]

    I. I. Guseinov, B. A. Mamedov, E. Çopuro ˘glu, Use of binomial coefficients in fast and accurate calculation of Clebsch–Gordan and Gaunt coefficients, and Wignern-jsymbols, J. Theor. Comput. Chem. 08 (2009) 251–259. doi:10.1142/s0219633609004782

    work page doi:10.1142/s0219633609004782 2009

  24. [24]

    Wei, New formula for 9-jsymbols and their direct calculation, Comput

    L. Wei, New formula for 9-jsymbols and their direct calculation, Comput. Phys. 12 (1998) 632–634. doi:10.1063/1.168745

    work page doi:10.1063/1.168745 1998

  25. [25]

    J. F. Shriner, W. J. Thompson, Angular-momentum coupling coefficients: New and improved algorithms, Comput. Phys. 7 (1993) 144–148. doi:10.1063/1.4823156

    work page doi:10.1063/1.4823156 1993

  26. [26]

    Schulten, R

    K. Schulten, R. G. Gordon, Exact recursive evaluation of 3j- and 6j-coefficients for quantum-mechanical coupling of angular momenta, J. Math. Phys. 16 (1975) 1961–1970. doi:10.1063/1.522426

    work page doi:10.1063/1.522426 1975

  27. [27]

    Schulten, R

    K. Schulten, R. G. Gordon, Recursive evaluation of 3jand 6jcoefficients, Comput. Phys. Comm. 11 (1976) 269–278. doi:10.1016/0010-4655(76)90058-8

    work page doi:10.1016/0010-4655(76)90058-8 1976

  28. [28]

    Schulten, R

    K. Schulten, R. G. Gordon, Recursive evaluation of 3jand 6jcoefficients, Comput. Phys. Commun. 35 (1984) C–377–C–379. doi:10.1016/s0010-4655(84)82592-8

    work page doi:10.1016/s0010-4655(84)82592-8 1984

  29. [29]

    J. H. Luscombe, M. Luban, Simplified recursive algorithm for Wigner 3jand 6jsymbols, Phys. Rev. E 57 (1998) 7274–7277. doi:10.1103/physreve.57.7274

    work page doi:10.1103/physreve.57.7274 1998

  30. [30]

    Xu, Fast evaluation of Gaunt coefficients: recursive approach, J

    Y .-l. Xu, Fast evaluation of Gaunt coefficients: recursive approach, J. Comput. Appl. Math. 85 (1997) 53–65. doi:10.1016/s0377-0427(97)00128-3

    work page doi:10.1016/s0377-0427(97)00128-3 1997

  31. [31]

    Xu, Improved recursive computation of Clebsch–Gordan coefficients, J

    G. Xu, Improved recursive computation of Clebsch–Gordan coefficients, J. Quant. Spectrosc. Radiat. Transfer 254 (2020) 107210. doi:10.1016/j.jqsrt.2020.107210. 26

    work page doi:10.1016/j.jqsrt.2020.107210 2020

  32. [32]

    P. G. Burke, A program to calculate a general recoupling coefficient, Comput. Phys. Commun. 1 (1970) 241–250. doi:10.1016/0010-4655(70)90040-8

    work page doi:10.1016/0010-4655(70)90040-8 1970

  33. [33]

    P. G. Burke, A program to calculate a general recoupling coefficient, Comput. Phys. Commun. 35 (1984) C–30. doi:10.1016/s0010-4655(84)82298-5

    work page doi:10.1016/s0010-4655(84)82298-5 1984

  34. [34]

    A. P. Yutsis, I. B. Levinson, V . V . Vanagas, Mathematical Apparatus of the Theory of Angular Momentum, Is- rael Program for Scientific Translations, Jerusalem, 1962. Translated from the Russian original (Mintis, Vilnius, 1960)

    1962

  35. [35]

