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Manuscript Title: SU(3) reduced matrix element package.
Authors: C. Bahri, J.P. Draayer
Program title: SU3RME
Catalogue identifier: ACVI_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 83(1994)59
Programming language: Fortran.
Computer: IBM 3090/600J.
Operating system: MVS/XA, UNIX, AIX 3.2.5.
Keywords: SU(3), SU(2), SO(3), Reduced matrix elements, Vector coupling coefficients, Wigner coefficients, Clebsch-Gordan Coefficients, Isoscalar factors, Coefficients of Fractional parentage, Second quantization, Tensor operators, Nuclear structure, Model shell, Algebraic model, Collective model, Harmonic oscillator, Deformed nuclei, Weighted search tree, Dynamical symmetry, Pseudo-spin, Pseudo-SU(3), Pseudo-SU(2).
Classification: 4.2, 17.18.

Nature of problem:
It is well-known from mean-field studies that a three-dimensional harmonic oscillator is a reasonable zeroth-order approximation for the common potential experienced by nucleons in nuclei. The choice of SU(3) as a basic symmetry is therefore natural since it is the exact symmetry group of the spherical oscillator, and furthermore, it is the dynamical symmetry group of the deformed oscillator when, as is usually the case, the deformation is generated by quadrupole interactions [1-3]. Nonetheless, for real nuclear systems SU(3) is only an approximate symmetry and therefore the calculation of SU(3) reduced matrix elements (RMEs) is important for including symmetry breaking terms in the Hamiltonian as well as for evaluating matrix elements of transition operators that are not SU(3) scalars.

By exploiting logical operations and bit manipulation techniques - which enter naturally because a second-quantized formulation of the basic theory is employed - in conjunction with standard numerical methods, SU(3) RMEs are calculated for SU(3)-coupled products of various combinations of creation and annihilation operators. The procedure is implemented in the SU(3)supsetSU(2)timesU(1) coupling scheme since it is conceptually and mathematically the simplest one to use (normally called the canonical coupling scheme in the literature). The evaluation of matrix elements in the SU(3)supsetSO(3) scheme (usually referred to as the non-canonical but physically important coupling scheme - at least for nuclear physics applications) can be obtained by multiplying these RMEs with the appropriate coupling coefficients which are available through the SU3 package [4].

Solution method:
The raising operators of SU(3) as well as those of spin, SU(2), are used to generate SU(3)timesSU(2) highest-weight states (|alphaHW>). Unit tensor operators of the appropriate particle rank, which can be considered as inside-out products of basis states (|betaOP><alphaOP|), are then generated using logic that is similar to what is used in constructing the highest-weight states but augmented with appropriate lowering operations to achieve the required (non-highest weight) realization of the SU(3)timesSU(2) symmetries. Matrix elements of these operators are then determined by sandwiching them between the sets of required initial and final highest-weight states (<betaHW|betaOP><alphaOP|alphaHW>). Unlike similar work of some years ago [5,6], the current implementation exploits logical operations and bit manipulation techniques in conjunction with the use of weighted search trees that allow for the storage of intermediate results. The use of this companion weighted search tree (WST) package means that repeated and sometimes lengthy calculation can be avoided [7].

There are no intrinsic program limitations other than those relating to the structure of the operators - they can only be of the a!, a!a!, a, aa,a!a, a!a!a, a!aa and a!a!aa types. This cn be extended either indirectly through the use of SU(3) and SU(2) coupling and recoupling procedures applied to the existing forms, or directly by extending the existing progrm logic to incorporate additional creation and annihilation operator combinations. As dimension overflow conditions are possible, they are flagged and can be fixed by following the directions that are given as part of the error message.

Unusual features:
The user must select the appropriate form for the implicit statement at the beginning of each program and subroutine. For example, for applications running under the IBM MVS/XA compiler the implicit real*8 option has to be selected whereas for some other compilers the corresponding implicit double precision statement must be used. On systems not having an extended precision option these statements must both be commented out.

Running time:
The program executes very efficiently because the basic procedures involve simple logical operations and bit manipulation techniques. For example, a test case consisting of four particles in the gds-shell, all possible one and two-body operators, and three pairs of bras and kets take only a few seconds to execute.

[1] J.P. Elliott, Proc. Roy. Soc. London Ser. A245, 128 (1958).
[2] J.P. Elliott, Proc. Roy. Soc. London Ser. A245, 562 (1958).
[3] J.P. Elliott and M. Harvey, Proc. Roy. Soc. London Ser. A272, 557 (1963).
[4] Y. Akiyama and J.P. Draayer, Comp. Phys. Comm. 5, 405 (1973).
[5] D. Braunschweig, Comp. Phys. Comm. 14, 109 (1978).
[6] D. Braunschweig, Comp. Phys. Comm. 15, 259 (1978).
[7] S.C. Park, C. Bahri, J.P. Draayer and S.-Q. Zheng (to be published in Comp. Phys. Comm.); see also S.C. Park, J.P. Draayer and S. Zheng, Time-Space Optimal Numerical Database for Large-Scale Scientific Applications, in International Computer Symposium 1990, December 17-19, 1990 (Hsinchu, Taiwan, ROC, 1990).