COLLECTIVE EXCITATIONS IN METAL CLUSTERS

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COLLECTIVE EXCITATIONS IN METAL CLUSTERS
Ph. Chomaz, D. Gambacurta, N.V.Giai, M. Grasso,
G. Piccitto, M.Sambataro, Ll. Serra,
C.Yannouleas, F.C.
in various moments
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Delocalized valence electrons moving in
the background of positive ions, with
some spillout Existence of magic
numbers Very collective dipole
excitation
c. of m. of electrons against
ionic background Simplest model
Jellium model Uniformly distributed
positive charge electrons
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INTERACTIONS
Jellium-electrons Coulomb from a
uniform spherical charge
Derived from
energy density functional
Electron-electron or
Bare
Coulomb with exchange Except for
the ionic background ( jellium) , like nuclei
BUT MUCH
SIMPLER



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only one kind of particles

spin-orbit not very important

KNOWN
INTERACTION
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A lazy man, as I am, can apply all the well known
machineryHartree-Fock, RPA, Boson expansions,.

• Very useful laboratory to test extensions
  of RPA since
• realistic many body system
• no adjustable parameter in the
  interaction
• Of course many important physical ingredients
  are
• missing, related to the structure of ions and
  their motion

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d
1st attempt F. Catara, Ph. Chomaz, N. Van
Giai, Phys. Rev.B48(1993)18207 Two-plasmon
excitations in metallic clusters All the
above indicated machinery BUT 2nd order boson
expansion not enough ?very huge
anharmonicities
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C. Yannouleas, F. Catara, N. Van Giai, Phys. Rev.
B51(1995)4569Ground state correlations and
linear response of metal clusters Careful
analysis of ground state properties within RPA

• Occupation numbers

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Probability for the presence of 2n p-h
excitations in ground state
GROUND STATE 0 2 4 ..2nph
ALSO SPIN, see next
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TRACED BACK TO LONG RANGE
F. Catara, Ll. Serra, N. Van Giai, Z.Phys.
D39(1997)153 Ground state correlations of jellium
metal clusters in local spin-density approximation
EVEN LARGER DEVIATIONS FROM HF ?
BEYOND RPA
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Necessary to avoid quasi boson approximation To
derive equations of motion, use is made of 0gt as
vacuum of Q. Finally, substituted with HF gt.
F. Catara, G. Piccitto, M. Sambataro, N. Van Giai
Towards a self consistent RPA for Fermi
systems Phys. Rev. B54(1996)17536
RENORMALIZED RPA Plus a few
more (also with Marcella), some more steps
Most recent D. Gambacurta, F. Catara Particle-hole
excitations within a self consistent
RPA Submitted
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As in RPA, introduce the collective ph operators
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Equations of motion method leads to
NOTE 0 gt contains correlations
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If 0 gt? HF gt standard RPA is obtained
For example
Quasi boson approximation
In general one and two-body density matrices
appear in the double commutators Linearization
of the equations of motion in the commutator
Contractions in a reference state
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Again if contraction in HF gt?standard RPA If
in 0 gt? ERPA Problem reduced to one body
density matrix Number operator method (Rowe)
NON LINEAR PROBLEM ITERATIVELY
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In the A matrix of ERPA the quantities
appear
They reduce to HF single particle energies
By diagonalizing? a kind of generalized s.p.
energies Spectrum much more compressed Reduced
gap between last occupied and first empty level.
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One body density
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ENERGY WEIGHTED SUM RULES
Identity IF eigenstates of H
both particles and holes in general
IF 0 gt ? HF gt RPA
BUT in the r.h.s. not only ph components, unless
0 gt ? HF gt Thouless theorem
IN ERPA (only ph components in Q)
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Exactly satisfied if one takes only ph components
of F
and if linearizes the commutator H,F
Self consistent in ph space To have a
completely self consistent approach, with the
complete F operator one has to generalize
and linearize
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Already used in M.Grasso, F. Catara, Phys.
Rev. C 63, 014317 (2000) 3 level Lipkin
model EWSR exactly preserved BUT SPURIOUS STATE
(not corresponding to any exact eigenstate) In
the present approach it should be possible to
eliminate Zero eigenvalues of metrics G