    Massot, E

    J.-N. Massot, E. El-Baz, J. Lafoucrière, A general graphical method for angular momentum, Rev. Mod. Phys. 39 (1967) 288–305. doi:10.1103/RevModPhys.39.288

    work page doi:10.1103/revmodphys.39.288 1967

  36. [36]

    Feyn Calc: Computer-algebraic calculation of Feynman amplitudes,

    J. Shapiro, Arbitrary 3n-jsymbols for SU(2), Comput. Phys. Commun. 1 (1970) 207–215. doi:10.1016/0010- 4655(70)90007-x

    work page doi:10.1016/0010- 1970

  37. [37]

    Ingelman, A

    J. Shapiro, Arbitrary 3n-jsymbols for SU(2), Comput. Phys. Commun. 35 (1984) C–25. doi:10.1016/s0010- 4655(84)82292-4

    work page doi:10.1016/s0010- 1984

  38. [38]

    N. S. Scott, A. Hibbert, A more efficient version of the weights and NJSYM packages, Comput. Phys. Commun. 28 (1982) 189–200. doi:10.1016/0010-4655(82)90054-6

    work page doi:10.1016/0010-4655(82)90054-6 1982

  39. [39]

    Bar-Shalom, M

    A. Bar-Shalom, M. Klapisch, NJGRAF—an efficient program for calculation of general recoupling co- efficients by graphical analysis, compatible with NJSYM, Comput. Phys. Commun. 50 (1988) 375–393. doi:10.1016/0010-4655(88)90192-0

    work page doi:10.1016/0010-4655(88)90192-0 1988

  40. [40]

    V . Fack, S. N. Pitre, J. Van der Jeugt, Calculation of general recoupling coefficients using graphical methods, Comput. Phys. Commun. 101 (1997) 155–170. doi:10.1016/s0010-4655(96)00170-1

    work page doi:10.1016/s0010-4655(96)00170-1 1997

  41. [41]

    V . Fack, S. Lievens, J. Van der Jeugt, On rotation distance between binary coupling trees and applications for 3n j-coefficients, Comput. Phys. Commun. 119 (1999) 99–114. doi:10.1016/s0010-4655(99)00216-7

    work page doi:10.1016/s0010-4655(99)00216-7 1999

  42. [42]

    Koike, Explicit formulae of angular momentum coupling coefficients, Comput

    F. Koike, Explicit formulae of angular momentum coupling coefficients, Comput. Phys. Commun. 72 (1992) 154–164. doi:10.1016/0010-4655(92)90147-q

    work page doi:10.1016/0010-4655(92)90147-q 1992

  43. [43]

    P. D. Stevenson, Analytic angular momentum coupling coefficient calculators, Comput. Phys. Commun. 147 (2002) 853–858. doi:10.1016/s0010-4655(02)00462-9

    work page doi:10.1016/s0010-4655(02)00462-9 2002

  44. [44]

    Fritzsche, Maple procedures for the coupling of angular momenta I

    S. Fritzsche, Maple procedures for the coupling of angular momenta I. data structures and numerical computa- tions, Comput. Phys. Commun. 103 (1997) 51–73. doi:10.1016/s0010-4655(97)00032-5

    work page doi:10.1016/s0010-4655(97)00032-5 1997

  45. [45]

    Fritzsche, Maple procedures for the coupling of angular momenta

    S. Fritzsche, Maple procedures for the coupling of angular momenta. an up-date of the racah module, Computer Physics Communications 180 (2009) 2021–2023. doi:10.1016/j.cpc.2009.06.018

    work page doi:10.1016/j.cpc.2009.06.018 2009

  46. [46]

    Deveikis, A

    A. Deveikis, A. Kuznecovas, The analytical Scheme calculator for angular momentum coupling and recoupling coefficients, Comput. Phys. Commun. 172 (2005) 60–67. doi:10.1016/j.cpc.2005.06.003

    work page doi:10.1016/j.cpc.2005.06.003 2005

  47. [47]

