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.
7. References [I] see, e.g., J.B.McGrory
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