8.2 Benchmarking of No-Core Gamow Shell Model Against Other Methods
375
using the Lawson method. One considers therein the s 1/2 , p 1/2 and p 3/2 proton and
neutron partial waves as Berggren basis states, whereas all other partial waves, for
which 2 ≤ ≤ 10, are described by the harmonic oscillator functions. The Berggren
basis in 3 H is generated by a Woods–Saxon potential [94]. The single-particle pole
states for 3 H in this basis are 0s 1/2 for protons and neutrons. Bound 0p 3/2 proton
and neutron pole states are added for the 4 He calculations. All poles of the Berggen
basis are bound, as the ground states of 3 H and 4 He are well bound so that the
continuum couplings are negligible. The resonances appearing in the partial waves
of Berggren basis for ≥ 2 are very high in energy, so that it is sufficiently precise
to take real-energy contours for the Berggren basis.
In standard shell model using a basis of harmonic oscillator states, the N ¯
hω
many-body basis space is defined by taking all the Slater determinants having
a harmonic oscillator energy of N ¯
hω or less. In order to effectively fulfill this
condition in the Berggren basis calculation, one assigns equivalent harmonic
oscillator energies to valence Berggren shells. For this, the bound 0s 1/2 and 0p 3/2
one-body states have an equivalent harmonic oscillator energy of 0 ¯
hω and 1 ¯
hω,
respectively. Scattering s 1/2 Berggren basis shells are assigned a harmonic oscillator
energy of 2 ¯
hω. As there are no Berggren basis poles besides 0s 1/2 states for 3 H, the
scattering p 3/2 and p 1/2 Berggren basis contours are given an energy of 1 ¯
hω. One
does not have any 0p 1/2 Berggren basis pole states for 4 He as well. Hence, the p 1/2
Berggren basis scattering states are also assigned an energy of 1 ¯
hω. 0p 3/2 bound
states which have an energy of 1 ¯
hω are present in the Berggren basis of 4 He, so that
the scattering p 3/2 Berggren basis contour are given an energy of 3 ¯
hω. By assigning
the ¯
hω-dependent values to scattering one-body basis states, one insures that N ¯
hω
spaces are implicitly included in the Berggren basis.
Let us consider the 1s 1/2 harmonic oscillator state as an example. Due to the
completeness of the Berggren basis, it can be expanded using basis states on the
s 1/2 contour. As the s 1/2 scattering states have an energy of 2 ¯
hω, the 1s 1/2 harmonic
oscillator state is then always implicitly included in the Gamow shell model basis
space by demanding N ¯
hω ≥ 2 ¯
hω.
For 3 H, the contours in Berggren basis are truncated at k max = 4 fm −1 and
discretized with 22 states per contour. For 4 He, k max = 3 fm −1 and one takes 15
states per contour. As the Berggren basis is complete, the harmonic oscillator N ¯
hω
many-body states are effectively included in the Berggren many-body basis spanned
by Slater determinants of equivalent harmonic oscillator energy smaller or equal to
N ¯
hω. Consequently, one can expect the Lawson method to converge if a sufficiently
large constant λ is used in the Hamiltonian ˆ
H +λ ˆ
H CM defined in Eqs. (8.1) and (8.6).
For example, the convergence of the expectation value of H CM to zero is achieved
for λ ≥ 5. The harmonic oscillator length has been adjusted so as to provide the
smallest expectation value of ˆ
H CM and is, respectively, equal to 1.6 fm and 1.5 fm
for the ground states of 3 H and 4 He. The N ¯
hω spaces used run from N = 2 to
N = 8 what is sufficient to quantify the center-of-mass properties.
The ground state energies of 3 H (left panel) and 4 He (right panel) are depicted
in Fig. 8.2, while the expectation values of ˆ
H CM , associated with these energies, are
shown in Fig. 8.3. One can see that the ground state energies, which are calculated
375
using the Lawson method. One considers therein the s 1/2 , p 1/2 and p 3/2 proton and
neutron partial waves as Berggren basis states, whereas all other partial waves, for
which 2 ≤ ≤ 10, are described by the harmonic oscillator functions. The Berggren
basis in 3 H is generated by a Woods–Saxon potential [94]. The single-particle pole
states for 3 H in this basis are 0s 1/2 for protons and neutrons. Bound 0p 3/2 proton
and neutron pole states are added for the 4 He calculations. All poles of the Berggen
basis are bound, as the ground states of 3 H and 4 He are well bound so that the
continuum couplings are negligible. The resonances appearing in the partial waves
of Berggren basis for ≥ 2 are very high in energy, so that it is sufficiently precise
to take real-energy contours for the Berggren basis.
In standard shell model using a basis of harmonic oscillator states, the N ¯
hω
many-body basis space is defined by taking all the Slater determinants having
a harmonic oscillator energy of N ¯
hω or less. In order to effectively fulfill this
condition in the Berggren basis calculation, one assigns equivalent harmonic
oscillator energies to valence Berggren shells. For this, the bound 0s 1/2 and 0p 3/2
one-body states have an equivalent harmonic oscillator energy of 0 ¯
hω and 1 ¯
hω,
respectively. Scattering s 1/2 Berggren basis shells are assigned a harmonic oscillator
energy of 2 ¯
hω. As there are no Berggren basis poles besides 0s 1/2 states for 3 H, the
scattering p 3/2 and p 1/2 Berggren basis contours are given an energy of 1 ¯
hω. One
does not have any 0p 1/2 Berggren basis pole states for 4 He as well. Hence, the p 1/2
Berggren basis scattering states are also assigned an energy of 1 ¯
hω. 0p 3/2 bound
states which have an energy of 1 ¯
hω are present in the Berggren basis of 4 He, so that
the scattering p 3/2 Berggren basis contour are given an energy of 3 ¯
hω. By assigning
the ¯
hω-dependent values to scattering one-body basis states, one insures that N ¯
hω
spaces are implicitly included in the Berggren basis.
Let us consider the 1s 1/2 harmonic oscillator state as an example. Due to the
completeness of the Berggren basis, it can be expanded using basis states on the
s 1/2 contour. As the s 1/2 scattering states have an energy of 2 ¯
hω, the 1s 1/2 harmonic
oscillator state is then always implicitly included in the Gamow shell model basis
space by demanding N ¯
hω ≥ 2 ¯
hω.
For 3 H, the contours in Berggren basis are truncated at k max = 4 fm −1 and
discretized with 22 states per contour. For 4 He, k max = 3 fm −1 and one takes 15
states per contour. As the Berggren basis is complete, the harmonic oscillator N ¯
hω
many-body states are effectively included in the Berggren many-body basis spanned
by Slater determinants of equivalent harmonic oscillator energy smaller or equal to
N ¯
hω. Consequently, one can expect the Lawson method to converge if a sufficiently
large constant λ is used in the Hamiltonian ˆ
H +λ ˆ
H CM defined in Eqs. (8.1) and (8.6).
For example, the convergence of the expectation value of H CM to zero is achieved
for λ ≥ 5. The harmonic oscillator length has been adjusted so as to provide the
smallest expectation value of ˆ
H CM and is, respectively, equal to 1.6 fm and 1.5 fm
for the ground states of 3 H and 4 He. The N ¯
hω spaces used run from N = 2 to
N = 8 what is sufficient to quantify the center-of-mass properties.
The ground state energies of 3 H (left panel) and 4 He (right panel) are depicted
in Fig. 8.2, while the expectation values of ˆ
H CM , associated with these energies, are
shown in Fig. 8.3. One can see that the ground state energies, which are calculated