Recent news from VAMPIR

frog.

Vol. 38, pp. 149-158, 1997 0 1997 Elsevier Science Ltd in Great Britain. All rights reserved 0146-6410197 $32.00 + 0.00

Parr. Nucl. Phys..

Pergamon

Printed

SOl46-6410(97)00020-3

Recent News from VAMPIR K. W.

SCHMID

lnstifut fiir Theoretische Physik, D-72076 Tiibingen. Germany

Universitiit

zu Tiibingen.

Auf der Morgensrellr

14,

1. Introduction Many

nuclear

structure

problems

large to allow for a complete

the SCM

to truncate

space

according

yields unsatisfying are of a more to extract

Configuration-!vIixing

to the unperturbed

convergence

[2].

degrees

ambiguities

of the

this avenue

spaces)

approach

[3,8],

[6,7],

and finally

elaborate

In the VAMPIR

single

of each V-AMPIR

same symmetry. projected

HFB

wave function. single projected Finally

advantage

that

projected

variation

the different

does

another

which

avenue

of the various

states,

trying

via variational

to the dynamics.

Projection

and the

possibilities

In Realistic

the EXCITED FED

tried

however.

spectrum,

Hamiltonian

via the EXCITED

lowest

(“yrast”)

parity

and

after

angular

to

model

V_4UPIR

VAMPIR

method

state

HFB

VAMPIR

successively

more

149

simply

excited

completely

approach,

together that

states

with

different

again by a

has the

from that of the symmetry-

It determines

with the configuration each further

as test

are constructed.

uses several

state.

the

symmetry-

is taken state

is

EXCITED

This procedure

finally,

of each

states

by a

description

The

a second

of the first excited

all these solutions.

In this way it is ensured

t,he wave flmct,ion

for the excited

the higher

fixed

mean-field

the optimal picture.

system

description

with a structure

FED

underlying

yields

to the yrast-solution

of only one for the description

calculations.

not, dist,urh

method

(i.e.

is approximated

The

This

quasi-particle

of this

in between

states

transformations

momentum)

of the considered

then the optimal

excited

with a given symmetry

vacuum.

(HFB)

the projection.

extension

The EXCITED

instead

state

independent

is diagonalized

one can describe

of variational

det,ermina.nt,

Out

of excited

In the same way afterwards

interaction

underlying

nuclear

forced to

has been

prescription,

in the nuclear

is left entirely

Gram-Schmidt-orthogonalized yields

determinant.

vacua

It often

This

Mean-field

correlations

definite

calculation

being

yrast-state.

HFB

After

for the description

Here for the first excited

the residual

corresponding

(V ariation

is the straightforward

The

the

are avoided.

Hartree-Fock-Bogoliubov

vacuum

states

Hamiltonian

One is therefore

in the last years

from

schemes

in a symmetry-projected

approach

for those

directly

of additional

neutrons,

by a variational

yrast-state

of the configurations.

the energetically

symmetry-projected

determined

a chain

and

energies

[l].

are far too

ones.

approach

of protons

app roach

which

many-nucleon

of configurations.

of the configurations

the VAMPIR its extension

(SCM)

especially

truncation

the inclusion

[9] are the most

number

[3-j],

chosen effective

We did follow therefore

In this way the selection

basis-systems,

number

of freedom

procedures.

traditional

particle

feasible

properties

nature

relevant

explore

of a suitably

to a numerically

collective

the

the use of single

diagonalization

as it is done in the Shell-model truncate

require

each

mixing

of via

symmetry-projected

t,ha.n t,he last, one a7.ddetl nrevioIislv.

These

150

K. W. Schmid

methods

have

phenomena

been

applied

in the .J -

still had imposed

[lOI with

TO mass-region.

time-reversal

(see Ref.

