1993-RevModPhys-metal Clusters Self-Consistent Jellium Model

The physics of simple metal clusters: self-consistent jellium model
and semiclassical approaches

Matthias Brack
Insti tut fur Theoretlsche Physik, Uni versi tat Regensburg, 0-93040 Regensburg, Germany
The jellium model of simple metal clusters has enjoyed remarkable empirical success, leading to many
theoretical questions. In this review, we first survey the hierarchy of theoretical approximations leading
to the model. We then describe the jellium model in detail, including various extensions. One important
and useful approximation is the local-density approximation to exchange and correlation effects, which
greatly simplifies self-consistent calculations of the electronic structure. Another valuable tool is the semi-
classical approximation to the single-particle density matrix, which gives a theoretical framework to con-
nect the properties of large clusters with the bulk and macroscopic surface properties. The physical prop-
erties discussed in this review are the ground-state binding energies, the ionization potentials, and the di-
pole polarizabilities. We also treat the collective electronic excitations from the point of view of the clus-
ter response, including some useful sum rules.
CONTENTS
1. Semiclassical density-variational calculations 714
2. Liquid-drop model expansion of the energy 714
I. Introduction 677 3. Asymptotic behavior of ionization potentials and
II. From the Quantal Many-Body Problem to Semiclassical electron amenities 716
Jellium Drops: A Hierarchy of Approximations 679 VI. Summary and Conclusions 718
A. The quantal many-body problem 679 Acknowledgments 719
B. Quantum chemistry 680 Appendix A: Mean-Field Theories 720
C. Molecular dynamics and static pseudopotential mod- 1. Hartree-Fock theory 720
els 681 2. Density-functional theory 721
D. Jellium model 682 a. Hohenberg-Kohn theorem and density-variational
E. Phenomenological shell models 682 equations 721
F. Semiclassical and classical approaches 683 b. Kohn-Sham equations 723
III. The Self-Consistent Jellium Model 684 c. Local-density approximation 724
A. Basic concepts 684 d. Pseudopotentials 724
B. Kohn-Sham-LDA calculations 685 e. Car-Parrinello equations 725
1. Spherical jellium model 686 Appendix B: Linear-Response Theory 725
2. Deformed jellium model 688 1. RPA and TDLDA 725
3. Finite-temperature effects 691 2. Sum rules and relations to classical hydrodynamics 726
C. Beyond LDA 692 a. Sum-rule expressions 726
1. Hartree-Fock calculations 692 b. Scaling model interpretation of moments m3 and
2. Explicit evaluation of long-range correlations 692 ml 727
3. Self-interaction correction 694 c. Local-current RPA, fluid dynamics, and normal
4. Weighted-density approximation 694 hydrodynamics 728
5. Gradient expansions of the xc functional 694 References 729
D. Extensions of the jellium model 695
1. Simple patches 695
2. Structureless pseudopotential models 695 INTRODUCTION
IV. Electric Dipole Response of Metal Clusters 696
A. Linear-response theory 696 The theoretical description of metal clusters has re-
B. Linear-response calculations 697 cently undergone a sort of phase transition. Until less
1. Static dipole polarizabilities 697 than a decade ago, metal clusters were either small mi-
2. Dipole resonances and the dynamic response 699
3. General discussion 701
cromolecules consisting of a few atoms, treated with
C. Sum-rule approach 702 molecular physics and quantum-chemical methods, or
1. Sum-rule relations and classical limits 702 small particles in the mesoscopic domain, which were
2. Coupling of surface and volume plasmons 704 essentially pieces of bulk metal and could be described
V. Large Clusters: A Step Towards the Bulk? 707 using the approaches of solid-state and statistical physics
A. Shells and supershells 707 (Kubo, 1962; Denton et al. , 1973; for a review, see
1. Shell effects in finite fermion systems 707
2. Electronic supershell structure in large alkali clus-
Halperin, 1986). With the discovery of electronic shell
ters 709
structure in free alkali clusters by Knight et al. (1984,
3. From electronic to ionic shells 713 1985), a new era has started in which emphasis is put on
B. Semiclassical theory and large-N expansions: links to the quantized motion of the delocalized valence electrons
the macroscopic world 713 in the mean field created by the ions. The detailed ionic
Reviews of Modern Physics, Vol. 65, No. 3, July 1993 0034-6861 /93/65(3) /677(56) /$1 0.60 1993 The American Physical Society 677
678 Matthias Brack: The physics of simple metal clusters
structure often does not seem to affect very much the numbers X. As a matter of fact, in order to describe the
properties of alkali and other simple metal clusters. Car- structure of clusters with many thousands of atoms one is
ried to the extreme, this behavior suggests the jellium strongly encouraged by the sheer numerical size of the
model, which is defined by a Hamiltonian that treats the quantum calculations to use statistical or quasiclassical
electrons as usual but the ionic cores as a uniform posi- methods. Besides, it is our belief that even for medium-
tively charged background. This naturally leads to a sized systems in which microscopic effects play an impor-
description of the electron density in terms of single- tant role, semiclassical approaches often allow for a more
particle wave functions that extend over the entire clus- transparent interpretation of many phenomena than the
ter. The mean field of the electrons can either be calcu- purely microscopic theories and therefore offer better in-
lated self-consistently in the simple spherical jellium sights into the important physical mechanisms.
model (Ekardt, 198Sa, 198Sb), yielding the correct shell- The jellium model has proven to be an almost ideal
closing numbers of electrons in many cases, or be phe- theoretica1 instrument for approaching the above goal: it
nomenologically parametrized including the effects of de- is simple enough to be applied to spherical metal clusters
formations (Clemenger, 198Sa, 198Sb) in analogy to the containing up to several thousand atoms, but still allows
nuclear shell model. In this way, using relatively simple- for a self-consistent microscopic description of the aver-
minded approaches to the many-body problem, a wealth age field felt by the valence electrons, correctly rendering
of experimental data on simple metal clusters can be many of the observed shell structures. At the same time
classified and often be theoretically reproduced at least it allows one to extract parameters from finite clusters
semiquantitatively (see de Heer, Knight et a/. , 1987, for a that can be directly compared to those of the bulk or of
review). plane metal surfaces for which it has been applied al-
Metal clusters today provide a convenient and relative- ready over twenty years ago (Lang and Kohn, 1970,
ly inexpensive tool for studying the properties of finite 1971, 1973). Its success in describing the "supershell"
fermion systems with increasing sizes all the way from structure in very large alkali clusters, for which none of
atomic to mesoscopic dimensions, and hopefully soon to the more structural models have any chance to compete,
the macroscopic domain as well. The recent observation speaks for itself.
by Pedersen et al. (1991) and other groups of the so- The obvious Aaw of the jellium model is its complete
called supershe11 structure in alkali clusters with up to neglect of ionic structure. In order for the model to work,
three thousand atoms represents a milestone in this de- several conditions must be satisfied. First, the valence
velopment. electrons must be strongly delocalized, a condition met in
This review article is devoted to some of the theoreti- meta1s that are good conductors. Second, the ionic back-
cal approaches used for the description of simple metal ground must respond very easily to perturbations, to per-
clusters. It has been conceived and prepared in close mit the electronic single-particle energies to determine
contact with Walt de Heer, whose review on experimen- the structure. This is obviously most likely to be satisfied
tal techniques and results appears as an adjacent article when the valence electron has an s-wave character with
in this issue. De Heer also discusses many theoretical respect to the ionic cores, since then there is no direc-
models and physical pictures and compares their results tionality to the binding. ' Thus the jellium model has its
to the experimental data. The present article is meant to main applicability in the group Ia metals, particularly
provide some background of the theory, mainly based on sodium, potassium, and the heavier alkalis, and to some
the microscopic mean-field and density-functiona1 ap- extent the Ib metals such as copper and silver.
proaches. We shall be discussing the results of different Even in elements that slow the jellium behavior most
theoretical models and approximations and, of course, we clearly, the model cannot compete quantitatively with ab
also have to look at experimental data in order to assess initio quantum-chemical methods or molecular dynam-
the differences. But for the detailed comparison between ics in explaining many finer experimental details of mi-
experiment and theory, we recommend that the reader croclusters with some 20 or fewer atoms where these
consult de Heer's review. structural methods can be applied. For detailed accounts
We cannot possibly give an account of all theoretical of the techniques and achievements of some of these
aspects of metallic clusters. This would be a truly inter- methods, we refer the reader to the recent literature:
disciplinary task, involving atomic, molecular, and solid- Bonacic-Koutecky, Fantucci, and Koutecky (1991) have
state physics, quantum chemistry, and many aspects of presented an extensive and very valuable review on
nuclear physics as well. The author of this article is, in quantum-chemical methods, and a comprehensive review
fact, a nuclear physicist and a certain bias in the selection on the molecular dynamics method has been given by
of the presented material cannot be denied. Galli and Parrinello (1991).
One of our aims here is to build bridges: from sophisti-
cated quantum mechanics to simple phenomenological This is certainly not a sufhcient condition; cf. the structure of
models on the one hand, and from atoms and "simple" elemental hydrogen.
molecules to the infinite bulk metal on the other hand. 2We use the term ab initio meaning a calculation using the full
We also put some emphasis on semiclassical approxima- electron Hamiltonian with the unmodified Coulomb interac-
tions and asymptotic expansions for very large particle tion.
Rev. Mod. Phys. , Vol. 65, No. 3, July 1 993

Matthias Brack: The physics of simple metal clusters 679
The jellium model on one side and the quantum chemi- linear-response theory. Results of microscopic and semi-
cal approaches on the other side represent two almost di- classical jellium model calculations are compared to
ametrically opposed points of view, a situation that has those of the quantum chemical and to the structural
led to a great deal of debate. Even their relative degrees models employing pseudopotentials. We also establish
of difhculty can be debated. Quantum chemistry has a links between microscopic theories and their classical
clear and simple concept of treating the many-body prob- counterparts, and discuss the coupling of collective sur-
lem, but in practice is computationally very complex and face and volume modes.
requires sophisticated approximations even for small The final section, Sec. V, is devoted to very large metal
molecules. The jellium model is an easy-to-use and rather clusters. In Sec. V.A we discuss, after some general con-
effective tool for the largest clusters also, but its limita- siderations on shell structure, one of the most fascinating
tions are hard to exceed and its physical applicability is aspects: the so-called supershell structure and its ex-
dificult to judge. Even though researchers on both sides planation in terms of a semiclassical quantization of the
seem to agree that either approach has its merits and its electronic orbits. Average cluster properties and their
limitations, the question remains when and where to use asymptotic behavior are discussed in Sec. V.B, in order to
them for those cluster sizes in which both methods can provide links from the microscopic to the macroscopic
be applied. We tend to believe that, ultimately, this world. Starting from self-consistent semiclassical calcu-
dispute can only be resolved empirically. Careful analy- lations, we show how to extract the asymptotic large-N
ses of the ab initio results in terms of mean-field quanti- expansion of the energy and other observables, and how
ties are certainly also very useful in partially settling some of the coefficients in these expansions are directly
these questions. linked to properties of the bulk metal.
In Sec. II we try to given an overview of some of the It should be clear from our emphasis that we have not
facets of the quantum many-body problem posed by the been able to review and discuss all relevant theoretical
phenomena seen in metal clusters. We shall go through papers on simple metal clusters. In particular, concern-
the successive approximations and simplifications used in ing the structural approaches, we have simply cited
the various approaches, starting from the purely micro- several important references without a detailed discus-
scopic ab initio description and ending up with semiclas- sion. We apologize to all whose work is not assessed ap-
sical mean-field theory. We hereby hope to give the unin- propriately or not mentioned at all.
itiated reader a guide to the various theoretical ap-
proaches that one meets when scanning through the
II. FROM THE QUANTAL MANY-BODY PROBLEM
literature on metal clusters. Some of the general and TO SEMICLASSICAL JELLIUM DROPS:
more formal aspects of mean-field and linear-response A HIERARCHY OF APPROXIMATIONS
theory have been put into appendices.
Section III deals extensively with the microscopic jelli-
um model and its results. We hope to demonstrate that
This section is an introduction and guide to the
different levels of sophistication of theoretical approaches
in its recent deformed versions, it yields results that are
strikingly close to those of quantum
to metal clusters and to the various approximations used
chemistry and
in different contexts. We also take the occasion here to
molecular dynamics. An important approximation for
review some selective literature on those approaches that
dealing with the jellium Hamiltonian is the local-density
will not be further discussed in the remaining sections.
approximation (LDA). We also review briefly some at-
tempts to go beyond the LDA and some extensions of the
jellium model that aim at a partial inclusion of the ionic A. The quantal many-body problem
structure without sacrificing its simplicity.
There is a wealth of interesting experimental data on Let us start by writing down the exact Hamiltonian for
the electric dipole response of metallic clusters, and we a neutral cluster consisting of 1V nuclei with Z electrons
devote the whole of Sec. IV to their description, using each:
I
2
N P2 z P~, x z
H=g 1 (Z )2 Ze +1 (2. 1)
R
where M, P,R are the mass, momenta, and coordi- actly known, it is impossible to solve the corresponding
nates, respectively, of the nuclei, m,
l
p,
r are those of1
Schrodinger equation.
the electrons in the o, th atom, and self-interactions must Luckily, the different scales of nuclear and electronic
be left out of the double sums. This constitutes a system masses allow a rather clear separation of their treatment.
of N(Z+1) charged particles interacting via the According to the Born-Oppenheimer hypothesis, the dy-
Coulomb forces. Although the Hamiltonian (2. 1) is ex- namics of the nuclei may either be neglected altogether,
Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

as in the quantum-cnemicat approaches (see Sec. II.B), or least formally, the scheme for a complete microscopic
treated classically as a slow, adiabatic motion (see Sec. treatment of the many-electron problem and is the basis
II.C), whereas the electrons must be treated quantum of the quantum-chemical approaches (see Sec. II.B).
mechanically. An alternative version of the mean-field approach is
Considerable simplification is achieved by explicitly obtained in the framework of the density-functional
treating only the m valence electrons of each atom quan- theory, in which exchange effects and correlations going
tum mechanically and including the core electrons with beyond the HF approximation can be included approxi-
the nuclei as ions of charge +me. The assumption mately in a local mean field. A major part of the micro-
"atom=ion+ w valence electrons" generally works quite scopic calculations for metal clusters so far has been done
well, even for materials in which the valence electrons are using density-functional theory by solving the so-called
not strongly delocalized, and provides the basis for a Kohn-Sham equations (see Appendix A. 2 for a brief out-
large proportion of calculations for molecules and clus- line of density-functional theory). In principle, density-
ters. The total Hamiltonian then is reduced to that of X functional theory applies only to static ground-state
interacting ions (Hj) and wX interacting valence elec- properties. But some information about excitation pro-
trons (H, i) in the external field VI provided by the ions: cesses, like ionization potentials and electron amenities,
can be gained from static calculations just by combining
II =III+II (2.2) energy differences of clusters with different numbers of
with electrons.
The mean-field concept allows one to describe most of
P ~+1 N 2 the electronic shell effects and many other static proper-
(we)
2 p(~+)=, IR —
(2.3) ties of metallic clusters, at least semiquantitatively, and
2M Rpl provides the common basis for the approaches sketched
p. 2 in Secs. II.C —II.E below and discussed more extensively
+ VI(r, )+-
wN wN
1
in the remaining sections III —V.
2&i 2 J(Q) li rj
i Collective excitations of the valence electrons in metal
where the ionic potential clusters have been both observed and theoretically dis-
cussed extensively. The microscopic framework for their
(2. 5) description within mean-field theory is the random-phase
approximation (RPA); its practitioners call it the time-
dependent local-density approximation (TDLDA) in the
couples the electronic and ionic degrees of freedom. Al- context of density-functional theory. See Sec. IV.A. and
though the core electrons are no longer explicit degrees
Appendix B for the formal basis of this theory. In
of freedom in the wave function, they still infItuence the essence, it is a linear-response theory using particle-hole
valence electrons by screening and Pauli exclusion effects. excitations from the determinantal ground state as the
The ion potential V, in Eq. (2.5) is a smoothed function basic degrees of freedom. RPA theory was originally
that includes the inAuence of the core electrons, and is developed for infinite electronic systems in solid-state
called a "pseudopotential" in physics and an "effective
physics, but has been used successfully also for finite fer-
core potential" in chemistry. (See Appendix A. 2.d for mion systems in molecular and nuclear physics, where
further discussion. ) collective excitations play an important role. It has re-
Even when the nuclear (or ionic) part of H is ignored cently been extensively used for analyzing the optical
or treated by classical equations of motion (Sec. II.C), the response of metal clusters, starting from ab initio
electron-electron interactions in Eq. (2. 1) or Eq. (2.4) still quantum-chemical or density-functional results for the
constitute an unsolvable quantal many-body problem. ground state. In Sec. IV.B we review the corresponding
The most common method for dealing with it is the literature and compare the different predictions.
mean-field approximation, which has been widely used
for treating many-fermion systems in all branches of
physics: one determines self-consistently an average po- B. Quantum chemistry
tential in which the electrons move as independent parti-
cles. The ambitious goal of the quantum-chemical ab initio
One starts from a Slater determinant built of electronic approach is to treat all the electronic degrees of freedom
single-particle wave functions and determines these by an in Eq. (2. 1) fully quantum mechanically. This can only
energy variational principle. This leads to the familiar be done at the cost of "freezing" the positions R of all
Hartree-Fock (HF) approximation (see Appendix A. l). nuclei. The Born-Oppenheimer approximation thus is
Thus the average part of the electronic repulsion is in- used to vary adiabatically the positions of the nuclei, let-
cluded in the mean field (or potential), which, due to the ting the electrons adjust their motion at any time to the
nonzero range of the Coulomb interaction, is nonlocal. instantaneous external field of the nuclei, until the total
Extensions of the HF approximation are obtained by in- static energy is minimized. This is a strict zero-
clusion of many-particle/many-hole excitations and their temperature treatment; no zero-point motion of the nu-
interactions in perturbation theory. This constitutes, at clei is included.

