216
5 Formulation and Implementation of the Gamow Shell Model
also developed new features absent from the many-body fermion dynamics code,
which deal with preconditioning (see Sect. 5.7) and angular momentum projection
(see Sect. 5.8.4), in order to accelerate the convergence the used eigensolver.
5.8.1 Slater Determinants and Partitions in Shell Model Codes
Configurations (also called partitions) and Slater determinants form the many-body
basic objects to build the Gamow shell model many-body basis. A Slater determinant is defined from the occupied one-body states by valence nucleons, while
a configuration enumerates its occupied shells, independently of the m quantum
numbers of the one-body states occupied in Slater determinants. For example,
if |0s 1/2 (−1/2) 0s 1/2 (1/2) 0p 3/2 (−3/2) 0p 3/2 (3/2) is a Slater determinant, its
associated configuration is [0s 2
1/2 0p 2
3/2 ]. One then has fewer configurations than
Slater determinants.
This separation leads to many advantages. Firstly, configurations must be
generated in a sequential manner due to the truncation imposed to the valence space.
This step is quick as configurations are in small number. As Slater determinants
of different configurations are independent and there are no truncations inside a
configuration, it is straightforward to parallelize routines involving the construction
of the Slater determinants of a fixed configuration. Looking for Slater determinant
indices, which demands the use of binary search, is also much faster, as binary
search is effected at the level of configuration first, and of Slater determinant
afterward. This also allows to save memory, as all indices depending on shells only
can be stored in arrays involving configurations only, while those depending on the
m quantum number only are handled in arrays dealing with Slater determinants. As
a consequence, the consideration of configuration first and its Slater determinants
afterward is always done when building the ˆ
H matrix.
In the Gamow shell model, contrary to standard shell model, basis configurations
and Slater determinants are built from few valence particles and many one-body
shells and states. Therefore, one chose to use a different implementation to represent
configurations and Slater determinants as in standard shell model. In the standard
shell model, the method of choice is to store Slater determinants via the bit
storage [35]. Indeed, due to the Pauli principle, a state can be occupied by one
nucleon at most, so that one can build a one-to-one correspondence between
Slater determinants and binary numbers. For example, |1001100000 is a Slater
determinant where the states 1, 4, and 5 are occupied, while the states 2, 3, and 6,
. . . , 10 are not occupied. All operations on Slater determinants can then be reduced
to operations on bits, so that the action of creation/annihilation operators becomes
very fast. The downside of this method is that it becomes inefficient if many states
are unoccupied. Indeed, as one integer possesses 32 bits, it is most efficient if one
has at most than 32 basis one-body states.
Typically, the sp partial waves are represented by Berggren basis contours in
the Gamow shell model, discretized with 15–30 points each, whereas the d partial
waves are built from 10 states generated by a harmonic oscillator potential. As a
5 Formulation and Implementation of the Gamow Shell Model
also developed new features absent from the many-body fermion dynamics code,
which deal with preconditioning (see Sect. 5.7) and angular momentum projection
(see Sect. 5.8.4), in order to accelerate the convergence the used eigensolver.
5.8.1 Slater Determinants and Partitions in Shell Model Codes
Configurations (also called partitions) and Slater determinants form the many-body
basic objects to build the Gamow shell model many-body basis. A Slater determinant is defined from the occupied one-body states by valence nucleons, while
a configuration enumerates its occupied shells, independently of the m quantum
numbers of the one-body states occupied in Slater determinants. For example,
if |0s 1/2 (−1/2) 0s 1/2 (1/2) 0p 3/2 (−3/2) 0p 3/2 (3/2) is a Slater determinant, its
associated configuration is [0s 2
1/2 0p 2
3/2 ]. One then has fewer configurations than
Slater determinants.
This separation leads to many advantages. Firstly, configurations must be
generated in a sequential manner due to the truncation imposed to the valence space.
This step is quick as configurations are in small number. As Slater determinants
of different configurations are independent and there are no truncations inside a
configuration, it is straightforward to parallelize routines involving the construction
of the Slater determinants of a fixed configuration. Looking for Slater determinant
indices, which demands the use of binary search, is also much faster, as binary
search is effected at the level of configuration first, and of Slater determinant
afterward. This also allows to save memory, as all indices depending on shells only
can be stored in arrays involving configurations only, while those depending on the
m quantum number only are handled in arrays dealing with Slater determinants. As
a consequence, the consideration of configuration first and its Slater determinants
afterward is always done when building the ˆ
H matrix.
In the Gamow shell model, contrary to standard shell model, basis configurations
and Slater determinants are built from few valence particles and many one-body
shells and states. Therefore, one chose to use a different implementation to represent
configurations and Slater determinants as in standard shell model. In the standard
shell model, the method of choice is to store Slater determinants via the bit
storage [35]. Indeed, due to the Pauli principle, a state can be occupied by one
nucleon at most, so that one can build a one-to-one correspondence between
Slater determinants and binary numbers. For example, |1001100000 is a Slater
determinant where the states 1, 4, and 5 are occupied, while the states 2, 3, and 6,
. . . , 10 are not occupied. All operations on Slater determinants can then be reduced
to operations on bits, so that the action of creation/annihilation operators becomes
very fast. The downside of this method is that it becomes inefficient if many states
are unoccupied. Indeed, as one integer possesses 32 bits, it is most efficient if one
has at most than 32 basis one-body states.
Typically, the sp partial waves are represented by Berggren basis contours in
the Gamow shell model, discretized with 15–30 points each, whereas the d partial
waves are built from 10 states generated by a harmonic oscillator potential. As a