5.5 Optimization of the Gamow Shell Model One-Body Basis
211
|Ψ GS , and then calculating U MSDHF and its eigenstates similarly to a Hartree-Fock
procedure (see Exercise VI for numerical illustrations).
Exercise VI
One will show in a numerical example that the multi-Slater determinant coupled
Hartree-Fock method allows to generate a basis potential with which Gamow shell
model calculations are stable and precise.
A. Run the two-particle Gamow shell model code with the modified surface
Gaussian interaction (see Eq. (9.174)) for 6 He, 6 Be, and 6 Li using bases
generated either by a Woods-Saxon potential mimicking the core or by the
multi-Slater determinant coupled Hartree-Fock potential. Notice that obtained
energies and widths are almost the same in both cases for 6 He and 6 Be and
explain why.
B. Show that calculations are problematic in the case of 6 Li when using a basis
generated by the Woods-Saxon potential of the core, especially for unbound
states.
On the contrary, by using the multi-Slater determinant coupled Hartree-Fock
potential, show that one obtains precise eigenenergies for the 6 Li spectrum.
Explain this result by considering the overall behavior of Hamiltonian
matrix elements with both basis-generating potentials.
Conclude by stating why optimization of the basis potential is necessary
in the Gamow shell model.
|Ψ GS has to be redefined if no , j state is occupied in the multi-Slater
determinant coupled Hartree-Fock ground state. Indeed, an , j state has to be
occupied in |Ψ GS to be able to apply the multi-Slater determinant coupled HartreeFock method. This occurs, for example, for the neutron p 1/2 partial wave when
considering the 6 He nucleus, as the 0p 1/2 neutron state is not occupied. In order
to solve this problem, it is sufficient to have a 1p-1h excitation to the initial multiSlater determinant coupled Hartree-Fock ground state. In the example of 6 He with
a 4 He core, |Ψ GS = |[0p 3/2 0p 1/2 ] J =1 can be used to optimize the neutron p 1/2
partial wave.
If, however, the considered , j partial wave does not possess any pole in the
basis, then its multi-Slater determinant coupled Hartree-Fock potential cannot be
defined. But the , j partial wave only occurs through the continuum coupling, so
that its effect will be much less important than the effect of pole states. It is then
sufficient to use a Woods-Saxon or harmonic oscillator potential to generate the
basis states of this particular partial wave.
211
|Ψ GS , and then calculating U MSDHF and its eigenstates similarly to a Hartree-Fock
procedure (see Exercise VI for numerical illustrations).
Exercise VI
One will show in a numerical example that the multi-Slater determinant coupled
Hartree-Fock method allows to generate a basis potential with which Gamow shell
model calculations are stable and precise.
A. Run the two-particle Gamow shell model code with the modified surface
Gaussian interaction (see Eq. (9.174)) for 6 He, 6 Be, and 6 Li using bases
generated either by a Woods-Saxon potential mimicking the core or by the
multi-Slater determinant coupled Hartree-Fock potential. Notice that obtained
energies and widths are almost the same in both cases for 6 He and 6 Be and
explain why.
B. Show that calculations are problematic in the case of 6 Li when using a basis
generated by the Woods-Saxon potential of the core, especially for unbound
states.
On the contrary, by using the multi-Slater determinant coupled Hartree-Fock
potential, show that one obtains precise eigenenergies for the 6 Li spectrum.
Explain this result by considering the overall behavior of Hamiltonian
matrix elements with both basis-generating potentials.
Conclude by stating why optimization of the basis potential is necessary
in the Gamow shell model.
|Ψ GS has to be redefined if no , j state is occupied in the multi-Slater
determinant coupled Hartree-Fock ground state. Indeed, an , j state has to be
occupied in |Ψ GS to be able to apply the multi-Slater determinant coupled HartreeFock method. This occurs, for example, for the neutron p 1/2 partial wave when
considering the 6 He nucleus, as the 0p 1/2 neutron state is not occupied. In order
to solve this problem, it is sufficient to have a 1p-1h excitation to the initial multiSlater determinant coupled Hartree-Fock ground state. In the example of 6 He with
a 4 He core, |Ψ GS = |[0p 3/2 0p 1/2 ] J =1 can be used to optimize the neutron p 1/2
partial wave.
If, however, the considered , j partial wave does not possess any pole in the
basis, then its multi-Slater determinant coupled Hartree-Fock potential cannot be
defined. But the , j partial wave only occurs through the continuum coupling, so
that its effect will be much less important than the effect of pole states. It is then
sufficient to use a Woods-Saxon or harmonic oscillator potential to generate the
basis states of this particular partial wave.