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
deformed Slater determinant, a number of particle-hole
excited configurations were considered under some selec-
tion criteria (see Eq. (23) of Ref. ) to keep the total
numebr of basis states manageable. Having the deformed
basis of Slater determinants chosen, one can use standard
spin projection techniques to solve the associated eigen-
values problem. The method proved to be very accurate
in 0?ω model spaces, such as sd and pf where one can
easily keep track of different deformed orbitals. The dif-
ficulties usually appear for quasi-spherical nuclei, such as
56Ni, when special attentions has to be given to the se-
lection of the basics states |κj, 0 >. A simlar problem
arises for the case of mixed parity valence space, such
f5pg9 (see below), due to difficulties in tracking fixed
configurations of nucleons filling the s.p. orbitals around
the level-crossing deformations.
The paper is organized as follow. Section II presents a
brief outline of the PCI formalism that was expanded in
Ref. . The new algorithm to select the PCI basis is
discussed in section III. Section IV analyzes the efficiency
of the new method in the case of the quasi-spherical nu-
cleus56Ni. Section V is devoted to the study of several
nuclei that can be described using the mixed parity va-
lence space f5pg9. Conclusions and outlook are given in
CONFIGURATION INTERACTION (PCI)
THE METHOD OF THE PROJECTED
The model Hamiltonian used in CI calculations can be
of the spherical harmonic oscillator, eiand Vijklare one-
body and two-body matrix elements that can be obtained
from effective interaction theory, such as G-Matrix plus
core polarization , which can be further refined using
the experimental data [8, 20].
One can introduce the deformed single particle (s.p.)
basis, which can be obtained from a constraint Hartree-
Fock (HF) solution, or from the Nilsson s.p. Hamiltonian
. The deformed s.p. creation operator is given by the
iand ciare creation and annihilation operators
where the matrix elements Wki= ?bk|ci? are real in our
calculation. The Slater Determinant (SD) built with the
deformed single particle states is given by
|κ? ≡ |s,ǫ? ≡ b†
where s refers to the Nilsson configuration, indicating the
pattern of the occupied orbits, and ǫ is the deformation
determined by the quadrupole ǫ2, hexadecupole ǫ4as in
Ref. , but also octupole ǫ4, etc.
The general form of the nuclear wave function is taken
as a linear combination of the projected SDs (PSDs),
tor. The energies and the wave functions [given in terms
of the coefficients fσ
IKκin Eq.(4)] are obtained by solving
the following eigenvalue equation:
MKis the angular momentum projection opera-
Kκ,K′κ′ − Eσ
IKκ′ = 0,(5)
the Hamiltonian and of the norm, respectively
Kκ,K′κ′ and NI
Kκ,K′κ′ are the matrix elements of
Kκ,K′κ′ = ?κ|HPI
Kκ,K′κ′ = ?κ|PI
More details about the formalism can be found in Ref.
III.CHOICE OF THE PCI BASIS
The analysis made in Ref.  indicated that one of the
most important problem of the PCI method is a proper
selection of the PCI basis. As introduced in our previous
work , the general structure of the PCI basis is
0p − 0h,
np − nh
where |κi,0? (i = 1,...N) is a set of starting states of dif-
ferent deformations. Assuming that we’ve found these
|κ,0? SDs (skipping the subscript i to keep notations
short), relative np-nh SDs, |κ,j?, on top of each |κ,0?
are selected using the constraint 
(E0− Ej)2+ 4|V |2) ≥ Ecut, (9)
where E0 = ?κ,0|H|κ,0?, Ej = ?κ,j|H|κ,j?, V
?κ,0|H|κ,j? and Ecutis a parameter.
