DUSM

DUSM-Novosleky-Vallières

The Drexel University Shell-Model (DUSM) code


The algorithm used in the DUSM code was introduced 10 years ago [Novoselsky,88a]-[Chen,89]. It addresses the difficult task of building the Hamiltonian matrix of a nuclear many-body system in antisymmetrized multi-shell configurations in multiple angular momentum coupling schemes like l-s (orbital angular momentum-spin) coupling, or j-t (spin-isospin) coupling. The multi-shell states in these coupling schemes arise normally from one of the subspaces, e.g., l in l-s and j in j-t. We use novel permutation group concepts to build the final antisymmetric quantum states in steps: we first construct {\it arbitrary} permutational symmetry states for nucleons in each shell of the multi-shell subspace, e.g., in each j shell in spin-isospin coupling. This recursive buildup is accomplished by diagonalizing the second order Casimir operator of the permutation group to obtain the coupling coefficients, and is a very stable algorithm [Novoselsky,88a]. The same algorithm is used to build single-shell states in the complementary subspace (e.g., the isospin subspace in spin-isospin coupling). A similar diagonalization method is used to calculate the outer-product and inner-product coupling coefficients (isoscalar) factors of the permutation group [Chen,89]. These coefficients allow the buildup of multi-shell states in the corresponding subspace (e.g., j) and to couple conjugate symmetry states from the complementary subspace (e.g., t) to form final antisymmetric states using a sum over paths method.

This approach of building the Hamiltonian matrix in steps has several advantages. It is numerically stable because our recursive schemes to calculate coupling coefficients are based on diagonalization methods. It reduces the I/O since the coupling coefficients and matrix elements of elementary operators only need to be stored in the individual subspaces (and not in a coupled form); for instance only 60 MB of input files are needed to perform calculations in the fp-shell for up to 12 particles. It generates a complete and orthonormal basis with explicit permutational symmetry. Finally, it produces a deterministic and fast algorithm, i.e., one which never needs a search. Therefore, this method solves the difficulties for constructing totally antisymmetric states in the multi-shell basis, mentioned in the previous subsection. In addition, it produces a fast and stable algorithm to build the many-body Hamiltonian matrix elements.

Since the wavefunctions are labeled by the arbitrary permutational symmetry of each subshell, additional permutational selection rules automatically apply. As a result the large Hamiltonian matrices are found to be very sparse; for matrices of order of hundreds of thousands only a few percent of the elements are non-zero. This is extremely important since the most time-consuming task of calculating the matrix elements is proportional to the number of non-zero elements. Other shell model methods lead to matrices which are significantly less sparse.

Another advantage of using our symmetry-adapted basis is that the Hamiltonian matrix can be decomposed into submatrices according to the quantum numbers generated by the symmetries. Each submatrix consists of the angular momentum states that belong to a given subshell symmetry. This greatly simplifies the calculation of the matrix elements since all the calculations that are related to the permutation symmetries of the subshells are common to all the states in the submatrix. The calculation of these independent submatrices can be done on different processors of a parallel machine. This is the key point that makes our method especially efficient for parallel computers.

The DUSM computer code [Vallieres,93] implemented our new approach for performing shell model calculations in the j-t coupling scheme, where the nucleons are distributed among several subshells. It outperforms other state-of-the-art Shell-Model codes by a factor of 4-5 in CPU usage with much reduced disk space and I/O requirements for sd-shell calculations. For fp-shell nuclei the improvements are even much better.


References


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