Since the quantal many-electron problem is too com- density-functional theory, most frequently in the local-
plicated for an exact solution, one starts from the density approximation (see Appendix A. 2), and the core
Hartree-Fock (HF) approximation (see Appendix A. l) electrons are represented by pseudopotentials (see Ap-
for the electronic wave functions to obtain a self- pendix A. 2.d).
consistent mean field; the correlations are then included On large computers, the MD method is usually corn-
perturbatively in a hierarchy of n-particle/n-hole bined with the simulated annealing technique of Kirkpa-
configuration interactions or, alternatively, by superpos- trick et al. (1983): the finite temperature is used as a
ing several Slater determinants ("multiconfigurational technical means to allow the system to relax into the
HF"). The pure ab initio treatment of all electrons is lim- lowest minimum of its Born-Oppenheimer energy surface
ited to very small clusters (X ~ 10); for larger systems fur- and thus to determine the optimal ground-state geometry
ther simplifications, using density-functional methods of the ions. This is very time consuming but crucial in
and/or pseudopotentials, must be made. view of the fact that the number of isomeric minima to
A detailed discussion of the quantum-chemical ap- which the simpler steepest-descent method can lead in-
proaches and their results is outside the scope of our creases very rapidly with the number of ions (Hoare and
present review. We refer the reader to a very recent and McInnes, 1983). In view of the above remarks, we shall
exhaustive review article by Bonacic-Koutecky, Fantuc- refer to these calculations as pseudopotential calcula-
ci, and Koutecky (1991) on the theory and application of tions, emphasizing the approximation in the Hamiltoni-
quantum-chemical methods for the description of metal an, rather than as Car-Parrinello or MD calculations,
clusters. As a few selected references for the history and which would emphasize the search for the stable struc-
development of these methods for metal clusters, let us tures.
just mention here Marinelli et al. (1976), Beckmann As examples of MD calculations for the ionic structure
et al. (1980), Fantucci et al. (1984), Garcia-Prieto et al. of small metal clusters and their thermal properties, we
(1984), Martins et al. (1985), Rao and Jena (1985), Bous- mention that Ballone et ai. (1989) studied Na2O and
tani et al. (1987), and Bonacic-Koutecky et al. Na&OK]o clusters; Na& clusters with N ~ 20 were investi-
(1988—1991). Later in this review we shall compare some gated by Rothlisberger and Andreoni (1991). Jones
ab initio predictions to the results of other approaches. (1991) has investigated the ground-state geometries of
small Al clusters with N up to 10. The fission of a small
doubly charged sodium cluster was investigated in MD
calculations by Barnett et al. (1991).
C. Molecular dynamics and static pseudopotential models The pseudopotential approach represents today
perhaps the most efFective tool for treating molecules and
In this subsection we briefIy review density-functional clusters containing up to several tens of atoms with their
calculations which investigate explicitly the ionic full ionic structure, in the adiabatic limit also for the in-
geometry of metal clusters, making use of pseudopoten- clusion of their dynamics. For a recent comprehensive
tials. We start with the most up-to-date, fully dynamic review on the pseudopotential approach we refer the
theory, and then discuss its static limits and some of its reader to Galli and Parrinello (1991). A short status re-
precursors. It is important to note that, for many pur- port on MD calculations for small metal and semicon-
poses, and indeed for our applications here, the dynamics ductor clusters was given by Andreoni (1991).
of the ionic motion is irrelevant to the quantities calculat- Generally, it can be said that the ionic ground-state
ed. We shall discuss only properties of the clusters that structures of small metallic clusters obtained in MD cal-
are static with respect to the ionic motion. The quality culations are practically identical to those found in ab in-
of the results will depend on the form of the Hamiltonian i tio quantum-chemical calculations.
and the approximations used to treat the electronic part. Another approximation scheme to include electron-ion
In principle, all that is required here of the dynamics is to correlations dynamically is the so-called "efective medi-
locate the stable structures. um theory" developed some time ago by N@rskov and
The molecular-dynamics (MD) method, developed by Lang (1980) and Ndrskov (1982). The effect of an
Car and Parrinello (1985), includes the dynamics of the "embedding" electron density p, on the binding of an
ions by solving their classical Newton equations, coupled
to the quantum-mechanical Kohn-Sham equations for
the electrons (see Appendix A. 2.e). This goes beyond the "
The name "ab initio molecular dynamics, which is often
quantum-chemical approaches in that it is able to treat
used in the literature, should. not be mistaken as indicating an
the systems dynamically (although only in the adiabatic
all-electron theory, which in this dynamical form would be im-
limit due to the basic restrictions of density-functional
possible even for small clusters. It is merely used to indicate
theory; see Appendix A. 2.). It also allows one to extract that ab initio pseudopotentials, derived from first-principles
thermodynamic properties of complex molecular sys- quantum-chemical calculations, have been employed.
tems, at least approximately. The price one pays is that 4A modified version of the simulated annealing method was
not all electrons can be treated fully quantum mechani- used by Manninen (1986b) to calculate the ionic ground-state
cally. Instead, the valence electrons are treated in structures of Na microclusters with % ~ 8.

atom is first studied microscopically in density-functional D. Jellium model

theory with LDA, leading to an energy functional of the
atom that depends on p, as an external parameter. Then The most dramatic but eKcient simplification is to ig-
this functional is used for a metal to include in each nore the ionic structure totally, replacing the charge dis-
Wigner-Seitz cell the efFects of the (superposed and aver- tribution of the ions by a constant background charge in
aged) electronic density tails of all neighboring cells. In a finite (spherical or deformed) volume. This is the
this model, the Wigner-Seitz radius and the cohesive en- three-dimensional, finite-size version of the jelliurn model
ergy of a given metal can be explicitly calculated. which was successfully used long ago for the description
So far the effective medium theory has been applied of metallic bulk and surfaces properties (Lang and Kohn,
mostly to metallic bulk and surface properties, particu- 1970—1973). The self-consistent mean field of the elec-
larly to the process of melting. For a recent review, we trons can be calculated microscopically in this model in-
refer the reader to Jacobsen and Nefrskov (1988). cluding the shell effects due to their quantization. It re-
Christensen et al. (1991) have presented the first applica- quires, however, the density of the ions (or, correspond-
tion of the theory to small Cu clusters. The electronic ingly, the Wigner-Seitz radius r, ) as an external parame-
shell effects were calculated from a tight-binding Hamil- ter, which characterizes the nature of the metal.
tonian and included in a way that is reminiscent of the The total neglect of the ionic structure is better
shell-correction theory of Strutinsky (1968) (see end of justified than one might think at first sight: the pseudo-
Sec. V.B.1). Their model allows one to describe in a self- potentials (see Appendix A. 2.d) have no singularities and
consistent, albeit approximate, way the interplay of ionic their sum in Vi [Eq. (2.S)] is, indeed, a rather smooth
and electronic shell effects. Nielsen et al. (1992) have function. This is the combined effect of screening and
used it to simulate the melting of a Cu cluster with 16 727 the Pauli principle, coming from the inner core electrons
atoms. This method appears rather promising, but no that fill the lowest orbitals in the Coulomb-like potentials
calculations for simple metal clusters have been reported of the individual nuclei. We refer the reader to textbooks
so far. on solid-state physics (e.g. , Ashcroft and Mermin, 1976)
As precursors to MD or simulated annealing calcula- for a more detailed discussion.
tions, many static investigations of the structure of small For finite clusters, a wealth of papers initiated by
clusters have been performed over the last decade, using Ekardt (1984a, 1984b) and independently by Beck (1984a,
more or less sophisticated pseudopotentials. Since the 1984b) has shown that a self-consistent and essentially
ionic structure requires a fully three-dimensional solution parameter-free microscopic jellium model calculation can
of the electronic Kohn-Sham equations, and the optimal account qualitatively, and in many cases even quantita-
geometry of the ions must, in principle, be searched sys- tively, for many experimentally observed properties of
tematically by minimizing the total energy, even purely metal clusters, in particular those of alkali metals (see the
static investigations are very complex and time consum- review of de Heer, 1993). Deformations of the jellium
ing. With the present-day generation of computers, this background (axial or triaxial) or a finite temperature of
is most elegantly done by the simulated annealing tech- the electrons can be included at reasonable cost in the jel-
nique. lium model. The self-consistent jellium model will be re-
We discuss some of the results of static structural cal- viewed in Sec. III and some of its applications in Secs. IV
culations when comparing them to other calculations in and V.
later sections. Let us mention here some approximate in- The justification of the jellium model for the descrip-
vestigations in which the ionic structure has been tion of clusters is, and probably will remain, an object of
simplified. Small cubic crystals were studied by Iniguez much debate and research. However, its undoubted vir-
et al. (1986). Martins et al. (1981) and Baladron et al. tue is that it can also be applied to large clusters with
(1985) introduced a spherical averaging of the pseudopo- many hundreds or thousands of atoms, where the more
tentials in order to have spherical symmetry. Manninen structural models cannot be used for practical reasons.
(1986a) imposed spherical symmetry only on the elec- The most beautiful example is the explanation of the
tronic density; he minimized the classical part of the total "supershells" in large alkali clusters, which we shall dis-
energy (the "Madelung energy"), varying the full three- cuss in Sec. V.A.
dimensional geometry of the ions. A systematic series of
studies with spherically averaged pseudopotentials (the
"SAPS model" ) was started by Iniguez et al. (1989, 1990) E. Phenomenological shell models
and Lopez et al. (1990) and recently extended to include
simulated annealing (Borstel et al. , 1992). For the appli- Many shell and single-particle effects do not depend
cations of the spherically averaged pseudopotential mod- very crucially on the microscopic self-consistency of the
el to metal clusters, we refer the reader to the recent re- total mean field, such as is obtained by iteratively solving
views of Balba, s and Rubio (1990) and of Borstel et al. the Hartree-Fock or the Kohn-Sham equations. One
(1992). Finally, we note that a semiclassical version of may therefore give up the self-consistency by simply
the spherically averaged pseudopotential model using parametrizing the total average potential in an easy-to-
variational trial densities (see Sec. V.B) was developed by use form and then solving just once the Schrodinger
Spina and Brack (1990). equation, in order to obtain single-particle spectra and

wave functions. This leads, of course, to a considerable structure, which we shall discuss in Sec. V.A. They
gain in numerical simplicity. The cost of such simplicity determined the parameters of their potential by fitting it
is a less fundamental description, since contact with the directly to the Kohn-Sham potentials obtained in the
two-body interaction is lost and the parameters of the self-consistent jellium model calculations of Ekardt
model have to be determined by fits to experimental ob- (1984b) for Na clusters with X ~ 192. Clemenger (1991)
servables. The advantage is a greater flexibility, allowing has adapted the Woods-Saxon potential to a variety of
closer contact with the measured data. metals and studied the scaling properties of the super-
The prototype of such a phenomenological shell model shell structure as a function of the Wigner-Seitz parame-
is the Woods-Saxon potential, which was successfully inter r, F. rauendorf and Pashkevich (1993) have adapted
troduced into nuclear physics by Maria Goeppert-Mayer the Woods-Saxon potential to deformed clusters with
(1949) and independently by Haxel, Jensen, and Suess X ~ 300.
(1949). After inclusion of a strong spin-orbit coupling The omission of the spin-orbit term in all these shell
term, it became possible for the first time to explain the models for metal clusters seems to be empirically justified
so-called magic numbers of nucleons responsible for the by the fact that no strong evidence of spin-orbit splittings
extra stability of certain nuclei like 82 Pb' (cf. Sec. has been observed so far. This can be theoretically un-
V.A. 1). The nuclear shell model has been successfully derstood considering the relativistic effects in a jellium
used to explain many single-particle properties of spheri- potential. First, relativistic effects in atoms and mole-
cal nuclei, despite the lack of an underlying Hartree- cules are generally much smaller than in nuclei. Second-
Fock basis in the context of realistic nucleon-nucleon in- ly, the Thomas term, which one obtains from a nonrela-
teractions. After the discovery of nuclear deformations tivistic reduction of the Dirac equation for fermions
through low-lying rotational excitations, Nilsson (1955) moving in a spherical electrostatic potential V(r), is pro-
introduced a shell model for deformed nuclei, which is portional to (1/r)dV/dr. In atoms, this is strongest near
based on an axially deformed harmonic-oscillator poten- the center where the potential is proportional to 1/r and
tial including a spin-orbit coupling term and an attractive thus affects most strongly the electrons in the lowest or-
term proportional to the square I of the single-particle bits. In clusters, V (r) is liat in the interior so that d V/dr
angular momentum operator that simulates a steeper is practically zero. In the surface, where the gradient is
wall. The Nilsson model was very successful in explain- large, the extra factor 1, /r gives an extra suppression,
ing the ground-state deformations of many nuclei and which varies as X ' . Detailed relativistic Kohn-Sham
their single-particle excitations (Mottelson and Nilsson, calculations (Schone, 1991) confirm, indeed, the smallness
1955; see also Bohr and Mottelson, 1975). It is rather re- of spin-orbit splittings within the jellium model.
markable that one is able to predict shapes with a model
that uses only a single-particle Hamiltonian, and a very
F. Semiclassical and classical approaches
oversimplified one at that. The explanation for nuclear
physics seems to be that self-consistency in shape be-
tween particle density and the potential field is a very One more simplification can be made that leads to a
powerful constraint, and that shell closure effects can considerable gain of efficiency in treating very large sys-
occur as a function of deformation as well as of particle tems: the neglect of shell effects. This is done automati-
number (see also Sec. V.A. 1). cally by the explicit use of semiclassical approximations
Clemenger (1985a, 1985b) adapted the Nilsson model to the kinetic-energy functional T, [p] (see Appendix
to small axially deformed Na clusters by dropping the A. 2.a). The density-functional formalism can then be ex-
spin-orbit term and readjusting the coefficient of the l ploited for direct density-variational calculations: one no
term. This model has since been frequently used in metal longer varies many electronic single-particle wave func-
cluster physics and is usually referred to as the tions, but one single function, the electronic density p(r)
Clemenger-Nilsson model. The model seems to work (or, if relevant, two spin densities). By doing this one
quite well in reproducing the overall shapes of clusters as sacrifices the single-particle structure, and thus shell
calculated by pseudopotential or ab initio methods. We effects, but the advantage is an enormous gain in simpli-
shall not discuss it here any further; it is well explained city and calculational speed, and this can still give
and extensively used in the reviews by de Heer, Knight significant results for average properties of the con-
et al. (1987b) and de Heer (1993). Reimann et al. (1993) sidered system. The famous prototype of such a model is
have proposed an extension of the Clemenger-Nilsson the Thomas-Fermi (TF) model of the atom.
model, fitted to spherical Kohn-Sham spectra, and ap- Extensions of the Thomas-Fermi model (the TFW
plied it to large deformed sodium clusters with % ~ 800 model, ETF model, etc. ; see Appendix A. 2.a) were
(see Sec. V.A. 1). developed long ago and have been successfully used for
The three-dimensional harmonic-oscillator potential finite fermion systems in many branches of physics. The
without l term was used by Saunders (1986) to describe variation of the density p(r) can either be done exactly,
triaxially deformed clusters. leading to an Euler-Lagrange type (integro-) difFerential
Nishioka et al. (1990) have introduced a spherical equation, or in restricted variational spaces using trial
Woods-Saxon potential for Na clusters particularly in density functions. In fact, the first self-consistent jellium
connection with calculations of the so-called supershell model calculations for spherical metallic clusters were

carried out using such a semiclassical density-variational The gross structure of the shell effects in the single-
method by Cini (1975). particle spectrum and the total energy of a finite fermion
Many average properties of metal clusters can be de- system can also be described rather well by semiclassical
scribed in such density-variational calculations, which we techniques that are based on the quantization of classical
shall review in Sec. V.B. A more formal and fundamen- trajectories. The well-known prototype of this idea is the
tal interest of this approach is the possibility of connect- old Bohr-Sommerfeld quantization rule. In its modern
ing the microscopic models to purely classical ones. In version, it has become very successful in explaining the
the large-N limit, the clusters become classical spheres or beating pattern of the so-called supershells in large alkali
drops, and a systematic expansion, the so-called "lepto- clusters. This will be discussed in Sec. V.A.
dermous" expansion, can be developed to determine the
surface and curvature energies from semiclassical theory.
This leads to the self-consistent foundation of a liquid-
III. THE SELF-CONSISTENT JELLIUM MOOEL
drop model similar to that in nuclear physics (see, for ex-
ample, Myers and Swiatecki, 1969). Likewise, the
asymptotic behavior of electronic ionization potentials A. Basic concepts
and aftinities and their classical limits, which have re-
ceived n. uch attention in the literature, can be studied The basic idea of the self-consistent jellium model is to
rigorously using this technique. These recent develop- replace the distribution of the ionic cores by a constant
ments will also be reviewed in Sec. V.B. positive background or jellium density pro in a finite
Just as for the static energetics of finite fermion sys- volume and to treat only the valence electrons explicitly
tems, so for their linear-response behavior one can obtain in the mean-field approximation, either microscopically,
links to classical models by the use of semiclassical and as described in this section, or semiclassically (see Sec.
large-X limits of the RPA method. An approximate V. B). The jellium background may be spherical, ellip-
theory extracted from the RPA under the assumption of soidal, or arbitrarily deformed.
local currents (Reinhard et al. , 1990) leads to the connec- Almost all jellium calculations so far have been per-
tion with classical hydrodynamics (or, to use a more ap- formed within density-functional theory, the formal as-
propriate term, Fermi-fluid dynamics). From this point pects of which we have summarized in Appendix A. 2.
of view, several aspects of surface plasmons can be dis- (For some recent Hartree-Fock calculations, see Sec.
cussed qualitatively, and the physics becomes more trans- III.C. 1.) In density-functional theory the total energy of
parent than in the purely microscopic RPA approach. the cluster is expressed as a functional of the local elec-
Section IV.C is devoted to these topics. tron density p(r):
E[p]=T, [p]+E„[p]+ f VI(r)p(r)+ —

1
p(r) e f (3.1)
Here T, [p] is the (noninteracting) kinetic-energy density trons per atom).

and E„,[p] the exchange-correlation energy density (see If the electron density is written in terms of single-
Appendix A. 2); the fourth term is the direct (Hartree) particle wave functions y;(r) as
Coulomb energy of the electrons. VI(r) is the ionic back-
ground potential, related to the background jellium p(r)= g=1 ~q;(r)l' (3.4)
charge density pI(r) by i
—which is always possible for physical (i.e. , non-

(3.2) negative and normalizable) densities due to the so-called
—
Coleman theorem (Gilbert, 1975) then the noninteract-
The jellium density is usually assumed to be uniform, i.e. , ing kinetic-energy functional T, [p] is
pI(r)=pIO inside the cluster and zero outside. The po- mX
g2
tential (3.2) here replaces the ionic potential (2.5) used in
the pseudopotential models. The EI in Eq. (3. 1) is the
T, [p) = f ~(r)d'r = f 2m
~
g IVq, (r) l' d'r . (3.5)
electrostatic energy associated with the jellium back-

ground. It does not depend on the electron density but is
included so that E[p] (3. 1) represents the total binding
energy of the cluster. The density p(r) must be normal-
ized to the total number mX of valence electrons: ~Throughout this paper, we denote by p the particle densities
and not the charge densities. The charges (in multiples of e) ap-
f p(r)d r=wN, (3.3) pear explicitly in all formulae with their correct signs.
where w is the valence factor (number of valence elec-

Matthias Brack: The physics of simple metal clusters
By a variation of the energy E[p] (3. 1) with respect to justified for large clusters. In the jellium model, however,
the y,'. (r), one then arrives at the so-called Kohn-Sham we have no way to determine the finite-size variation of r,
equations, whose solutions will be discussed in Sec. III.B. theoretically, so that the simplest choice is that of the
A direct variation of the energy with respect to the bulk value.
function p(r) is also possible if one is satisfied with semi- A remark about the internal consistency of the jellium
classical approximations for the kinetic-energy functional model might be appropriate here. If we consider the
T, [p]. This is at the cost of neglecting the shell effects, inner part of a large neutral cluster and neglect the sur-
but has the advantage that one is varying only one func- face effects, the energy per electron e(p) as a function of
tion p(r) instead of wX wave functions y;(r), and still al- the (constant) density p is given by
lows one to obtain average cluster properties self- f2
consistently. We shall discuss this approach and the cor- e(p) = ap +e„,(p), (3.7)
2&i
responding calculations in Sec. V.B.
Most applications of the jellium model to metal clus- where the first part is the kinetic energy per electron [see
ters so far have been restricted to the local-density ap- Eq. (A19) of Appendix A. 2.a]. The Coulomb energies
proximation (LDA) for the exchange-correlation (xc) cancel exactly if the density p is chosen to be equal to the
functional E„,[p] As iscussed in Appendix A. 2.c, this
d. jellium density pro, which must be done to ensure charge
approximation consists in using locally the exchange- neutrality. If we now search for a minimum of e(p) in
correlation energy per electron e„,(p) obtained in many- Eq. (3.7) with respect to varying p (and, with it, p~o), we
body calculations for an infinite system of electrons with find it for values r, =4 —4. 3 a.u. , depending somewhat on
constant density p, i.e., e„,(p) is taken at the local value the detailed LDA exchange-correlation functional used.
p=p(r) everywhere in the finite system. By construction, For example, for the functional of Gunnarsson and
this approximation is exact in those regions of space Lundqvist (1976) [see Eq. (A.33) in Appendix A. 2.c],
where the density p(r) is constant, and it is badly justified which has often been used for metal cluster calculations,
where the density varies strongly, such as in the surface this minimum is at r, =4. 08 a.u. Very similar values are
region of metal clusters. In spite of its simplicity, the obtained with all other exchange-correlation functionals
LDA in connection with the Kohn-Sham approach has in the literature. In such a variational calculation there
met with considerable success in almost all branches of is, of course, only one metal that has a stationary value of
physics. its density in the bulk region. It is perhaps no coin-
Metal clusters (besides metal surfaces) present perhaps cidence that the jellium model works best for alkali met-
one of the most crucial testing grounds for the LDA, als, in particular sodium, with r, values close to this
since their surfaces are typically much steeper than those minimum. It has, indeed, already been observed by Lang
of atoms or small molecules. However, the success of the and Kohn (1970) that good surface energies are obtained
LDA in conjunction with the jellium model in describing in the jellium model only for metals with r, ~4 a. u. (see
surface energies and work functions for metal surfaces also Sec. V. B.2). In fact, the model completely breaks
(Lang and Kohn, 1970, 1971; Monnier et gl. , 197g) down for r, ~ 2. 3 a.u. , in that it gives negative surface en-
least for alkali-like metals —
has encouraged, and to some ergies for the corresponding metals (e.g. , aluminum).
extent also justified, its application to metallic clusters. This should not be forgotten when applying the jellium
We shall brieAy discuss some extensions of the LDA in model to finite clusters, in particular when their shape is
Sec. III.C below and, in particular, report there on an en- not kept spherical. A negative surface energy means that
couraging test of the exchange part of the LDA function- the cluster is not stable against deformation, so that, in
al by means of a Hartree-Fock calculation for finite metal the jellium model, the energetically most stable
clusters (Sec. III.C.2). configuration for a finite aluminum system is not a sphere
One essential point of the jellium model is that it con- but alu-foil!
tains only one single parameter, namely, the Wigner-Seitz
radius r„which characterizes the metal. It is related to
B. Kohn-Sham-LDA calculations
the jellium density pro by
i
4m The variation of the energy E[p] (3. 1) with respect to
Pro S (3.6) the single-particle wave functions y, (r), *.
6 =0
Otherwise, the model is completely free of parameters,
since the electron density is determined variationally. 5(p, '(r ) E[p(r) ] (3.8)
Usually, one takes the bulk value for r, corresponding to

with the subsidiary condition that the y, (r) be normal-
the ionic lattice in the crystal. Naturally, this is only
ized, leads to the Kohn-Sham equations (see Appendix
A. 2)
6See, for example, Jones and Gunnarsson (1989) or Dreizler
I T+ +Ks(r) ] I '(r) (3.9)
and Gross (1990) for its applications to electronic systems, and
Sprung (1972) for a discussion of the LDA in nuclear physics. T is the kinetic-energy operator; the local potential

