374
8 No-Core Gamow Shell Model
waves for protons and neutrons. The basis-generating potential is that of the multiSlater determinant coupled Hartree–Fock which binds the proton and neutron 0s 1/2
states with energies −10.417 MeV and −11.982 MeV, respectively. The scattering
continua i{s 1/2 }, i{p 3/2 }, i{p 1/2 } are chosen along the real k−axis and each of them
is discretized with 18 points. Here i denotes the number of discretization points
which ranges from 1 to 18 for the s-wave and from 0 to 18 for the p-waves.
The harmonic oscillator length is fixed at b = 1.5 fm in order to provide with
energies closest to experimental values. ¯
hω is then equal to 18.5 MeV. d-waves are
represented by five harmonic oscillator states, which from now on are denoted as
5d. The physical argument behind this choice lies in the fact that for > 1 the
centrifugal barrier is large enough to confine the single-particle states in the potential
well and, hence, the harmonic oscillator basis becomes a realistic alternative. For f
and g partial waves, one used three harmonic oscillator states. In total, the singleparticle basis consists of 154 partial waves. The no-core Gamow shell model energy
of 3 H is then: E NCGSM = −8.39 MeV, whereas the result of Fadeev calculation [95]
is: E Fadeev = −8.47 MeV.
Figure 8.1 serves as an illustration of the rate of convergence of the energy with
the angular momentum of basis states. An exponential fit of the no-core Gamow
shell model results for different values of the single-particle basis allows to
extrapolate energies in the non-truncated model space. The extrapolated energy is:
E extrp = −8.449±0.087 MeV, where the fit function is: E = E extrp +b ×e −cc . One
sees that the inclusion of partial waves with angular momentum larger than = 4,
should have a very small contribution, of the order of 50 keV, to the eigenenergy.
In the above no-core Gamow shell model calculation, the features related to the
center-of mass have not been assessed. Thus, we will now calculate the expectation
value of the center-of-mass ˆ
H CM operator (see Eq. (8.6)) of the 3 H ground state.
Following the assertion of Ref. [93], one assumes that the no-core Gamow shell
model wave function in a sufficiently large model space factorizes and the centerof-mass wave function is that a harmonic oscillator 0s state. In the largest model
space used in Fig. 8.1, the expectation value of the ˆ
H CM is approximately 7 keV.
Consequently, one can confirm that in this case the ground state wave function of
3 H indeed separates into an intrinsic wave function and harmonic oscillator centerof-mass 0s wave function.
However, although this example has proven effectiveness of Berggren basis in the
no-core Gamow shell model applications, it cannot be considered as representative
of the general case. Indeed, while the ground state binding energy of 3 H calculated
using the local Argonne υ 18 interaction converged fast to its exact value given by
a Faddeev calculation, that convergence is much slower using the nonlocal N 3 LO
interaction. Consequently, we will use the N 3 LO interaction to quantify the centerof-mass properties in the ground states J π = 1/2 + in 3 H and J π = 0 + in 4 He.
The momentum cutoff of the N 3 LO interaction in this calculation is Λ = 1.9 fm −1 .
In this study, one uses three different methods of calculation. The first method is
that of the standard shell model, as the use of the harmonic oscillator basis states
in this model allows to separate exactly many-body wave functions in intrinsic and
center-of-mass parts. The second method is that of the no-core Gamow shell model
8 No-Core Gamow Shell Model
waves for protons and neutrons. The basis-generating potential is that of the multiSlater determinant coupled Hartree–Fock which binds the proton and neutron 0s 1/2
states with energies −10.417 MeV and −11.982 MeV, respectively. The scattering
continua i{s 1/2 }, i{p 3/2 }, i{p 1/2 } are chosen along the real k−axis and each of them
is discretized with 18 points. Here i denotes the number of discretization points
which ranges from 1 to 18 for the s-wave and from 0 to 18 for the p-waves.
The harmonic oscillator length is fixed at b = 1.5 fm in order to provide with
energies closest to experimental values. ¯
hω is then equal to 18.5 MeV. d-waves are
represented by five harmonic oscillator states, which from now on are denoted as
5d. The physical argument behind this choice lies in the fact that for > 1 the
centrifugal barrier is large enough to confine the single-particle states in the potential
well and, hence, the harmonic oscillator basis becomes a realistic alternative. For f
and g partial waves, one used three harmonic oscillator states. In total, the singleparticle basis consists of 154 partial waves. The no-core Gamow shell model energy
of 3 H is then: E NCGSM = −8.39 MeV, whereas the result of Fadeev calculation [95]
is: E Fadeev = −8.47 MeV.
Figure 8.1 serves as an illustration of the rate of convergence of the energy with
the angular momentum of basis states. An exponential fit of the no-core Gamow
shell model results for different values of the single-particle basis allows to
extrapolate energies in the non-truncated model space. The extrapolated energy is:
E extrp = −8.449±0.087 MeV, where the fit function is: E = E extrp +b ×e −cc . One
sees that the inclusion of partial waves with angular momentum larger than = 4,
should have a very small contribution, of the order of 50 keV, to the eigenenergy.
In the above no-core Gamow shell model calculation, the features related to the
center-of mass have not been assessed. Thus, we will now calculate the expectation
value of the center-of-mass ˆ
H CM operator (see Eq. (8.6)) of the 3 H ground state.
Following the assertion of Ref. [93], one assumes that the no-core Gamow shell
model wave function in a sufficiently large model space factorizes and the centerof-mass wave function is that a harmonic oscillator 0s state. In the largest model
space used in Fig. 8.1, the expectation value of the ˆ
H CM is approximately 7 keV.
Consequently, one can confirm that in this case the ground state wave function of
3 H indeed separates into an intrinsic wave function and harmonic oscillator centerof-mass 0s wave function.
However, although this example has proven effectiveness of Berggren basis in the
no-core Gamow shell model applications, it cannot be considered as representative
of the general case. Indeed, while the ground state binding energy of 3 H calculated
using the local Argonne υ 18 interaction converged fast to its exact value given by
a Faddeev calculation, that convergence is much slower using the nonlocal N 3 LO
interaction. Consequently, we will use the N 3 LO interaction to quantify the centerof-mass properties in the ground states J π = 1/2 + in 3 H and J π = 0 + in 4 He.
The momentum cutoff of the N 3 LO interaction in this calculation is Λ = 1.9 fm −1 .
In this study, one uses three different methods of calculation. The first method is
that of the standard shell model, as the use of the harmonic oscillator basis states
in this model allows to separate exactly many-body wave functions in intrinsic and
center-of-mass parts. The second method is that of the no-core Gamow shell model