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Binder ... use JAC immediately with Binder in the cloud.

Jena Atomic Calculator (JAC) for the computation of atomic representations, processes and cascades

What is JAC?

We here provide a first public version of JAC, the Jena Atomic Calculator and an open-source Julia package for doing atomic computations. JAC is a (relativistic) electronic structure code for the computation of (atomic many-electron) interaction amplitudes, properties as well as a large number of excitation and decay processes for open-shell atoms and ions across the whole periodic table. In forthcoming years, moreover, JAC will -- more and more -- facilitate also studies on atomic cascades, responses to external fields and particles, the time-evolution of atoms and ions as well as selected symbolic computations of expressions from Racah's algebra.

A primary guiding philosophy of JAC was to develop a general and easy-to-use toolbox for the atomic physics community, including an interface that is equally accessible for working spectroscopiests, theoreticians and code developers. Beside of its simple use, however, I also wish to provide a modern code design, a reasonable detailed documentation of the code as well as features for integrated testing. In particular, many typical computations and the handling of atomic data should appear within the code similar to how they would appear in spoken or written language. Shortly speaking, JAC aims to provide a powerful platform for daily use and to extent atomic theory towards new applications or, in short, a community platform for Just Atomic Computations.

Remark: Although major efforts have been undertaken during the past four years, JAC is still in an early state of development and includes various features that are only partly implemented or have not yet tested in good detail. Despite of possible failures and deficiencies of the present code, however, I here annouce JAC and kindly ask potential users and developers for response, support and encouragement.

Kinds of computations

In some more detail, JAC distinguishes and aims to support (partly still within the future) nine kinds of computations which can be summarized as follows (Figure):

  1. Atomic computations, based on explicitly specified electron configurations: This kind refers to the computation of level energies, atomic state representations and to either one or several atomic properties for selected levels from a given multiplet. It also help compute one selected process at a time, if atomic levels from two or more multiplets are involved in some atomic transition.
  2. Atomic representations: This kind concerns different representations of atomic wave functions; in particular, it includes systematically-enlarged restricted active-space (RAS) computations of atomic states and level energies due to a pre-specified active space of orbitals as well as due to the (number and/or kind of) virtual excitations that are taken to be into account. Such RAS computations are normally performed stepwise by making use of the (one-electron) orbital functions from some prior step. Other atomic representations refer to approximate atomic Green functions and, in the future, combined techniques with concepts from close-coupling, (exterior) complex scaling, DMRG or perturbation theory.
  3. Interactive computations: Here, the (large set of) methods of the JAC program are applied interactively, either directly from the REPL or by using some short Julia script in order to compute and evaluate the desired observables (atomic parameters), such as energies, expansion coefficients, transition matrices and amplitudes, rates, cross sections, etc. An interactive computation typically first prepares and applies (certain instances of) JAC’s data types, such as orbitals, configuration-state functions (CSF), atomic bases, levels, multiplets, and others. And like Julia, that is built on many (high-level) functions and methods, JAC then provides the required language elements for performing specific atomic computations at different degree of complexity and sophistication.
  4. Atomic cascade computations: A cascade typically includes ions of an element in three or more charge states that are connected to each other by different atomic processes, such as photoionization, dielectronic recombination, Auger decay, radiative transitions, and where the relative level population of these charge states is determined by the set-up and geometry of the given experiment. Cascade computations are usually based on some predefined (cascade) approach that enables one to automatically select the state-space of the ions, to choose the atomic processes to be considered for the various steps of the cascade, and to specify perhaps additional restrictions in order to keep the computations feasible.
  5. Atomic responses: With this kind, I wish to support in the future computations that help analyze the response of atoms to incident beams of light pulses and particles, such as field-induced ionization processes, high-harmonic generation and several others. For these responses, the detailed structure of the atoms and ions has often not yet been considered until today but will become relevant as more elaborate and accurate measurements will become feasible.
  6. Atomic time-evolution of statistical tensors: We here wish to simulate the population and coherences of (atomic) levels using the Liouville equation, when atoms and ions are irradiated by (intense) light pulses. For these computations, however, we always assume that the level structure of the atoms is kept intact. Further (decay) processes of the excited atoms and ions can be takent into account by some loss rate, but without that the atoms can leave the pre-specified space of sublevels. In particular, I here plan to consider the interaction of atoms and ions with pulses of different shape, polarization strength and duration.
  7. Atomic descriptors: In building models for machine learning, the definition of proper (atomic) descriptors or feature transformations is central for predicting the physical properties and behaviour of atoms and ions; here, we shall provide various of such descriptors useful for atomic physics.
  8. Semi-empirical estimates of atomic properties, such as cross sections, stopping powers, asymptotic behaviour, etc. These estimates often refer to simple model computations or the use of fit functions. They are only implemented when data are needed but no ab-initio computations of the properties are feasible otherwise.
  9. Symbolic evaluation of expressions from Racah's algebra: This kind refers to the algebraic transformation and simplification of (Racah) expressions, which may generally include any number of Wigner n-j symbols of different kind as well as (various integrals over) the spherical harmonics, the Wigner rotation matrices and the Kronecker and triangular deltas. Of course, the complexity of such Racah expressions increases very rapidly as more Wigner symbols are involved. A symbolic evaluation of these expressions is naturally based on the knowledge of a large set of special values, orthogonality relations and sum rules that may include rules with a (multiple) summation over dummy indices, cf. the monography by Varshalovich et al (1988).