[5] and references

Recently

these last restrictions

HFB

calculations

tained. approach,

however,

elaborate

versions

2. Outline

vacuum

This

of completely

ourselves

mentioned

[ll].

shapeecoexistence

versions

on the underlying

of t,hese methods HFB

For the first time results

unrestricted

in the present

the mathematical

formahsm

above

of the VAMPIR

quasi-particle

contribution

we

transformations

of symmetry-projected

determinants

have been ob-

to the one determinant

as well as its numerical

are ready for application,

approach

implementation

V.UvfPIR for the more

too.

has been discussed

in detail

elsewhere

[7,8,9]

and will hence be

in the following. by a finite

I

{cy, c:, . ..}M~ /O > is defined

introduced

even in the most recent

and axial symmetry

could be removed

our modelkspace

operators

complex

therein).

restrict

only briefly

\Ve define

to the rather

e.g.,

of the Theory

The theory scetched

success.

However,

invariance

on the basis

We shall

good

and

!blb-dimensional

{c,, ck, . ..}J~~

by c,/O >r

set of Fermion

for spherical

0 for all i = 1,

single

....iVb.

nucleon

General

creation states.

and The

quasi-particle

annihilation

corresponding

creators

are then

by

expression

and the one for the corresponding

annihilators

can be combined

easily

to a single

matrix-equation

where

the (2i%fb x 2Mb)-matrix

quasi-particles.

Eqs.(2)

transformation

conserving

sen finite

single

particle

where the product

vation still

basis.

are violated

conserved

HFB

in order

relations,

The corresponding

orbital

angular

excitation),

vacuum

> different

IF > is neither

and, in general,

is the so-called

even or wit,h odd t,ota.l nil&on

the vacuum

“number-parity” numbers

A.

1;.

[12].

character

It is the most

which can be constructed ]F > can be represented

numbers total

characterizing angular

an eigenstate

Furthermore

(3) has no definite

, i.e.

Fermion

for the

general within

linear

the cho-

as

from zero.

momentum,

i” nor of its 33component

to ensure

transformation

(2) sums over all the quantum

and the radial operator

the famous

the anti-commutation

(isospin-projection,

of the latter, momentum

define

runs over all (Y with a,]0

Since the transformation sis states

F has to be unitary

the single particle

momentum,

the 33projection

of the square of the total

particle

number

parity

IF > contains

either.

either

ba-

and charge

angular conser-

The only symmetry

only

components

with

Recent From the vacllum .i ? AZ-;I”

(3) one can construct

I1sing the operator

from VAMPIR

News

configllrations

IS1

with thr tl~~siretl s!-mmrtry

cluantllm

n,lml,ers

[5]

!A) where

I? is the parity-,

=I the nucleon

numb-.

:? and 2 the neutron

the usual rot,ation-operator,

and D’nrI,-(R! its representation

\-ia the A-quantum

the configuration

vacuum

number

(3) does still

dependence

as physical rotational

depend

is eliminated

by taking

configurations.

Even

symmetry

with the intrinsic

the linear

if only

thus introduces

degrees

obtained

on the orientation

rmmber

momentum

eigenstntes.

by actin g with the oprrator

of the intrinsic

quantisation

Ii

(4) on thr HFB

axis.

This

lmphysical

combir ations

a single

additional

of freedom

in angular

and proton

determinant

is considered,

configuration-mixing

of the underlying

HFB

the restoration

coefficients

transformation

f.

of the

which

together

will have to be determined

by variation. As already extension

mentioned, to linear

we shall restrict

combinations

code we have constructed the underlying discussed

fields

in the Refs.

[5,9].

3. Unrestricted

of several

can handle

mean

versus

nucleon

quantum

Symmetry

numbers,

not the case

in the older

underlying The

HFB

of axially

3komponent

assumption

versions

of the

of time-reversal

Consequently,

the resulting

describe

of doubly-even

states

are accessible.

too.

Details

mixin g by chains

[9] and the

of the procedure of variational

(5‘1. The

to determine

calculations

are

Methods

transformation

F mixes all states regardless

and prot,on or neutron

the S-dimensional

of the form

of this type is straightforward

option,

Restricted

doubly-odd

wave functions

projection

and odd nuclei

of the approach

where

origin.

