Semiclassical versus quantal time-dependent mean-field descriptions of electron dynamics
in ion-cluster collisions
L. Plagne* and J. Daligault
De ´partement de Recherche Fondamentale sur la Matie `re Condense ´e, CEA-Grenoble, 17, rue des Martyrs,
F-38054 Grenoble Cedex 9, France
K. Yabana
Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan
T. Tazawa
Department of Physics, Yamaguchi University, Yamaguchi 753-8511, Japan
Y. Abe and C. Guet
†
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
Received 22 July 1999; published 15 February 2000
Both quantal and semiclassical mean-field theories based on the density-functional theory have recently been
applied to describe the dynamical evolution of electrons during and after the collision between highly charged
ions and metallic clusters. We here compare and assess both methods. The quantal calculation solves the
time-dependent Kohn-Sham equations, whereas the semiclassical one solves the Vlasov equation with the same
exchange-correlation functional. Predicted observables such as electron transfer and residual electronic exci-
tations are in excellent agreement with each other. Ranges of applicability of both methods are discussed and
complementarities are emphasized.
PACS numbers: 36.40.-c
I. INTRODUCTION
Delocalized electrons in metallic clusters experience large
amplitude dynamics on a femtosecond time scale in such
physical processes as either irradiation by an intense ul-
trashort laser pulse 1 or the passage of a highly charged ion
2,3. Let us consider the latter process. During a collision of
a highly charged ion with a metal cluster at medium velocity
i.e., around the Fermi velocity, there is a large range of
impact parameters for which several electrons are transferred
from the cluster to the ion while some others are emitted in
the continuum. After collision, that is after a few femtosec-
onds, a highly charged and electronically excited cluster has
been formed as well as a hollow ionized or neutral atom.
Conventional quantum collision theories such as the closed-
coupling approach clearly cannot cope with such compli-
cated processes which require too huge number of configu-
rations even for modern supercomputers. Instead classical
models have been put forward with fair success 4,5. It is,
however, desirable to go beyond such purely phenomeno-
logical classical models and account for the quantum many-
body electronic dynamics to some extent.
Time-dependent density functional theory TDDFT is ap-
pealing in this respect for review see Ref. 6. TDDFT is an
extension of the static density functional theory DFT which
has been impressively successful as to the description of
ground-state properties of many-electron systems, even in
the local density approximation LDA. Although it is the
density alone which is at the heart of DFT, it is convenient to
express it in terms of single-particle wave functions associ-
ated to an effective local one-body potential, the Kohn-Sham
KS potential. The static problem then reduces to solving a
set of differential equations, the KS equations which re-
sembles the Hartree-Fock HF equations.
The time-dependent extension in the local density ap-
proximation TDLDA leads to the time-dependent Kohn-
Sham TDKS equations. These equations describe the time
evolution of single-particle wave functions in their mean
field. In TDLDA, one usually assumes that the time-
dependent exchange-correlation potential has the same local
density functional form as in ground state.
The Vlasov equation is nothing but the semiclassical ap-
proximation to the TDLDA 7. It is a mean-field theory with
the same density-dependent exchange-correlation potential.
Formally one can derive the Vlasov equation through a
expansion of the Wigner transform of the equation of motion
of the TDLDA density matrix and keeping the first term
only. The corresponding static solution is the Thomas-Fermi
approximation. Although the dynamics are purely classical,
the Vlasov equation still complies with the Pauli principle.
This was pointed out by Bertsch 8 as this equation is just
Liouville’s theorem with a self-consistent potential. As a
mean-field equation with no explicit collision term, the Vla-
sov equation does not create or destroy correlations. Thus it
preserves the Pauli correlations built in the initial Thomas-
Fermi one-body phase space distribution. In the small ampli-
tude limit, the analogy between the Vlasov equation and the
TDLDA equation whose limit is the random phase approxi-
mation has been thoroughly studied in the case of collective
*Permanent address: Institut fu ¨r Theoretische Physik A, RWTH,
D-52056 Aachen, Germany.
†
Permanent address: De ´partement de Recherche Fondamentale sur
la Matie `re Condense ´e, CEA-Grenoble, 17, rue des Martyrs,
F-38054 Grenoble Cedex 9, France.
PHYSICAL REVIEW A, VOLUME 61, 033201
1050-2947/2000/613/0332017/$15.00 ©2000 The American Physical Society 61 033201-1