Angular coefficients for symmetry-adapted configuration states in jj-coupling

Abstract

In atomic structure and collision theory, the efficient spin-angular integration is known to be crucial and often decides, how accurate the properties and behavior of atoms can be predicted numerically. Various methods have been developed in the past to keep the computation (and implementation) of the spin-angular integration feasible for complex shell structures, including open d- and f-shell elements. To support such computations, we here provide a new implementation of the angular coefficients for jj-coupled and symmetry-adapted configuration states that is entirely built upon the quasi-spin formalism. The module SpinAngular is based on Julia, a new programming language for scientific computing, and supports a simple access to all (completely) reduced tensors, coefficients of fractional parentage for subshells with $j\le 9/2$ as well as the re-coupling coefficients from this formalism. Moreover, this module has been worked out for multiple purposes, including 1) the accurate calculation of atomic properties, 2) further studies on spin-angular integration theory, 3) the development of new or existing computer programs as well as 4) the manipulation of reduced matrix elements from this theory. The present implementation will therefore help advance the algebraic evaluation of many-electron (transition) amplitudes and to apply the theory to newly emerging research areas.

Program summary

Program title: SpinAngular

CPC Library link to program files: https://doi.org/10.17632/jjpff3pysn.1

Code Ocean capsule: https://doi.org/10.24433/CO.3772334.v1

Licensing provisions: MIT License

Programming language: Julia

Nature of problem: For symmetric one- and two-particle operators, the (many-electron) matrix elements between symmetry-adapted configuration states can always be written as a sum of angular coefficient x interaction strength. While the interaction strengths describe the physical interaction and just depends on the (one-electron) orbitals of the active shells, the angular coefficients arise from the (many-electron) spin-angular integrals and are determined geometrically by the occupation and coupling of all electrons. These angular coefficients need to be calculated efficiently and often require special care.

Solution method: The spin-angular approach [1] is implemented for (tensorial coupled) one-particle operators of arbitrary rank, scalar two-particle operators as well as for the manipulation of reduced matrix elements from this theory.

References

• G. Gaigalas, Z. Rudzikas, C. Froese Fischer, J. Phys. B, At. Mol. Opt. Phys. 30 (1997) 3747.

Introduction

In atomic structure theory, both the efficient decomposition and implementation of many-electron matrix elements are very crucial as they often decide, how much electron-electron correlations can be taken into account into the computations and how accurate predictions can be made for any particular atomic property or process. Therefore, different approaches have been worked out [1], [2], [3], [4] in the past decades in order to decompose the matrix elements for (symmetric) one- and two-particles operators between open-shell, and especially symmetry-adapted, configuration state functions (CSF). Well-known approaches from the literature are based, for example, on the coefficients of fractional parentage [5], [6], the calculus of unit tensors [7], [8], the seniority notation [5], [9], the re-coupling coefficients [3], [10] along with their graphical evaluation [11], [12], or the classification of the anti-symmetric subshell states by Grant and others [12], [13], to name just the most relevant ones. Although many of these approaches have been applied (very) successfully in Grasp [14], [15] and other structure codes [16], [17], [18], these implementations were often built quite ad hoc upon previously implemented coefficients and parts of existing codes.

Beyond the standard coupling of the (total) angular momenta, a further reduction and simplification of the spin-angular integration can be achieved within the (so-called) quasi-spin formalism. In this formalism, the power of Racah's algebra is exploited also with regard to the occupation of the subshells. This additional (formal) reduction of the matrix elements is often termed complete and makes the formalism readily applicable to atoms and ions with complex shell structures, including open d- and f-shell elements, as well as to the modeling of atomic cascades [19], [20], [21], [22]. Here, we provide a re-implementation of the angular coefficients in the quasi-spin formalism for jj-coupled and symmetry-adapted configuration states within the framework of Julia and Jac, the Jena Atomic Calculator [23], [24]. Apart from the computation of these angular coefficients for symmetric one-particle operators of any rank and scalar two-particle operators, the newly developed SpinAngular module also supports a simple access to all (completely) reduced tensors, the coefficients of fractional parentage for subshells with $j\le 9/2$, the matrix elements of the $T^{(k)}$ and $W^{(k_q{k}_j)}$ operators in jj-coupling as well as the associated recoupling tensors. Moreover, this module has been worked out for multiple purposes, including 1) the accurate calculation of atomic amplitudes, 2) further studies on the spin-angular integration theory as well as 3) manipulation of various reduced matrix elements from this theory. All these features might help improve the further decomposition and algebraic manipulation of (many-electron) matrix elements, and to apply them in newly emerging research fields, such as the modeling of astrophysical light curves [25], atomic cascades [26], [27], [28], or strong-field ionization processes [29], [30].

In the next section, we first recall the concept of the isospin formalism by introducing the central notations as well as the tensors in angular-momentum and quasi-spin space. This section also explains the decomposition of the matrix elements for (symmetric) one- and two-particle operators, and how it naturally leads to the definition of the spin-angular coefficients. Subsection 3.5, especially, summarizes all the building blocks that are central to this decomposition and that can be obtained from the implementation below. Details of this implementation within the framework of Julia are given in Section 3, together with a short comparison with other available codes in Maple and Fortran, respectively. Apart from the basic data structures for keeping all tensor components, we here explain how these angular coefficients can be computed for both, a list (array) of CSF as well as dynamically constructed configuration states. Section 3.3 shows a few simple examples for dealing with the functions from the SpinAngular module. Finally, a summary and short outlook is given in sections 4.