222
5 Formulation and Implementation of the Gamow Shell Model
row master node then casts the part 2 of the input Gamow shell model vector to
its row and column.
• Using message passing interface overlapping, one multiplies at the same time the
considered ˆ
H square by the input Gamow shell model vector part to obtain a part
of the output Gamow shell model vector. The latter is then reduced on the row
via the row communicator.
• Using message passing interface overlapping, one multiplies at the same time
the considered ˆ
H square by the storage vector mentioned above, which takes
care of the symmetric part of ˆ
H , not stored. A table of n c vectors is used therein
for output, where n c is the number of threads per node, in order to avoid a race
condition, i.e., a modification of the same variable by several threads at the same
time, which leads to unpredictable results (see Ref. [34]). The race condition
would occur because input and output indices are exchanged when one uses ˆ
H
symmetry.
• All resulting parts of ˆ
H squares times input Gamow shell model vector parts are
summed among the n c threads and then reduced on the column via the column
communicator. The output Gamow shell model vector is then obtained.
The cost of message passing interface communications is then that of the transfer
of 4 d/n d complex numbers, with two transfers out of four which overlap with
multiplications.
5.8.4 Treatment of the Angular Momentum Projection
The basis of Slater determinants is not rotationally invariant. While the total
angular momentum projection M is conserved, the total angular momentum J
is not. As explained in Ref. [36], both the requirements of rotational invariance
and antisymmetry at basis level considerably increase complexity in building ˆ
H .
Moreover, despite the smaller dimensions involved in a J -conserving basis, the
relative number of nonzero matrix elements is much larger. Hence, the use of a
basis of Slater determinants is more efficient computationally.
As [H, J] = 0, applying ˆ
H to a linear combination of Slater determinants
coupled to J still provides a vector coupled to J . However, in practice, ˆ
H and J
do not exactly commute due to numerical inaccuracy, so that the J quantum number
is eventually lost after several matrix-vector multiplications.
It is possible to recover rotational invariance by suppressing the Gamow shell
model vector components with J = J . For this, one uses the Löwdin operator [37]:
ˆ
P J =
J =J
ˆ
J 2 − J (J + 1)
J (J + 1) − J (J + 1)
,
(5.71)
5 Formulation and Implementation of the Gamow Shell Model
row master node then casts the part 2 of the input Gamow shell model vector to
its row and column.
• Using message passing interface overlapping, one multiplies at the same time the
considered ˆ
H square by the input Gamow shell model vector part to obtain a part
of the output Gamow shell model vector. The latter is then reduced on the row
via the row communicator.
• Using message passing interface overlapping, one multiplies at the same time
the considered ˆ
H square by the storage vector mentioned above, which takes
care of the symmetric part of ˆ
H , not stored. A table of n c vectors is used therein
for output, where n c is the number of threads per node, in order to avoid a race
condition, i.e., a modification of the same variable by several threads at the same
time, which leads to unpredictable results (see Ref. [34]). The race condition
would occur because input and output indices are exchanged when one uses ˆ
H
symmetry.
• All resulting parts of ˆ
H squares times input Gamow shell model vector parts are
summed among the n c threads and then reduced on the column via the column
communicator. The output Gamow shell model vector is then obtained.
The cost of message passing interface communications is then that of the transfer
of 4 d/n d complex numbers, with two transfers out of four which overlap with
multiplications.
5.8.4 Treatment of the Angular Momentum Projection
The basis of Slater determinants is not rotationally invariant. While the total
angular momentum projection M is conserved, the total angular momentum J
is not. As explained in Ref. [36], both the requirements of rotational invariance
and antisymmetry at basis level considerably increase complexity in building ˆ
H .
Moreover, despite the smaller dimensions involved in a J -conserving basis, the
relative number of nonzero matrix elements is much larger. Hence, the use of a
basis of Slater determinants is more efficient computationally.
As [H, J] = 0, applying ˆ
H to a linear combination of Slater determinants
coupled to J still provides a vector coupled to J . However, in practice, ˆ
H and J
do not exactly commute due to numerical inaccuracy, so that the J quantum number
is eventually lost after several matrix-vector multiplications.
It is possible to recover rotational invariance by suppressing the Gamow shell
model vector components with J = J . For this, one uses the Löwdin operator [37]:
ˆ
P J =
J =J
ˆ
J 2 − J (J + 1)
J (J + 1) − J (J + 1)
,
(5.71)