    Xiang, L

    S. Xiang, L. Wang, Z.-C. Yan, H. Qiao, A program for simplifying summation of Wigner 3j-symbols, Comput. Phys. Commun. 264 (2021) 107880. doi:10.1016/j.cpc.2021.107880

    work page doi:10.1016/j.cpc.2021.107880 2021

  48. [48]

    SymPy: symbolic computing in Python

    A. Meurer, C. P. Smith, M. Paprocki, O. ˇCertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, Š. Rouˇcka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, A. Scopatz, Sympy: symbolic computing in...

    work page doi:10.7717/peerj-cs.103 2017

  49. [49]

    N. S. Scott, P. Milligan, H. W. C. Riley, The parallel computation of Racah coefficients using transputers, Comput. Phys. Commun. 46 (1987) 83–98. doi:10.1016/0010-4655(87)90037-3. 27

    work page doi:10.1016/0010-4655(87)90037-3 1987

  50. [50]

    V . Fack, J. Van der Jeugt, K. S. Rao, Parallel computation of recoupling coefficients using transputers, Comput. Phys. Commun. 71 (1992) 285–304. doi:10.1016/0010-4655(92)90015-q

    work page doi:10.1016/0010-4655(92)90015-q 1992

  51. [51]

    R. M. Baer, M. G. Redlich, Multiple-precision arithmetic and the exact calculation of the 3-j, 6-jand 9-j symbols, Commun. ACM 7 (1964) 657–659. doi:10.1145/364984.365075

    work page doi:10.1145/364984.365075 1964

  52. [52]

    Srinivasa Rao, V

    K. Srinivasa Rao, V . Rajeswari, C. B. Chiu, A new Fortran program for the 9-jangular momentum coefficient, Comput. Phys. Commun. 56 (1989) 231–248. doi:10.1016/0010-4655(89)90021-0

    work page doi:10.1016/0010-4655(89)90021-0 1989

  53. [53]

    R. E. Tuzun, P. Burkhardt, D. Secrest, Accurate computation of individual and tables of 3-jand 6-jsymbols, Comput. Phys. Commun. 112 (1998) 112–148. doi:10.1016/s0010-4655(98)00065-4

    work page doi:10.1016/s0010-4655(98)00065-4 1998

  54. [54]

    R. M. Dodds, G. Wiechers, Vector coupling coefficients as products of prime factors, Comput. Phys. Comm. 4 (1972) 268–274. doi:10.1016/0010-4655(72)90019-7

    work page doi:10.1016/0010-4655(72)90019-7 1972

  55. [55]

    A. J. Stone, C. P. Wood, Root-rational-fraction package for exact calculation of vector-coupling coefficients, Comput. Phys. Comm. 21 (1980) 195–205. doi:10.1016/0010-4655(80)90040-5

    work page doi:10.1016/0010-4655(80)90040-5 1980

  56. [56]

    D. F. Fang, J. F. Shriner, A computer program for the calculation of angular-momentum coupling coefficients, Comput. Phys. Commun. 70 (1992) 147–153. doi:10.1016/0010-4655(92)90097-i

    work page doi:10.1016/0010-4655(92)90097-i 1992

  57. [57]

    Lai, Y .-N

    S.-T. Lai, Y .-N. Chiu, Exact computation of the 3-jand 6-jsymbols, Comput. Phys. Commun. 61 (1990) 350–360. doi:10.1016/0010-4655(90)90049-7

    work page doi:10.1016/0010-4655(90)90049-7 1990

  58. [58]

    Lai, Y .-N

    S.-T. Lai, Y .-N. Chiu, Exact computation of the 9-j symbols, Comput. Phys. Commun. 70 (1992) 544–556. doi:10.1016/0010-4655(92)90115-f

    work page doi:10.1016/0010-4655(92)90115-f 1992

  59. [59]