2
VKs(r) is a sum of three terms: wXe
VKs ( r ) VKS I .
P( r ) l = V. [p( r ) ] + VH [p( r ) ] + VI ( r ) I
(3. 10) wNe
for I" &RI, (3. 13)
of which the first is due to the exchange and correlation
contributions: and the ionic energy EI is simply
(3.11) 3 (w%e)
(3.14)
R
VH[p] is the Hartree potential of the electrons, given in
the square brackets in Eq. (3. 1) above, and V~ is the jelli- If the electron density is also assumed to have spheri-
um potential (3.2). The constants E, in Eq. (3.9) are the cal symmetry, the total Kohn-Sham potential (3.10) is
Lagrange multipliers used to fix the norm of the ith state; spherical and the single-particle states y, (r) will have
their interpretation as single-particle energies is not good angular momentum quantum numbers I;, I;, their
justified in general as discussed in Appendix A. 2. angular parts being given by Y&t t (0, $) in polar coordi-
Since VH and V„, depend on the density p(r), the nates (r, 0, $). Equations (3.9) can then be reduced to ra-
Kohn-Sham equations (3.9) are nonlinear in the y, and dial Schrodinger equations for the radial parts R„ t (r) of
t
must be solved by iteration until self-consistency is the wave functions and solved numerically on a one-
reached (i.e., un. il the results do not change any more dimensional mesh in the variable r (Beck, 1984a, 1984b;
upon iteration). In general, (3.9) are partial differential Ekardt, 1984a, 1984b). As a variational precursor of this
equations in the three spatial coordinates and their solu- model we mention that of Martins et al. (1981). In their
tion is not trivial. However, if symmetries are assumed treatment, which is not fully self-consistent, they re-
or imposed, the problem simplifies considerably. A large placed the Kohn-Sham potential (3. 10) by a simple varia-
majority of Kohn-Sham calculations so far have been tional square-well potential.
performed assuming spherical symmetry of the clusters. During the last eight years, many papers have been
The calculation then becomes one-dimensional and is rel- published with results of spherical jellium model calcula-
atively easy to do, even for clusters with up to %=3000 tions. We cannot cite them all here; many of them are
electrons (Genzken and Brack, 1991). We review spheri- referred to when we compare their results with experi-
cal Kohn-Sham calculations in Secs. III.B.1; their exten- mental data or with results of other models. The results
sion to finite temperatures is discussed in Sec. III.B.3 and of several groups have been summarized by Balbas and
their application to very large clusters in Sec. V.A. 2. Rubio (1990). All authors, except those cited in Sec.
When major electronic shells are only partially filled, III.C, used LDA functionals for the exchange-correlation
the mean field tends to be deformed, as is well known part of the energy.
from nuclear physics (see also Sec. V.A. 1). The spherical As an illustration, we show in Fig. 1 (upper part) the
jellium model therefore must be generalized to include Kohn-Sham potentials of three alkali clusters with
deformed ionic background densities, in order to mini- X =40, obtained in some early jellium model calculations
mize the total energy of such "nonmagic" systems. In an by Chou et al. (1984). Their shape is similar and scales
average sense, this simulates the nonspherical distribu- essentially with the Wigner-Seitz radius r, . The
tions of the ions known from ab initio and molecular- minimum near the surface is related to the fact that the
dynamics calculations. In Sec. III.B.2, jellium calcula- electronic density always has a maximum there as a
tions in two and three dimensions are reviewed, corre- consequence of the Friedel oscillations (Lang and Kohn,
sponding to axially and nonaxially deformed clusters, re- 1970). Some of the Kohn-Sham single-particle levels E;
spectively. are shown in the lower part of Fig. 1 directly as functions
of r, ; their positions for the respective metals are marked
1. Spherical jellium model by the dashed vertical lines.
The successes and failures of the spherical jellium
model have been discussed extensively by de Heer (1993)
In the spherical jellium model, the ionic background
while comparing its results with experimental data. Let
density pt(r) is that of a uniformly charged sphere with
us summarize here some general trends and add a few re-
radius RI ..
marks:
(i) Spherical shell closings-: The most prominent "mag-
ic numbers" observed in mass abundances, ionization po-
tentials, and electron amenities correspond to the filling of
in which the value of RI is fixed by the number X of ions. major spherical shells and are in general correctly repro-
The ionic potential Vt(r), which replaces the sum of indi- duced for alkali metals, some noble metals, and, to some
vidual ionic potentials (2.5), is then easy to evaluate and extent, also for Al clusters. A problem exists with the
is given by atomic numbers 34 and 40: The self-consistent jellium

reduction of the 2p lf -gap.

(ii) Deformed sh-ell closings: In principle, the imposed
)CD
spherical symmetry does not allow one to treat clusters in
the regions between the major spherical shells. From the
analogous situation in nuclei, we know that such clusters
CD
o -4 are deformed (see also the general discussion on shell
CL
CD
eff'ects in Sec. V.A. 1). This is, indeed, the case and will be
-6 discussed in the next subsection.
O
CD
(iii )I'onization potentials (IP) and electron affinities
LLI
8
(N=40) (EA): The jellium model with LDA gives a reasonable
-10 ~ I qualitative description of these quantities, namely their
0 10 20 30 approaching the bulk work function 8 like 1/R for large
Radius R (a. u) N and their sawtooth-like behavior at major shell clos-
-2
(~) ' ings. (For an extensive discussion concerning the asymp-
totic slopes of IP and EA plots versus 1/Rl, see Sec.
)CD
-3 V. B.3.) However, in Na and K, the average values of the
IP are too large. This is related to the bulk values 8'
CD
which are too large in the jellium model, as is well known

) -4
CU
CD
since the pioneering calculations of Lang and Kohn
CD
LLI (1971). It can partially be remedied by inclusion of the
ionic structure via pseudopotentials (Lang and Kohn,
1971; Monnier et al. 1978). Moreover, the finer details of
,'Na IK IPs and EAs are not correctly explained in the spherical
e I I
jellium model; the experimentally observed odd-even

r, (a. u. ) staggering is missing and the amplitude of the shell Auc-
tuations is exaggerated by a factor of up to two. For
FIG. 1. Potentials and electronic single-particle energies c.; for
small clusters (N ~ 12), the ab initio and the pseudopoten-
three alkali clusters with %=40, obtained in the self-consistent
spherical jellium model by Chou et al. (1984): upper part, total
tial models with local-spin-density approximation
Kohn-Sham potentials versus radius r; lower part, electronic (LSDA) clearly give a better quantitative description.
Kohn-Sham levels c.; of the 1d —1g shells vs Wigner-Seitz radius Some of the fine structure is partially improved in the
r, . Values for the three metals Li, Na, and K, respectively, are spheroidal model discussed below. [An improvement
indicated by the vertical dashed lines. over the LDA results, both for neutral and for the partic-
ularly critical negatively charged clusters, was obtained
model for Na clusters (and, to a lesser extent, for Li clus- in jellium model calculations using a nonlocal weighted-
ters) gives %=34 as a stronger spherical shell than density approximation to the xc energy (WDA; see Sec.
%=40, contrary to experiment. This is due to too large a III.C.4) by Rubio et al. (1989, 1991a) and Balbas et al.
f
splitting between the 1 state and the 2p state lying above
it (see Fig. 1 above). Some remedies for this failure are
(1989, 1991). On the other hand, very recent Hartree-
Fock plus perturbation-expansion calculations by Guet
discussed in Sec. III.D below. The phenomenological et al. (1993) seem to confirm the validity of the LDA for
Clemenger-Nilsson model does better here (see also de both exchange and correlation effects, by yielding almost
Heer, 1993), but this is just due to a suitable choice of the identical ionization potentials to those of the Kohn-
l term. Similarly, a %'oods-Saxon-type potential is Sham-LDA approach for a series of Na clusters. ]
better, since it is Oat in the inner part and does not exhib- (iv) Dipole polarizabilities and photoabsorption cross sec
it the "Friedel dip" near the surface (see Fig. 1), which in tions: We discuss these quantities and their description
the self-consistent model leads to some extra attraction in the jellium model extensively in Sec. IV.
for higher-l states and thus pulls the 1 state down more
than the 2p state. (Similar discrepancies are found for
f It is clear that in clusters with up to % -20 atoms the
jellium model is not quantitatively competitive with
larger spherical shells and subshells; we refer the reader quantum-molecular methods or spin-dependent LDA cal-
to Sec. V.A. 2 in connection with very large alkali clus- culations with pseudopotentials. For very large clusters
ters. ) In general, it can be said that the detailed ordering with hundreds or thousands of atoms, however, the jelli-
and spacing of single-particle levels near the Fermi sur- um model will presumably remain the only tractable way
face, and therefore the correct prediction of "magic num- to make theoretical predictions. The agreement obtained
"
bers, can be rather sensitive to the radial shape of the recently with the experimentally observed supershell
potential. Of course, the ionic structure also plays a role structure (see Sec. V.A) is, indeed, very encouraging. It
when it comes to these details. Indeed, Kohn-Sham cal- is therefore interesting to ask to how small a size the jelli-
culations by Borstel et al. (1992) including the ionic um model can be extrapolated.
structure schematically in the so-called spherically aver- Some points of comparison with respect to ab initio
aged pseudopotential model (see Sec. II.C) also yield a and pseudopotential theory will be mentioned here.

First, the basic starting assumption of delocalized single- expect it to have the correct deformation behavior. ' The
particle wave functions seems to be rather well satisfied errors introduced by just summing the occupied E; in-
for alkali-metal clusters. For example, in the crease with the number of particles; this gives an upper
configuration-interaction (CI) calculation of Na6 by limit to the usefulness of the simple Clemenger-Nilsson
Bonacic-Koutecky et al. (1991), the leading ground-state model in determining the correct ground-state deforma-
configuration has an amplitude of 0.91 in the CI wave tions. This situation is well known in nuclear physics:
function. This is despite the fact that correlation eIIFects the breakdown of the Nilsson model in describing the de-
are known to be crucial to obtain binding of alkali-metal formation energies of heavy nuclei, in particular at the
clusters. Evidently the LDA is a rather good approxima- large deformations occurring in the fission process, has
tion. Secondly, the electron densities in these more mi- made it necessary to correct for its missing self-
croscopic calculations are seen to concentrate on the consistency. An approximate but very powerful tool for
interstices of the ionic lattice rather than on the atoms or achieving this was introduced by Strutinsky (1968); his
on the lines between neighboring atoms. This shows a shell-correction theory is mentioned briefly at the end of
degree of delocalization not present in ordinary chemical Sec. V.A. 1.
binding. Also, the large-scale nodal structure of the It is therefore of basic theoretical interest to verify the
single-particle wave functions exactly duplicates that of phenomenological potential of the Clemenger-Nilsson
the jellium model. This is particularly striking in the model by microscopic, self-consistent calculations in the
highly symmetric clusters such as Nas(Td ), where the oc- framework of density-functional theory. In the deformed
cupied s and p jellium states as well as the empty s and d jellium model, one uses the spirit of the Born-
orbitals can be identified with the corresponding molecu- Oppenheimer approach underlying the quantum-
lar orbitals. Finally, Rothlisberger and Andreoni (1991) chemical and molecular-dynamics calculations (see Secs.
have compared the spherically averaged densities and II.B, II.C): one varies the shape of the jellium density
mean potentials of their pseudopotential calculations distribution and lets the electrons adjust themselves in
with those of the spherical jellium and phenomenological the corresponding deformed ionic potential (including
models. We shall come back to this comparison in the their interaction and exchange-correlation efFects in
context of the deformed jellium models discussed in the LDA, as usual). The ground-state configuration is then
next section. found by minimizing the total energy with respect to the
jellium shape. Practically, one parametrizes the shape of
the jellium density in terms of one or several deformation
variables. These variables take the roles of the ionic posi-
2. Deformed jellium model tions R of the ab initio approaches [see Eq. (2. 1)]. Al-
lowing for a sufticient number of shape degrees of free-
Some of the shortcomings of the spherical jellium mod- dom for the jellium therefore should bring this model
el can be removed, or at least reduced, by relaxing the closer to the more realistic approaches, the main
spherical shape of the clusters. Indeed, as discussed ex- di6'erence being that here only the averaged geometry of
tensively by de Heer (1993), there is good evidence that the ions is varied.
clusters are deformed in regions between the major One technical problem in the deformed jellium model
spherical shell closures. The deformed shell model of is that the background jellium potential Vl(r) is no
Clemenger (1985a, 1985b) allows one to interpret the fine longer a simple analytical function as in the spherical
structure of mass abundance spectra and the splitting of case; cf. Eq. (3. 13). It must therefore be calculated nu-
the dipole resonances in sodium clusters in the mass re- merically either by direct integration over the jellium
gions 8 &N & 18 and 20 &% &40. In this model (see the density or by solving the Poisson equation. Similarly, the
Appendix of de Heer, 1993, for details) the potential de- Kohn-Sham equations become more complex with de-
pends on a deformation parameter, and the equilibrium creasing symmetry of the cluster and have to be solved in
(ground-state) shape of each cluster is calculated simply two or three spatial dimensions explicitly.
by minimizing the sum of occupied single-particle ener- The existence of axially deformed equilibrium shapes
gies with respect to this parameter. Qbviously, such a within the framework of the self-consistent jellium model
model is not self-consistent in two respects: First, the has been confirmed in Kohn-Sham calculations for
density distribution of the electrons is not guaranteed to spheroidal clusters by Ekardt and Penzar (1988, 1991;
have the same shape as that of the potential (although Penzar and Ekardt, 1990). These authors use an axially
this is approximately the case at the shapes of minimal symmetric ellipsoidal jellium density with constant
total energy; see, for example, Bohr and Mottelson,
1975).
Second, the sum of single-particle energies is far from 7In a pure harmonic-oscillator model, from which the
representing the total binding energy of an interacting Clemenger-Nilsson model does not much differ, one can obvi-
system. As is well known from HF theory [cf. Eq. (A15) ously exploit the virial theorem to correct for this double count-
in Appendix A. l], the sum of occupied E, ; contains the ing. But realistic self-consistent potentials are quite different
potential energy twice, and there is a priori no reason to from harmonic-oscillator potentials.

volume and half-axes zo, po given in terms of a single de-

formation parameter 5, restricted by — 2 & 6 & 2, and the
cluster radius Rl (3. 12)
2/3 1/3
2—
zo= 2+5 RI
5
Rr . (3.15)
) ) po 2+6
The electronic density is assumed to have axial symme-
try, too, and the Kohn-Sham equations are solved in
spheroidal coordinates, again using the LDA functional
of Gunnarsson and Lundqvist (1976) for the exchange-
correlation energy [see Eq. (A33) in Appendix A. 2.c]. I
The total energy of the cluster must be calculated for 0.3 0, 2 0. 3 0.& 0.5 0.6 0.7 (y = 0'j
each deformation 6, and the ground state is found by

minimizing the resulting energy with respect to 6. The FIG. 2. Deformation energy surface of Na&6, obtained by Lau-
shell structure obtained by this model for the IPs and ritsch et al. (1991) in the triaxially deformed self-consistent jelli-
EAs of Na and Cu clusters is somewhat reduced by the um model. P and y are the Hill-Wheeler quadrupole deforma-
deformation effects, but stiH exaggerated with respect to tion parameters. The contour lines correspond to increments of
experiment. An interesting result is that the odd-even 0.01 Ry of the total energy of the cluster. The lowest energy
staggering in these quantities, a striking feature observed (E = —1.393 Ry) is found for P=O. 38, y = 33, corresponding to
in clusters up to X = 92, can to some extent be explained a triaxially deformed ground state.
by polarization effects of the odd electrons. The results
of Ekardt and Penzar (1991) for collective photoabsorp- 1991). In fact, this difference has been found to be very
tion spectra obtained in the axially deformed model are sensitive to hexadecapole deformations (Hirschmann
discussed in Sec. IV.B. et al. , 1993). The same transition, prolate~triaxial
Nonaxial deformations of small sodium clusters have ~oblate, is again expected on the grounds of the phe-
recently been investigated in the self-consistent Kohn- nomenological shell models for 20&1V &40 while filling
Sham framework by Lauritsch et al. (1991). Here, a tri-
axial ellipsoid with constant volume was assumed for the
f
the 1 and 2p shells.
In recent photoabsorption measurements on positively
jellium density, ' for reasons discussed in Sec. III.D. 1 charged Na clusters by Liitzenkirchen et al. (1993), the
below, the jellium density was given a diffuse surface with transition from prolate to oblate shapes in the regions
a width of one atomic unit. The Kohn-Sham equations 12&% &20 and 20&% &40 has been observed. For
(in the LDA) were then solved numerically on a three- )
N 40 (i.e., filling the lg shell), however, the shapes seen
dimensional mesh for each given deformation of the jelli- to be oblate again (see also Borggreen et al. , 1993); it is
um background. Potential-energy surfaces of these triax- a challenge to explain this in further self-consistent calcu-
ially deformed clusters were presented as functions of the lations.
two quadrupole degrees of freedom P and y introduced An interesting result of Lauritsch et al. (1991) is the
by Hill and Wheeler (1953): P~O measures the overall occurrence of several almost degenerate isomers with
quadrupole deformation (P=O corresponding to spherical different spherical, axial, or nonaxial shapes, separated by
shape), and y measures the axis ratios of the ellipsoid barriers of -0.
S —I eV. Indeed this shape isomerism re-
(y=O' giving prolate axial, @=60 oblate axial, and calls that found in ab initio quantum-chemical and
0'(y (60' triaxial shapes). molecular-dynamics calculations, although the heights of
As an example we show in Fig. 2 the Born- the barriers between are overestimated in the jellium
Oppenheimer energy surface of Na, 6 in the (P, y) space, model due to the missing residual interactions. Never-
obtained with the model of Lauritsch et al. (1991). It has theless, these results show that a lot of qualitative, aver-
its minimum at P=0. 38, @ =33, corresponding to a tri- aged features of the ab initio approaches can also be ob-
axial shape. This supports the predictions of Saunders tained in the self-consistently deformed jellium model
(1986) using the triaxial harmonic-oscillator potential even for small clusters.
[see also Fig. 4(a) of de Heer, 1993]. The shapes of the In Fig. 3 we show the energy per particle for the region
other clusters with 8 & X & 20 of that model are of sodium clusters with 8 ~ X ~ 20, obtained in self-
confirmed, too, in the self-consistent jellium model: Na, 2 consistent Kohn-Sham-LDA calculations in three succes-
has a triaxial shape, whereas Na, o, Na, 4, and Na, 8 are ax- sive approximations with decreasing symmetry. The up-
ially deformed. Thus while filling the 1d shell, there is a permost line corresponds to the spherical jellium model.
transition from spherical (N =8) to prolate (10) to oblate The next-lower line gives the energies obtained in the axi-
(18) shapes; the crossover goes through 0'( y (60, lead- ally symmetric model, whereas the lowest line is that ob-
ing to triaxial shapes (12, 16). The case N=14 is rather tained in the triaxial jellium model. The latter two differ
critical: the two axial minima (prolate and oblate) are only for the two nonaxial clusters with %=12 and 16.
nearly degenerate and their energy difference depends (Odd values of N have no significance in these calcula-
crucially on details of the model (see Lauritsch et al. , tions, in which the spin degrees of freedom were not in-