Documentation & News

A detailed User Guide, Compendium & Theoretical Background to JAC is available that describes the use and underlying atomic theory of the JAC code. News about recent developments of JAC are summarized here.

Licence & Reference

The code in this repository is distributed under the MIT licence. The associated User Guide, Compendium & Theoretical Background to JAC is distributed under the Creative Commons Attribution 4.0 International (CC BY 4.0) license.

For reference to (using) this code, please, use the Computer Physics Communications publication on JAC:

See also CITATION.bib for the relevant references(s).


In Julia, you can install the JAC package like any other package by by just entering the package manager (with ]) and by typing

pkg> add

Moreover, to run the tutorials, you will need to have Jupyter notebooks running.typing

If you haven't used Julia and Jupyter before, you can also run under Linux or Windows either the script Install Julia, Jupyter & JAC via or ... via snap.

JAC also works for Apple's MAC system, though there is less experience from our side. Please, see the Installation Guides Mac or ...

If you found a good solution for still other operating systems and/or hardware architectures, where the hints above do not work, please, send useful comments and email to I will be happy too add them here to further facilitate the use.

Dependencies and external code used in JAC

The JAC code makes use of:

  • standard Julia packages, such as SpecialFunctions, FortranFiles, GaussQuadrature, GSL and QuadGK.
  • Matrix elements from G. Gaigalas and S. Fritzsche, Comp. Phys. Commun. 267, 108086 (2021).


The 'simplest access' to the JAC toolbox is by using Binder in the cloud. If you click here:


you will get a Jupyter notebook where you can call 'using JAC' in order to have Julia and JAC installed. Then you can run all examples and calls like on your own computer, just a bit slower (by a factor 3..5). This will help you to run a few first examples (as shown in the example folder above) and in order to decide of whether you wish to install the code locally.

You can also directly access the Getting started with JAC tutorial, and similar for other tutorials that are distributed together with the code. Further details can then be found from the User Guide, Compendium & Theoretical Background to JAC. Make use the index or a full-text search to find selected items in this (.pdf) User Guide.