    Wei, Unified approach for exact calculation of angular momentum coupling and recoupling coefficients, Comput

    L. Wei, Unified approach for exact calculation of angular momentum coupling and recoupling coefficients, Comput. Phys. Commun. 120 (1999) 222–230. doi:10.1016/s0010-4655(99)00232-5

    work page doi:10.1016/s0010-4655(99)00232-5 1999

  60. [60]

    Wei, Direct and exact compute and table of entire 3j, 6j, and 9j symbols: Erratum to CPC 120 (1999) 222–230, Comput

    L. Wei, Direct and exact compute and table of entire 3j, 6j, and 9j symbols: Erratum to CPC 120 (1999) 222–230, Comput. Phys. Commun. 182 (2011) 1199. doi:10.1016/j.cpc.2011.01.008

    work page doi:10.1016/j.cpc.2011.01.008 1999

  61. [61]

    C. F. Vermaak, D. Vermaak, H. G. Miller, A packed storage program for angular momentum coupling coeffi- cients, Comput. Phys. Commun. 31 (1984) 41–46. doi:10.1016/0010-4655(84)90080-8

    work page doi:10.1016/0010-4655(84)90080-8 1984

  62. [62]

    Rasch, A

    J. Rasch, A. C. H. Yu, Efficient Storage Scheme for Precalculated Wigner 3j, 6j and Gaunt Coefficients, SIAM J. Sci. Comput. 25 (2004) 1416–1428. doi:10.1137/S1064827503422932

    work page doi:10.1137/s1064827503422932 2004

  63. [63]

    Pinchon, P

    D. Pinchon, P. E. Hoggan, New index functions for storing Gaunt coefficients, Int. J. Quantum Chem. 107 (2007) 2186–2196. doi:10.1002/qua.21337

    work page doi:10.1002/qua.21337 2007

  64. [64]

    I. I. Guseinov, B. A. Mamedov, Algorithm for the storage of Clebsch–Gordan and Gaunt coefficients with the same selection rule and its application to multicenter integrals, J. Mol. Struct.: THEOCHEM 715 (2005) 177–181. doi:10.1016/j.theochem.2004.08.036

    work page doi:10.1016/j.theochem.2004.08.036 2005

  65. [65]

    H. H. H. Homeier, E. O. Steinborn, Some properties of the coupling coefficients of real spherical harmonics and their relation to Gaunt coefficients, J. Mol. Struct.: THEOCHEM 368 (1996) 31–37. doi:10.1016/s0166- 1280(96)90531-x

    work page doi:10.1016/s0166- 1996

  66. [66]

    S. A. Yükçü, N. Yükçü, E. Öztekin, New representations for Gaunt coefficients, Chem. Phys. Lett. 735 (2019) 136769. doi:10.1016/j.cplett.2019.136769

    work page doi:10.1016/j.cplett.2019.136769 2019

  67. [67]

    S. Özay, S. Akdemir, E. Öztekin, New orthogonality relationships of the Gaunt coefficients, Comput. Phys. Commun. 298 (2024) 109118. doi:10.1016/j.cpc.2024.109118

    work page doi:10.1016/j.cpc.2024.109118 2024

  68. [68]

    S. A. Yükçü, ¸ S. Atalay, N. Yükçü, E. Öztekin, Calculation of the Gaunt coefficients over real spherical har- monics, Can. J. Phys. 103 (2025) 321–327. doi:10.1139/cjp-2024-0161. 28

    work page doi:10.1139/cjp-2024-0161 2025

  69. [69]

    Politis, Gaunt coefficients for complex and real spherical harmonics with applications to spherical array processing and Ambisonics, 2024

    A. Politis, Gaunt coefficients for complex and real spherical harmonics with applications to spherical array processing and Ambisonics, 2024. doi:10.48550/arXiv.2407.06847.arXiv:2407.06847

    work page doi:10.48550/arxiv.2407.06847.arxiv:2407.06847 2024

  70. [70]

