title: Couplings and recouplings of four angular momenta: Alternative 9j symbols and spin addition diagrams.
abstract: The Wigner 9j symbols of the first kind-also known as Fano X-coefficients-serve to connect different addition schemes of four angular momenta, widely known examples being the LS and the jj couplings in atomic, molecular, and nuclear spectroscopies. Here, we also consider alternative sequences of binary couplings of four angular momenta, which are dealt through the 9j symbols of the second kind, and are explicitly given by the pentagonal (or Biedenharn-Elliott) identity. These coefficients are essential ingredients in the quantum-mechanical treatments of rotational and polarization phenomena in reaction dynamics and photoinduced processes. We also emphasize the combinatorial structure underlying the extended construction of a previously introduced truncated icosahedral "abacus", and provide extensions useful for algebraical manipulations, semiclassical interpretations, and computational applications, including all the 120 addition schemes.
authors.affiliation: Instituto de Física, Universidade Federal da Bahia, Campus Universitario de Ondina, 40170-115, Salvador, BA, Brazil.|Instituto Federal de Alagoas, Campus Piranhas, Avenida Sergipe s/n, 57460-000, Piranhas, AL, Brazil.
authors.forename: Robenilson F
authors.initials: RF
authors.lastname: Santos
authors.affiliation: Departamento de Física, Universidade Estadual de Feira de Santana, Feira de Santana, BA, Brazil.
authors.forename: Ana Carla P
authors.initials: AC
authors.lastname: Bitencourt
authors.affiliation: Departamento de Física, Universidade Estadual de Feira de Santana, Feira de Santana, BA, Brazil.
authors.forename: Mirco
authors.initials: M
authors.lastname: Ragni
authors.affiliation: Instituto de Física, Universidade Federal da Bahia, Campus Universitario de Ondina, 40170-115, Salvador, BA, Brazil.
authors.forename: Frederico V
authors.initials: FV
authors.lastname: Prudente
authors.affiliation: Dipartimento di Farmacia, Università G. d'Annunzio, Via dei Vestini, 66100, Chieti, Italy. ccoletti@unich.it.
authors.forename: Cecilia
authors.identifier: http://orcid.org/0000-0002-3609-290X
authors.initials: C
authors.lastname: Coletti
authors.affiliation: Dipartimento di Matematica, Università degli Studi di Pavia, Pavia, Italy.|INFN, Sezione di Pavia, via Bassi 6, 27100, Pavia, Italy.
authors.forename: Annalisa
authors.initials: A
authors.lastname: Marzuoli
authors.affiliation: Instituto de Física, Universidade Federal da Bahia, Campus Universitario de Ondina, 40170-115, Salvador, BA, Brazil.|Dipartmento di Chimica, Biologia e Biotecnologie, Università di Perugia, via Elce di Sotto 8, 06123, Perugia, Italy.|Istituto di Struttura della Materia, Consiglio Nazionale delle Ricerche, 00016, Rome, Italy.
authors.forename: Vincenzo
authors.initials: V
authors.lastname: Aquilanti
keywords: Biedenharn–Elliott identity; Quantum angular momentum theory; Spin networks; Wigner 6j and 9j symbols