8.2 Benchmarking of No-Core Gamow Shell Model Against Other Methods
377
using the Lawson method, with either the harmonic oscillator basis or the Berggren
basis, or without any center-of-mass correction using the Berggren basis only, are
similar and converge to their exact value. Typically, the largest ground state binding
energy is obtained using the no-core Gamow shell model without the center-of-mass
correction. Indeed, using the Lawson method in the no-core Gamow shell model
wave functions, one effectively projects ground state wave functions in the largest
N ¯
hω model space included in the Gamow shell model space. Therefore, the ground
state binding energy is smaller than in the absence of the Lawson method.
The ground state of 4 He in the no-core Gamow shell model is also slightly more
bound than obtained in the no-core shell model with harmonic oscillator basis. This
comes from the use of a Woods–Saxon potential which is very close to the Hartree–
Fock potential for the 4 He ground state, so that the no-core Gamow shell model
results converge faster when N increases, as compared to the no-core shell model
in the harmonic oscillator basis. As can be seen from Fig. 8.3, expectation values
of ˆ
H CM are already very small in 2 ¯
hω spaces and eventually become smaller than
50 keV in 8 ¯
hω spaces when the Lawson method is applied to no-core Gamow shell
model. This implies that it is possible to have a separation of intrinsic and center-ofmass parts in many-body wave functions, as long as the model space is sufficiently
large.
As already seen from Fig. 8.2, the ground state energy becomes closer and closer
to its exact value even when the Lawson method is not applied. Indeed, due to
the generalized variational principle, ground state eigenenergy is stationary when
enlarging Gamow shell model basis spaces, so that it has to converge to the exact
eigenenergy at the limit of infinite space, where eigenstates are separated in centerof-mass and intrinsic parts. Hence, the spuriosity of eigenstates decreases when
the size of Gamow shell model space increases, to vanish identically at the limit
of infinite space. It is then not necessary to apply the Lawson method in order to
compute the ground state energy. In fact, one can obtain a ground state energy close
to the exact energy if the model space used is sufficiently large.
However, the generalized variational principle in general does not apply to
excited states. In order to demonstrate this statement, let us consider the case of
the first two eigenstates of 4 He, which are 0 + states. If one diagonalizes the Gamow
shell model Hamiltonian without consideration of the center-of-mass properties of
the calculated 0 + states, it is possible that the two first 0 + states of 4 He converge
to the same 0 + ground state. As the Hamiltonian is intrinsic, both 0 + states can
indeed become degenerate at the limit of infinite space, with their intrinsic parts
being that of the ground state, whereas their center-of-mass parts are orthogonal.
Consequently, the generalized variational principle can only be applied to ground
states of subspaces with fixed angular momentum and parity. For example, one can
calculate the binding energies of the 3/2
−
1 and 1/2
−
1 eigenstates of 5 He using the
generalized variational principle, because both these states are lowest energy states
for the considered quantum numbers.
Another method to solve this problem would be to work in a fully intrinsic ab
initio coupled-channel formalism, as in the no-core shell model with continuum
[31–40]. In this approach, the cluster + target composite is built with a harmonic
377
using the Lawson method, with either the harmonic oscillator basis or the Berggren
basis, or without any center-of-mass correction using the Berggren basis only, are
similar and converge to their exact value. Typically, the largest ground state binding
energy is obtained using the no-core Gamow shell model without the center-of-mass
correction. Indeed, using the Lawson method in the no-core Gamow shell model
wave functions, one effectively projects ground state wave functions in the largest
N ¯
hω model space included in the Gamow shell model space. Therefore, the ground
state binding energy is smaller than in the absence of the Lawson method.
The ground state of 4 He in the no-core Gamow shell model is also slightly more
bound than obtained in the no-core shell model with harmonic oscillator basis. This
comes from the use of a Woods–Saxon potential which is very close to the Hartree–
Fock potential for the 4 He ground state, so that the no-core Gamow shell model
results converge faster when N increases, as compared to the no-core shell model
in the harmonic oscillator basis. As can be seen from Fig. 8.3, expectation values
of ˆ
H CM are already very small in 2 ¯
hω spaces and eventually become smaller than
50 keV in 8 ¯
hω spaces when the Lawson method is applied to no-core Gamow shell
model. This implies that it is possible to have a separation of intrinsic and center-ofmass parts in many-body wave functions, as long as the model space is sufficiently
large.
As already seen from Fig. 8.2, the ground state energy becomes closer and closer
to its exact value even when the Lawson method is not applied. Indeed, due to
the generalized variational principle, ground state eigenenergy is stationary when
enlarging Gamow shell model basis spaces, so that it has to converge to the exact
eigenenergy at the limit of infinite space, where eigenstates are separated in centerof-mass and intrinsic parts. Hence, the spuriosity of eigenstates decreases when
the size of Gamow shell model space increases, to vanish identically at the limit
of infinite space. It is then not necessary to apply the Lawson method in order to
compute the ground state energy. In fact, one can obtain a ground state energy close
to the exact energy if the model space used is sufficiently large.
However, the generalized variational principle in general does not apply to
excited states. In order to demonstrate this statement, let us consider the case of
the first two eigenstates of 4 He, which are 0 + states. If one diagonalizes the Gamow
shell model Hamiltonian without consideration of the center-of-mass properties of
the calculated 0 + states, it is possible that the two first 0 + states of 4 He converge
to the same 0 + ground state. As the Hamiltonian is intrinsic, both 0 + states can
indeed become degenerate at the limit of infinite space, with their intrinsic parts
being that of the ground state, whereas their center-of-mass parts are orthogonal.
Consequently, the generalized variational principle can only be applied to ground
states of subspaces with fixed angular momentum and parity. For example, one can
calculate the binding energies of the 3/2
−
1 and 1/2
−
1 eigenstates of 5 He using the
generalized variational principle, because both these states are lowest energy states
for the considered quantum numbers.
Another method to solve this problem would be to work in a fully intrinsic ab
initio coupled-channel formalism, as in the no-core shell model with continuum
[31–40]. In this approach, the cluster + target composite is built with a harmonic