arXiv:0906.3756v1 [nucl-th] 19 Jun 2009
Improved basis selection for the Projected Configuration Interaction method applied
to heavy nuclei
Zao-Chun Gao1,2, Mihai Horoi1, and Y. S. Chen2
1Department of Physics, Central Michigan University, Mount Pleasant, Michigan 48859, USA
2China Institute of Atomic Energy P.O. Box 275-18, Beijing 102413, China
(Dated: June 19, 2009)
In a previous paper we proposed a Projected Configuration Interaction method that uses sets of
axially deformed single particle states to build up the many body basis. We show that the choice
of the basis set is essential for the efficiency of the method, and we propose a newly improved
algorithm of selecting the projected basis states. We also extend our method to model spaces that
can accomodate both parities, and can include odd-multipole terms in the effective interaction, such
as the octupole contributions. Examples of52Fe,56Ni,68Se,70Se and76Se are calcualted showing
good agreement with the full Configuration Interaction results.
PACS numbers: 21.60.Cs,21.60.Ev,21.10.-k
The full configuration interaction (CI) method [1, 2]
using a spherical single particle (s.p.) basis and realis-
tic Hamiltonians, also know as the nuclear shell model,
has been very successful in describing various properties
of the low-lying states in light and medium nuclei. The
main limitations of this method are the exploding dimen-
sions with the increase of the number of valence nucleons,
or/and with the increase of the valence space. Although,
there are continuous improvements to the CI codes [3, 4]
and computational resources, the exploding CI dimen-
sions significantly restrict the ability to investigate heavy
nuclei, especially those which exhibit strong collectivity.
The deformed mean-field approaches, however, have the
ability to incorporate the collective effects at the single
particle level. The mean-field description in the intrinsic
frame naturally takes advantage of the spontaneous sym-
metry breaking. This approach provides some physical
insight, but the loss of good angular momentum of the
mean-field wave functions makes the comparison with the
experimental data difficult. The CI calculations in spher-
ical basis provide the description in the laboratory frame.
The angular momentum is conserved, but the physical in-
sight associated with the existence of an intrinsic state
is lost. One important aspect of the CI approach is its
ability of using all components of effective interactions
compatible with a given symmetry, but restricted to a
chosen valence space. Examples of realistic Hamiltoni-
ans, such as the USD [1, 5]in the sd shell, the KB3 ,
FPD6  and GXPF1  in the pf shell, have provided a
very good base to study various nuclear structure prob-
The recent history of projection techniques combined
with CI particle-hole configurations includes the Pro-
jected Shell Model(PSM) [9, 10] and the Deformed Shell
Model (DSM) proposed in reference . PSM uses a de-
formed intrinsic Nilsson+BCS basis projected onto good
angular momentum, and a multipole-multipole Hamilto-
nian that diagonalized in the space spanned by the pro-
jected states. The Nilsson model  has been proven
to be very successful in describing the deformed intrin-
sic single particle states, and the quadrupole force was
found to be essential for describing the rotational mo-
tion . PSM was proven to be a very efficient method
in analyzing the phenomena associated with the rota-
tional states, especially the high spin states, not only for
axial quadrupole deformation, but also for the octupole
 and triaxial shapes [14, 15]. However, its predic-
tive power is limited because the mulitpole-multipole plus
pairing Hamiltonian has to be tuned to a specific class of
states, rather than an region of the nuclear chart. The re-
cently proposed DSM is using the same realistic effective
Hamiltonian as the full CI method, and a Hartree-Fock
procedure to select the deformed basis. One can only as-
sume that this procedure would not be very accurate for
quasi-spherical nuclei. The main difficulties for all these
models is a proper selection of the deformed basis. Their
accuracy can only be assessed by comparison with the ex-
act results provided by the full CI method using the same
effective Hamiltonian, and not by direct comparison with
the experimental data. Other model using similar tech-
