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Manuscript Title: Two-program package to calculate the ground and excited state wave functions in the Hartree-Fock-Dirac approximation.
Authors: L.V. Chernysheva, V.L. Yakhontov
Program title: RSCFHF AND RFCHF
Catalogue identifier: ADJZ_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 119(1999)232
Programming language: Fortran.
Computer: PC.
Operating system: Windows 3.1/95/NT 3.51, VMS, Linux, Unix.
RAM: 8M words
Word size: 32
Peripherals: disc.
Keywords: Atomic physics, Structure, Atomic energy levels, Dirac hamiltonian in, Atomic theory, Correlations in atoms, Ls- and jj-coupling in, Atoms, Nuclear mass effect.
Classification: 2.1.

Nature of problem:
Description of atomic spectra - discrete atomic energy levels, total energy of an atom/ion, and phase shifts of continuum spectrum single- particle atomic wave functions - using a "fully relativistic" approach.

Solution method:
Single-particle atomic orbitals are assumed to be four-component spinor eigenstates of the total angular momentum operator j = l + s, its projection jz, and the parity operator Phi = beta phi. Approximate atomic ground state wave functions are obtained by taking either the single Slater determinant constructed from the bispinors (e.g., for closed-shell atoms which are described by the 1S0-term), or a special linear combination of similar determinants so as to obtain the eigenfunctions of the total angular momentum operators J**2, Jz (in the case of one or more open shells). The program RSCFHF, the first part of the package described, generates the necessary coefficients automatically by analysing both the occupation numbers and total angular momenta of individual subshells. The "big" and "small" radial parts of the S single-particle bispinors, with S being the number of atomic subshells, are determined then by numerically solving the self-consistent single-configuration Hartree-Fock-Dirac equations that result upon varying approximate ground state atomic wave functions of the given form so as to obtain the extremum of the variational functional. Together with S self-consistent subshell potentials, the S "big" and S "small" solutions obtained are used then as input data to the RFCHF code, the second program of the present package. The latter is intended to generate excited (discrete and continuum) one-particle relativistic wave functions in a fixed field induced by the "ground state" atomic configuration. In analogy with the "ground state", discrete state excited one-particle wave functions obey the zero boundary conditions, are normalised and are assumed to possess not less than (ns-ls-1) nodes at finite distances from the nucleus. For continuum spectrum orbitals, the two last restrictions are replaced by a chosen asymptotic condition.

Restrictions:
All "subshell" single-particle orbitals that share the quantum numbers nlj are assumed to have the same radial dependence: Fnlj(r), Gnlj(r). In the course of self-consistent calculations with the RSCFHF code, the orbitals with different values of the quantum numbers (nlj) are assumed to be orthogonal. So are excited state orbitals (epsilon ls js) obtained with the RFCHF code, and the "core" ones, (nc lc jc), such that (ls js) /= (lc jc). A user-defined option available in the input to the RFCHF program controls the orthogonality of excited and core states with ls = lc, js = jc.
Presently, both programs cannot handle the situation when three or more atomic subshells are unfilled, as the codes are unable to generate appropriate angular coefficients. In this case, the so-called "averaged configuration calculation", i.e. where an atom/ion is not described by any definite term, is carried out automatically. Alternatively, a user can explicitly specify in the above case the required coefficients in the input to the programs to facilitate calculations under certain "term"-conditions.

Unusual features:
Following are the two most important features distinguishing the programs under consideration from the known codes [1-6].
  1. The system os 2S (S=1 in the case of a frozen core) first order integro-differential equations of the HFD approximation is reduced to the system of S second order ones, by eliminating the "small" radial components. Each equation of the system obtained in such a way has a form similar to that of the non-relativistic HF approximation, except for a potential term. This enables one to preserve the same program structure which has formerly been used in [7,8] to obtain its numerical solution. Once the "large" radial component and self-consistent potential are found for each subshell, the corresponding "small" components are calculated accordingly (see below). This method both reduces considerably the CPU time (without loss of accuracy) and makes the memory requirements much less restrictive, thus enabling the programs to be run on even a PC of the standard configuration.
  2. In the RSCFHF program, we have incorporated, following [9], an option to treat charged particles different from the electron (e.g., negative and/or positive muons and positrons) thus extending the scope of physical processes.

Running time:
The test cases took < 2 min.

References:
[1] M.A. Coulthard, Proc. Royal Soc. 91 (1967) 44.
[2] F.C. Smith, W.R. Johnson, Phys. Rev. A 91 (1967) 136.
[3] J.P. Desclaux, D.F. Mayers, O'Brien, J. Phys. B: Atom. Mol. Phys. 4 (1971) 631; Comp. Phys. Comm. 9 (1975) 31.
[4] I.M. Band, V.I. Fomichev. The complex of programs: REINE. Parts I, II. Report No. 498 of the Leningrad O.B. Konstantinov Nuclear Physics Institute, Academy of Sciences of the USSR, Leningrad (1979) (in Russian).
[5] V.A. Dzuba, O.P. Sushkov, V.V. Flambaum, The complex of programs to calculate the atomic wave functions and the energies. Report No. 82-89 of the Novosibirsk Nuclear Physics Institute, Academy of Sciences of the USSR, Novosibirsk (1982).
[6] I. Lindgren, A. Rosen, Case Studies in Atomic Physics 4, No 3 (1974) 105.
[7] L.V. Chernysheva, N.A. Cherepkov, V. Radojevic, Comp. Phys. Comm. 11 (1976) 57.
[8] L.V. Chernysheva, N.A. Cherepkov, V. Radojevic, Comp. Phys. Comm. 18 (1979) 87.
[9] L.V. Chernysheva, S.K. Semenov, M.Ya. Amusia, N.A. Cherepkov, V.F. Orlov, The program system for atomic calculations: ATOM. Program XXIII. The program to calculate electron wave functions in the frozen core Hartree-Fock-Dirac approximation. Report No. 1319 of the A.F. Ioffe Physical-Technical Institute, Academy of Sciences of the USSR, Leningrad (1989) (in Russian).