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Manuscript Title: An atomic program for energy levels of equivalent electrons: lanthanides and actinides.
Authors: S. Edvardsson, D. Aberg
Program title: Lanthanide
Catalogue identifier: ADMZ_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 133(2001)396
Programming language: C.
Computer: SUN SPARC Ultra 10, IBM SP, PCs running Solaris.
Operating system: SUN Solaris 2.6 & 2.7, IBM AIX 4, Linux.
RAM: 8M words
Word size: 32
Keywords: Rare-earth ions, Crystal field energy levels, Configuration interaction, Electron correlation, Effective operators, Magnetic field, Three-particle operators, Determinantal product states, Solid state physics.
Classification: 7.3.

Nature of problem:
Diagonalizations of various one-, two- and three-particle operators that result in energy eigenvalues and their eigenfunctions. The various operator parameters have to be calculated ab initio or taken from experiment; these are used as input to the program. The approach is non-relativistic (it is empirically known that the main relativistic and correlation effects can be absorbed in the input parameters) making the program mainly useful for systems such as studies of energy structure of open valence shells of rare-earth or actinide atoms.

Solution method:
First a basis set of determinantal product states are generated for a pure electronic configuration according to the exclusion principle of Pauli. Using this basis, the matrix elements of the upper triangle of the Hermitean matrix are calculated. The matrix elements of all operators are added. The complete matrix is subsequently diagonalized to give the various eigenvalues and eigenvectors. Configuration interaction effects are accounted for by using various standard two- and three-particle operators [1-3].

The hamiltonians are all classical so this program is inappropriate (without appropriate modifications) for relativistic systems with energy levels that have angular dependencies that cannot be absorbed into the used classical operators. Other practical restrictions are mainly three: computing time, matrix size and disk storage. Other than that there are no remaining limitations built into the software. An example of these three practical limitations are the electron configuration f7. In this case the size of the energy matrix is 3432x3432 which places a great demand on primary memory (approx. 200 Mb in total) and takes several hours to diagonalize on a fast workstation. However the main problem is not the diagonalization, in fact the calculation of the three-particle matrix elements is even more demanding. After several optimizations (application of several symmetries, code and compiler optimizations), this part of the calculation still takes approximately a day on a SUN Ultra 10. Fortunately, the calculations of the three-particle matrix elements only need to be made once and are then saved to the local disk. Other files are storage of the hermitean matrix and subsequently its eigenvectors; these can become quite large. This problem is solved within the program since these files are written in a compressed format (zlib) causing a size reduction of approximately a factor of ten.

Unusual features:
The linear-algebra operations (zhpeev: complex packed format diagonalization routine) are performed by the standard numerical libraries CLAPACK and CBLAS [4]. Several files that otherwise could be huge are saved in gz-format using the library zlib [5]. Rules for fast calculations of three-particle matrix elements are derived (described below) and implemented in the program. The program does not need or use any tables of Racah coefficients of fractional parentage.

Running time:
The running times range from a few minutes (f2) up to ~ a day (f7), i.e., depending on the problem and whether reading of previously stored matrix elements is used or not.

[1] K. Rajnak and B.G. Wybourne, Phys. Rev. 132 (1963) 280.
[2] K. Rajnak, J. Opt. Soc. Am. 55 (1965) 126.
[3] B.R. Judd, Phys. Rev. 141 (1966) 4.
[4] http://www.netlib.org/clapack/
[5] http://sunsite.doc.ic.ac.uk/packages/zip/zlib/zlib.html