5.3 Truncation of the Many-Body Berggren Basis in the Gamow Shell Model
203
5.3
Truncation of the Many-Body Berggren Basis
in the Gamow Shell Model
The presence of numerous scattering states in the Berggren ensemble generates very
large valence space dimensions in the Gamow shell model. Thus, the Hamiltonian
matrix diagonalization cannot be handled with methods relying on full matrix times
vector operations such as the Lanczos or Davidson methods [15, 16], unless drastic
truncations are imposed.
In fact, the dimensionality problem in Gamow shell model is even more acute
than in standard shell model [15], as one typically needs 30–50 Berggren basis
states per partial wave to attain convergence, compared to the 5–10 harmonic states
typically needed in no-core shell model. Consequently, in practice, one usually
truncates the Gamow shell model space according to the number of occupied
scattering states in Slater determinants, which is typically 2, 3, or 4 at most. By
limiting the number of particles in the continuum, truncated Gamow shell model
spaces are usually tractable when only a few partial waves are represented in the
Berggren basis. Moreover, it is not necessary to use Berggren basis for the proton
part of the wave function in neutron-rich nuclei, because protons are well bound and
do not participate significantly in the asymptote of nuclear wave functions. This is
also the case for a partial wave of large angular momentum, as its centrifugal barrier
prevents nucleons from entering the asymptotic region. Thus, it is sufficient to use
a basis of harmonic oscillator states for the latter cases, whereas the Berggren basis
is utilized for ≤ 1, 2.
5.3.1 Natural Orbitals
Another powerful method to diminish the size of Gamow shell model space without
loss of numerical precision could be implemented using the natural orbitals [17]
defined as the eigenvectors of the scalar one-body density matrix:
ρ
mn = =Ψ
|
a
†
m ˜
a n
0
0
|Ψ
,
(5.60)
where m, n are indices of Berggren basis states, and |Ψ is an approximation of the
final many-body state (see Exercise IV for numerical examples of the use of natural
orbitals). Operator ˜
a in (5.60) refers to the modified annihilation operator. Contrary
to the creation operator a † , the annihilation operator a n is not a spherical tensor [18].
Therefore, it cannot be coupled with other creation and/or annihilation operators to
a given value of the angular momentum. In fact, the annihilation operator a must be
multiplied by a phase factor of its angular part in order to become a spherical tensor
[18]. It is standard to denote this newly defined modified annihilation operator by
˜
a n [18]. Note that this tilde has a different signification from that introduced in
Sect. 5.1.1 to emphasize the biorthogonality of complex-energy states.
203
5.3
Truncation of the Many-Body Berggren Basis
in the Gamow Shell Model
The presence of numerous scattering states in the Berggren ensemble generates very
large valence space dimensions in the Gamow shell model. Thus, the Hamiltonian
matrix diagonalization cannot be handled with methods relying on full matrix times
vector operations such as the Lanczos or Davidson methods [15, 16], unless drastic
truncations are imposed.
In fact, the dimensionality problem in Gamow shell model is even more acute
than in standard shell model [15], as one typically needs 30–50 Berggren basis
states per partial wave to attain convergence, compared to the 5–10 harmonic states
typically needed in no-core shell model. Consequently, in practice, one usually
truncates the Gamow shell model space according to the number of occupied
scattering states in Slater determinants, which is typically 2, 3, or 4 at most. By
limiting the number of particles in the continuum, truncated Gamow shell model
spaces are usually tractable when only a few partial waves are represented in the
Berggren basis. Moreover, it is not necessary to use Berggren basis for the proton
part of the wave function in neutron-rich nuclei, because protons are well bound and
do not participate significantly in the asymptote of nuclear wave functions. This is
also the case for a partial wave of large angular momentum, as its centrifugal barrier
prevents nucleons from entering the asymptotic region. Thus, it is sufficient to use
a basis of harmonic oscillator states for the latter cases, whereas the Berggren basis
is utilized for ≤ 1, 2.
5.3.1 Natural Orbitals
Another powerful method to diminish the size of Gamow shell model space without
loss of numerical precision could be implemented using the natural orbitals [17]
defined as the eigenvectors of the scalar one-body density matrix:
ρ
mn = =Ψ
|
a
†
m ˜
a n
0
0
|Ψ
,
(5.60)
where m, n are indices of Berggren basis states, and |Ψ is an approximation of the
final many-body state (see Exercise IV for numerical examples of the use of natural
orbitals). Operator ˜
a in (5.60) refers to the modified annihilation operator. Contrary
to the creation operator a † , the annihilation operator a n is not a spherical tensor [18].
Therefore, it cannot be coupled with other creation and/or annihilation operators to
a given value of the angular momentum. In fact, the annihilation operator a must be
multiplied by a phase factor of its angular part in order to become a spherical tensor
[18]. It is standard to denote this newly defined modified annihilation operator by
˜
a n [18]. Note that this tilde has a different signification from that introduced in
Sect. 5.1.1 to emphasize the biorthogonality of complex-energy states.