-0.0
—0 140 I I I, total energy
I I I I I I I
-0.1—
—0. 142—
-0.2
~ ~
—0. 144—
-0.3 =
—0. 146— —04—
CJ
-0. 148— I I I I
-0. 150— 2 6 8 10
(D
—0. 152— —0.0
I i I I
(D I I
I
-0. 54—
1
—0. 1—
Na-20
jellium
0 156 I I I I I I I I I
8 10 12 14 16 18 20 —P.3—
particle number
/
FIG. 3. Total energy per electron F. /X vs X for sodium clusters 05

with 8~N ~20, obtained by Hirschmann et al. (1993) in the
8 10
self-consistent triaxial diffuse jellium model of Lauritsch et al.
(1991): X, spherical model; 0, axial (spheroidal) model; tri- +, r [a.u. ]
axial (ellipsoidal) model; A, semicalssical spherical model (cf.
Sec. V.B). FIG. 4. Spherically averaged total Kohn-Sham potentials for
the sodium clusters Nalo and Na&0.. dashed lines, results of
eluded explicitly; the results therefore do not exhibit any molecular-dynamics (MD) calculations by Rothlisberger and
Andreoni (1991);solid lines, self-consistent jellium model results
odd-even eFects. ) One observes that quite an appreciable
by Hirschmann et al. (1993), using the diffuse-surface jellium
gain in energy is brought about by the deformation of the mode1 of Lauritsch et al. (1991). Note that Na2o is spherical in
clusters, except close to the spherical closed-shell num- the jellium model, whereas Na&o is prolate axially deformed.
bers 8 and 20. Also shown in Fig. 3, by the dotted curve,
is the result of a semiclassical density-variational calcula-
tion (see Sec. V.B). It is seen to interpolate nicely be- um model of Lauritsch et al. (1991), and those of the
tween the microscopic curves and represents, in fact, a MD results of Rothlisberger and Andreoni (1991) using
rather good approximation to these results including de- the Car-Parrinello method. The jellium model results
formation, although in the semiclassical approximation agree astonishingly well with the pseudopotential results
all clusters remain spherical. where the ionic structure is included.
It is of interest at this point to compare the jellium An interesting detail that can be observed in Fig. 6 is
model results to those of pseudopotential calculations the relative closeness of the 1p and 1d type levels around
that include the ionic structure. In Fig. 4 we show the X = 14. Since they have opposite parity, a static octupole
spherically averaged total Kohn-Sham potentials ob- (or more generally, any left-right asymmetric) deforma-
tained by Rothlisberger and Andreoni (1991) for the clus- tion would mix these levels, which could lead to a lower-
ters Na]p and Na&p and compare them to those obtained ing of the energy. Such situations are known also from
by Hirschmann et al. (1993) in the difFuse jellium model
atomic nuclei. Hamamoto et al. (1991) have shown that
of Lauritsch et al. (1991). Although the pseudopotential an octupole instability in typically quadrupole-deformed
results show important oscillations, their average trend is regions are a rather general trend of finite fermion sys-
rather well reproduced by the jellium model. tems and can lead to shell e6'ects of comparable impor-
In Fig. 5 we compare the spherically averaged elec- tance to those induced by the nonaxial quadrupole defor-
tronic densities obtained in the two approaches for the matlons.
clusters Na» (which is oblate axially deformed in the jel- In conclusion we can say that the jellium model is able
lium model) and Na20, The pseudopotential (left part) are to describe surprisingly well the average trends of poten-
given for three di6'erent temperatures; the jellium results
tiaIs, energy levels, and densities in the structural pseudo-
potential calculations, when a sufficiently flexible defor-
(right part) at T=o only. The agreement between the
T =0 results is almost quantitative, which is rather mation is included, even for such small clusters as
surprising in view of the potentials shown in Fig. 4 Na8 —Na2p. This is encouraging for the application to
above. It shows that the electrons average out the ionic larger systems in which molecular dynamics becomes
structure of the pseudopotentials by their motion, so that more and more time consuming and one is bound to rely
their density distribution is almost unaA'ected and is very on the jellium model predictions.
well reproduced by the jellium model.
The single-particle energies for Na clusters in the re- 8A similar study of Kohn-Sham levels, obtained in pseudopo-
gion 8 ~ X ~ 20 are shown in Fig. 6. Here we compare tential calculations for Mg clusters with 8 ~ X & 20, was recent-
the Kohn-Sham levels c., obtained in the spheroidal jelli- ly presented by Delaly et al. (1992), who also confirm the gen-
um model of Ekardt and Penzar (1988), the triaxial jelli- eral trends of the jellium model predictions.
Rev. Mod. Phys. , Vol. 65, No. 3, Juiy 1993

(a) (b)
K
60 K
FIG. 5. Spherically averaged electron densi-
P3
70 K
ties for the sodium clusters Na&8 an a:
azo. left
cf part (a), results of molecular-dynamics (MD)
tD
calculations b y Rothlisberger and Andreoni
C)
C)
C) (1991) at various temperatures; right part b,
X results of the self-consistent jellium model as in
Na '
.4 b at zero temperature. (Note that
5 Na» is oblate axially deformed in thee jeelliumium
4, model. )
1
I I I I I I I
0 2 4 6 8 10 12 14 i6 0 2 4 6 8 10 12 14 16 18
r [a.u. ]
r (a. u. )
3. Finite-temperature effects II A f de Heer, 1993). The manifesta-

tion of shell structure in the abundance spectra o
a p roduced at finite
In many experiments, clustersrs are cluster-beam experiments is thoug t to bee a result of eva-
temperatures o f up too several hundred Kelvin or even poration of neutral atoms by the hot clusters: the closer
number of valence electrons g ets to a number corre-
tth enum
—0, 12:— !
I I I I I I I
spoIl d'iIlg to0 a filled major shell, the more s table
a the clus-
(a) ter wi*11 becanand thee smaller the probability forr eeva p oration
— 2s
of a further atom, so that Anally at thee time of
—0.20:—
=
1d
detection —when t h e b earn —
am has cooled off the closed-
—0.28 =
1p
shell species are th e most m abundant (Bjdrnholm et al. , a.
1993).
—0.36 = = ls Th e quuestion therefore arises to w hat a extent a finite
I & I
8 10 12 14 16 18 20 temperature a ffec t s th e magnitude of the electron'nic shell

N
effects themselves. Shell effects are weakened at finite
I
I
I I I I
I I
I I ns. Most of the excitation en-
—0. 12'=
ergy wii11 beein in thee sstructural degrees of freedom,om name
namely,y,
in the v»'0 rations, distortions, and liquefaction o f the clus-
'
-0.20: : 2s
1d
ter. T Th e amoun ount of phase space associated wit ig-
—0.28'= r sha es that produce electronic shell clo sures
wiill tthenen beere
reduced. Independent of this, thee occu p ation
—0. 36: o f t h e eecron'
1 t nic orbitals will be smeared out as in the
I
8
& I
10 12 14 16 18 20 '-D' distribution. This will also smoo th outou shell

N
effects, as is well known from nuclear p ysics o
I ~
I
&
I
&
I &
I
Mottelson, 1975; Brack and Quentin, 1981).
—0. 12 =
At first sight, one might think that even several hun-
—0.20:: :
.
=1d
2S dred degrees are small on the scalee of the electronic
single-particle energies, so that a this second effect should
—0. 28: be neg 1'igi'bl e.. This is certainly true for clusters wit with
V
fewer than a h un d re d a toms in which the main spacing
—0.36:—
I
between electronic levels correspo s onds to several thousand
8 10 12 14 16 18 20 degrees. However, in very r lar g e clusters in the mass re-
N ion %=1000—3000, which now have become a e
available
me avai
' in expansion sources, thee spectra are much more
FICx . 6. Kohn-Sham single-particle leve.lss in Na clusters with '
8 ~ X ~ 20, calculated in difFerent mo e s. compr esse d an and thee temperatures
e ave a
in questionn do have
dynamics (MD) results { {Rothlisberger
o is and Andreoni, 1991; ec. Furthermore, the detailed s ape
noticea bl e e ffect.
spheroidal jellium modeodel ((Ekardt and Penzar, , 1988)'; (a)
a triaxial abundance spec t ra d e pends in a rather subtle way on first
i u -
difFuse-surface
'
jellium mode l (Lauritsch et al. , 1991). From second differences of the total free energies wi
Hirschmann et a/. , 1993. respect to the atomic number, so that thee tern e p erature

smearing effects can, indeed, become visible (Bjdrnholm only to the total HF energy. However, the inclusion of
et al. , 1991; Pedersen et al. , 1991). LDA correlations along with a HF treatment of the ex-
The effects of occupation-number smearing were ad- change goes against the empirical "rule of thumb" that
dressed recently in finite-temperature Kohn-Sham-LI3A exchange and correlation effects should always be kept
calculations for sodium clusters in the spherical jellium together at the same level of approximation (see, for ex-
model (Brack, Genzken, and Hansen, 199 la, 1991b; ample, Jones and Gunnarsson, 1989). To be more
Genzken and Brack, 1991). The electrons were treated as specific: once one goes beyond the HF approximation,
a canonical system in the heat bath of the ions, and the Pauli and other types of correlations (also of RPA type;
appropriate density-functional theory for T)
0 (see Ap- see Appendix B) cannot be disentangled, and therefore
pendix A. 2.) was used. The canonical partition function adding a correlation functional to HF risks double count-
was calculated exactly in terms of the Kohn-Sham ing.
single-particle energies, and the relevant thermodynami- HF calculations for spherical sodium clusters within
cal quantities were derived from it self-consistently. the jellium model were performed by Hansen (1989),
Since the most important results of these calculations Guet and Johnson (1992), and by Hansen and Nishioka
concern the supershell structure in very large alkali clus- (1993). Hansen (1989) investigated the addition of corre-
ters, we shall discuss them in Sec. V.A. 2 below. lations in LI3A. Since these authors addressed them-
Only the temperature of the electrons can be treated selves mainly to the electric response properties of sodi-
rigorously in the jellium model; the ions are not accessi- um clusters, we shaH discuss their results in Sec. IV.B
ble microscopically. This is a serious restriction, since a below.
dominant fraction of the thermal energy in cluster beams
is carried by the ions. However, the ionic part of the 2. Explicit evaluation of long-range correlations
thermal energy can be assumed to be a smooth function
of the particle number. Therefore the shell-structure os- Starting from the HF approximation, one can in prin-
cillations coming from the valence electrons and their
ciple include all correlations that go beyond the simple
temperature dependence can also be studied in the jelli- exchange systematically in perturbation theory, as is
um model. done in the quantum-chemical approaches (see Sec. II.B).
In an infinite system, the results can be expressed in
C. Beyond LDA terms of an effective mass of the electron, which has both
a momentum and an energy (or frequency) dependence.
In this section we review a few approaches that go The momentum dependence ("k mass") is due mainly to
beyond the local-density approximation. The discussions the nonlocality of the exchange (Fock) potential (see Ap-
here are brief and outline only the basic ideas. Results pendix A. l), which is due to the finite range of the
for experimentally measured observables, as far as they Coulomb force and tends to reduce the effective mass.
have been obtained, are discussed and compared to those The energy dependence ("co mass") comes predominantly
of Kohn-Sham-LDA calculations in Secs. III.B and IV.B. from the coupling of the electron to collective vibrations
We first review two sets of calculations that were per- through the long-range correlations and tends to increase
formed explicitly to test the LDA by computing the ex- the effective mass. Pictorially speaking, the electron is
change exactly in the Hartree-Fock approximation and "dressed" by a plasmon cloud that increases its inertia.
by computing explicitly the leading correlations in finite The two effects have a tendency to cancel each other in
clusters. infinite systems, so that the net effective mass can become
close to the free mass (see, for example, Mahan, 1981).
1. Hartree-Fock calculations For finite systems there is no unique way to define an
effective mass; it is more appropriate to speak of the self-
Since the Coulomb exchange is treated exactly in energy of the electron and of the screened Coulomb in-
Hartree-Fock theory, the standard LI3A exchange func- teraction. The effect of the correlations can also be stud-
tional Eq. (A. 31) can, in principle, be easily tested with ied by looking at the density of the "dressed" single-
HF calculations. It is just a question of numerical effort particle levels, the so-caHed quasiparticle energies. In the
to treat the jellium model in a HF approximation; due to pure HF approximation without long-range correlations,
the nonlocal nature of the Fock potential V~ (see Appen- this density is known to be too low at the Fermi energy,
dix A. 1), the calculations become considerably more resulting in too large an energy gap between the highest
complicated and time consuming than in the Kohn-
Sham-LDA approach.
The problem is that the HF approximation is known to
be rather poor for metallic systems, since the correlations 9Note that this e6'ect is automatically included in a full
are responsible for an appreciable part of their binding, Hartree-Pock calculation. The pseudopotentials are often an-
and one must include corrections. A compromise, al- gular momentum dependent, which introduces an additional
ready suggested by Kohn and Sham (1965), consists in momentum dependence to the delocalized electron energies.
adding a LDA (or LSDA) functional for correlations This effect is not accounted for in the jellium model.

occupied and the lowest unoccupied orbitals. The corre- overestimates the correlation eff'ect by as much as 50%%uo.
lations tend to close this gap. Saito et al. (1990) also compared their results to those
We shall not go into the gory details of many-body obtained with the self-interaction correction (see Sec.
theory or use any diagrams here. Let us just mention III.C.3 below).
that an appreciable part of the long-range correlations Bernath et al. (1993) also addressed the question of the
can be obtained, in both homogeneous and —
with con- density of quasiparticle energies in Na4o. Their approach
siderably more effort —
inhomogeneous systems, by sum- is more phenomenological in the sense that they used an
ming an infinite series of bubble diagrams, i.e., by iterat- approximate RPA method to obtain the lowest-order
ing the process of creating particle-hole pairs around the contribution to the self-energy, using a fitted separable
Fermi surface and destroying them again. (This is exact- interaction and selecting its leading multipolarities.
ly the basic excitation process used in the RPA method; (Physically speaking, this corresponds to an explicit cou-
see Appendix B.l.) These RPA correlations are the dom- p1ing of the valence electrons to surface dipole, quadru-
inant part of the correlation effect included in most LDA pole, and octupole vibrations. ) They found that the origi-
exchange-correlation energy functionals. A systematic nal HF single-particle spectrum, as obtained by Hansen
approach to the inclusion of these correlations in the and Nishioka (1993), is compressed — without, however,
electron propagator is given by the so-called GW approx- reducing much the "bandgap" at the Fermi surface — in
imation (Hedin and Lundquist, 1969). the direction of the Kohn-Sham-LDA spectrum. They
Reinhard (1992) has recently checked the LDA by also extracted an co-dependent effective mass that was
computing explicitly the RPA-correlation contributions found to be of the order of 40%%uo larger than the free-
to the ground-state energies and rms radii of closed-shell electron mass.
Na clusters (N =8, 20, 40, and 80) in the jellium model Here we should also mention recent work by Koskinen
using a pseudopotential-folded positive charge density et al. (1992), who did a configuration-mixing calculation
(see Sec. D.2 below). He started from the Kohn-Sham in the je11ium model for small Na clusters with X + 10,
approach including only the exchange-energy LDA func- using nuclear shell-model codes. The jellium Hamiltoni-
tional (thus omitting explicitly the correlation energy). an including the full electron-electron interaction was di-
He compared his results to those obtained in the stan- agonalized in a (partially truncated) many-particle space
dard Kohn-Sham-LDA approach, employing the correla- constructed from a set of harmonic-oscillator single-
tion energy functionals of Gunnarsson and Lundqvist particle states, including the orbits 1s, 1p, 2s, ld, 2p, 1f,
(1976) and of von Barth and Hedin (1972) (see Appendix 2d, and 1g. Up to %=4 the full space, containing up to
A. 2.c), thus using exactly the same methods for treating 6164 configurations, could be diagonalized. This ap-
the RPA correlations as the respective authors of these proach goes beyond both HF and Kohn-Sham-L(S)DA; it
two functionals. The difFerences were smaller than 10%%uo becomes exact in the limit of an infinite set of single-
for Nas and smaller than l%%uo for all larger clusters. This particle orbits. Both ionization potentials and total bind-
surprising agreement seems to be a strong confirmation ing energies were calculated and their convergence upon
of the validity of the LDA, at least for global properties increasing the configuration space was demonstrated. In
such as energy and radii, even for very small jellium those cases where convergence could be approximately
spheres. reached, a close agreement with Kohn-Sham-LSDA re-
The jellium-HF calculations of Guet and Johnson sults was obtained. Koskinen et al. (1992) also calculat-
(1992) have recently been extended (Guet et al. , 1993) to ed the photoabsorption cross sections for Na8 —Na&0 with
include the RPA correlations in the ionization potentials similar results to those for the Kohn-Sham-LDA-RPA
of alkali clusters with %=9, 21, and 41. Here the elec- calculations discussed in Sec. IV.B.2. In particular, they
tronic self-energies were calculated with the fully obtained a splitting of the resonance in Na9 and Na&0,
screened Coulomb interaction by iterating the Dyson showing that a sufIiciently large single-particle
equation to all orders. Again, the results were in the configuration space, even in a spherical basis, can de-
same good agreement with those of Kohn-Sham-LDA scribe the effects of a deformed mean field.
calculations, i.e., within a few percent for X =9 and less All these recent results yield a rather' positive test of
than one percent for the larger systems (see also Sec. the Kohn-Sham approach using the LDA, even for very
III.B.1). In a similar earlier work, Saito et al. (1990) small jellium spheres. The question therefore remains to
evaluated the electronic self-energies only to first order in what extent one should expect these correlation effects
the screened Coulomb interaction and investigated the amongst the valence electrons to depend on the presence
quasiparticle energies in closed-shell alkali clusters up to of the ionic structure or the core electrons, or how much
%=40. They concluded that their treatment gave a the results might be modified by additional correlations
better agreement of the quasiparticle energies with exper- between the valence electrons and the other charges.
imental ionization potentials than the Kohn-Sham-LDA As far as the jellium model is concerned, one is en-
results. The iteration to all orders performed by Guet couraged to conclude that extensions beyond the LDA
et aI. (1993) reduces the discrepancy with the Kohn- might not be important. At least, it seems that the errors
Sham-LDA results appreciably. In fact, these authors made in the LDA to the exchange-correlation energy are
show that the inclusion of the lowest-order diagram alone much smaller than those due to overall neglect of the ion-