The |κ,0? SDs need to be properly chosen in order
to get good accuracy.In our previous work , we
have chosen the SDs with the lowest unprojected expec-
tation energy for each configuration, and we used the
same basis for all the spins. That approach proved to
work well for quite deformed nuclei, limiting its range
of application. For instance, the description of56Ni with
GXPF1A  exhibits a spherical ground state minimum
which is selected as a basis SD. This spherical SD has a
good spin I = 0, which won’t be useful if we calculate
I ?= 0 states. This example suggests that a more effi-
cient method may involve choosing different basis sets
for different spins. Another problem with this method of
selecting the |κ,0? basis states is that for each configu-
ration, only one SD was selected. Therefore some states,
such as the β−vibrational states, cannot be described
unless two or more shapes for the same configuration are
artificially included outside of any algorithm.
To address these problems, we developed a new
method of finding an efficient set of |κ,0? states. The
deformed single particle states are generated from the
Nilsson Hamiltonian shown in Eq. (2) of Ref. . For
simplicity, we set
Ei= ei. (10)
In a first step, at each deformation ǫ = (ǫ2,ǫ3,ǫ4), we
build many Slater determinants(SDs) denoted by |s,ǫ?,
where s denotes a configuration of nucleons occupying
the deformed single particle orbitals. These SDs are pro-
jected onto good angular momentum I, and the projected
energy is calculated
We then identify the configuration sa which has the
lowest Eexp(I,s,ǫ) at each shape ǫ. Searching over all
possible deformations ǫ, we obtain the energy surface of
Eexp(I,sa,ǫ) as a function of ǫ. The SD |sa,ǫa? which
has the lowest Eexp(I,sa,ǫ) is chosen as the first |κ,0?
state denoted as
|κ1,0? = |sa,ǫa?. (12)
The next step is to find the second |κ,0? state. We try
all possible |s,ǫ?, and for each |s,ǫ?, we build the 2 × 2
Matrix pair (A,B),
with |i(j) = 1? = |κ1,0?,|i(j) = 2? = |s,ǫ?.
Solving the generalized eigenvalue problem
Ax = λBx,(15)
we get two eigenvalues, λ1and λ2, and their sum,
The SD |sb,ǫb? with the lowest S2is selected as the second
|κ,0? denoted as
|κ2,0? = |sb,ǫb?. (17)
The process of finding more |κ,0? SDs can be continued
in a similar manner. Assuming that we have found the
(n − 1)−th |κ,0? SD, |κn−1,0?, then |κn,0? is chosen as
the |sx,ǫx?, corresponding to the lowest Sn. Here,
+ ··· + λ(n)
are eigenvalues of Eq.(15) with
H11 H12 ... H1n
H21 H22 ... H2n
Hn1 Hn2 ... Hnn
N11 N12 ... N1n
N21 N22 ... N2n
Nn1 Nn2 ... Nnn
|i(j)? = |κi(j),0?, if i(j) = 1,2,··· ,n − 1;
|i(j)? = |s,ǫ?, if i(j) = n.
Sometimes we may only use part of the Snsum over
the lowest λi’s as a selection criteria, i.e.,
+ ··· + λ(n)
k,(1 ≤ k ≤ n), (21)
Evaluating the projected energies for all SDs takes a
long time to calculate, and not all of them may be neces-
sary. Therefore, we enforce additional truncations. First,
the HF energy, EHF, is calculated. Next, at each shape ǫ,
the SDs having all particles occupying the lowest Nilsson
orbits are considered as the 0p-0h SDs. All particle-hole
excitations up to 4p-4h built on these 0p-0h SDs are cre-
ated, and their expectational energies
n= Snby definition.
Eexp(s,ǫ) = ?s,ǫ|H|s,ǫ? (22)
are calculated. Those SDs satisfying Eexp(s,ǫ) − EHF<
Eexpup, where Eexpupis a input parameter, are saved. Fi-
nally, the projected energies Eexp(I,s,ǫ) of the saved SDs
are evaluated, and compared with the lowest projected
energy Eexp(I,sa,ǫa) available. We keep those SDs sat-
isfying Eexp(I,s,ǫ) − Eexp(I,sa,ǫa) < Epjup(I), where
Epjup(I) is a input parameter, and we discard the oth-
ers. The values of the parameters Eexpup and Epjup(I)
must be large enough so that the |κ,0? can be properly
chosen, but too large Eexpupand Epjup(I) value may re-
sults in wasted computation without any improvement in
Here we summarize the advantages of the new method
of selecting the basis states |κ,0?.
mentioned, the method proposed in Ref.  uses the
same |κ,0? states for all spins, however, certain |κ,0?
states may not bring any contribution to certain spins.