Thus

of the total

after

of their angular

projection

spin any type

of parity. of state

can

already

via a single

determinant.

This

is

certain

symmetries

were imposed

on the

transformations.

requirement

to the

parity

and finally

in doubly-even,

configurations

this general

complex

numbers,

he described

here to test

and the configuration

In a given basis the unrestricted, momentum

ourselves

Though

c.ollnlinzzs are inclllderl.

symmetric total

invariance test

HFB

angular

transformations momentum

introduces

wave functions

or doubly-odd

folu-

in addition are restricted

nuclei.

by the use of essentially nart,icldar

induces

operator

a two-fold

alld more-nllckon

the vacua

degeneracy

EC =

number

all possible

ilrp missinp

0.

The

into the system.

not, even all states

t,ransformations collnlines

are eigenstates

eigenvalues

to even nucleon

Furthermore.

complex

that

IL with

15.721

and can only in these nuclei two-nucleon

: b-cl natllral

K. W. Schmid

152 (or unnatural) one natllral

pa.rity pairs

cannot

a.nd one unnatural

excitations

which are dominated

substructures

cannot

be coIlpled

parity

to an unnatural

pair not to a natural

by configurations

be described

even within

parity parity

containing

follr nucleon

four nucleon

such “missing

the up to now most

wave function

and

stat,e.

Consequently.

couplings”

as irreducible

advanced

COMPLEX

Vtl4IPIR

approach. In the earlier

calculations

transform&ions admitted. various

: proton-neutron-

Consequently, so called

restricted

beginning

states

couplings

calculated

in terms

since also axial

in fact

makes

made explicit variables

variables

basis spaces, in choosing Newton more

(one

definite

mixing

drops

the number

modern

accessible

by the

is still conserved

method,

and all the corresponding is included however,

right from the

is only suited

HFB

for

vacuum.

Instead

E.g.,

VAMPIR

of freedom.

than hand

even for general

wa.ve flmction

itself.

angular

approach

just

momentum

to

I these are

calculations

in larger

and some care has to be taken

the inverse

Hessian

method,

given

replaced

the Quasi-

was updated)

by a

by Gill and Murray

more stable.

vacua of the form (3). number.

for t,he svstem

independent

case we have furthermore

\Ve therefore

This

which

we have 20 variables

For unrestricted

a few thousands

and

momentum.

approaches

and 552 in the unrestricted

For given total

numbers

of linear

basis,

are now

Furthermore,

angular

the number

In the unrestricted

numerically HFB

representation

(2)).

in the previous

in an sd-shell

of the code (there

with even or with odd total nucleon

(eqs.

of the intrinsic

used for the minimisation.

however,

there is no a priori

all the matrixelements

transformation

On the other

the Hessian

transformation

in using the canonical

of normalisation).

used in the older versions

Number

on the HFB

simpler

will easily reach

updating

a multi-configuration

the sums run over all the quantum

alone.

out because

procedure

narit,v in t,he start,inc

were

were

even on the basis of symmetry-

approach,

approach.

considerably.

degrees

fast as the old version,

mlmher

nuclei

solution

value of the 3-component

required.

[13], is equally parity

HFB

and only real mean-fields

from that of the underlying

implementation

of variables

implementation

only components

MONSTER

of the HFB

56 in the COMPLEX

the numerical

method

The

of the VAMPIR

does increase

(K-)

approach,

there is no advantage

of the HFB-transformation

add the configuration

on the

Performance

Thus

the numerical

VAMPIR,

restrictions

In this way. &mixing

is not any more imposed

versions

with

in the variation

for the variation

21 extra

and

use of the symmetries

in the REAL

can be overcome

is not any more required,

not only over subspaces This

are avoided.

of the A- and B-matrices

symmetry

in doubly-even

excitations.

in the system.

as it was done in the earlier

symmetry

in the space of the VAMPIR

is not too different

invariance

degeneracy

states

is done in the MO?JSTER

Implementation

Since time-reversal

severe

were forbidden

that these deficiencies

This

whose structure

4. Numerical

parity

two-quasi-particle

and missing

more

approaches.

the Hamiltonian

symmetry-projected

two-fold

VAMPIR

transformations.

which diagonalizes

natural

however,

even

and parity-mixing

only

REAL

It should be stressed,

exited

we had imposed

Obviously

Thus they contain

one has to ensure

lmder ronsidera.tion.