ic structure in the total energy functional. the two-electron-correlation) function. It can be con-
For completeness, however, we shall review below structed (Przybylski and Borstel, 1984a, 1984b) to give
three extensions of the LDA that have been studied ex- the correct asymptotic 1/r falloff of the Kohn-Sham po-
tensively for atoms within the Kohn-Sham approach and tential for a neutral atom or cluster. We refer the reader
partially applied recently to metal clusters. to Dreizler and Gross (1990) for a discussion of the
WDA and its application to atomic and molecular sys-
3. Self-interaction correction tems.
Using the WDA with an approximate pair-correlation
One serious breakdown of the local-density approxima- function developed by Chacon and Tarazona (1988),
tion to the exchange energy affects the asymptotic Balbas et al. (1989, 1991) and Rubio et al. (1989, 1991a)
behavior of the Kohn-Sham potential for Coulombic sys- studied the static properties of metal clusters, in particu-
tems. In HF theory, where the Coulomb exchange is lar for negative ions in which the LDA is known to be
treated exactly, it is well known that the mean field doubtful. Both for ionization potentials (see Sec. III.B.1)
asymptotically falls off like 1/r far outside the surface of and for static electric polarizabilities (see Sec. IV.B.1),
a spherical system; this is simply the field of the remain- they obtained a considerable improvement in the agree-
ing spherical charge distribution seen by one electron ment with experiment over the results obtained in the
that is taken far away. This is no longer so in the Kohn- LDA.
Sham theory when the exchange is treated in the LDA.
As stated at the end of Appendix A. 1, the Hartee poten- 5. Gradient expansions of the xc functional
tial VH (A9) contains spurious self-interaction contribu-
tions of the electrons, which are exactly canceled when
Another way of including nonlocal effects for exchange
the Fock potential (A10) is added to it. However, with
and correlations consists in expanding the exact, nonlocal
the LDA one makes a crude approximation to the Fock
energy functional in terms of gradients and higher-order
potential, whereas the Hartree potential is left intact, so
derivatives of the density p(r). This was proposed from
that this cancellation no longer takes place. As a conse-
the beginning of the density-functional theory by Hohen-
quence, one obtains too much screening and the Kohn-
berg and Kohn (1964). (Applied to the kinetic-energy
Sham potential falls oIF much faster than I/r.
functional T, [p] this leads to the extensions of the
A "self-interaction correction" to remedy this failure
Thomas-Fermi model discussed at the end of Appendix
of the LDA has been proposed by Perdew (1979) and fur-
A. 2.a. ) They also suggested that one study the partial
ther elaborated and tested for atomic systems by Perdew
resummation of the series obtained by the gradient ex-
and Zunger (1981) with considerable success. It makes
pansion; this idea has received much attention in the
the Kohn-Sham potential state-dependent and thereby
literature and led to the so-called generalized gradient
complicates the self-consistent calculations appreciably;
approximations (GGA).
the Kohn-Sham orbitals y;(r) are no longer orthogonal
Although nonlocal effects are included in this ap-
and must, at least in principle, be reorthogonalized.
proach, it has the advantage that it still leads to a local
Since the exchange corrections apply only to electrons
xc potential in the Kohn-Sham equations. We refer the
with parallel spins, one must start from the spin-
reader again to Dreizler and Gross (1990) for an exten-
dependent LDA to use the self-interaction correction
sive review of a large variety of gradient expansions and
properly. the techniques used to derive them. Some recent versions
The self-interaction correction scheme cannot be sys-
of generalized gradient-approximations that have been
tematically improved or extended in terms of perturba-
used for atoms, small molecules, and metal surfaces were
tion theory; it is rather an ad hoc prescription that one
proposed by Langreth and Mehl (1983), Perdew (1986),
must take or leave. We refer the reader to Dreizler and
Perdew and Wang (1986), Becke (1988), Engel et al.
Gross (1990) for a detailed discussion and some varia-
(1992), and Perdew et al. (1992); for a recent review see
tions of the scheme, and to Moullet and Martins (1990)
Perdew (1991a, 1991b). The asymptotic fallofF of the xc
for some comparisons in atoms and diatomic molecules.
potential obtained in the generalized gradient approxima-
For metal clusters the self-interaction correction has so
tion has recently been discussed by Ortiz and Ballone
far been used only in the LDA-jellium model; cf. StampfIi
(1991).
and Bennemann (1987), Saito et al. (1990), and Pacheco
No results with generalized gradient approximation
and Ekardt (1992) (cf. Sec. IV.B).
functionals seem to be available yet for jellium model cal-
culations of finite clusters. Delaly et al. (1992) recently
4. Weighted-density approximation used the functionals of Perdew (1986) and Becke (1988) in
pseudopotential calculations for magnesium clusters with
The weighted-density approximation (WDA), intro- X ~20.
duced by Alonso and Girifalco (1977) and Gunnarsson A more systematic use of generalized gradient approxi-
et al. (1977), makes use of an explicitly nonlocal func- mation functionals for metal clusters would be highly
tional for the exchange-correlation energy E„,[p] in desirable, since their steep surfaces give a crucial test for
terms of approximated forms of the pair-correlation (i.e. , the validity of density-gradient expansions. Some prelim-

inary studies (Brack, 1988) using the semiclassical the jellium density with a diffuse surface and determined
density-variational method (see Sec. V.B) seemed to indi- the diffuseness parameter by minimization of the total en-
cate that the variational use of such functionals which— ergy for each cluster. The resulting diffuseness was found
often have been tested only perturbatively using, for ex- to be of the order of —1 a.u. for all clusters in the range
ample, HF densities —
can lead to instabilities of the sur- 8 ~ lV + 40. The effects of the increased surface
face of small jellium clusters. However, Perdew et al. diffuseness of the clusters are very beneficial to several of
(1992) with their new generalized gradient approximation their properties (see also Balbas and Rubio, 1991). As in
functional found stable solutions in semi-infinite jellium. the case of Lange et al. (1991), the shell-closing situation
We note in this context that the density-gradient expan- is improved in the right direction, 34~40. At the same
sion of the kinetic energy up to fourth order gives no time the ionization potentials of Na clusters, which are
problems and has been quite successfully used in varia- systematically too high in the standard jellium model (see
tional calculations for both metal clusters and atomic nu- Sec. III.B.1), are reduced. Furthermore, the increased
clei (see Sec. V.B). spillout of the electrons increases the static dipole polar-
izabilities and reduces the surface-plasmon energies;
these effects, too, bring the theoretical results closer to
D. Extensions of the jellium model experiment (cf. Sec. IV.B).
The introduction of a diffuse surface of the jellium den-
The most significant shortcoming of the jellium model sity also has the technical benefit of easing the numerical
is its lack of ionic structure. We review here very briefly calculations in the ellipsoidally deformed self-consistent
a few attempts to include schematically some effects of jellium model (see Sec. III.B.2). Lauritsch et al. (1991)
the ionic geometry without losing the simplicity of the used a Fermi function with a constant diffuseness of 1
jellium model. Some of them are simple phenomenologi- a.u.
cal ad hoc patches (Sec. III.D. 1) and some have a more Although the diffuse jellium surface can be determined
solid basis built on pseudopotential theory (Sec. III.D.2). variationally, as was done by Rubio et al. (1991b), so that
no fit parameter is needed, it has not yet been given a mi-
croscopic justification.
1. Simple patches
Lange et al. (1991), in an attempt to simulate the effect 2. Structureless pseudopotential models
of an oxygen ion embedded in Na&O clusters, introduced
a modification of the jellium density by adding a bump at A step towards the inclusion of pseudopotential effects
its center. They observed that in Kohn-Sham-LDA cal- has been taken by Reinhard et al. (1992) and Genzken
culations this improves the situation of the spherical shell et al. (1993), who relate the diffuseness of the jellium den-
closings that we discussed above in Sec. III.B.1: the gap sity to the ionic pseudopotentials. They use a convolu-
f
between the 1 and 2p levels is reduced and thereby the
stability of the iV=34 cluster is reduced in favor of the
tion of the steplike jellium density with the ionic density
distribution corresponding to a "soft-core" pseudopoten-
X =40 cluster. Similarly, the stability for %=186 is re- tial of the type introduced by Ashcroft (1966) [see Eq.
duced in favor of %=196 or 198. A partial remedy for (A34) in Appendix A. 2.d], which is just a surface delta
some of the systematic failures of the spherical jellium function peaked at the radius r =r, . This leads to analyt-
model can thus be achieved. Essentially, the trick con- ical expressions for the jellium density and the back-
sists in an increase of the surface diffuseness of the ground potential VI(r) that can easily be incorporated in
Kohn-Sham potential, which pushes states with higher spherical Kohn-Sham-LDA calculations. The empty-
angular momentum I upwards with respect to those with core radii r, determined from bulk and surface properties
smaller /. are taken from the literature, and thus no new parame-
Yannouleas and Broglia (1991b) introduced a similar ters have to be determined either variationally or by ad
perturbative correction of the jellium potential, hoc fits. The results of this pseudopotential-folded jelli-
representing the "nonjellium behavior" of small clusters, um model for alkali clusters are similar to those found by
which could be fitted to the position of the surface Rubio et al. (1991b) mentioned above: one obtains at the
plasmon. This helped to repair the lack of red shift of same time an increase in the static electronic polarizabili-
the dipole resonance systematically obtained in all ty and a corresponding decrease in the dipole resonance
jellium-LDA calculations (see Sec. IV.B.2). energies (cf. Sec. IV.B), as well as a reduction of the ion-
Such simple patches represent little more than punctu- ization potentials and electron amenities.
al remedies obtained by fitting ad hoc parameters and A similar and more systematic approach has been pro-
have no predictive power at all. They show, however, posed by Perdew et al. (1990) in their "stabilized jellium
that some of the results of the jellium model are very sen- model, " which is an extension of the "pseudojellium
sitive to modifications of the jellium density distribution. model" introduced earlier by Utreras-Diaz and Shore
A more systematic variational approach was taken by (1984, 1989). The effects of the ionic cores, represented
Rubio et al. (199lb) in spherical Kohn-Sham-LDA cal- again by the empty-core pseudopotential of Ashcroft
culations. These authors introduced a parametrization of (1966), are included here in a modified exchange-

correlation energy functional. This approach retains all (4.2)

the simplicity of the jellium model and has been used suc-
cessfully for metal surface properties, yielding work func- In Eq. (4. 1), i runs over the number Z of "active" elec-
tions and surface energies in good agreement with experi- trons: in ab initio approaches these would be all elec-
ment (cf. Sec. V.B.2). Due to an explicit ionic structural trons; in pseudopotential or jellium models, Z = wN is the
term in the energy functional, this model also yields the number of valence electrons. By solving Eq. (4.2), one
correct cohesive energy of bulk metal —
a quantity that is obtains the "constrained" ground state +& from which
not accessible in the simple jellium model. Brajczewska the polarizability tensor a (made diagonal by choosing a
et al. (1993) have recently calculated binding and ioniza- suitable coordinate system) is found either from the term
tion energies of Al, Na, and Cs clusters with 1 ~X ~20 linear in A, of the induced dipole moment (hence "linear
using the stabilized jellium model and solving the Kohn- response") or from the quadratic term in the total ener-
Sham-LSDA equations. In particular for aluminum, gy
which is not accessible in the standard jellium model,
they found a reasonable agreement with experimental
ionization energies.
IV. ELECTRIC DIPOLE RESPONSE (4. 3)

QF METAI CLUSTERS
If one is dealing with a spherical system, the above
A. Linear-response theory definition has the formal inconvenience that the external
field breaks the spherical symmetry, which complicates
Linear-response theory is the most convenient tool for the solution of the variational equation (4.2). However,
studying the interaction of a system with an external, not since only the linear response at A, =O is required, this
too strong field. In connection with metal clusters, it has difficulty can be circumvented by a multipole expansion
been extensively used to calculate static dipole polariz- of the wave function. The linear response, i.e., the
abilities and photoabsorption cross sections. In the lowest-order change in the ground-state wave function
present section we address calculations that have been (or density), will always have the same multipolarity as
done using the random-phase approximation (RPA) or the external field, and therefore it is sufficient to consider
the equivalent time-dependent local-density approxima- only the corresponding multipole (here: the dipole) com-
tion (TDLDA). (The formal aspects of these theories are ponent of the wave function or density, for which the
presented in Appendix B.) In addition to discussing jelli- variational equation still can be written in the spherical
um model results, we shall also review some ab initio and variable r. For the calculation of atomic dipole polariz-
pseudopotential model calculations. abilities, this technique was used by Mahan (1980), who
In Sec. IV.B we review the recent literature and discuss modified the equations originally derived from the RPA
agreements and discrepancies between the theoretical by Sternheimer (1957; see also earlier references quoted
and experimental results for the static polarizabilities and therein).
the resonances in the photoabsorption cross sections. Alternatively, a can be obtained from the moment
Section IV.C is devoted to a presentation of RPA sum-
rule relations and the classical limits of the RPA, leading
m, (D j ) of the RPA dipole strength function [see Sec.
to the well-known results of the Mie theory (1908) for IV.C below and Eq. (B13) in Appendix B.2]. This is the
surface plasmons in metallic spheres, and of a transpar- most convenient way if one starts from a microscopic
ent physical picture of the coupling of surface and RPA calculation. (Both ways may be combined to obtain
volume plasmons. a rather sensitive numerical test of the numerical
The two observables of metallic clusters that so far methods; similarly, other sum rules discussed in Sec.
have been investigated by these methods and compared IV.C may be used for this purpose. )
to experiment are the static electric dipole polarizability In the long-wavelength limit, which is well fulfilled for
and the photoabsorption cross section. The static dipole small- and medium-sized clusters, the photoabsorption
polarizability of a microscopic system is defined as in cross section o (co) is dominated by dipole absorption. It
classical physics: one applies an external, static electric can thus be obtained directly from the RPA dipole
field Eo and expands the total energy up to second order strength function Sg(E) given in Appendix B.2, Eq. (B6),
in Eo. The coefficient of the quadratic term is then the evaluated for Q =D and averaged over the spatial direc-
polarizability. Formally this is achieved by including the tions:
electric dipole operator D
z
D =e g r(i) (4. 1) ' We assume here that the unperturbed ground state has no
permanent dipole moment. Furthermore, we treat only the
in the variational equation via the Lagrange multipliers electronic response; in structural models, the ionic contribu-
A, k, , ):
=(A, „, A, tions must be added separately.

calculations that 5 in Eq. (4.7) is proportional to the elec-

o (o) ) =
4&co
g SD A.
(E = fico) . tronic spillout AZ, defined as the number of electrons
outside the jellium edge:
This can also be expressed in terms of the imaginary part hZ=4~
of the dynamic polarizability a(co): R
r p rdr . (4. 8)
O'ITco As discussed by de Heer (1993), the experimental po-

1m[a(~)] . (4.5) larizability of alkali clusters is much larger than the clas-
C
sical formula (4.6) and is not quantitatively reproduced
In the usual formulation of the TDLDA (see, for exam-
by the jellium response. " In fact, the polariz-
ple, Ekardt, 1984a), one calculates the dynamic polarizabilities obtained in the jellium model are systematically
ability a(co) directly. too small by —15 —20% for Na and K clusters in the re-
In principle, one encounters a fundamental problem in gion 2(1V(40. For Al clusters the situation is less
the calculation of photoabsorption cross sections within clear: here the jellium model tends to overestimate the
the framework of density-functional theory. As is well polarizabilities for X ~ 40, whereas for 40 ~ 2V ~ 60 there
known, the density-functional theory gives a priori no ac- seems to be a reasonable agreement (see Fig. 23 of de
cess to excited states. Nevertheless, the TDLDA has Heer, 1993).
been quite successful for the calculation of the dipole
The lack of polarizability of alkali clusters found in the
response of atoms (see, for example, Stott and Zaremba,
microscopic jellium-Kohn-Sham-LDA calculations has
1980; Zangwill and Soven, 1980). As discussed at the end often been attributed to the LDA treatment of the ex-
of Appendix B.1, the time-dependent formulation of change energy: the noncancellation of self-interactions
density-functional theory is a highly nontrivial problem; (see Appendix A. 1) leads to too much screening and thus
the TDLDA should therefore be used with some caution.
a too fast falloff of the self-consistent Kohn-Sham poten-
Gross and Kohn (1990) have proposed an explicitly
tial, which in turn gives rise to an underestimation of the
frequency-dependent exchange-correlation energy func-
density tail and thus of the electronic spillout, . Indeed,
tional that can be used for TDLDA calculations. To our the self-interaction correction (see Sec. III.C.3), which
knowledge, this functional has not been used for finite was introduced in order to correct this shortcoming of
systems so far. the LDA, was found by Stampfii and Bennemann (1987)
to increase the polarizabilities of small Na clusters con-
B. Linear-response calculations siderably, thus removing a good part of the discrepancy
with experiment. This has recently been confirmed by
Before discussing the dynamic response predicted by Pacheco and Ekardt (1992).
RPA or TDLDA calculations, we examine the static di- The weighted-density approximation (WDA), which is
pole response. This already shows some of the inherent tailored to yield the correct asymptotic —1/r falloff of
limitations of the jellium model. the Kohn-Sham potential (see Sec. III.C.4), has also been
reported by Rubio et al. (1991a) and by Balba, s and Ru-
1. Static bio (1990) to increase the polarizabilities.
dipole polarizabilities
The relevance of the above self-interaction correction
It is convenient to compare the dipole polarizability and WDA results can be explicitly tested in Hartree-
calculated from the quantum theory with the classical Fock (HF) calculations, in which there is no problem of
polarizability of a conducting sphere. This is given by spurious self-interaction contributions or of a wrong fall-
off of the average potential, since the exchange is treated
cx i=A (4.6) exactly here.
Guet and Johnson (1992) have performed HF+ RPA
The static response in the jellium model is predicted to be calculations in the spherical jellium model for closed-
larger. The main reason for the increase is the so-called shell Na clusters with X up to 92. They used the un-
electronic spillout, as observed by Snider and Sorbello correlated HF ground state and solved the RPA equa-
(1983a, 1983b) and by Beck (1984b). Snider and Sorbello tions (including approximately the continuum contribu-
(1983b) showed in Thomas-Fermi-Weizsa. cker density- tions). They found that the static polarizabilities were in-
variational calculations (cf. Sec. V.B.l) that the dipole
polarizability of a spherical metal cluster is given by
a=(XI+5) (4.7)
Kresin (1989—1992) obtained a very good agreement of cal-
where 6 in the limit AI — + ~ goes to a constant 6 that is
culated polarizabilities with experimental values. His method
the position of the image plane relative to the jellium makes use of approximate Thomas-Fermi solutions for the elec-
edge for an infinite plane metal surface in an external tronic densities and thus is an approximation to the microscopic
electric dipole field (Lang and Kohn, 1973). Therefore a Kohn-Sham-LDA-RPA approach, which fails as described
approaches its classical value (4.6) like a/a„—+ 1 above. The good agreement must therefore be considered as a
+35~/RI. Beck (1984b) showed in Kohn-Sham-LDA coincidence.