The new method improves the efficiency of the PCI basis,
by choosing different |κ,0? states for different spins.
Secondly, the |κ,0? chosen by the new method may in-
clude two or more shapes for the same configuration s.
Therefore, the present method explicitly includes the idea
Firstly, as already
imbedded in the Generator Coordinate Method (GCM)
, and may be used to describe some collective vibra-
tions, such as the β−vibration.
Thirdly, there are no limitations on the K values. In
Ref. , we only selected basis states with relatively
small-K values to keep the basis dimensions manageable.
This limitation could be a problem in the case of the high
spin states, for which the high-K configurations could
be close to the yrast line. The new method described
here selects basis states with all possible SDs satisfying
|K| ≤ I.
Finally, and perhaps more importantly, the drawback
of getting large overlaps between different basis SDs,
which is typical for an uncorrelated selections of the basis,
is avoided by construction in the new method. Getting
basis states with large overlaps leads to many spurious
states due to the zero eigenmodes of the norm matrix.
The significance of these zero modes is that some of the
basis states that exhibit large overlaps bring insignifi-
cant contribution to the solutions of the Hill-Wheeler
Eq. (5), while unnecessarily increasing the dimensions
of the problem. Those useless SDs can be automatically
filtered out by the present method because the overlap
problem has been fully considered step-by-step when the
generalized eigenvalue equations (15) are solved.
IV. CALCULATIONS OF56NI
Using the new method, we recalculated the nuclei56Ni
with GXPF1A interaction. Let’s first consider the
case of I = 0. Both ǫ2and ǫ4span the interval from −0.45
to 0.45 in steps of 0.03. The first basis state, |κ1,0?, hav-
ing the lowest projected energy, corresponds to I = 0. In
Fig. 1, the surfaces of the Eexp(I,sa,ǫ) (See Eq. (11))
and Eexp(sa,ǫ) (See Eq. (22)) are plotted as functions
of ǫ2and ǫ4. Our calculation shows that the configura-
tion sahas all 16 valence particles in56Ni occupying the
orbits coming from the 1f7/2subshell. The unprojected
minimum, Fig. (1)(a) is at −203.800 MeV, and its shape
is spherical, consistent with the HF result. However, the
projected energy surface presents a quite different pic-
ture. There are four minima around the spherical shape,
the lowest one has −204.473 MeV and a small oblate de-
formation, ǫ2 = −0.09 and ǫ4 = −0.09. This energy is
673 keV lower than the HF energy, and 1.236 MeV above
the exact CI ground state energy of −205.709 MeV. This
|κ,0? state is a good candidate for the ground state.
The second basis state, |κ2,0?, has the same configura-
tion as the first one, but a different shape characterized
by ǫ2= −0.24 and ǫ4= −0.15. Comparing with Fig. 1,
this shape is quite different from any of the remaining 3
minima. The reason is that the SDs at those 4 minima
are highly overlapping each other after the angular mo-
mentum projection. Once the lowest one is picked up, the
others will automatically be filtered out by the present
method. The |κ2,0? corresponds to the first excited 0+
state, which might be called a β−vibrational state.
FIG. 1: (Color online) Unprojected energy surface (left panel)
and the projected energy surface with I = 0 (right panel) for
the ground state of56Ni with the GXPF1A interaction. The
lowest energy is marked by ’⊕’
FIG. 2: (Color online) Si− Si−1 values of |κi,0? at I = 0 for
56Ni as a function of ǫ2. ǫ4 are included in the calculation.
|κ1,0? is the lowest one.