This

either

the right

is achieved

hv

153

Recent News from VAMPIR l~locking cnw orbit, if odd A systems overlap-matrix Obviously the three

more involved

integrations symmetry.

numbers

the 3-axis

induced Hence

have

c.nlculations,

at least

vector--computers.

since

the

CPU

On the other

computers.

One can distribute

combination

the integration. sponding

hand

model time

Since

gradients)

does

almost

of the projected

a constant

achieved

How nicely

overhead

96

this

using

code is plotted

behaviour

which

5. Results

and

The quality

of the unrestricted

of

of n~m~bc~r of iI)-

for multi~proc,~ssox

to be performed

the results elements

on cnch grirl

at the end and pt,rform

(overlap.

part of the -program

or ~‘VPII

(more

energy

and <-err,‘-

t,han 99 perceut

).

192

224

Fortran

256

77 package

can be seen from CPU

of the number

performance

I60

processors

works,

the inverse

as function

is the optimal

128

the Cray

this procedure

of 10 seconds)

V.4MPIR

VAhlPIR

matrix

integration5

the power suited

as

entirely

Number

we have

with

operat,ions collect

aro11n~l

and ovcrlap~matricc~s

numerical

are particularly

available.

on qoo~l

be p(>rformc,rl on sequential

essentially

mathematical

processors

64

in Edinburgh.

increase

integrations

the identical

the calculation

can hardly

hy the projection

of energy-

:\vo of

analytically

Now also the t\vo rotations

always five-fold

is the by far most time consuming

the code can be parallelized

In practice

involves spaces,

multi-fold

over the different

induced

the calculation

applications

could be performed

had to be performed.

vectors

in larger

in the earlier

projection

with the two integrations

so that

of thr rotatl,tl

bllt still str;rightfor~vartl.

transformations:

momentum

integrations

gradient

tegrations.

point

together

to be done numerically

well as of t,he corresponding Such

by the angular

only three-fold

In this cast’ t,he ralcldation

as for the even A-case

we pa?; a price for the use of unrestricted

due to axial nucleon

is slightly

arp to be d(~scribctl.

time

needed

of processors

one can reach

1, where

T3D

(after

by the unrestricted

used.

in parallel

on the Cray

Fig.

We observe

computer subtracting GENERAL

a perfectly

linear

computing.

Discussions VAMPIR

IGC:V\ in the followinp

approach

wa.s tcsktl

which we shall denote

hv selert6d

as GENERi\L

annlicn t,ions in an IsOd--shell

COnIPLES mod?1 sna(‘e.

K. W. Schmid

154 This allows a direct more

rcstrirtetl

IsOd-shell

The

single

?Je\:)

VALIPIR

where

the number

have been

present

(e(Odj,2)

force

= cc.4

configuration

MeV,

of earlier

out of the middle

spaces

are considerably

-3.28

hleV> and E(Od3,2)

of the

larger

than

approach.

~(ls1p)

=

[13] has been used, except has b een chosen

the mass-dependent

for the fact

as i instead

that

= +0.93 version

the exponent

of

01 of the

of 0.3.

yrast-spectrum

1.d -‘,1 5’

-.i_

6’ ‘B

RM

H”

nuclei

[14]. A s e ff ec t.IVF: interaction

= 18)( 9)”

6* p,

as well as with the results

\VP chose

in the GCV

= -4.15

from experiment

and Wildenthal ?(.A)

calclllations.

of the shell-model

variables

energies taken

SC&I diagonalizations

and MONSTER

of variational

scaling-factor

with exact

the dimensions

particle

the Chung

comparison

EXOCi

CCY

CM

C”