creased over the Kohn-Sham-TDLDA results towards the exact treatment of the exchange does not increase the
the experimental values, but by a much lesser amount spillout of the electrons, contrary to general expectations.
than through the self-interaction correction. However, Consequently the static dipole polarizability also stays
the increase with respect to the LDA results is due only the same. Indeed, applying a static external dipole field,
to neglect of correlations (other than exchange) in the Hansen and Nishioka (1993) found the polarizability of
ground state: the correlations lead to an increased bind- Na8 in HF to be only marginally larger than in the
ing of the electrons and therefore to a reduction of their Kohn-Sham-LDA approach without correlations.
polarizability. This had been shown explicitly in earlier In summary, it appears from these HF results that the
results by Hansen (1989), who did HF calculations for local-density approximation for the exchange is surpris-
Nas and Nazo with and without explicit inclusion of an ingly good, even for small Na clusters. A similar con-
LDA functional for the correlation energy. In fact, it is clusion could also be drawn by Reinhard (1992) and Guet
more correct to compare the HF results to Kohn-Sham- et al. (1993) for the correlation contributions, especially
LDA calculations without correlations. of RPA type, to binding and ionization energies (see Sec.
Hansen and Nishioka (1993) fully confirmed these re- III.C.2). This contradicts the above findings with the
sults. They performed HF calculations in the spherical self-interaction correlation and WDA, and rather sug-
jellium model for Na clusters with X up to 58 and com- gests the conclusion that the failure of the jellium model
pared their results to Kohn-Sham-LI3A calculations with to yield the correct polarizabilities (and redshifts of the
exchange only. They showed that the HF treatment, due photoabsorption resonances; see below) is due to the
to the nonlocal and strongly state-dependent mean field, neglect of the ionic structure.
leads to a considerably stronger binding of the single- Unfortunately, no ab initio quantum-molecular calcu-
particle states, particularly the lowest ones, but at the lations have been done so far for the static polarizabilities
same time suppresses the inner part of their wave func- of metal clusters. However, density-functional results
tions. The two eQ'ects have a tendency to cancel, and the with pseudopotentials including the ionic structure are
resulting densities are close to the Kohn-Sham-LI3A den- available. Moullet et al. (1990a, 1990b), who optimized
sities. the ionic structure of Na2 —Na9 in a local-spin-density
This is shown in Fig. 7, where we display the densities (LSDA) treatment using nonlocal pseudopotentials, ob-
obtained by Hansen and Nishioka (1993), in the HF, the tained very good values for the dipole polarizabilities, in-
Hartree (no exchange), and the Kohn-Sham-LDA (ex- cluding the fine structure of the experimentally observed
change only) approximations. It is very interesting to values (e.g. , the dip at % =4; see Fig. 22, Sec. V. C. 1, of de
note that, in spite of the correct asymptotic l. /r falloA of Heer, 1993). They showed that the comparison of experi-
the state-dependent HF potentials, the HF density tails mental and calculated values of o. can be used to decide
cannot be distinguished from the Kohn-Sham-LDA which isomeric form of the ionic geometry is present in
tails— at least in the region shown in the figure. Thus the ground state, which is not always possible on grounds
I I I I I 1 I I l I I I I I I I I
1.0
0.6
FIG. 7. Electronic densities (in units of the jel-
0.4 lium density pyp) of spherical Na clusters: (a)
N=8; (b) %=20; (c) %=40; (d) %=58. Solid
0.0 1 1 I
00 l lines, Hartree-Fock results; dotted lines,
0 2 4 6 8 10 12 14 % 18 20 0 2 4 6 8 10 12 14 % 18 20 Kohn-Sham results in LDA with exchange
1.8 18 I I
only; dashed lines, Hartree results (no ex-
I I ( ) I I
1.6
(c) change at all). The square profiles show the
1.4 jellium densities. From Hansen and Nishioka,
1992.
1.0
0.8
0.6 —0.6
0.4
02
00 I I 1 I I I I I I I
0 2 4 6 8 10 12 14 % 18 20 0 2 4 6 8 10 12 14 % '8 20
r (cLu. )

of a minimization of the total energy alone. The results 1990, 1993), and in full HF+RPA calculations by Cruet
of Moullet et al. (1990a, 1990b) depend to some extent and Johnson (1992).'
on the choice of the pseudopotential, but clearly show The dipole absorption cross sections of spherical alkali
that the inclusion of ionic structure improves the agree- clusters obtained in all these jellium calculations usually
ment with experiment appreciably. It should be noted exhibit a dominant peak that exhausts some 75 —90 % of
that no self-interaction correction was included in their the dipole sum rule and is redshifted by 10—20% with
calculation. respect to the Mie formula (4.9). As will be discussed in
These results seem to suggest that a non-negligible part Sec. IV. C below, the centroid of the RPA strength distri-
of the observed polarizabilities comes from the nonlocal bution tends towards the Mie resonance in the limit of a
effects associated with the pseudopotentials. Note that macroscopic metal sphere. Its redshift in finite clusters is
the ionic core polarizability will also contribute and is a quantum-mechanical finite-size effect that is closely re-
not included in any of these treatments. lated to the electronic spillout.
Earlier results with structural models using pseudopo- Some 10—25 % of the dipole strength is typically found
tentials had already indicated the above improvement. at higher energies and can be interpreted as a reminis-
Manninen (1986b), using a local pseudopotential, found cence of a strongly fragmented volume plasmon (see Sec.
reasonably good agreement of the average polarizabilities IV.C below). Often, the dominant peak is also fragment-
a of Na2 —Na8 with the experimental values. He used, ed into two (e.g. , Na2O) or more lines (e.g. , Na40). The
however, only an approximate expression for the energy, fragmentation of collective strength in spherical clusters
derived perturbatively by minimizing the classical can be attributed to an interference of specific particle-
Madelung energy. Furthermore, the absolute values of a hole (or more complicated) excitations with the predom-
were improved by the choice of an unusually large pseu- inant collective mode (Yannouleas et al. , 1989, 1993;
dopotential parameter r, =4. 0 a.u. He did not reproduce Yannouleas and Broglia, 1991a). This fragmentation
the dip for Naz. Similarly, Rubio et al. (1990) found the may be compared to Landau damping in the solid, al-
spherically averaged pseudopotential model to improve though there it refers to a collective state lying in a
the polarizabilities of Al clusters with X (40 over the jel- single-particle continuum. '
lium model results. When compared to experiment, all jellium calculations
The simple extensions of the jellium model that simu- yield an insufhcient redshift of the Mie resonance. This
late a part of the ionic structure (see Sec. III.D) also indi- is directly connected to the lack of polariz-
cate this trend: Rubio et al. (1991b) (see also Balbas and ability via the sum-rule estimate E, (see Sec. IV. C below).
Rubio, 1990) and Lauritsch et al. (1991) noticed that the Therefore a finite surface diffuseness of the jellium densi-
introduction of a diffuse surface of the jellium back- ty (cf. Sec. III.D), or other corrections found to improve
ground density helps to increase the static polarizabilities the polarizability, will also improve the position of the di-
obtained in jellium-LDA calculations. In fact, by a vari- pole resonance.
ational determination of the diffuseness of the jellium sur- There are other deviations from the single-resonance
face, Rubio et al. (1991b) obtained good agreement with Mie formula that are reproduced by the jellium model
the experimental polarizabilities of Na clusters. Finally, calculations. In open-shell clusters one finds a further
recent Kohn-Sham-LDA-RPA calculations by Genzken splitting of the dipole resonance, which is a consequence
et al. (1993) with a pseudopotential-folded diffuse jellium of their static deformation and can easily be described in
density (cf. Sec. III.D.2) also yielded the same results. the phenomenological Clemenger-Nilsson model (Selby
et al. , 1991; Bernath et al. (1991);see also Sec. VIII of de
2. Dipole resonances and the dynamic response Heer, 1993). It has also been obtained self-consistently in
the spheroidal jellium model with TDLDA calculations
by Ekardt and Penzar (1991). Very recently, the double-
The classical theory of dynamic polarizability predicts peak feature in the photoabsorption cross section of posi-
a single dipole resonance at a frequency given by (Mie, tively charged clusters has been observed for L&+& by
1908) Brechignac et al. (1992a), for Ag clusters in the region
1 j2
10&% & 16 by Tiggesbaumker et al. (1992), and for Na
Z@2e 2 clusters with 14 & X &48 (except %=21 and 41) by
CO Mi
I (4.9) Liitzenkirchen et al. (1992) and Borggreen et al. (1993).
which, with R =r, Z'~, is equal to 1/&3 times the bulk

plasma frequency. A self-consistent spherical jellium-Kohn-Sham-LDA+RPA
The linear response obtained in the jellium model fol- code was made available by Bertsch (1990).
lows the Mie result, but only in a qualitative way. This The same kind of fragmentation also occurs for the nuclear
was shown in self-consistent Kohn-Sham-TDLDA calcu- giant resonances. It is much stronger there due to the spin-
lations by Ekardt (1984a, 1985a, 1985b) and Penzar et al. orbit interaction. For a comparison of the situations in nuclei
(1990), in LDA+RPA calculations using semiclassical and atomic clusters, we refer the reader to Reinhard et al.
potentials (cf. Sec. V.B.2) by Yannouleas et al. (1989, (1992).
Re~. Mod. Phys. , VoI. 65, Np. 3, Juiy 1993

The latter results confirm the transition of oblate to pro- collective electronic vibrations are expected at consider-
late ground-state deformations obtained in the deformed ably higher energies (see Brack, 1989; Serra et al. , 1989a,
jellium model when filling the ld shell for 8 (N ~ 18 (see 1989b) and that their coupling to the dipole mode would
also the discussion in Sec. III.B.2). In larger clusters, require multi-particle-hole excitations, this mechanism is
however, it is not easy to disentangle the effects coming not expected to give an important contribution to the ob-
from static deformations and those from the fragmenta- served width (Bertsch and Tomanek, 1989).
tion mechanism discussed above. The only processes that would give a true coherent
It can thus be said that the microscopic jellium model width due to decay into a continuum are the processes (i)
with TDLDA or RPA calculations is able to describe the and (ii). However, electron emission (i) is not possible in
correct qualitative trends of the observed resonances in most (small) clusters, where the ionization threshold is
the photoabsorption cross sections of small clusters, in- typically 1 —2 eV higher than the observed plasmon peak.
cluding effects of fragmentation and deformation split- (This is dift'erent from the case of nuclear giant reso-
ting. The interpretation of the resonances as surface nances, which lie high up in the nucleon continuum, so
plasmons, weakly coupled to volume plasrnons, will be that their widths include a large contribution from the
discussed in Sec. IV. C below. evaporation of a nucleon. ) Evaporation of a monomer (i),
Some differences from experiment have been explained with a typical dissociation energy of about 1 eV, is ener-
by phenomenological corrections to the jelliurn model: getically possible and, in fact, believed to be the actual
Blanc et al. (1991) showed that the use of an "e8'ective decay channel of the observed surface plasmons. Howev-
mass" of the electron, taken to be the known value for er, its contribution to their width is expected to be on the
bulk lithium, can fit the unusually large red shift of the order of, at most, a few millielectron volts, if standard es-
dipole resonance of Li8 in terms of a corrected Mie fre- timates of evaporation times are used (see, for example,
quency. Similarly, an ad hoc "core-polarizability" Selby et al. , 1991), which are of the order of the inverse
correction in the jellium-RPA calculation can explain the Debye frequency and therefore cannot explain the ob-
deviation of the resonances observed in large potassium served plasmon widths of about -0. 3 —0. 5 eV.
clusters by Brechignac et al. (1992b; see de Heer, 1993, An appreciable contribution to the width can be ex-
Sec. VIII). Such corrections introducing empirical bulk pected from the coupling to collective ionic vibrations
parameters into finite systems have not, however, been (v), although the energy scale of the latter is of the order
microscopically justified so far and therefore have little of meV only. As we have discussed above, static defor-
predictive power. mations of a cluster split the dipole peak into two or
The observed widths of the resonance peaks are even three subpeaks. Therefore an incoherent superposition of
more difficult to explain microscopically than their posi- thermal (or quantum-mechanical zero-point) vibrations
tions. Several processes can in principle contribute to the of the ions will lead to an effective broadening of an oth-
width of the plasmon peak: erwise sharp dipole plasmon. A fully microscopic
(i) Emission of an electron, i.e. , autoionization ("escape description of this mechanism is outside the scope of an
width"). This is only possible if the plasmon energy lies RPA calculation, and one must therefore resort to simple
above the ionization threshold. phenomenological models in order to estimate this effect.
(ii) Evaporation of a single neutral atom. To this end Bertsch and Tomanek (1989) proposed a
(iii) Interference of the collective state with specific method that has been successfully used to estimate
particle-hole states that lie close in energy (fragmenta- spreading widths of nuclear collective vibrations (Gallar-
tion; cf. "Landau damping" in the solid). do et al. , 1985; Bertsch and Broglia, 1986; see also
(iv) Coupling of the dipole oscillation to other collec- Bertsch, Bortignon, and Broglia, 1983). The ionic vibra-
tive electronic modes (for nuclear giant resonances called tion was assumed here to be of (axially symmetric) quad-
"spreading width"). rupole type. The coupling to the electronic motion was
(v) Coupling of the collective electronic vibration to described by parametrizing the static deformation energy
collective ionic vibrations (cf. phonons in a lattice). in terms of the empirical surface energy of the bulk met-
Of these processes, only (i) and (iii) can be described in al. The thermal fluctuations of the cluster surface, which
the usual RPA, which includes one-particle/one-hole led to a broadening of the electronic dipole plasmon
(Ip-1h) configurations only. The coupling to other col- through deformational splitting, were estimated adibati-
lective electronic modes (iv) would require at least a cally via statistical Boltzmann factors. The resulting
2p-2h, and more generally an np-nh, treatment which be- width was found to be of the order of -0. 4 eV for small
comes numerically very involved. Coupling to ionic sodium clusters at room temperature, in reasonable
motion [(ii) and (v)] is strictly not possible within the jelli- agreement with the observed linewidths of dipole
um model. The fragmentation (iii) has already been dis- plasmons (see Sec. VIII.L of de Heer, 1993).
cussed above; in small clusters like Nazo, where a corre- This model was taken up by Pacheco and Broglia
sponding splitting has been experimentally resolved, it (1989) and further refined in a series of papers, taking the
cannot be made responsible for the linewidth. Rather lit- zero-point shape vibrations into account as well (see
tle is known so far about the coupling to other electronic Pacheco et al. , 1991, and references quoted therein). The
vibrations (iv) To the xtent that all higher-multipole
e. quadrupole motion was extended to include nonaxial de-
Rev. IVlod. Phys. , Vol. 65, No. 3, July 1993

formations, and the P, y deformation energy surfaces (cf. 1990, 1991). These calculations typically predict more
Sec. III.B.2) of the deformed clusters were calculated fragmentation of the strength function than the jellium
within the Clemenger-Nilsson model. Penzar, Ekardt, model, ' although in the "magic" 8-electron system the
and Rubio (1990) treated the same effects self- results are similar, with a single dominant peak. It is also
consistently in the spheroidal jellium model (see also found for this system that the RPA treatment essentially
Ekardt and Penzar, 1991). reproduces the response obtained with more accurate
The mechanism of thermal line broadening due to configuration-mixed wave functions.
shape vibrations of the whole cluster predicts a tempera- The systematic lack of dipole polarizability and the
ture (T) dependence of the width of the form I ~ &T. corresponding absence of redshift in the dipole reso-
Experimentally, the temperature dependence of the nances, found in all jellium-LDA calculations for alkali
linewidth is, however, too poorly known to test this pre- clusters, are closely related to each other by general
diction. Similarly, the form of the resonance would be sum-rule arguments (see Sec. IV. C below). The origin of
predicted to have a Gaussian falloff. Some experimental this failure is not easy to pin down quantitatively, al-
dipole resonances, particularly for charged clusters, can though it is clear from the quantum-chemical calcula-
be fitted rather well with a Lorentzian shape; in other tions that the ionic core needs to be better treated. The
cases the falloff seems to be steeper (see Sec. VIII. L of de wrong asymptotic falloff of the Kohn-Sham potential due
Beer, 1993). In general, however, the experimental infor- to the LDA treatment of the exchange is another possible
mation from photoabsorption measurements is too limit- error source. The self-interaction-corrected LDA results
ed to decide on the precise line form of the resonances. and the WDA calculations discussed in Sec. IV.B.1 above
In large matrix-supported clusters the experimental ought to give a partial answer to the problem of the LDA
widths I of the dipole absorption lines are nearly temper- exchange, but they seem to be contradicted by Hartree-
ature independent and can be fitted by an inverse-radius Fock results on the one hand and by the pseudopotential
law (see, for example, Kreibig and Genzel, 1985): model results, which include the ionic structure, on the
other hand. It is therefore very important to pursue
(4.10) these theoretical investigations, both testing the LDA
and studying the role of the ionic structure more sys-
where R is the radius of the cluster and UF the Fermi ve- tematically.
locity of the valence electrons. This fit does not, howev- Clearly, the experimental details of the electric
er, extrapolate to the observed widths in free clusters response of metal clusters serve as a crucial testing
with % ~ 50, which are considerably smaller. ground for the theory. The calculated dipole strength
Equation (4. 10) had been predicted by Kawabata and and its fragmentation depend rather sensitively, however,
Kubo (1966) from semiclassical response theory. Their on details of the models, such as the self-consistency of
coefFicient 3, however, does not fit the experimental one, the potential, the exchange-correlation-energy density
which seems to depend on the embedding matrix functional, or the pseudopotential used. A numerical
(Kreibig and Genzel, 1985). Recently Yannouleas and source of uncertainty stems from the fact that the space
Broglia (1992) have rederived Eq. (4. 10) with a larger of particle-hole configurations included in the calculation
coefIicient A. They used the so-called wall-dissipation must be restricted for practical reasons. This concerns,
mechanism (Blocki et al. , 1978), which has been studied in particular, the electronic states lying in the continuum,
in nuclear physics in connection with fission and heavy- which are often treated only approximately. Moreover,
ion dynamics; it is equivalent to Landau damping in the the use of a restricted Gaussian basis set in ab initio and
solid and corresponds in large clusters to the fragmenta- pseudopotential calculations might easily lead to a nu-
tion mechanism (iii) above. merical underestimation of the electronic density tail and
In summary, it must be said that the decay mecha- thus of the polarizability. ' Therefore more systematic
nisms of the collective dipole resonances in metal clusters and rigorous investigations of all these approximations
are, both theoretically and experimentally, still rather are definitely called for.
poorly understood. More experimental information on Finally, a few words concerning the terminology used
their temperature dependence and the detailed line form in the predominant jellium model literature might be ap-
is required to shed light on this problem and to test the propriate here. Terms like "surface plasmon, ""volume
simple theoretical models developed so far. plasmon, " "
"effective mass, or "Landau damping" are
strictly defined for infinite systems only, and their usage
3. General discussion
' For the "nonmagic" clusters, the character of the fragmenta-
On a quantitative level, calculations including ionic tion depends on the assumed ionic configuration to a degree
structure have achieved greater accuracy than jellium that allows one to determine the configuration from the empiri-
calculations in reproducing the experimental dipole cal dipole strength function.
response. Unfortunately, ab initio calculations of the dy- ~5One numerical test of possible truncation errors is to check
namic dipole response are only available so far for Na the fulfillment of the energy-weighted sum rule; see Eq. (4. 13)
and Li clusters with Z 8 (see Bonacic-Koutecky et al. , below and Eq. (B7) in Appendix B.2.

for small clusters may raise objections. However, the We discuss in the following the most important mo-
qualitative behavior of the electric response of small al- ments mk obtained for the electric dipole operator (4. 1).
kali clusters, even with N & 20, is dominated by the same The linear moment m, is the simplest case; it is model in-
physics as the condensed-matter phenomena denoted by dependent, and the RPA value is exact. It is proportion-
these names, even if significant quantitative differences alto Z:
exist (such as the relative positions of the unperturbed
= f2
2
particle-hole excitations, the plasmon peaks, and the con- Z
tinuum threshold). Clearly, the use of these terms for
fPl1
2' (4. 13)
small systems underscores the similarity of the physical so that one obtains the famous Thomas-Reiche-Kuhn or
phenomena, but it should not cover the differences. "f-sum" rule:
In purely microscopic language, the electronic
response is made up by (multi-) particle-hole excitations.
However, to the extent that individual particle-hole tran-
f o (E)dE =2~ ROC
Z . (4. 14)
sitions often have little physical meaning, the use of mac- Thus, theoretically, the integrated photoabsorption cross
roscopic or semiclassical pictures can help one gain a section just measures the total number of electrons taking
better physical understanding. We hope to illustrate this part in the collective motion. Hence the experimental
point of view in the following section. determination of this integral helps to identify the collec-
tive nature of a resonance. The observed resonances in
C. Sum-rule approach alkali clusters typically account for at least 60% of the
total dipole strength (see Sec. VIII of de Heer, 1993).
The results of microscopic RPA and TDLDA calcula- The moment I
3 of the RPA response can be shown
(see A.ppendix B.2) to be the restoring force parameter
tions are obtained in rather involved numerical codes and
are not always easy to interpret. Many of the global for translational oscillations of the electrons against the
response properties can, however, be analyzed and under- ionic background. For the electronic Hamiltonian (2.4),
stood in a transparent way in terms of sum rules that ap- the only contribution to the restoring force of dipole os-
ply to the RPA response. cillations can come from the external ionic potential VI,
since both the kinetic energy and the Coulomb interac-
tion between the electrons are translationally invariant.
1. Sum-rule relations and classical limits
The corresponding expression for m3 can easily be de-
rived using the techniques discussed in Appendix 8.2 and
The sum-rule approach (Bohigas et al. , 1979) allows
reads
one to estimate the global features of an RPA spectrum,
such as its centroid and variance, in terms of simple in — 2 $2
2
d2
some cases even analytical —
expressions (see Appendix m3(D, )=
2 I f V~(r)
dx.
p(r)d r, (4. 15)
B.2 for the formal details). This approach has been wide-
ly used in nuclear physics, particularly in connection where p(r) is the density of valence electrons. For a
with giant nuclear resonances (see, for example, Bohigas spherical density p(r), Eq. (4. 15) can easily be
et al. , 1979; Gleissl et al. , 1990). It has also been applied transformed using the Poisson equation for VI to yield a
to metal clusters by Bertsch and Ekardt (1985) and later simple overlap integral of the electron density with the
by many others (Brack, 1989; Serra et al. , 1989a, 1989b, ionic density pt (Brack, 1989):
1990; Reinhard et al. , 1990; Lipparini and Stringari, 2
1991), Reinhard and Gambhir (1992). Sorbello (1983) e' A2
discussed a "dipole force sum rule" that is closely related

m3=
2 m
4w
3 f pi(r)p(r)d'r . (4. 16)
to the RPA sum rules discussed here.
We define the RPA moment mk as Note that this formula holds for any form of ionic densi-
ty distribution, as long as the electron density is spheri-
m& =—
1
f dE E"1m[a(E) ], (4. 1 1) cal, and is therefore not limited to the jellium model. For
the latter, using Eqs. (3.6) and (3. 12), one can rewrite Eq.
where 1m[a(E)] is the imaginary part of the dynamic po- (4. 16) further as
larizability function as calculated in RPA. Physically, 2
e Z
these moments can be related to energy-weighted
ments of the photoabsorption cross section cr(E) by
mo-
I 2p
(4. 17)
where b, Z is the electronic spillout defined in Eq. (4.8)