The third basis state, |κ3,0?, has a prolate shape with
ǫ2 = 0.27 and ǫ4 = 0.06. Its configuration can be ob-
tained starting from |κ1,0?, but with 4 particles jumping
from the |Ω| = 7/2 (1f7/2) orbits to the |Ω| = 1/2 (2p3/2)
orbits. |κ3,0? can generate a deformed rotational band,
which has been observed in experiments . Here, we
only create the band head, which is the third 0+state.
Informations about higher |κ,0? SDs are shown in Fig.
2. The value Si− Si−1 indicates the energy position of
each |κi,0? state.
Once we have selected the |κ,0? SDs, we perform the
PCI calculations. There are two parameters used in PCI:
one is the number of |κ,0? SDs, n, and the other is the
Ecutused in Eq. (9) to select the number of particle-hole
excitations on top of each |κ,0? .
It is interesting to study how many |κ,0? are needed
to describe the low-lying states. Fig. (3)(a) shows the
FIG. 3: (Color online) Left panel, I = 0 PCI energies for56Ni
as functions of n with Ecut = 1 keV. Right panel, I = 0 PCI
energies for56Ni as functions of Ecut = 1 with n = 8. Full CI
results are also shown as the open circles by comparison.
PCI energies as functions of n for Ecut = 1 keV. For
n = 1 the PCI dimension is only 180, yet the first 0+
PCI energy is −205.409 MeV, only 300 keV above the
exact full CI value. |κ2,0? must be included to describe
the second 0+state, and |κ3,0? reproduces the third 0+
state. To accurately describe more excited states more
|κ,0? SDs are need. For the lowest 5 states in56Ni a good
approximation can been achieved starting with n = 7.
If one keeps on increasing n, then those 5 energies will
become closer and closer to the CI values. For instance,
with n = 15, the the ground state PCI energy becomes
−205.603 MeV, just 100 keV above the exact value.
The PCI energies are also affected by the Ecutparam-
eter. For Ecut= 1000 keV no particle-hole excited SDs
are included, and the PCI dimension is the same as the
number of |κ,0? SDs included, n = 8 in this case. There-
fore the PCI energies are exactly the values of λiin Eq.
(18). More particle-hole excited SDs can be included by
reducing the value of Ecut. For example, by decreasing
from Ecut= 1000 keV to Ecut= 1 keV, the PCI energies
drop ∼ 1.0÷2.3 MeV for the lowest 5 states, and become
close to the full CI values. By decreasing from Ecut= 1
keV to Ecut= 0.2 keV the energy drop becomes slower,
and is around ∼ 100 ÷ 200 keV. The PCI energies for
I ?= 0 states are also calculated, and shown in Fig. 4, the
number of |κ,0? is n = 15 and Ecut= 1 keV. One can
observe good agreements between the PCI and the CI re-
sults, including the states in the rotational band starting
at about 5 MeV.
V.CALCULATIONS IN f5pg9 VALENCE SPACE
We have also extended our calculations to the f5pg9
valence space, which includes the 1f5/2, 2p3/2, 2p1/2, and
the 1g9/2spherical shells. 1g9/2orbital has positive par-
ity, and the other fp orbitals have negative parity. There-
fore, SDs with both parities can be built for any num-
FIG. 4: (Color online)The lowest 5 energies at each spin for
56Ni calculated using PCI (open circles) and full CI (filled
ber of nucleons. The positive parity SDs are those with
even number particles occupying the fp orbitals, while
the negative parity SDs have odd number particles oc-
cupy the fp orbitals. The angular momentum projection
does not change the parity. The parity of the projected
states remain the same as that of the original SDs. There-
fore, one can split the PCI basis into the positive parity
part, I+, and the negative parity part, I−, at each spin
I. With the present method, different |κ,0? SDs can be
generated separately for the I+basis and I−basis.
The interaction for f5pg9 shell space was taken from
Ref. . It includes, besides the usual qqudrupole, hex-
adecupole, and pairing terms, octupole contributions and
monopole corrections. In all cases, 20 |κ,0? SDs are taken
for each Iπbasis and Ecutwas fixed to 1 keV. The first
example we analyze is the N=Z nucleus68Se, which is
known to be deformed with competing oblate and pro-
late deformations. The energies of the 20 |κ,0? SDs for
both Iπ= 0+basis and Iπ= 0−basis are shown in Fig.
