Table Of ContentECOLE DE PHYSIQUE DES HOUCHES - UJF & INPG - GRENOBLE
a NATO Advanced Study Institute
LES HOUCHES
SESSION LXXIII
2-28 July 2000
Atomic clusters and nanoparticles
Agrkgats atomiques et nanoparticules
Edited by
C. GUET, P. HOBZA, F. SPIEGELMAN and F. DAVID
sI1
SCIENCES
Springer
Les Ulis, Paris, Cambridge Berlin, Heidelberg, New York,
Barcelonu, Hong Kong, London
Milan, Paris, Tokyo
Published in cooperation with the NATO Scientific Affair Division
Preface
Atomic clusters are aggregates of atoms or molecules with a well-defined size
varying fiom a few constituents to several tens of thousands. Clusters are
distinguished fiom bulk matter in so far as their properties are strongly affected
by the existence of a surface involving a large ktion of the constituents and by
the discreteness of their electronic excitations. On the one hand, the fmite nature
of the number of constituents leads to novel structural and thermodynamic
properties with no equivalent in the bulk. On the other hand, clusters bridge the
physics of atoms to the physics of bulk matter. Studying thoroughly the physical
properties of clusters may shed new light on elementary excitations in solids and
liquids.
The physics and chemistry of atomic clusters and nanoparticles constitute a
broad interdisciplinary domain. The field is growing hst, as witnessed by the
number of publications and conferences or workshops. The directions of growth
are numerous. Some involve pure basic science, others are motivated by more
applied material science considerations. Within the fast development of new
technologies one observes an unavoidable trend toward miniaturization in micro-
and nano-electronics. One wishes to technologically master devices of sizes so
small that their quantum specific properties are the most important. Specific
magnetic properties of clusters or dots on surfaces might offer new materials for
applications in high-density recording and memory devices. Similar trends are
observed in other fields, such as catalysis, energy storage, and control of air
pollution.
The present school was focused on basic science. It gathered lecturers with
research practice in atomic and molecular physics, condensed matter physics,
nuclear physics, and chemistry and physical chemistry, not to mention
computational physics.
The field has developed rapidly thanks to a strong coupling between theory
and experiment. Two predominantly experimental courses provided the
opportunity to appreciate what is and will be experimentally possible. Patrick
Martin presented a large experimental review, and Hellmut Haberland lectured
on recent experimental observations of phase transitions in metal clusters.
The theoretical courses covered three main domains: (i) electronic properties
of metallic clusters and nanostructures, (ii) phases and phase changes of small
systems, and (iii) chemical processes in nanoscale systems. George Bertsch gave
a series of quite interactive lectures that introduced the students to basic
phenomena in metal clusters, quantum dots, fullerenes and nanotubes. He
presented the basic theoretical tools to study cluster properties, particularly
electronic excitations. He emphasized the importance of simple models that
large-scale ab initio calculations validate, as well as the universal features of
finite quantum systems.
The Density Functional Theory stands at a central position as a quantum
mechanical method for practical studies of large molecules and clusters. Today,
the combination of quantum mechanics and molecular mechanics using classical
force fields allows us to understand biological systems to a large extent. Denis
Salahub gave the state of the art of this theory, which he illustrated with
chemical applications throughout his course.
Metal clusters and nanosystems are excellent physical objects for
illuminating the links between the quantum and classical worlds. Matthias Brack
reviewed semiclassical methods of determining both average trends and quantum
shell effects in the properties of finite fermionic systems. His course was
centered on two important theoretical themes: (a) the Extended Thomas-Fermi
Model, and @) the Periodic Orbit Theory (POT).
Pairing correlations in atomic nuclei are intimately related to the phenomena
of superconductivity (and superfluidity) in macroscopic systems. Recent
experiments on small metallic clusters also reveal pairing correlations. Hubert
Flocard devoted his course to a thorough investigation of pairing correlations in
finite systems within the state of the art theoretical models.
Cluster and nanoparticle physics is also part of condensed matter physics.
Matti Manninen focused his course on the physics of nanosystems from the point
of view of condensed matter physics, emphasizing concepts and mesoscopic
features that are common between finite systems and low-dimensional systems.
Magnetic properties of small systems are very sensitive to the size, structure,
and composition of the system. Within a pedagogical four-lecture course
Gustavo Pastor provided a detailed account of the most powerful theoretical
methods for eficiently describing the magnetic properties of clusters,
particularly transition-metal clusters. The spin-fluctuation theory is quite
appropriate for clusters since changes or fluctuations of structure are important.
Scattering processes are also useful to gain insight on the many-body properties
of finite systems, as demonstrated by Andrey Solov'yov in his two-lecture
course. He emphasized the essential role of surface and volume plasmon modes
in the formation of electron energy loss spectra.
