5.8 Memory Management and Two-Dimensional Partitioning
217
whole, 340 one-body states are generated, so that one would need 11 integers in
order to store one Slater determinant. This would be inefficient from both memory
and speed points of view, as the computing efficiency of bit algebra would be
hindered by the numerous look-ups of integers in arrays. Therefore, it has been
preferred to store configurations and Slater determinants as integer arrays, so that
the Slater determinant of the example considered in the previous paragraph becomes
{1, 4, 5}. It has been noticed that calculations are nevertheless rather fast when using
this storage scheme. Added to that, the parallelization of the routines using basis
configurations and Slater determinants is straightforward.
One has seen that it would be impossible to store all the data related to the
neutron space in the Gamow shell model if the considered nucleus has a large
neutron-to-proton ratio (from symmetry arguments, the situation is analogous if
one has more valence protons than valence neutrons). This arises due to the use
of the Berggren basis, because it is customary to have 100–200 neutron valence
shells on the discretized scattering contours, whereas the no-core shell model
has, for example, 30 neutron shells in a 8 ¯
hω space. The recalculation of manybody neutron matrix elements would also take too much time. Consequently, one
devised memory optimization schemes in order to store many-body matrix elements
between neutron Slater determinants. A short denomination of the latter matrix
elements are the phases, because one has to store matrix elements of the form
SD f |a †
α |SD i SD f |a †
α a β |SD i , equal to ±1. Clearly, the indices of involved onebody shells and states, as well as those of configurations and Slater determinants,
must also be included in the storage.
Another useful memory optimization comes from m-reversal symmetry, where
ˆ ¯
T |j m = (−1)
j −m
|j − m
by definition for a one-body state of total angular quantum number j and angular
quantum projection m. In the case of the storage of phases, if one applies ˆ ¯
T to
all states present in SD f |a †
α |SD i SD f |a †
α a β |SD i , . . . , the obtained phase and
associated indices can be easily recovered from the initial phase. This allows to
provide an additional memory gain factor of about 2 for the storage of phases. One
will also see in the following that m-reversal symmetry allows to halve the cost of
computation of Hamiltonian times vector for even-even nuclei.
5.8.2 Memory Management of the Hamiltonian
Due to the rapid growth of matrix dimension with the number of valence nucleons,
memory utilization must be carefully assessed. In order to compare standard shell
model and Gamow shell model for that matter, one will consider N v valence
nucleons in a many-body space generated by N s states. As one aims at giving
an overall order of magnitude for used memory, antisymmetry, parity and M
projection will be neglected in the following discussion, as their impact is small
217
whole, 340 one-body states are generated, so that one would need 11 integers in
order to store one Slater determinant. This would be inefficient from both memory
and speed points of view, as the computing efficiency of bit algebra would be
hindered by the numerous look-ups of integers in arrays. Therefore, it has been
preferred to store configurations and Slater determinants as integer arrays, so that
the Slater determinant of the example considered in the previous paragraph becomes
{1, 4, 5}. It has been noticed that calculations are nevertheless rather fast when using
this storage scheme. Added to that, the parallelization of the routines using basis
configurations and Slater determinants is straightforward.
One has seen that it would be impossible to store all the data related to the
neutron space in the Gamow shell model if the considered nucleus has a large
neutron-to-proton ratio (from symmetry arguments, the situation is analogous if
one has more valence protons than valence neutrons). This arises due to the use
of the Berggren basis, because it is customary to have 100–200 neutron valence
shells on the discretized scattering contours, whereas the no-core shell model
has, for example, 30 neutron shells in a 8 ¯
hω space. The recalculation of manybody neutron matrix elements would also take too much time. Consequently, one
devised memory optimization schemes in order to store many-body matrix elements
between neutron Slater determinants. A short denomination of the latter matrix
elements are the phases, because one has to store matrix elements of the form
SD f |a †
α |SD i SD f |a †
α a β |SD i , equal to ±1. Clearly, the indices of involved onebody shells and states, as well as those of configurations and Slater determinants,
must also be included in the storage.
Another useful memory optimization comes from m-reversal symmetry, where
ˆ ¯
T |j m = (−1)
j −m
|j − m
by definition for a one-body state of total angular quantum number j and angular
quantum projection m. In the case of the storage of phases, if one applies ˆ ¯
T to
all states present in SD f |a †
α |SD i SD f |a †
α a β |SD i , . . . , the obtained phase and
associated indices can be easily recovered from the initial phase. This allows to
provide an additional memory gain factor of about 2 for the storage of phases. One
will also see in the following that m-reversal symmetry allows to halve the cost of
computation of Hamiltonian times vector for even-even nuclei.
5.8.2 Memory Management of the Hamiltonian
Due to the rapid growth of matrix dimension with the number of valence nucleons,
memory utilization must be carefully assessed. In order to compare standard shell
model and Gamow shell model for that matter, one will consider N v valence
nucleons in a many-body space generated by N s states. As one aims at giving
an overall order of magnitude for used memory, antisymmetry, parity and M
projection will be neglected in the following discussion, as their impact is small