    S. Özay, S. Akdemir, E. Öztekin, The coupling coefficients with six parameters and the generalized hypergeo- metric functions, Computer Physics Communications 315 (2025) 109656. doi:10.1016/j.cpc.2025.109656

    work page doi:10.1016/j.cpc.2025.109656 2025

  71. [71]

    Haegeman, WignerSymbols.jl: a Julia package for computing 3j, 6j, and Racah symbols,https://github

    J. Haegeman, WignerSymbols.jl: a Julia package for computing 3j, 6j, and Racah symbols,https://github. com/Jutho/WignerSymbols.jl, 2024. Accessed 3 May 2026

    2024

  72. [72]

    Fraux, wigners: a Rust+Python library for Wigner 3j and Clebsch–Gordan coefficients,https://github

    G. Fraux, wigners: a Rust+Python library for Wigner 3j and Clebsch–Gordan coefficients,https://github. com/Luthaf/wigners, 2025. Accessed 3 May 2026

    2025

  73. [73]

    com/0382/CGcoefficient.jl, 2026

    0382, CGcoefficient.jl: Wigner 3j, 6j, 9j, CG, Racah W, and Moshinsky brackets in Julia,https://github. com/0382/CGcoefficient.jl, 2026. Accessed 3 May 2026

    2026

  74. [74]

    Moshinsky, Transformation brackets for harmonic oscillator functions, Nucl

    M. Moshinsky, Transformation brackets for harmonic oscillator functions, Nucl. Phys. 13 (1959) 104–116. doi:10.1016/0029-5582(59)90143-9

    work page doi:10.1016/0029-5582(59)90143-9 1959

  75. [75]

    Dumont, wignerSymbols: a C++port of the Schulten–Gordon Wigner 3j/6j/9j routines,https://github

    J. Dumont, wignerSymbols: a C++port of the Schulten–Gordon Wigner 3j/6j/9j routines,https://github. com/joeydumont/wignerSymbols, 2024. Accessed 3 May 2026

    2024

  76. [76]

    Fujii, py3nj: vectorized Python interface to the SLATEC Wigner 3j/6j/9j routines,https://github.com/ fujiisoup/py3nj, 2025

    K. Fujii, py3nj: vectorized Python interface to the SLATEC Wigner 3j/6j/9j routines,https://github.com/ fujiisoup/py3nj, 2025. Accessed 3 May 2026

    2025

  77. [77]

    Li, WignerFamilies.jl: bulk evaluation of Wigner 3j and 6j families via Luscombe–Luban recurrences in Julia,https://github.com/xzackli/WignerFamilies.jl, 2023

    Z. Li, WignerFamilies.jl: bulk evaluation of Wigner 3j and 6j families via Luscombe–Luban recurrences in Julia,https://github.com/xzackli/WignerFamilies.jl, 2023. Accessed 3 May 2026

    2023

  78. [78]

    Boyle, spherical: numerical WignerD-matrices, spin-weighted spherical harmonics, and 3j symbols in Python/Numba,https://github.com/moble/spherical, 2025

    M. Boyle, spherical: numerical WignerD-matrices, spin-weighted spherical harmonics, and 3j symbols in Python/Numba,https://github.com/moble/spherical, 2025. Accessed 3 May 2026

    2025

  79. [79]

    O. C. Gorton, wigner: modern Fortran library for Wigner 3j, 6j, and 9j symbols with OpenMP and lookup tables,https://github.com/ogorton/wigner, 2022. Accessed 3 May 2026

    2022

  80. [80]

    H. T. Johansson, FASTWIGXJ: pre-tabulated and dynamic-hash Wigner 3j, 6j, and 9j lookup, companion to WIGXJPF,https://fy.chalmers.se/subatom/fastwigxj/, 2023. Accessed 3 May 2026

    2023

Showing first 80 references.