The thermodynamical properties of clusters are certainly of major
importance. They require theoretical approaches to the concepts of melting,
fkeezing and phase changes in finite systems and their dependence on size. The
notions of "phase-like" forms, coexistence, solid-liquid equilibrium, phase
diagrams and their specific formulation in terms of the thermodynamical
variables and functions in various ensembles require the development of new and
sophisticated algorithms.
xxiii
Computer simulations of cluster dynamics and thermodynamics have become
a major tool for predicting and understanding the finite temperature behavior of
clusters. David Wales' course was concerned with concepts and recent methods
of achieving topological analysis and sampling of complex multi-dimensional
potential energy landscapes.
Sergei Chekmarev described a novel approach to the computer simulation
study of a finite many-body system that allows one to gain detailed information
about this system, including its potential energy surface, equilibrium properties
and kinetics. Biomolecules can now be investigated with a high degree of
confidence.
The chemical processes in or with gas phase clusters and nanoscale particles
are a growing field of cluster science. The key to understanding the properties of
various families of atomic clusters lies in the determination and description of
the chemical bond. Concepts such as valence and valence change, bond
directionality, hybridization, hypervalence, electronic population distribution and
charge fluctuation are essential. The course of Pave1 Hobza addressed the above
concepts, methods and mechanisms. Lucjan Piela focused his lecture (not
published here) on cooperativity effects in quantum chemistry. This subject is
indeed topical for molecular clusters.
In addition to the main courses that are the contents of this book, there were
seminars not published herein. These seminars, that triggered stimulating
discussions, were given by Jacqueline Belloni, Stephen Berry, Catherine
Brdchignac, Vlasta Bonacic-Koutecky, Frank Hekking, Joshua Jortner, Vitaly
Kresin, Richard Lavery, Eric Suraud, and Ludger Woeste.
During the four weeks the lecturers and students got to know each other
pretty well. Hopefully every student discovered and shared the excitement that
his (her) colleagues &om other fields experienced. More practically, one would
have realized that some as yet unfamiliar methods could be of great interest for
one's own research. The overall organization of the school provided the best
conditions to meet these goals. The daily lecture program was kept light, with no
more than three main courses and sometimes an extra seminar. This schedule
allowed plenty of time for discussions and organizing small working groups. We
even observed that some collaborative research work had started. On behalf of
all the lecturers and students we would like to dedicate this summer school to the
late Professor Walter D. Knight. Walter Knight and his group at Berkeley
pioneered the experimental field of cluster beams. His experiments in the 1980s
led to major discoveries relating to finite size effects and to a strong revival of
cluster physics.
This School would not have been possible without:
-
the help and the support of the Scientific Board of the "6cole de
Physique";
-
the staff of the "~coled e Physique": Ghyslaine &Henry, Isabel Lelievre
and Brigitte Rousset;
-
the fmancial support of the Scientific and Environmental Affairs
Division of NATO, the Research Directorate of the European
Commission, and the Formation Pennanente of CNRS; and
-
the support to the "~coled e Physique" by the Universitk Joseph Fourier,
the French Ministry of Research, CNRS and CEA.
C. Guet
P. Hobza
F. Spiegelman
F. David
CONTENTS
Lecturers xi
Prkface xvii
Preface xxi
Contents XXV
Course 1. Experimental Aspects of Metal Clusters
by T.P. Martin
Introduction
Subshells, shells and supershells
The experiment
Observation of electronic shell structure
Density functional calculation
Observation of supershells
Fission
Concluding remarks
Course 2. Melting of Clusters
by H. Haberland
1 Introduction
2 Cluster calorimetry
2.1 The bulk limit . . . . . . . . . . . .
2.2 Calorimetry for free clusters . . . . .
3 Experiment
3.1 The source for thermalized cluster ions .
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4 Caloric curves
4.1 Melting temperatures . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Latent heats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Other experiments measuring thermal properties of free clusters . .
5 A closer look at the experiment
5.1 Beam preparation . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Reminder: Canonical versus microcanonical ensemble . . .
5.1.2 A canonical distribution of initial energies . . . . . . . . . .
5.1.3 Free clusters in vacuum, a microcanonical ensemble . . . .
5.2 Analysis of the fragmentation process . . . . . . . . . . . . . . . .
5.2.1 Photo-excitation and energy relaxation . . . . . . . . . . . .
5.2.2 Mapping of the energy on the mass scale . . . . . . . . . . .
5.2.3 Broadening of the mass spectra due to the statistics
of evaporation . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Canonical or microcanonical data evaluation . . . . . . . . . . . . .
6 Results obtained from a closer look
6.1 Negative heat capacity . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Unsolved problems
8 Summary and outlook
Course 3 . Excitations in Clusters
by G.F. Bertsch
1 Introduction
2 Statistical reaction theory
2.1 Cluster evaporation rates . . . . . . . . . . . . . . . . . . . . .
2.2 Electron emission . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Radiative cooling . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Optical properties of small particles
3.1 Connections to the bulk . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Linear response and short-time behavior . . . . . . . . . . . . . . .
