208
5 Formulation and Implementation of the Gamow Shell Model
5.5
Optimization of the Gamow Shell Model One-Body Basis
While the Berggren basis is theoretically complete for any finite-range potential,
one cannot use an arbitrary potential to generate the one-body basis of the Gamow
shell model. A typical example of this problem is 8 He, in a model space of 4 He core
plus four valence neutrons, as the 0p 3/2 single-particle states of 5 He have a sizable
width, whereas 8 He is weakly bound. As the single-particle approximation of the
8 He ground state has a very large width, close to 500 keV, a very strong continuum
coupling involving the four valence neutrons occurs to bind the 8 He ground state.
On the contrary, using a Berggren basis in which the 0p 3/2 single-particle state is
slightly bound leads to stable calculations, where particle-hole excitations in the
continuum are small, so that truncating at 2p-2h level is generally sufficient.
This effect shows that the Berggren bases generated by different potentials are not
equivalent in practical calculations, even though they are all theoretically complete.
This arises from a necessary discretization of Berggren basis contours and truncation
of Gamow shell model spaces. Indeed, the matrix elements provided by different
Berggren basis sets of one-body states can vary significantly. For calculations to
be most stable, it is then necessary to devise a Berggren basis with which 1p1h excitations to the continuum are as small as possible. Moreover, while it is
sufficient in principle to enclose all resonances of the basis between the L + contour
of complex-energy scattering states and the real k-axis (see Sect. 3.5), the presence
of a resonance too close to the L + contour creates numerical instabilities. Indeed,
the rapid change of the phase shift of complex-energy scattering states close to
the resonance generates large numerical cancellations between off-diagonal matrix
elements involving these scattering states.
A simple and effective contour which is made of three segments can be devised
to avoid this problem. The first segment starts from k = 0 and ends at k peak ,
where (k peak ) (k res ) and (k peak ) −−(k res ), with k res the complex linear
momentum of the resonant state in the Berggren basis. The second segment starts
from k peak and ends in k middle 0.5–1 fm −1 , while the third segment starts from
k middle and ends in k max 2–5 fm −1 . The numerical advantage of this form of
contour is twofold. On the one hand, it encompasses the resonance state of the
Berggren basis and is sufficiently far from it not to generate numerical instabilities.
On the other hand, as the phase shift of scattering states varies slowly at high
energy, one can take fewer points on the third segment without inducing numerical
inaccuracies.
Let us now consider the more difficult problem of optimization of the basisgenerating potential. The Hartree-Fock potential provides with a very good singleparticle basis in the case of closed-shell nuclei, as it is spherical and all 1p-1h
excitations from the Hartree-Fock ground state vanish. However, it is no longer
the case for open-shell nuclei, as its Hartree-Fock potential is no longer spherical.
Deformation in Hartree-Fock potentials arises in open-shell nuclei because the
one-body states of an open shell, of different angular momentum projections, are
unevenly occupied, so that spherical symmetry is broken.
5 Formulation and Implementation of the Gamow Shell Model
5.5
Optimization of the Gamow Shell Model One-Body Basis
While the Berggren basis is theoretically complete for any finite-range potential,
one cannot use an arbitrary potential to generate the one-body basis of the Gamow
shell model. A typical example of this problem is 8 He, in a model space of 4 He core
plus four valence neutrons, as the 0p 3/2 single-particle states of 5 He have a sizable
width, whereas 8 He is weakly bound. As the single-particle approximation of the
8 He ground state has a very large width, close to 500 keV, a very strong continuum
coupling involving the four valence neutrons occurs to bind the 8 He ground state.
On the contrary, using a Berggren basis in which the 0p 3/2 single-particle state is
slightly bound leads to stable calculations, where particle-hole excitations in the
continuum are small, so that truncating at 2p-2h level is generally sufficient.
This effect shows that the Berggren bases generated by different potentials are not
equivalent in practical calculations, even though they are all theoretically complete.
This arises from a necessary discretization of Berggren basis contours and truncation
of Gamow shell model spaces. Indeed, the matrix elements provided by different
Berggren basis sets of one-body states can vary significantly. For calculations to
be most stable, it is then necessary to devise a Berggren basis with which 1p1h excitations to the continuum are as small as possible. Moreover, while it is
sufficient in principle to enclose all resonances of the basis between the L + contour
of complex-energy scattering states and the real k-axis (see Sect. 3.5), the presence
of a resonance too close to the L + contour creates numerical instabilities. Indeed,
the rapid change of the phase shift of complex-energy scattering states close to
the resonance generates large numerical cancellations between off-diagonal matrix
elements involving these scattering states.
A simple and effective contour which is made of three segments can be devised
to avoid this problem. The first segment starts from k = 0 and ends at k peak ,
where (k peak ) (k res ) and (k peak ) −−(k res ), with k res the complex linear
momentum of the resonant state in the Berggren basis. The second segment starts
from k peak and ends in k middle 0.5–1 fm −1 , while the third segment starts from
k middle and ends in k max 2–5 fm −1 . The numerical advantage of this form of
contour is twofold. On the one hand, it encompasses the resonance state of the
Berggren basis and is sufficiently far from it not to generate numerical instabilities.
On the other hand, as the phase shift of scattering states varies slowly at high
energy, one can take fewer points on the third segment without inducing numerical
inaccuracies.
Let us now consider the more difficult problem of optimization of the basisgenerating potential. The Hartree-Fock potential provides with a very good singleparticle basis in the case of closed-shell nuclei, as it is spherical and all 1p-1h
excitations from the Hartree-Fock ground state vanish. However, it is no longer
the case for open-shell nuclei, as its Hartree-Fock potential is no longer spherical.
Deformation in Hartree-Fock potentials arises in open-shell nuclei because the
one-body states of an open shell, of different angular momentum projections, are
unevenly occupied, so that spherical symmetry is broken.