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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

I. INTRODUCTION

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 [6],

FPD6 [7] and GXPF1 [8] in the pf shell, have provided a

very good base to study various nuclear structure prob-

lems microscopically.

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 [11]. 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 [12] 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 [13]. 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

[10] 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 [16],

and the Quantum Monte Carlo Diagonalization (QMCD)

method [17].

In a previous paper [18] 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. [18] 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

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deformed Slater determinant, a number of particle-hole

excited configurations were considered under some selec-

tion criteria (see Eq. (23) of Ref. [18]) 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. [18]. 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

section VI.

II.

CONFIGURATION INTERACTION (PCI)

THE METHOD OF THE PROJECTED

The model Hamiltonian used in CI calculations can be

written as:

H =

?

i

eic†

ici+

?

i>j,k>l

Vijklc†

ic†

jclck, (1)

where, c†

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 [19], 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

[18]. The deformed s.p. creation operator is given by the

following transformation:

iand ciare creation and annihilation operators

b†

k=

?

i

Wkic†

i, (2)

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†

i1b†

i2...b†

in|?, (3)

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. [18], but also octupole ǫ4, etc.

The general form of the nuclear wave function is taken

as a linear combination of the projected SDs (PSDs),

|Ψσ

IM? =

?

Kκ

fσ

IKκPI

MK|κ?, (4)

whereˆPI

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-

?

K′κ′

(HI

Kκ,K′κ′ − Eσ

INI

Kκ,K′κ′)fσ

IKκ′ = 0,(5)

where HI

the Hamiltonian and of the norm, respectively

Kκ,K′κ′ and NI

Kκ,K′κ′ are the matrix elements of

HI

NI

Kκ,K′κ′ = ?κ|HPI

Kκ,K′κ′ = ?κ|PI

KK′|κ′?,

KK′|κ′?.

(6)

(7)

More details about the formalism can be found in Ref.

[18].

III.CHOICE OF THE PCI BASIS

The analysis made in Ref. [18] 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 [18], the general structure of the PCI basis is

0p − 0h,

|κ1,0?,

|κ2,0?,

....................

|κN,0?,|κN,j?,···

np − nh

|κ1,j?,··· ,

|κ2,j?,··· ,

. (8)

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 [18]

∆E =1

2(E0− Ej+

?

(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 [18], 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 [20] exhibits a spherical ground state minimum

which is selected as a basis SD. This spherical SD has a

=

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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. [18]. 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

Eexp(I,s,ǫ) =?s,ǫ|HPI

KK|s,ǫ?

KK|s,ǫ??s,ǫ|PI

. (11)

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),

A =

?H11 H12

H21 H22

?

,B =

?N11 N12

N21 N22

?

,

where

Hij= ?i|HPI

with |i(j) = 1? = |κ1,0?,|i(j) = 2? = |s,ǫ?.

MK|j?,Nij= ?i|PI

MK|j?, (13)

(14)

Solving the generalized eigenvalue problem

Ax = λBx,(15)

we get two eigenvalues, λ1and λ2, and their sum,

S2= λ(2)

1

+ λ(2)

2. (16)

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,

Sn= λ(n)

1

+ λ(n)

2

+ ··· + λ(n)

n.(18)

and λ(n)

1,λ(n)

2,..., λ(n)

n

are eigenvalues of Eq.(15) with

A =

H11 H12 ... H1n

H21 H22 ... H2n

...................

Hn1 Hn2 ... Hnn

,B =

N11 N12 ... N1n

N21 N22 ... N2n

...................

Nn1 Nn2 ... Nnn

,

and

Hij= ?i|HPI

|i(j)? = |κi(j),0?, if i(j) = 1,2,··· ,n − 1;

|i(j)? = |s,ǫ?, if i(j) = n.

MK|j?,Nij= ?i|PI

MK|j?,(19)

(20)

Sometimes we may only use part of the Snsum over

the lowest λi’s as a selection criteria, i.e.,

Sk

n= λ(n)

1

+ λ(n)

2

+ ··· + λ(n)

k,(1 ≤ k ≤ n), (21)

and Sn

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

accuracy.

Here we summarize the advantages of the new method

of selecting the basis states |κ,0?.

mentioned, the method proposed in Ref. [18] 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

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imbedded in the Generator Coordinate Method (GCM)

[21], 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. [18], 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[20]. 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 [22]. 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? [18].

It is interesting to study how many |κ,0? are needed

to describe the low-lying states. Fig. (3)(a) shows the

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?

?

??

?

?

??

?

?

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

circles).

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. [23]. 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.

5.

It is known that68Se nucleus exhibits shape coexis-

tence features. The constrained HF calculations of Ref.

[23] 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