Qk 1~
4~2 above.
To our knowledge, the experimental photoabsorption
However, one should bear in mind that some of the mo- cross sections have not been analyzed so far in terms of
ments may physically diverge, even though they are finite their quadratic-energy weighted moments which are pro-
in RPA. portional to m3. Such an analysis by means of the above

equations might give some information about the overlap

of the electrons with the ionic charge distribution.
Ei(D )=+A' e Zlma (4.21)
However, these results for m3 and m can also be ex- & One can say rigorously that a lower limit for the centroid
ploited in a different way. An estimate of the peak ener- E of the strength function is given by the inequality
gy can be constructed from the ratio m3/m &,
E3=+m3/m, . (4. 18) The physical significance of this lower limit of E is that
of a slow, adiabatic motion of the electrons, which adjust
As is known from general sum-rule relations (Bohigas, their density (and with it their mean field) at any moment
et al. , 1979), this gives an upper limit of the centroid E of to the external dipole field. It is important to note that
any strength function, which for a narrow collective reso- the terms "diabatic" and "adiabatic" here concern the
nance is close to its peak energy (see Appendix B.2). motion of the electrons only and are independent of the
From Eqs. (4. 13) and (4. 17) one gets for the spherical jel- adiabaticity of the ionic motion in the Born-
lium the spillout formula Oppenheimer sense.
1/2 If E& is identified with the peak position of the dipole
e AZ resonances, Eq. (4.21) is surprisingly well fulfilled by the
E3=
m p
S
Z experimental results. It is therefore often used to predict
1/2 dipole resonance peak energies in terms of measured po-
1
—Ac&) AZ larizabilities (see Sec. VIII of de Heer, 1993). Note that
i 1 (4. 19)
3 Eq. (4.21) is exact for the peak energy of a Lorentzian
form of the cross section, for which E coincides with the
&
here Ace is the bulk plasma frequency. Note that in the
&
mean value E.
limit hZ=0, E3 becomes exactly equal to the classical The apparent difference between the two physical pic-
Mie frequency (4.9) of the dipole surface plasmon, as dis- tures, leading to the rapid diabatic limit E3 and the slow
cussed further below for the general multipole case. adiabatic limit E&, seems to be in contradiction with the
Within the jellium model one thus obtains, through the fact that both energies are close to the measured peak en-
identification of E3 with the energy of a surface dipole ergies of the plasmon resonances (apart from splittings
plasmon, a simple and transparent explanation of its red- and the overall absence of redshift). However, as has
shift with respect to the Mie frequency: it is due to the been noticed in the theoretical calculations (Bertsch and
quantum-mechanical spillout of the electrons over the jel- Ekardt, 1985; Brack, 1989; Yannouleas et al. , 1989,
lium surface. A more refined picture, in which couplings Reinhard et al. , 1990), these two energies are surprising-
of surface and volume plasmons are included, will be dis- ly close for the dipole plasmons of alkali clusters; in fact,
cussed further below. Note, however, that the direct their difference is smaller than the difference of either of
connection of the redshift with the electronic spillout is them from the measured peak energies. This shows us
justified only in the simple jellium model in which the that when the strength function is strongly concentrated
positive charge distribution has a sharp edge. in a single resonance there is no distinction between adia-
The energy E3 has been widely used for estimating the batic and diabatic motion of the electrons with respect to
energies of giant resonances in nuclei. The physical the ionic background.
meaning of this upper estimate E3 of E, as discussed also For collective modes of higher multipolarity, the situa-
in Appendix B.2, is that of a rapid, diabatic oscillation of tion may become quite different. In the case of nuclear
the valence electrons against the ions. The oscillation is collective quadrupole vibrations, for example, there is a
so fast that the self-consistent mean field of the electrons difference of almost an order of magnitude between E&,
remains that of the ground state; one therefore also which can be approximately identified with the low-lying
speaks of the "sudden approximation" for the collective shape vibrations, and E3, which is known to yield an ex-
electronic motion. The numerical evaluation of E3 leads cellent description of the high-lying quadrupole giant res-
one to values that are only marginally higher than the onances (see, for example, Lipparini and Stringari, 1989;
dominant peak energies of TDLDA or RPA results Gleissl et al. , 1990).
(Bertsch and Ekardt, 1985; Yannouleas et al. , 1989; An interesting result is obtained if one takes the classi-
Reinhard et al. , 1990; Lipparini and Stringari, 1991). cal limit of the energy E3 for a macroscopic metal
Another very useful moment is m,
. Unlike m& and sphere, i.e. , the limit Z — +~ of a spherical cluster. In
m3, there is no closed formula for m &, but since it is this limit, the jellium model is certainly a good approxi-
just one-half of the static polarizability [see Eq. (B13) in mation for the ionic density, and the electrons will have a
Appendix B.2] it is easy to calculate in the RPA. We use step-function-like density with the same radius. There-
m, and m to construct another estimate of the reso-
& fore their spillout (which is a purely quantum-mechanical
nance, E, : phenomenon) will be zero, and from the result (4. 19)
Ei —QM i Im (4.20) above one finds that E3 goes over into the classical Mie
frequency for the surface plasmon. Brack (1989) has
Using Eq. (4. 13), we can express E, for the dipole shown that the same limit holds for all electric multipole
response as vibrations described by the operators Ql = er PL (cosO):

1/2 2. Coupling of surface and volume plasmons

E3(QL ) ficoL "= Act)pl for Z ~~
The sin1ple sum-rule model using the two estimates E3
and E, is not quite satisfactory, in spite of n1any qualita-
tive successes and the limits discussed above, since it
in agreement with the expression derived by Mie (1908) works we11 only in the idealized case of a narrow reso-
for surface plasn1ons of multipolarity I, . nance that exhausts the ml sum rule for a given excita-
Similar considerations can be n1ade for the energy El tion operator (cf. the so-called plasmon-pole approxima-
in Eq. (4.21), which involves the static polarizability. If tion). As the experimental results and the fully micro-
we assume that the polarizability for all multipolarities L, scopic RPA and TDLDA calculations show, the reality
in the limit of large radii reaches its classical value is often more complicated and presents one with frag-
mented collective strength (see Sec. IV.B.2). One can,
~(L)
~class g I2L+1 (4.23) however, go one step further in the exploitation of sum-
rule relations and thereby come closer to the microscopic
and combine this with the expression (see, for example, RPA.
Brack, 1989) An approximate RPA treatment, which makes the as-
2
sumption of local currents (or velocity fields) and leads to
g2
m, (Q~)= L fr p(r)d r, (4.24) a secular equation for coupled harmonic vibrations de-
scribed in terms of local trial operators Q;(r), has recent-
ly been proposed by Brack (1989) and Reinhard et al.
we obtain the same limit as in (4.22) for the energy E i .. (1990) and used for the calculation of dipole plasmons
and polarizabilities of alkali clusters. The ground-state
E, ( QL ) ~ "=
Picot
I.
21. +1
%~1 forZ —+~ . densities obtained in the self-consistent jellium-Kohn-
Sham n1ethod are the only ingredients; this approach
does not require any adjustable parameters. It is, in fact,
(4.25) an extension of the Quid dynamics approach that has
We thus find that the two estimates E3 and El become been successfully used to describe nuclear giant reso-
nances (Bertsch, 1975; da Providencia and Holzwarth,
identical in the large-Z limit for spherical clusters. Since
their difference gives an upper bound of the variance o. 1985; see also Lipparini and Stringari, 1989, for a recent
review. Both approaches and their relation to classical
through the inequality given in Eq. (B9) of Appendix B.2,
hydrodynamics are brielly discussed in Appendix B.2.)
we learn from this result that the variance of the dipole
The static dipole polarizabilities obtained for sodium
strength should go to zero in the macroscopic limit. We
clusters in the local-current RPA by Reinhard et al.
shall further elucidate this point below in connection
with a discussion of the coupling between surface and di-
(1990) are in perfect quantitative agreement with those
resulting from the much more tin1e consuming micro-
pole plasmons.
A warning must be given here. The limit in "Z~ ~" scopic linear-response calculations by Beck (1984b),
Ekardt (1985a, 1985b), and Manninen et al. (1986). In a
Eqs. (4.22) and (4.25) should not be taken too literally,
semiclassical version of the 1ocal-current RPA, the
since for macroscopic clusters or metal spheres the long-
correct average values of the polarizabilities have been
wavelength limit and thus the use of a static dipole
obtained in terms of extended Thomas-Fermi variational
operator is no longer justified. The above discussion ap-
densities (Brack, 1989; cf. also Sec. V. B).
plies, therefore, only up to the limit in which clusters are
In the following, we shall use the local-current RPA
still smaller than the wavelength corresponding to the
picture to discuss some aspects of the physics of surface
observed surface plasmon frequency, which is typically
and volume plasmons in metal clusters. (We refer the
about 500 nanometers for Na, corresponding to Z = 10' .
reader to Appendix B.2 for the relevant formalism. ) For
In this respect we note that Serra et al. (1990) have
a given (electric) multipolarity L, the coupled intrinsic
studied the electronic multipole response of spherical
calculations using modes of the cluster may be described by the following
metal clusters in jellium-Kohn-Sham
the simple su'm- trial set of local operators:
operators of the type jL (qr) YLD(0) and
rule approach. For small values of the momentum Q (r)=r ILO(0) (4.26)
transfer q they recovered the surface mode systematics
discussed above, whereas for large q the response was where p, is an arbitrary real number ~ 1. By construct-
found to be mainly determined by particle-hole excita- ing the restoring force and inertial tensors (B28), (B27)
tions. For intermediate q values, bulk oscillations were corresponding to these modes and solving the secular
found and their connection with the hydrodynamical equation (B26), one obtains a spectrum of eigenmodes
model predictions were established. In the limit of a big from which the various sum rules can be evaluated.
sphere, they obtained an in1proved bulk-plasmon pole ap- Since the gradients of Q, are proportional to the velocity
proxin1ation for the dispersion relation, which includes fields, p; =I. is the mode corresponding to incompressible
non-negligible exchange and correlation effects. flow, with B,QL =b, (r I'Lo)=0. Thus the operator Qt

describes pure surface oscillations without compression. gets the dipole operator Qi (in the z direction); the scal-
In the large-X classical limit, this leads to the pure Mie ing transformation (B21) then is a simple translation: the
surface plasmon, as we see from the limits of E3(QI ) and dipole Mie plasmon is a pure translational osciHation of
E, (QI ) in Eqs. (4.22) and (4.25) above. Modes with all the electrons against the ionic background.
p;WL, on the other hand, lead to local compression of In Fig. 8 we show the dipole spectra obtained in the
the electron density. As shown by Brack (1989), the cou- local-current RPA approach from self-consistent Kohn-
pling of any number of modes with p; WL gives in the Sharn ground-state densities for spherical sodium clusters
classical limit a degenerate set of eigenmodes, all having of increasing sizes. The energies of the eigenmodes are
the volume-plasmon energy fico~i', the energy E3(Q; ) con- given as histograms; the height of the lines corresponds
sequently has the same limit: to the percentage of the dipole m sum rule carried by &
each state. At the top of the figure we see Nas; the exper-
E3(Q;)~Aco~i for Z~~ (p, AL) . (4.27) imental position of the surface plasmon (de Heer, Selby
et al. , 1987) is indicated by an arrow. The dominant
(As in the previous section, the symbol oo should in reali- -2.
peak of the theoretical spectrum is at 75 eV and car-
ty be a number not larger than about 10' .) Note that the ries 84%%uo of the dipole strength; it is redshifted with
surface and the volume modes are completely decoupled
respect to the classical Mie surface plasmon, which lies at
in this limit: the volume modes cannot be excited by the
3.4 eV and is indicated by a vertical dashed line in the
electric multipole operator (4.26) with p, =L. This result
figure. (The fact that the redshift is not strong enough to
is a generalization (to L&1) of that obtained by Jensen
reach the experimental position has already been dis-
(1937), who investigated the dipole eigenmodes of a metal cussed in Sec. IV.B above. ) The remaining strength of
sphere using classical hydrodynamics in a variational
16% is scattered over several states in the region
local-density scheme. -3. 5 —5 eV; only two of them carry more than one per-
In finite clusters, these two types of modes are coupled cent. These states contain the remaining strength of the
due to finite-size and quantum-mechanical efFects (spill- fragmented volume plasmon; their centroid is also sub-
out of the electron density; kinetic, exchange, and corre-
lation energies). This coupling leads to the following
changes from the above classical result: s(~) Nos exp
LJp
(i) The compressional volume-type modes are no longer 100-

I
degenerate; the volume plasmon is fragmented into a fo m) I
bunch of scattered eigenfrequencies. That part of the

volume plasmon which lies in the electron ionization con-
50- ~ I
I
redshift
I
I
I
m=====I
tinuum can, in fact, be found as a strongly Landau- I
I
I~ ~l I I
damped resonance, as shown by Ekardt (1985a, 1985b), 5 l6 ~(ev)

I
but it carries a negligible fraction ((10 ) of the total di-

pole strength.
(ii) The surface plasmon and parts of the fragmented NGi, i, o Mie ]
dip p]
volume plasmon are shifted away from their respective
classical Mie frequencies. In the case of the dipole %m)
1f
modes, both are redshifted. The same redshift is found 50-

for higher multipolarities in large clusters, for small clus- I
ters, however, the kinetic-energy contribution to the re- I
storing forces can lead to a substantial blueshift (Brack, 4 5 l6 ~(ev)

1989; Serra et al. , 1989a, 1989b). This is exactly the I
mechanism that shifts the nuclear quadrupole giant reso-

nance to higher energies and leads to a fundamental %m, (classical limit)
difference between normal hydrodynamics and fluid dy- 50—
namics (see Appendix B.2.c). I
In the microscopic particle-hole excitation (RPA) pic- I
ture, this coupling leads to fragmentation of the dipole 5 6 u(ev)

strength, discussed in Sec. IV.B above. The fragmenta-
tion of the surface-plasmon peak cannot always be FIG. 8. Collective dipole spectra for sodium clusters, obtained
correctly described in the local-current RPA picture; ob- in spherical jellium Kohn-Sham plus local-RPA calculations
(Brack, 1989; Reinhard et al. , 1990). Shown is the strength in
viously, the strong coupling effects between particular
percents of the total dipole m strength (normalized to 100%%uo).
particle-hole excitations must be connected with nonlocal &
The lowest spectrum (Na„) represents the classical limit, where

currents. But the remaining strength of —10—25 Jo lying 100%%uo of the strength lies in the surface-plasmon (frequency
above the surface peak can be interpreted as the cod;p') and the volume-plasmon (frequency co») has zero strength.
remainder of a strongly fragmented volume plasmon. For the finite clusters, the surface plasmon is redshifted and its
Let us illustrate the picture of coupled surface and missing strength is distributed over the remainder of the strong-
volume modes for the dipole case L =1. For p,. =1 one ly fragmented volume plasmon.

stantially redshifted with respect to the volume-plasmon tems. Nevertheless, the notion of the two most predom-
energy of 5.85 eV. [Note that in a microscopic RPA cal- inant physical types of collective oscillations, namely,
culation (Yannouleas et al. , 1989) almost identical spec- translational surface modes and compressional volume
tra are obtained. The distribution of the fragmented modes, remains a useful concept. '
volume-plasmon states is somewhat di6'erent, and the 1.0 1.0
surface mode can be fragmented too, but the moments
m3, I I&, and the variance o. agree to within less
&
than 5% for the two calculations (Reinhard et al. ,

0.6
0.4
NAms ETF p 6
-
NA~o28KS
1990).] At the bottom of Fig. 8, the limit is X~~ 0.2 0.2
shown, representing a macroscopic metal sphere with a

-0.2 -0.2
sharp surface: One surface plasmon at the Mie frequency
-0.4
fico &/&3 that carries 100% of the dipole strength, and
-0.6
one (infinitely degenerate) volume plasmon at fico, that is Q 8
I I I I I I
-0.8 I I I I I I
fully decoupled and therefore has no dipole strength. 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70

z {a.u. ) z (a.u. )
For clusters in between, the tendency is to reduce the
coupling between surface and volume modes and to in- I I I I I
crease the dipole strength of the surface mode with in-

creasing N. 04 -
NA254 ETF NAq54 KS
The distinction between volume and surface plasmons 0.2 0.2 /
is strictly possible only in the large-particle limit. For 0
smaller particles, their coupling and the increasingly -0.2 -0.2 'I
/
)I I/
dominant role of the surface mix the two types of modes. -0.4
(I
-0.4 V
However, a look at the transition densities 5p(r) corre- -0.6 (b)-

sponding to some of the eigenmodes shown in Fig. 8 will -0.8 I I I I I I
-0.8 I I I I I I
Q 5 10 15 20 25 30 35 0 5 10 15 20 25 30
reveal why it is still meaningful to speak of surface and z (a.u. ) z (a.u. )
volume modes even in relatively small clusters. The tran-
sition density 5p, (r) of the ith eigenmode defined by 0.6
0.4
5p;(r ) = —V'. (pu; ) (4.28) 0.2

M
in terms of its velocity field u; (see Appendix 8.2.b) tells p

I
-0.2 /
-0.2
us where the essential changes occur during the collective I
/
vibration in each eigenmode. -0.4 /
/ NA2o KS
NA ETF
Figure 9 shows the transition densities 6p, along the z 2o
(e) -0.6 (g)
axis obtained by Genzken (1992) for various cluster sizes -0.8 I I I I
-0.8 I I I I
0 4 8 12 16 20 0 8 16
and for two typical eigenmodes: the lowest, most collec- z (a.u. )
4
z (a.u. )
12 20
tive mode (shown by solid lines), and one of the fragment-

ed volume modes lying around 5 eV (shown by dashed
lines). These results were obtained both in semiclassical 0.4-
calculations, using the model of Brack (1989) and shown

in the left parts (a), (b), (e), (f) of the figure, labeled ETF p p
(extended Thomas-Fermi), and microscopically as by -0.2 -0.2

Reinhard et al. (1990), shown in the right parts (c), (d), -0.4- /
(g), (h) of the figure, labeled KS (Kohn-Sham). In the NAsETF / NAs ' KS (h)-
-0.6- -06- N
semiclassical results, where the densities p(r) are con- /
strained to be constant in the interior part of the cluster,

I I I I I I I I
'
0 3 6 9 12 15 0 3 6 9 12 15
the separation can be clearly observed even in the small- {a.u. ) z z (a.u. )
est cluster Na8: the lowest mode has all its transition FIG. 9. Dipole transition densities (4.28) along the z axis for
density peaked in the narrow surface region near the jelli- spherical sodium clusters: left parts (ETF): semiclassical local-
um edge, whereas the higher mode has an appreciable RPA results (Brack, 1989); right parts (KS): Kohn-Sham re-
sults in local RPA (Reinhard et al. , 1990); solid curves, lowest
nonzero transition density in the inner volume region. In
eigenstate (dipole plasrnon); dashed curves, state lying around 5
the microscopic Kohn-Sham results, the shell oscillations eV (belonging to the fragmented volume plasmon). From
of the electronic densities partially blur the situation, but Cxenzken, 1992.
the same trends can be observed at least in the larger
clusters. For microclusters like Nazo and Na8, it is no
longer possible to divide the electronic density distribu-
tion into a volume and a surface part due to the large 6Kresin (1991) has also used the picture of coupled surface
shell oscillations. Correspondingly, the two types of and volume plasmons, with the somewhat oversimplifying as-
modes are much more strongly mixed in these small sys- sumption of a single volume-plasmon frequency.

