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5 Formulation and Implementation of the Gamow Shell Model
Exercise IV
In a numerical example involving a nucleus with three valence nucleons, one
will show that the use of natural orbitals is efficient and precise in practical
applications.
A. One will consider resonance of the 7 He ground state in the approximation of
an inert 4 He core and 3 valence neutrons. The many-body basis generating the
ground state wave function of 7 He consists of the p 3/2 partial wave only.
Explain why it is sufficient from a physical point of view to have only basis
states belonging to the neutron p 3/2 partial wave.
Run the many-body Gamow shell model code using the modified surface
Gaussian interaction (see Eq. (9.174)) to calculate the 7 He ground state wave
function and its natural orbitals in the Berggren basis.
B. Run the Gamow shell model code in the same condition as in A, but
with natural orbitals instead of the Berggren basis. Show that one obtains
convergence using only a few natural orbitals, compared to the much larger
number of demanded Berggren basis states.
C. Redo the same exercise with 2p-2h truncations for 7 He and notice that using a
non-truncated basis space with a few natural orbitals does not improve results.
Explain the situation by comparison with the exact case considered in A and
B.
Compared to the Hartree-Fock states, natural orbitals offer a more adapted basis for
the diagonalization problem, on the assumption that |Ψ is close to the desired final
many-body state.
Natural orbitals have been applied with success in the contexts of variational
multiparticle-multihole configuration mixing method [19] and density matrix renormalization group approach [20]. In the present study, the approximate solution
|Ψ is obtained in a smaller configuration space in which only two particles are
allowed in the nonresonant continuum space. With the corresponding basis of
natural orbitals, 5 to 7 states per partial wave offer results which are of a similar
quality as those obtained in Berggren basis with about 30 states per partial wave.
The difference of calculated Gamow shell model eigenergies using the Berggren
basis or a basis of natural orbitals is in this situation smaller than 15 keV, so
that both bases can be deemed as equivalent. In this way, one may reduce the
sizes of matrices to be diagonalized by several orders of magnitude what enables
large-space Gamow shell model calculations. However, if one aims at calculating
nuclear states without truncating the Gamow shell model space, a basis of a few
natural orbitals is no longer sufficient. For that matter, it is more efficient to use the
density matrix renormalization group approach (see Sect. 5.9), based on a successive
renormalization of nonresonant degrees of freedom within the used many-body
basis.
5 Formulation and Implementation of the Gamow Shell Model
Exercise IV
In a numerical example involving a nucleus with three valence nucleons, one
will show that the use of natural orbitals is efficient and precise in practical
applications.
A. One will consider resonance of the 7 He ground state in the approximation of
an inert 4 He core and 3 valence neutrons. The many-body basis generating the
ground state wave function of 7 He consists of the p 3/2 partial wave only.
Explain why it is sufficient from a physical point of view to have only basis
states belonging to the neutron p 3/2 partial wave.
Run the many-body Gamow shell model code using the modified surface
Gaussian interaction (see Eq. (9.174)) to calculate the 7 He ground state wave
function and its natural orbitals in the Berggren basis.
B. Run the Gamow shell model code in the same condition as in A, but
with natural orbitals instead of the Berggren basis. Show that one obtains
convergence using only a few natural orbitals, compared to the much larger
number of demanded Berggren basis states.
C. Redo the same exercise with 2p-2h truncations for 7 He and notice that using a
non-truncated basis space with a few natural orbitals does not improve results.
Explain the situation by comparison with the exact case considered in A and
B.
Compared to the Hartree-Fock states, natural orbitals offer a more adapted basis for
the diagonalization problem, on the assumption that |Ψ is close to the desired final
many-body state.
Natural orbitals have been applied with success in the contexts of variational
multiparticle-multihole configuration mixing method [19] and density matrix renormalization group approach [20]. In the present study, the approximate solution
|Ψ is obtained in a smaller configuration space in which only two particles are
allowed in the nonresonant continuum space. With the corresponding basis of
natural orbitals, 5 to 7 states per partial wave offer results which are of a similar
quality as those obtained in Berggren basis with about 30 states per partial wave.
The difference of calculated Gamow shell model eigenergies using the Berggren
basis or a basis of natural orbitals is in this situation smaller than 15 keV, so
that both bases can be deemed as equivalent. In this way, one may reduce the
sizes of matrices to be diagonalized by several orders of magnitude what enables
large-space Gamow shell model calculations. However, if one aims at calculating
nuclear states without truncating the Gamow shell model space, a basis of a few
natural orbitals is no longer sufficient. For that matter, it is more efficient to use the
density matrix renormalization group approach (see Sect. 5.9), based on a successive
renormalization of nonresonant degrees of freedom within the used many-body
basis.