~

Method

Fig.2

displays

obtained

total

by 5 different

in the first (RV)

the

column

results

misses

of the REAL

the RV-vacuum quasi-particle unchanged particle

energies

spins

the

absolute

spin states

Furthermore,

and the odd spin of the SCM

energy

spectrum

states

core

of the yrast

them to the exact

approach

excitation

by more

which

By construction

the higher

in this

the relative

(RM)

I60

than

diagonalizes state

therefore get some,

In the next

and all corresponding the total though

energy

small,

can be obtained

as well.

can be well reproduced,

though

VAMPIR 3.

spectrum column

Hamiltonian

come

too,

of

two-

state

remains

the two-quasi-

of the RV calculation

For them,

the

in the space

of the O+ ground from

For

rather

symmetry-projected

contributions

in this case the symmetry-restrictions

presented

in section

of the SCM

the chosen

of 24Mg

of REAL

as discussed

energies

2.5 MeV.

spectrum

SCM results

from the left we first give the results

accessible

for the O+ ground

excitations. while

to the

and compares

Starting

are not

MONSTER

relative

methods

the RV reproduces

obtained

excitations.

overcome

approximate

Odd

calculations.

however,

energies

from the right.

the even spin states well,

binding

the relative

are

excitation

the order of the 4+ and 3+ excitations

is reversed. The third

column

odd snin stnt,es

from the left displays can he oht,ailled

then

the results from

of the COMPLEX

the I<=0

vacllum.

V.UIPIR

however.

(CV)

approach.

t,he firTIre rlearlv

Here

indicates

News from VAMPIR

Recent

den

spin states

t,he energy

t,he nbsol~~tc~ energy qound

is considerably

st,nte energy

otld spin states !c’\I)

approach.

Using

of Ref.

S~ontl

bllt

rc’lativc

energies

but

within

approach.

also

about

Yotc,

that

of that nucleus

has here

same

file lowast

t,o the latter

approal-h

energies

in the fourth

~ow~~Y~Y.

: Now hn SC’11

for the descript.lon

r multi-determinant

the relative

CO1IPLES

of

tilt,

SIOSSTER

of the even anti the odd spin -Taft>\

column

15385

only.

via the FED there

from the left, which

GCV

of both trivial

even

to only .558 linear SCM

expansion

Obviously.

V.UIPIR

tlisplav-

rh(J

SCM

a rather

dimensions

in Fig. similar

configurations).

pattern.

GCV

Again

The

in the GC’I-

could bt, ~~orrf~lntcfl presenled

herr

at least in “11g.

results

for the yrast

as in case of 24Mg are compared

the GCV (e.g..

So, even in the middle

of the esact

for 28Si.

configll-

is in the GC’\.

the results

correlations

ciul he

of SCX

variahltx

solution

;9]. H owever.

are obtained

larger

bpin states

of the wave function

3. The same methods

are slightly

Now not only- rllt,

and odd

independent

the “free”

method

calculations.

: e.g., the number

is not much space for such additional

are displayed

description

energies

is by no means

complicated

determinant

holds for doubly-odd vrast

binding

This

here the deviations SCM

of the (one-determinant)

as compared

the largest

We observe though

yiplcls an excellent The

that

the IsOd-shell

spectrum

both.

absolute

the rather

configurations

demonstrate

the

is 1968

by additional

each other.

respect

Again the shortcomings

by the correspondin,

100 IieV.

by a single

states

with

well as can bp sun

approximated

Within

the same qualIt?; as in the R\ calculation.

[lGj.

for the 3+ state

clearly

improved

this method,

equally

are of about

by less than 700 KeV.

last wvpshow the results

rpproduccd rations

is missed

can be overcome

~a11 bt> reprodxlced results

differencrs

IS5

about

results

agree

300 KeV

with

well with the SCM

for the 3’

of the shell the “free”

state

GCV

which

approach

solutions.

and odd nuclei.

st,a.tes of t,hr do~~blv~odtl nuclrlls

-4s an example 2”A1

we present

Herr no RV reslllts

in Fig.