It is known that68Se nucleus exhibits shape coexis-
tence features. The constrained HF calculations of Ref.
 as well as our results in Fig. 6 show that there are two
minima. Both of them are axially and reflection symmet-
ric. The lowest minimum has −40.718 MeV and oblate
shape, and the second one has −39.956 MeV and pro-
late shape. It is interesting that the results of our new
method presents the same picture as one can observe in
the left panel of Fig. 5, where the lowest |κ1,0? is oblate
with S1= Eexp(0,sa,ǫa) = −42.405 MeV and |κ2,0? is
prolate with S2−S1= −41.549 MeV. However, both en-
ergies are about 1.6 MeV lower than those obtained by an
HF procedure, due to the angular momentum projection.
For the 0−
basis the lowest |κ,0? SD lies at
−36.44MeV, which is relatively high (see Fig.5), and has
a prolate shape. The corresponding configuration is the
same as that of the second HF minimum, except that
FIG. 5: (Color online) Si− Si−1 values of |κi,0? for Iπ= 0+
(left) and Iπ= 0−(right) in68Se as a function of ǫ2. ǫ4 are
included in the calculation.
strained Hartree-Fock (CHF) calculations as a function of
(Color online) Energy surface provided by Con-
´2Y20 and q3 =
´3Y30. (b is the harmonic
one particle was excited from the Ω = 3/2(p3/2) orbital
to the Ω = 3/2(g9/2) orbital. Because68Se has N = Z,
this excited particle can be either a neutron or a proton,
and therefore there are two different |κ,0? SDs with the
same shape and the same low energy. Similar cases ap-
pear for other |κ,0? SDs. Therefore, in Fig. 5(b) each
symbol corresponds to two different |κ,0? SDs. The po-
sition of the second lowest symbol is only about 130 keV
above the lowest one, and its configuration is similar to
the lowest one, but with the odd particle excited from
the Ω = 1/2(p3/2) orbital to the Ω = 1/2(g9/2) orbital.
Using the |κ,0? states described in Fig. 5, one can
calculate the PCI energies for the 0+and 0−states, which
turned out to be very close to the CI results. For I ?= 0,
similar good results have also been achieved, as shown in
As indicated in the above discussions, the PCI method
spin/parity for68Se calculated using PCI (open circles) and
full CI (filled circles).
(Color online) The lowest 3 energies at each
TABLE I: PCI Dimensions compared with those of full CI for
π = +
π = −
6.7 × 106
2.0 × 107
3.2 × 107
4.2 × 107
5.0 × 107
5.5 × 107
5.8 × 107
5.9 × 107
5.7 × 107
6.7 × 106
2.0 × 107
3.2 × 107
4.2 × 107
5.0 × 107
5.5 × 107
5.8 × 107
5.9 × 107
5.7 × 107
not only provides a good approximation for the CI re-
sults, but it is also a convenient tool to gain some insight
into the physics of the nuclear states. One interesting
example is the lowest state in Fig. 7(b), which is a 3−
state. As shown in Fig. 8, the (lowest) |κ1,0? SD has
Kπ= 3−and oblate deformation. The configuration of
this |κ1,0? is the same as the oblate HF minimum, except
that one particle was excited from the Ω = 3/2(p3/2) or-
bital to the Ω = 9/2(g9/2) orbital to form a Kπ= 3−
SD. The second SD, |κ2,0?, has the same energy and the
same shape as |κ1,0? because N = Z, and due to the
isospin symmetry of the adopted Hamiltonian. If only
the particle-hole excitations built on |κ1,0? are included,
one obtains a PCI energy of −40.469 MeV. If the |κ2,0?
SD is further included, the PCI energy drops 300 keV
to −40.769 MeV, This energy is only 300 keV above the
exact value of −41.043 MeV. However, the PCI energy
for n = 20 is −40.843 MeV, only 70 keV lower than what
one can obtain with n = 2. Therefore, it is clear that the
lowest 3−state has mostly contributions from the lowest
2 oblate Kπ= 3−SDs, i.e. |κ1,0? and |κ2,0?.