3.3 Collective excitations . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Calculating the electron wave function
4.1 Time-dependent density functional theory
5 Linear response of simple metal clusters
5.1 Alkali metal clusters . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Silver clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xxvii
6 Carbon structures 89
6.1 Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.2 Polyenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.3 Benzene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.4 c60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.5 Carbon nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.6 Quantized conductance . . . . . . . . . . . . . . . . . . . . . . . . . 102
Course 4 . Density Functional Theory, Methods. Techniques.
and Applications
by S. Chrbtien and D .R . Salahub 105
1 Introduction 107
2 Density functional theory 108
2.1 Hohenberg and Kohn theorems . . . . . . . . . . . . . . . . . . . . 110
2.2 Levy's constrained search . . . . . . . . . . . . . . . . . . . . . . . 111
2.3 Kohn-Sham method . . . . . . . . . . . . . . . . . . . . . . . . . . 1 12
3 Density matrices and pair correlation functions 113
4 Adiabatic connection or coupling strength integration 115
5 Comparing and constrasting KS-DFT and HF-CI 118
6 Preparing new functionals 122
7 Approximate exchange and correlation functionals 123
7.1 The Local Spin Density Approximation (LSDA) . . . . . . . . . . 124
7.2 Gradient Expansion Approximation (GEA) . . . . . . . . . . . . . 126
7.3 Generalized Gradient Approximation (GGA) . . . . . . . . . . . . 127
7.4 meta-Generalized Gradient Approximation (meta-GGA) . . . . . . 129
7.5 Hybrid functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.6 The Optimized Effective Potential method (OEP) . . . . . . . . . 131
7.7 Comparison between various approximate functionals . . . . . . . . 132
8 LAP correlation functional 132
9 Solving the Kohn-Sham equations 134
9.1 The Kohn-Sham orbitals . . . . . . . . . . . . . . . . . . . . . . . 136
9.2 Coulomb potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 38
9.3 Exchange-correlation potential . . . . . . . . . . . . . . . . . . . . 139
9.4 Core potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 39
9.5 Other choices and sources of error . . . . . . . . . . . . . . . . . . 140
9.6 Functionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
xxviii
10 Applications 141
10.1 Ab initio molecular dynamics for an alanine dipeptide model . . . 142
10.2 Transition metal clusters: The ecstasy, and the agony. . . . . . . . . 144
10.2.1 Vanadium trimer . . . . . . . . . . . . . . . . . . . . . . . . 1 44
10.2.2 Nickel clusters . . . . . . . . . . . . . . . . . . . . . . . . . 145
10.3 The conversion of acetylene to benzene on Fe clusters . . . . . . . 149
11 Conclusions 154
Course 5 . Semiclassical Approaches to Mesoscopic Systems
by M . Brack 161
1 Introduction 164
2 Extended Thomas-Fermi model for average properties 165
2.1 Thomas-Fermi approximation . . . . . . . . . . . . . . . . . . . . . 165
2.2 Wigner-Kirkwood expansion . . . . . . . . . . . . . . . . . . . . . 166
2.3 Gradient expansion of density functionals . . . . . . . . . . . . . . 168
2.4 Density variational method . . . . . . . . . . . . . . . . . . . . . . 169
2.5 Applications to metal clusters . . . . . . . . . . . . . . . . . . . . . 173
2.5.1 Restricted spherical density variation . . . . . . . . . . . . . 173
2.5.2 Unrestricted spherical density variation . . . . . . . . . . . 177
2.5.3 Liquid drop model for charged spherical metal clusters . . . 178
3 Periodic orbit theory for quantum shell effects 180
3.1 Semiclassical expansion of the Green function . . . . . . . . . . . . 181
3.2 Trace formulae for level density and total energy . . . . . . . . . . 182
3.3 Calculation of periodic orbits and their stability . . . . . . . . . . . 187
3.4 Uniform approximations . . . . . . . . . . . . . . . . . . . . . . . . 190
3.5 Applications to metal clusters . . . . . . . . . . . . . . . . . . . . . 192
3.5.1 Supershell structure of spherical alkali clusters . . . . . . . 192
3.5.2 Ground-state deformations . . . . . . . . . . . . . . . . . . 194
3.6 Applications to two-dimensional electronic systems . . . . . . . . . 195
3.6.1 Conductance oscillations in a circular quantum dot . . . . . 197
3.6.2 Integer quantum Hall effect in the two-dimensional electron
gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
3.6.3 Conductance oscillations in a channel with antidots . . . . 200
4 Local-current approximation for linear response 202
4.1 Quantum-mechanical equations of motion . . . . . . . . . . . . . . 203
4.2 Variational equation for the local current density . . . . . . . . . . 205
4.3 Secular equation using a finite basis . . . . . . . . . . . . . . . . . 207
