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Manuscript Title: GRASP92: a package for large-scale relativistic atomic structure calculations.
Authors: F.A. Parpia, C.F. Fischer, I.P. Grant
Program title: GRASP92
Catalogue identifier: ADCU_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 94(1996)249
Programming language: Fortran.
Computer: IBM POWERstation 320H.
Operating system: IBM AIX 3.2.5+.
RAM: 64M words
Word size: 32
Peripherals: disc.
Keywords: Atomic physics, Structure, Atomic energy levels, Breit interaction in Atoms, Configuration Interaction methods for Atoms, Dirac hamiltonian in Atomic theory, Correlation in atoms, Jj-coupling for atomic Electrons, Mass polarisation in Atoms, Multiconfiguration Methods for atoms, Nuclear mass effects In atoms, Nuclear volume effects In atoms, Atomic oscillator Strengths, Radiative decay rates Of atoms, Relativistic Corrections in atoms, Transverse photon Interaction in atoms.
Classification: 2.1.

Nature of problem:
Prediction of atomic spectra - atomic energy levels, oscillator strengths, and radiative decay rates - using a 'fully relativistic' approach.

Solution method:
Atomic orbitals are assumed to be four-component spinor eigenstates of the angular momentum operator, j = 1 + s, and the parity operator Phi = betaphi. Configuration state functions (CSFs) are linear combinations of Slater determinants of atomic orbitals, and are simultaneous eigenfunctions of the atomic electronic angular momentum operator, J, and the atomic parity operator, P. Lists of CSFs are either explicity prescribed by the user or generated from a set of reference CSFs, a set of subshells, and rules for deriving other CSFs from these.
Approximate atomic state functions (ASFs) are linear combinations of CSFs. A variational functional may be constructed by combining expressions for the energies of one or more ASFs. Average level (AL) functionals are weighted sums of energies of all possible ASFs that may be constructed from a set of CSFs; the number of ASFs is then the same as the number, nc, of CSFs. Optimal level (OL) functionals are weighted sums of energies of some subset of ASFs; the GRASP92 package is optimised for this latter class of functionals.
The composition of an ASF in terms of CSFs sharing the same quantum numbers is determined using the configuration-interaction (CI) procedure that results upon varying the expansion coefficients to determine the extremum of a variational functional. Radial functions may be determined by numerically solving the multiconfiguration Dirac-Fock (MCDF) equations that result upon varying the orbital radial functions or some subset thereof so as to obtain an extremum of the variational functional. Radial wavefunctions may also be determined using a screened hydrogenic or Thomas-Fermi model, although these schemes generally provide initial estimates for MCDF self-consistent- field (SCF) calculations.
Transition properties for pairs of ASFs are computed from matrix elements of multipole operators of the electromagnetic field. All matrix elements of CSFs are evaluated using the Racah algebra.

The maximum size of a multiconfiguration (MC) calculation, as measured by the length of the configuration state function (CSF) list nc, is limited by numerical stability, processing time, or storage. Numerical stability typically decreases as the number of radial functions varied increases and as the number of open subshells increases. Processing time increases as some power of nc greater than 1 but generally appreciably less than 3. Lists of angular integrals, V(k)rs(abcd), distinguished by tensor rank, k, are written to disk; the available disk storage must be large enough to store all such lists together. Each list is subsequently read into memory and sorted by canonically-ordered Slater integral indices abcd; the available memory (including any available virtual memory) must be large enough to store the longest list before it is sorted. The lengths of the unsorted and sorted lists increase as some power of nc greater than 1 but generally less than the maximum of 2.
The maximum size of a configuration interaction (CI) calculation is limited by processing time and storage. Processing time increases as some power of nc greater than 1 but generally appreciably less than 2. A sparse representation of the lower triangle of the Hamiltonian matrix is written to disk; the available disk storage must be large enough to store this representation of the Hamiltonian matrix. The size of this representation of the Hamiltonian matrix increases as some power of nc greater than 1 but generally less than the maximum of 2.
All orbitals that share the quantum numbers nlj (i.e., all members of a subshell) are assumed to have the same radial dependence (Pnly(r), Qnlj(r). Orbitals with different values of the quantum numbers nlj are assumed to be orthogonal.
The tables of coefficients of fractional parentage used in GRASP92 are limited to subshells with j <= 7/2; occupied subshells with j = 9/2 are, therefore, restricted to a maximum of two electrons.

Unusual features:
The GRASP92 package comprises task-specific component programs for the specification of nuclear properties, the generation and manipulation of lists of configuration state functions (CSFs), the computation of radial wavefunctions, of approximate atomic state functions (ASFs), the computation of properties of electromagnetic transitions between ASFs, and for the conversion of data between machine-specific unformatted representations and universal formatted representations.
All component programs in the GRASP92 package have been designed for interactive use; the number of keystrokes required by the user is reduced by the provision of defaults appropriate to the types of calculations that are expected to be performed most frequently, and by the provision of interpretation for 'wild card' characters as sets of data items.
Several devices have been adopted to reduce computational effort and storage requirements: in multiconfiguration (MC) calculations, the list of angular integrals is presorted by tensor rank prior to sorting by canonically-ordered Slater integral indices; in configuration- interaction (CI) and transition property calculations, angular integrals are not stored and an ordered list of radial integrals is searched and augmented as required as the calculation progresses; in MC and CI calculations, the lower triangle of the Hamiltonian matrix is stored in a sparse representation; the Davidson-Liu algorithm [1] as implemented by Stathopoulos and Fischer [2] is used to extract the eigenvalues and eigenvectors of interest. Certain linear-algebraic operations are preformed using subprograms from the BLAS [3] and LAPACK [4] libraries. Angular-momentum recoupling coefficients are computed using the NJGRAF package of Bar-Shalom and Klapisch [5].
A minor revision of a preprocessor program due to K.G. Dyall [6] is used to automate the setting of array dimensions and the selection of installation-dependent features.

Running time:
CPU time required to execute test cases: 300 min

[1] E.R. Davidson, J. Comput. Phys. 17 (1975) 87; Comput. Phys. Commun. 53 (1989) 49; B. Liu, in Numerical Algorithms in Chemistry: Algebraic Methods,edited by C. Moler and I. Shavitt (Lawrence Berkeley Laboratory, Berkeley, California, 1978); C.W. Murray, S.C. Racine and E.R. Davidson, J. Comput. Phys. 103 (1992) 382.
[2] A. Stathopoulos and C. Froese Fischer, Comput. Phys. Commun. 79 (1994) 1.
[3] C.L. Lawson, R.J. Hanson, D. Kincaid, F.T. Krogh, ACM Trans. Math. Soft. 5 (1979) 308; J. Dongarra, ACM Trans. Math. Soft. 14 (1988) 1; J.J. Dongarra, J. Du Croz, S. Hammarling and R.J. Hanson, ACM Trans. Math. Soft. 14 (1988) 18; J.J. Dongarra, J. Du Croz, I.S. Duff and S. Hammarling, ACM Trans. Math. Soft. 16 (1990) 1, 18.
[4] E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov and D. Sorensen, LAPACK User's Guide (Society for Industrial and Applied Mathematics, Philadelphia, 1992).
[5] A. Bar-Shalom and M. Klapisch, Comput. Phys. Commun. 50 (1988) 375.
[6] K. G. Dyall, Comput. Phys. Cpmmun. 39 (2986) 141.