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
- [Novoselsky,88a] A. Novoselsky, J. Katriel and R. Gilmore, J. Math. Phys.
29, 1368 (1988).
- [Novoselsky,88b] A. Novoselsky, M. Vallières and R. Gilmore, Phys. Rev.
C 38, 1440 (1988).
- [Chen,89] J.-Q. Chen, A. Novoselsky, M. Vallières and R. Gilmore, Phys.
Rev. C 39, 1088 (1989).
- [Vallieres,93] M. Vallières and A. Novoselsky, Nucl. Phys. A 570,
345c (1993).