A very simple example has been discussed in the CPC reference above and just refers to the low-lying level structure and the Einstein A and B coefficients of the 3s 3p^6 + 3s^2 3p^4 3d -> 3s^2 3p^5 transition array for Fe^{9+} ions, also known as the spectrum Fe X. To perform such a computation within the framework of JAC, one needs to specify the initial- and final-state configurations by an instance of an Atomic.Computation, together with the specifier process=Radiative(). We here also provide a title (line), the multipoles (default E1) and the gauge forms for the coupling of the radiation field that are to be applied in these calculations:

    grid = Radial.Grid(true);   setDefaults("standard grid", grid)
    defaultsSettings = PhotoEmission.Settings()
    photoSettings    = PhotoEmission.Settings(defaultsSettings, multipoles=[E1, M1], gauges=[UseCoulomb, UseBabushkin], printBefore=true)
    comp = Atomic.Computation(Atomic.Computation(), name="Energies and Einstein coefficients for the spectrum Fe X",  
                              grid = grid, nuclearModel = Nuclear.Model(26.);
                              initialConfigs = [Configuration("[Ne] 3s 3p^6"), Configuration("[Ne] 3s^2 3p^4 3d")],
                              finalConfigs   = [Configuration("[Ne] 3s^2 3p^5")], 
                              processSettings = photoSettings ); 

This example is discussed also in one of the tutorials below or directly in the cloud.


The following IJulia/jupyter notebooks introduce the reader to JAC and demonstrate several features of this toolbox.
They can be explored statically at GitHub or can be run locally after the software repository has been cloned and installed. In order to modify the cell-output of the notebooks and to better print wide tables, you can create or modify the file ~/.jupyter/custom/custom.css in your home directory and add the line: div.output_area pre { font-size: 7pt;} .

Current limitations of JAC

Although JAC has been designed for all atoms and ions across the periodic table, a number of limitations occur:

  • All self-consistent-field computations are based on a local potential (e.g. core-Hartree, Kohn-Sham, Dirac-Hartree-Slater, ...) that can be controlled by the user.
  • Until the present, no serious optimization has been done for the code; this restricts most computations to CSF expansion with several hundred CSF.
  • All continuum orbitals are generated in a Dirac-Hartree-Slater potential of the ionic core, and without the explicit treatment of the exchange interaction.

Encouragement & Contribution

The scope of JAC is much wider than what I can (and plan to) implement myself here in Jena. With JAC's upload to GitHub, I therefore wish to encourage the users to fork the code and to report improvements, failures, bugs, etc. Non-trivial changes to the code can be made available via pull requests, i.e. by submitting code for review (by other users) prior to their merger with the master code.

In particular, I like to encourage contributions from the atomic physics community if the overall style of the package is maintained and if consensus exists how to add new features to the code. The goal should be to avoid duplication and inhomogeneity across the package as well as to implement (too) specific features that may cause issues in the future. External support by developers may include incremental improvements as well as multiple approaches for algorithms and modules in order to provide well-tested alternatives, for instance, if some particular approach does not work properly in all applications. Moreover, emphasis will be placed first on all those applications that receive enough attention by the community.

In contrast, I shall not support those developments which appears too sophisticated or detrimental to a long-term maintenance of the code. Other specialized parts might be incorporated later if the code has left its early stage of development and becomes robust enough.

Although a good number of tests have been made on JAC, this is still a very first implementation, and no code is error free. I shall therefore appreciate reports from the users if problems are encountered or, more helpful, if solutions are provided. One of the simplest way to start contributing to JAC is writing a tutorial, in addition to those provided above, in order to navigate others to the task of a new user. Also, new graphical user interface and plotting features on different outcomes of atomic computations will be very helpful for the community. A few further suggestions for extending and improving JAC can be found in section 1.7 in the User Guide, Compendium & Theoretical Background to JAC.



  • Birger Böning (HI Jena, Germany)
  • Gediminas Gaigalas (U Vilnius, Lithuania)
  • Jiri Hofbrucker (U Jena, Germany)
  • Stefan Schippers (U Giessen, Germany)
  • Andrey Surzhykov (U Braunschweig, Germany)
  • Martino Trassinelli (U Sorbonne Paris, France)
  • Andrey Volotka (HI Jena, Germany)


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