niques includes MONSTER, the family of VAMPIRs ,
and the Quantum Monte Carlo Diagonalization (QMCD)
In a previous paper  we proposed a new method
of calculating the low-lying states in heavy nuclei using
many particle-hole configurations of spin-projected Slater
determinants built on multiple sets of deformed single
particle orbitals. This Projected Configuration Interac-
tion (PCI) method takes advantage of inherent mixing
induced by the projected Slater determinants of varying
deformations with the many particle-hole mixing typical
for the Configuration Interaction (CI) techniques. Di-
rect comaparison between PCI and CI results are always
possible, provided that the deformed s.p. states are al-
ways obtained starting from a valence space of shperi-
cal orbitals. In Ref.  we use a simple mechanism
of selecting a number of basic deformed Slater determi-
nants in the sd and pf model space, denoted |κj, 0 >,
by searching for the minimum energy of fixed configu-
ratin of deformed s.p. orbitals. Starting from each basic
calculations [24, 25, 26] of the 2-neutrino and neutrino-
less DBD matrix elements have been carried out for some
DBD nuclei up to136Xe. However, for heavier DBD nu-
clei150Nd and238U, the huge CI dimensions make the full
CI calculation unmanageable. PCI can take full advan-
tage of the deformation, and an efficient truncation could
be obtained for well deformed nuclei, such as150Nd and
238U. As a first inroad into this problem, the low-lying
0+states76Ge and76Se are calculated using the present
version of the PCI, and are compared with full CI re-
sults in Fig. 10. Using only 6 |κ,0? SDs (n = 6) for
each nucleus, the PCI dimensions are 561 and 647 for
76Ge and76Se, respectively. The calculated PCI energy
of the lowest 0+state for76Se is 200 keV higher than the
exact value, and only 86 keV higher for76Ge. In addi-
tion, good approximations for the excited 0+states have
also been reached. Given these encouraging results, one
would hope that PCI calculations could be successfully
performed for the heavy deformed DBD nuclei, such as
150Nd and238U, in a not so distant future.
VI.CONCLUSIONS AND OUTLOOK
In this article we propose a newly improved algorithm
of selecting the basis of Slater determinants that can
be used with the Projected Configuration Interaction
method introduced in Ref. . The new algorithm de-
pends on a number of parameters that can be used to fine
tune its efficiency. Its main advantages over the original
method of selecting of the basis are summarized at the
end of Section III.
We used the new algorithm to revisit the calculation
of56Ni, quasi-shperical nucleus that has a relatively low-
lying rotational band. We were able to calculate its low-
lying states very efficiently, and with good accuracy, gain-
ing also insight into the physics of these states. We have
also use the new method to analyze some Se and Ge
isotopes in the f5pg9 valence space. Both natural and
unnatural parities can be accurately described for these
nuclei, even for cases with pronounced competing defor-
mations, such as70Se and70Se. The PCI dimensions are
significantly lower the corresponding CI dimensions, as
well as the corresponding computational effort. In addi-
tion, in most cases, the low-lying projected basis states
can provide some physical insight into the structure of
the low-lying states. Finally, we calculated with the new
method the low-lying 0+states in76Ge and76Se that are
relevant for the double beta decay of76Ge. The hope is
that this method could be used some day to study the
double beta decay of the strongly deformed150Nd and
Further improvements to the PCI method will include
the extension of the formalism developed in Ref.  to
calculate electromagnetic transition probabilities. The
new method uses different bases for different spins, which
introduces additional complications. Other observables,
such as spectroscopic amplitudes and DBD matrix ele-
ments have to be worked out. Further improvement of
the basis may be achieved for some cases that exhibit
significant octupole deformation, which will require full
projection on good parity.
The authors acknowledgesupport from the DOE Grant
No. DE-FC02-09ER41584. M.H. acknowledges support
from NSF Grant No. PHY-0758099. Z.G. acknowledges
the NSF of China Contract Nos. 10775182, 10435010and
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