1 the results

for

can he Lrivcan sinl,e this

K. W. Schmid

156 method

is rest,rickd

for the nucleus spectrum

to doubly-even

‘8Si.

It is seen that

nuclei.

The

RM

calculation

a.lso in this case the GCV

was

I~a.sed

approach

on

the Oi RV sc>lution

reproduces

the shell-model

very well. -102 j,]

CM

Method

However,

here remaining

for by additional lowest

states

correlations.

of *‘Al.

CV approach,

differences

so that

As an example

Because the

GCV

description

of an odd

CM results

have been obtained

based

on t,he mean-field

in absolute

nucleus.

Again

the

400 KeV would have to be accounted

for an odd nucleus

of time-reversal spectrum

energy of about

symmetry

presents

the first

agreement

with

here using only the complete

which was oht,a.ined with

these

we display states

the results

for the three

are inaccessible

by even the

symmetry-projected the

SCM

solutions

one-quasi-particle

t,hr CV annroach

one-determinant is excellent. configuration

for thp 0 + erolmd

state

The space of t,he

157

Recent News from VAMPIR neighhouring

doubly-even

‘“$2.

nucleus

Thus larger

deviations

are to be expcctrd

for the Chl rf4ts

as in case of the even -4 nuclei. 6. Conclusions We have entirely

and Outlook

reported

the

unrestricted

in an lsOd-shell

results

vacua

basis

This

holds

and furthermore,

even

problem.

Most

in the middle

model

Slater

importance

can be severly

affected

like *‘Ca before

three-fold

in the GCV

in much larger

body

for systems

major

problem

almost

160.

Recent

dimensions

(GCV)

approach

a well known

investigations

than

form factors,

response

of the center

of momentum

in trrms

by construction

this is an l/X

heavier

believe

are 1aro;est.

the wave functions

factors, treatment

effect

[17.18]

functions

but

and thus show that

and even energies

motion

even in cuclei

1ean invariance. of full G a 1’1

that the restoration

..free”

and odd nllclei.

we encounter

is fulfilled

that

thou%11

basis systems.

expand

argued

perfectly

V.4UPIR

shell.

principle

diagonalizations

doubly-odd

CO2vlPLEX

a single

It is usually

Spectroscopic

again

by projection

integration

approach.

Thus

or vector

demonstrated

in the

a correct

present

are particularly the mathematical

the center

of momentum

methods

to be performed

computers.

developing

feasible

than

losing

i.e., an essentially

the shell-model

for doubly--even,

done

obvi~~usl!;

is necessary.

day sequential

approach

where

ever

shell-model

determinant.

GENERAL

many

broken.

We therefore

the variation,

of complete

In this way the Pauli

by an incorrect

and beyond.

This can be achieved another

to the nuclear

at least

is not true.

larger

calculations

can be reproduced

but can be applied

becomes

is severely

this statement

of the shell

the unrestricted

determinants.

invariance

results

is of the same quality

spaces

as the basis

approaches

of (generalized)

the exact

HFB

only a single symmetry-projected

however,

to small

as soon

of minor

with the results

the agreement

However,

the Galilean

Comparison

has shown that

Unlike the shell-model. is not limited

symmetry-projected

[ll].

we have used for each state theory.

of the first

treatment

that

rest frame,

to the five-fold

of Galilean processing

the multi-fold

for parallel

apparatus

in a not very distant

in addition

For parallel

study

suited

[17]. The corresponding

data

invariance

integrations Since

needed for the projection we are confident

that

operator

integration is hardly

the situation

processing.

integral

is quite

involves

already

possible

present

on present

different.

We have

to be performed

in the

we have

succeeded

of general

this procedure

already

HFB

determinants

will become

GCV in into

numerically

future.

Acknowledgement The work reported

here has been done in collaboration

with Esko Hammasin

and Amand

Faessler.

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