We have also calculated states of both parities in70Se.
The results are shown in Fig. 9. Once again, the PCI
FIG. 8: (Color online) (a), Si−Si−1values of |κ,0? at Iπ= 3−
(Fill Circles) at Iπ= 3−in68Se. The open Symbols from left
to right refer to PCI calculations with n = 1,2 and 20.
68Se. (b), PCI energies (Open Symbols) and CI energies
FIG. 9: (Color online) The same as Fig. 7 but for70Se.
results are very close to those of full CI for both positive
parity and negative parity for a wide range of spin values.
However, much smaller dimensions of the PCI matrices
are necessary. The PCI dimensions correponding to the
Iπcalculations in Fig. 9 are compared in Table I with
the full coupled-I CI dimensions. The PCI dimensions are
small fractions, roughly 10−4, of the full CI dimensions.
As is well known, the most serious problem with full CI
method is the explosion of the dimensions as the number
of the single-particle valence states, and/or number of
valence nucleons. However, this problem seems to be less
of an issue for the PCI method. The total PCI dimension
can be estimated as the product of two numbers, n×m,
where n is the number of |κ,0? states and m is the number
of particle-hole excitations selected by Ecut in Eq. (9).
Our investigations indicate that n is related to how many
low-lying states of a given spin one wants to accurately
describe. For instance, if we are interested in only the
yrast state, quite often a good approximations can be
obtained with n = 1 or 2. n = 20 seems to be enough to
describe the lowest 3-5 states of each Iπin the present
FIG. 10: (Color online) Lowest 3 0+energies of76Ge and76Se
calculated by PCI (filled symbols) and full CI (open circles)
calculations. As regarding m, the |κ,j? SDs are limited
to 1p-1h and 2p-2h excitations according to Eq.
Note that |κ,j? has the same Kπas that of |κ,0?. As
the parameter Ecut is enforced via Eq. (9), m can be
significantly reduced. For instance, in the case of Iπ= 0+
for70Se, m2p2h= 801 for |κ1,0?, but only 138 SDs were
finally included if Ecut= 1 keV.
The exploding CI dimensions has as a consequence a
rapid increase of the computing time necessary for full CI
calculation. Using the modern coupled-I code NuShellX
, the full CI calculation of all states in Fig. 9 could
take almost one year when only one processor is used.
The calculation of the lowest 3 states of each Iπin70Se
would take in average about 20 days. For the same calcu-
lation, PCI takes around 5 hours for each Iπ. The main
computational workload in PCI is related to the calcula-
tion of the dense matrices H, and N in Eq. (6). It should
be metioned that extra time is needed to to search for the
optimized set of |κ,0? SDs. The computing time can be
affected by: (1) the number of mesh points used for the
shape paramters; (2) the values of the parameters Eexpup
and Epjup(Iπ) that decides how many SDs are considered
in the optimization process; (3) the total number n of
|κ,0? basis states selected; (4) the value of Ecut. For ex-
ample, in the calculation of Iπ= 0+in70Se, both ǫ2and
ǫ4run from −0.3 to 0.3 in steps of 0.02, Eexpup= 7 MeV
and Epjup(Iπ= 0+) = 5 MeV, and Ecut = 1. Under
these conditions, it will takes about 10 hours to obtain
20 |κ,0? SDs using one processor. For other Iπ, the com-
putational time ranges from few hours to 1 or 2 days.
However, the total time for a PCI calculations is at least
10 times shorter than that of the corresponding full CI
calculation for the case of70Se.
Finally, we used the new PCI method to calculate the
low lying 0+states in76Ge and76Se. The nuclear struc-
ture of these two nuclei is relevant for the double beta
decay (DBD) process of76Ge. DBD is one of the most
actively investigated nuclear physics problem, which may
reveal new physics beyond the Standard Model, includ-
ing the absolute values of the neutrino masses. Full CI
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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