It is again instructive to compare the analogous situa- V.A we focus on their shell structure and in Sec. V.B on
tion in nuclear physics. The famous nuclear giant isovec- their average properties. In both contexts, we use the
tor dipole resonance, a strongly collective vibration of self-consistent jellium model as a basis and semiclassical
protons against neutrons, can be interpreted in very simi- methods as important tools: on the one hand, the quanti-
lar terms. The two leading mechanisms are a compres- zation of classical trajectories helps one to understand
sional mode (corresponding to the volume plasmon) pro- the so-called supershell structure, and on the other hand,
posed by Migdal (1944) and by von Steinwedel and Jen- the semiclassical density-variational method allows one
sen (1950), and a purely translational mode (correspond- to obtain average cluster properties regardless of their
ing to the surface plasmon) proposed by Goldhaber and size and thus to study them in the asymptotic limit
Teller (1948). In a hydrodynamic description one obtains &~ oo.
a good fit to the average energies of the experimental gi-
ant dipole resonances by taking a suitable combination of
A. Shells and supershells
the two modes (Myers et al. , 1977). After explicit diago-
nalization of the two coupled modes, the lower of the two
eigenfrequencies fits the experimental resonances, 1. Shell effects in finite fermion systems
whereas the higher is larger by roughly a factor of two
and escapes experimental detection (Gleissl et a/. , 1990). Quantization of a system of particles in a finite spatial
The microscopic HF+RPA theory has difhculty explain- domain leads to discrete energy eigenvalues, which are
ing the finer details and the proper widths of these reso- usually grouped into bunches of degenerate or close-lying
nances (see, for example, Liu and Van Giai, 1976), but an levels, called shells. The amount of bunching depends on
analysis of transition densities supports the picture of the the symmetries and the integrability of the confining po-
above two classical modes. tential. For fermion systems obeying the Pauli principle,
Another measure for the coupling between surface and this leads to shell effects which are well-known in atoms
volume plasmons is given by the variance o. of the dipole and nuclei: local minima in the total binding energy per
strength distribution. As discussed in Appendix B.2, an particle versus particle number or deformation,
upper limit of o can be given in terms of E&(Q, ) and sawtooth-like behavior of the particle separation energy
F-, (Q, ); see Eq. (B9). Since both these energies go to the (ionization potential, electron affinity), or oscillations in
same limit, %co~&/&3 for large X, cr has to go to zero. the radial density distribution. These effects can be de-
scribed theoretically in terms of independent (or weakly
Note, however, that this variance should not be directly
identified with the experimentally measured linewidth I
interacting) fermions moving in a common potential. In-
of the surface plasmon that we discussed at the end of versely, the experimental observation of shell effects sug-
Sec. IV.B.2. gests the existence of a mean field in which fermions
The theoretical RPA prediction of some 10—15% of (more generally, some quasiparticles with fermionic na-
the dipole strength, lying well above the surface plasmon
ture) move more or less independently. In the case of
metal clusters, the observation of shell effects has been
and containing part of the fragmented volume pIasmon,
cannot be verified in the case of sodium due to a lack of very suggestive, indeed, of the single-particle motion of
the loosely bound valence electrons and has stimulated
experimental data in that energy range. However, recent
the development and refinement of the mean-Geld-type
photoabsorption measurements in small Ag clusters with
%=8—40 (Tiggesbaumker et al. , 1992) systematically re- models described earlier in this review.
Shells of single-particle levels are a global phenomenon
veal some dipole strength lying clearly above the dom-
in the sense that they depend more on the overall form of
inant surface-plasmon peak, in qualitative agreement
the mean field (e.g. , symmetry, steepness of the surface,
with the RPA prediction. To the extent that the jellium
deformation) than on the finer local details of its radial
model can be trusted for small Ag clusters, this may
confirm the above picture at least qualitatively. dependence (e.g. , oscillations which themselves can be
We finally point out that Barbera, n and Bausells (1985) connected to shell effects). For large alkali clusters, this
have discussed the coupling of surface and bulk plasmons means that the inclusion of the ionic structure on top of a
in connection with the inelastic scattering of electrons jellium model calculation need not modify appreciably
from small metal spheres. Ekardt (1987) discussed in this the shell situation, provided that the (spherical or de-
context wave-vector dispersion versus angular momen- formed) jellium density comes close to the averaged ionic
tum dispersion of the volume plasmons in small metal distribution and that the single-particle nature of the
electronic orbits is predominant. This may explain the
clusters using spherical jellium- TDI.DA-Kohn-Sham cal-
success of the jellium model in correctly explaining most
culations.
of the observed "magic numbers" corresponding to
spherical-shell closings, in particular of the very large al-
V. LARGE CLUSTERS: A STEP TOWARDS THE BULK? kali clusters, which we discuss in the following subsec-
tion.
In this section we deal with very large metal clusters That shells are not only peculiar to spherical systems
containing up to many thousands of atoms. We discuss has most clearly been formulated by Strutinsky (1968),
them from two complementary points of view: in Sec. who pointed out the close connection between the oscil-

lating part of the single-particle level density, 5g(E), and mers, which were correctly interpreted for the first time
the oscillating part of the total energy, i.e. , the "shell by Strutinsky (1968) using the shell-correction formalism.
correction" 5E [see Eq. (5. 1) belowj, both as functions of That shape isomers in metal clusters are found not only
particle number and of deformation I.n fact, the mere ex- in quantum-chemical and molecular-dynamics calcula-
istence of static nuclear deformations is due to shell tions, but also within the deformed jellium model, has
effects and could be accounted for theoretically by the been discussed already in Sec. III.B.2.
famous Nilsson model (Nilsson, 1955; Mottelson and One should therefore use the term "magic number"
—
Nilsson, 1955) which has been revived by Clemenger with some care: it may not be taken as a synonym for
(1985a, 1985b) in a simplified form (namely, leaving out sphericity. This was justified in the first years after the
the spin-orbit interaction). The mechanism, which leads discovery of the nuclear shell model (Goeppert-Mayer,
to a spontaneous deformation of the mean field even if — 1949; Haxel, Jensen, and Suess, 1949), when "magic num-
the basic two-body interaction is a central one is just — bers" like 82 and 126 had been recognized as the num-
another example of the Jahn-Teller elfect (Jahn and Tell- bers of nucleons corresponding to filled major spherical
er, 1937): when a spherical l shell is only partially filled, shells (where the strong spin-orbit coupling was an essen-
the system lifts the degeneracy of its ground state by al- tial ingredient). Since enhanced stability also occurs in
lowing the mean field to give up spherical symmetry, re- deformed systems, the corresponding numbers of parti-
sulting in an energy gain. However, the shell effects in cles can also have "magic" character, e.g. , the neutron
deformed systems are usually less pronounced than in number %=146, which is characteristic of the most
spherical ones. (See, for example, the smaller subpeaks in stable fission isomers in actinide nuclei having large de-
the cluster abundances of Knight et al. , 1985, which cor- formations corresponding to an axis ratio of -2:1 (see,
respond to deformed subshells, compared to the dom- for example, Bje(rnholm and Lynn, 1980).
inant spherical-shell peaks. ) Just recently, the Strutinsky shell-correction method
A very effective and successful method for investigat- has been applied to large deformed sodium clusters by
ing the shell structure in the total energy of a finite fer- two independent groups. Frauendorf and Pashkevich
mion system, as a function of both deformation and par- (1993) used a deformed version of the Woods-Saxon po-
ticle number, has been introduced by Strutinsky (1968). tential parametrized by Nishioka et al. (1990) to calcu-
According to his basic theorem, the energy of an in- late the ground-state deformations of Na clusters with
teracting fermion system can be divided into a smooth X ~ 300, including axial quadrupole, octupole, and hexa-
part E and an oscillating part, the energy shell correction decupole shapes. Reimann et al. (1993) improved the
5E- Clemenger-Nilsson model by fitting the l term to self-
consistent Kohn-Sham levels of spherical clusters. They
E =E+5E . (5. 1) calculated the equilibrium deformations of spheriodal Na
Whereas E varies slowly with particle number and with clusters with 50 ~ X ~ 850 and reanalyzed the experimen-
the deformation of the system, the shell correction 6E tal mass abundance spectra of Bjgfrnholm et al.
contains all the oscillations coming from the shell bunch- (1990,1991), finding good agreement between the calcu-
ing of energy levels. To a very good approximation, 5E lated and observed "deformed magic" numbers.
can be extracted from the sum of occupied single-particle Many aspects of shell structure can be qualitatively,
energies (or quasiparticle energies, see Bunatian et ai. , and sometimes even quantitatively, described by semi-
1972) e; of the averaged mean field, i.e. , from classical methods. A very powerful tool for investigating
E, =+~, E;, by subtracting its suitably defined averaged the gross shell structure in the single-particle level densi-
part (Strutinsky, 1968). Brack and Quentin (1981) have ty of a given potential in terms of classical trajectories
numerically tested this approximation using Hartree- has been developed by Gutzwiller (1971) and by Balian
Fock calculations with effective nuclear interactions, and and Bloch (1972; see also earlier papers cited in these two
also investigated its extension to finite temperatures. In articles). Strutinsky et al. (1977) generalized this method
practice, the average energy E can be taken from a phe- successfully for realistic nuclear potentials and explained
nomenological liquid-drop mode1, whereas 5E can be the behavior of the various isomeric valleys in contour
found from the single-particle energies c, of phenomeno- plots of the level density and the energy as functions of
logical shell-model potentials. [For an extensive review particle numbers and deformation (see also Strutinsky,
on the Strutinsky method and its application to nuclear 1975). Nishioka et al. (1990) applied the same kind of
fission barrier calculations, see Brack et al. (1972).t A analysis to metal clusters, using Woods-Saxon-type po-
modified form of the Strutinsky renormalization idea was tentials, and discussed the "supershell" structure, which
discussed by Brack et al. (1991b) in their calculation of will be the subject of the following subsection.
thermal electronic properties of metal clusters. We shall not present here the details of the
The regular oscillatory behavior of both the single- Cxutzwiller-Balian-Bloch theory and its applications to
particle level density and the shell-correction energy metal clusters, but refer the interested reader to a forth-
leads, in general, to shape isomerism: several local rnini- coming review article by Bje(rnholm et al. (1993). Some
ma can exist in the multidimensional energy surface. A of the simplest aspects and results of this approach will
famous example in nuclear physics are the fission iso- be referred to below in order to explain the observed shell

structure in large alkali clusters. Reimann and Brack oscillations rejecting the main shells are modulated by
(1993) have observed that the calculated ground-state de- an oscillating amplitude of lower frequency. Such a pat-
formations of spheroidal sodium clusters with tern was found by Balian and Bloch (1972) to be a very
50 5 N ~ 850 (Reimann et al. , 1993) can be explained in general feature of quantal eigenmodes in a cavity with
terms of a family of classical rhomboidal planar orbits of rejecting walls. For the case of a spherical cavity, they
a particle in a spheroidal cavity, exactly as it was pro- showed that this beating is the result of an interference of
posed by Strutinsky et al. (1977) in the context of nu- the two most important classical trajectories responsible
clear deformations on the basis of the Balian-Bloch for the oscillating part of the level density, namely, trian-
theory. gles and squares. (Other orbits contribute, too, but with
smaller amplitudes; they are important only near the in-
2. Electronic supershell structure in large alkali clusters terference minima. ) Their contribution to the level densi-
One of the salient features of the level density in a ty g(k) as a function of the wave number k =V2mE/A'
steep confining potential is a beating pattern: the regular equals (see also Strutinsky, 1975)
Qg(3+4)(k)
1
2
( kR /77) )g2
2nzR . 2 3sin kL3 +—
4
+ 3 ~sin kL4 + 3K (5.2)
where R is the radius of the cavity. This superpo- scribe difFerent metals, to discuss the supershell structure.
sition of two sin functions with comparable ampli- This supershell beating is also present in the total ener-
tudes 33 =(V3/2)'~, A„=(1/V2)'~, and wavelengths gy of the system. In Fig. 11 we show the energy shell
L3=3&3R, L4=4&2R leads to a beating of the level correction for spherical sodium clusters with N up to
density; up to a small term contributing less than 5%%uo, 3000 obtained in self-consistent Kohn-Sham calculations
one obtains by Genzken and Brack (1991) and Genzken et al. (1992)
at various temperatures T, plotted versus X' . The shell
2 R
5g(3+4)(k) = (kR /~)'~ g2
A 3cos(kL )cos kbL correction 5F(N) is defined here as the difference be-
tween the total free energy F(N) of a cluster with N
atoms and its average part F(N):
(5.3)
6F(N) =F(N) —F(N) . (5.5)
with
-=1
L =—(L3+L~), bL =1—(L3
= L4) . — (5.4)
Strutinsky (1968) introduced a numerical energy-
averaging procedure to calculate E(N) at T=O from the
single-particle energies c;, which can be extended to finite
Here the factor cos(kL ) gives the fast oscillations in en- temperatures (Brack and Quentin, 1981). A simple alter-
ergy, representing the main-shell oscillations, and the native way is to use a liquid-drop model expansion of the
second cos factor gives the beating amplitude. total free energy of the type discussed in Sec. V.B.2:
The same pattern is also found for the density of ener-
gy eigenvalues in smooth potentials, provided their sur- F(N) =FLDM(N) =ebN+a, N +a, N'~ (5.6)
face is steep enough. (The pure spherical harmonic os-
cillator and Coulomb potentials have no well defined sur-
face region; correspondingly, their level densities do not 2500
show any beating pattern. ) In Fig. 10 we show the elec-
tronic level density obtained by Nishioka et al. (1990) 2000— =300
for a spherical Na cluster. They used a phenomenologi- I
6)
cal Woods-Saxon potential fitted to the self-consistent I- 1500—
jellium-Kohn-Sham results of Ekardt (1984b) and extra- CO
UJ
polated it to a size of N=3000. To emphasize the gross 1000—
LU
shell structure, the discrete eigenvalue spectrum was UJ
folded with a Lorentzian having a width of about a fifth 500—

of the main-shell spacing. The shell oscillations around
the average level density and the beating pattern are evi- 0 I I t
dent in this figure. Nishioka et al. (1990) termed the 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
groups of main shells, separated by the interference mini- WAVE NUMBER (A )
ma, "supershells" and discussed their stability against FIG. 10. Electronic single-particle level density g(k) as a func-
variations in the radial dependence of the potential. tion of wave number k, evaluated in a spherical Woods-Saxon
Clemenger (1991) has also used Woods-Saxon potentials, potential corresponding to a Na cluster with %=3000, by
related by simple scaling considerations in order to de- Nishioka et al. (1990).

Matthias EIrack: The physics of simple metal clusters
[In obtaining Fig. 11, a, and a, were determined at each In order to make the beating pattern of the shell oscil-
temperature by a simple fit such that 5F(N) is oscillating lations visible in very large clusters, where thermal
around zero; the bulk energy was fixed at its theoretical suppression of the shell structure becomes important,
value e& = — 2. 2567 eV obtained for r, = 3. 96 a.u. Pedersen et al. (1991) multiplied the logarithmic deriva-
(Genzken and Brack, 1991).] The supershell beating is tives of the mass yields by a factor depending exponen-
clearly visible in 5F(N). Note, however, that the ampli- tially on X' . This is justified by the following argu-
tude of the shell oscillations is reduced with increasing ment. The temperature dependence of 5F(N) for a
temperature, and the minima become less sharp than at spherical closed-shell system can be schematically es-
T=O. The "magic numbers" corresponding to filled timated from the harmonic-oscillator model (Bohr and
spherical main shells are given in the curve for T=O at Mottelson, 1975) to vary as
the corresponding minima.
Neither the level density nor the shell-correction ener- 5F(T)=5F(0) . ', t=T (5.7)
sinht Ace
gy oF(N) are directly observable. As discussed by
Bj@rnholm et al. (1990, 1991), the mass abundances in Expanding for large temperatures and using Ace ~ X
expansion beams depend rather on the differences this gives, indeed, a temperature suppression factor
b, ,F(N) =F(N — 1) F(N) — and 62F(N) =F(N +1) ~ exp( —N'~ ). Therefore, scaling up the experimentally
+F(N —1) —2F(N), which are very sensitive to observed mass yields by the inverse factor just compen-
temperature-smoothing effects for the larger cluster sizes. sates for this thermal compression.
In fact, these latter quantities were shown by Brack et al. In the upper part of Fig. 12 we have reproduced the
(1991a, 1991b) to vanish almost completely for N ~ 600 at relevant figure of Pedersen et al. (1991), which shows the
temperatures T=600 K or above, putting in doubt the first differences of the logarithmic experimental yields,
observability of the supershell structure, which only A&lnI&, averaged over a range X+Ko with Ko =0.03K in
starts at %~900, in supersonic expansion experiments. order to eliminate statistical fluctuations and multiplied
Nevertheless, Pedersen et al. (1991) and Martin, by +N exp(cN'~ ). Here c is an adjusted constant con-
Bjprnholm et al. (1991) have experimentally put the ex- taining the effective temperature, and the extra factor
istence of supershells in Na clusters into evidence, and &N compensates for the decrease of the shell correction
Brechignac et al. (1992c) observed them in Li clusters. at T=O with increasing N (Bohr and Mottelson, 1975).
Note that in taking the differences b &1nI&, one focuses on
2 ~ ~
~ ~ i $ I ~ 1 / ~ \ I f I ~ I $ I ~ I J I ~ I ~ ~
the oscillating part of the mass yields, which is dominat-
ed by electronic effects, whereas the smooth ionic contri-
CU
0- butions, as well as possible systematic experimental er-
LL,
rors, are canceled.
—2 l I I I i I I I I ~ 1 i I t I
In the lower part of Fig. 12 we show the theoretical jel-

~ f ~ ~ ~ I ~ I ~ I I
8
lium model results by Genzken and Brack (1991) for the
2 ~ $ ~ ~ ~ f I ~ ~ $ ~ I I / I I I $ ~ ~ I / l ~ I ( ~
negative second difference of the total free energies,

GJ
0- b, zF(N), multiplie—d by the same enhancement factor
(with the value of c readjusted by —10%). In this quanti-
LL.
2 0 6 8 'l
0 12 10 ty, too, the ionic contributions, which are only crudely
represented in the jellium model anyhow, are practically
canceled. The similarity of the two curves shown in this
T= OK figure is striking. In comparing them, one makes the im-
plicit assumption
I~ —exp{ b, ,F(N) lkT I .— (5.8)
CD 0- Since b, , F (N) is the free dissociation energy of one neu-

tral atom [up to the constant F(1), which cancels when
c I
taking the second difference b, 2F], the Boltzmann factor
LA
on the right-hand side of Eq. (5.8) is a measure of the rel-
ative stability of the cluster N against evaporation of a
C3
monomer,
~ I L I ~ 'I I I I I I I I j ~ I I I I I I I I I I I I
2 /+ 6
'I/3
8 10 12 14-
Na~ ~Na+ i + Nai (5.9)
FICr. 11. (Free) energy shell correction (5.5) vs X' ' at three
in an evaporative ensemble at thermodynamic equilibri-
temperatures, obtained in spherical jellium-Kohn-Sham-LDA um. Although this is certainly a rather simplifying as-
calculations by G-enzken and Brack (1991). In the lowest part, sumption, which neglects dynamic and nonequilibrium
the magic numbers" corresponding to filled major spherical effects of the evaporation process (see Bj6rnholm et al. ,
shells are indicated. 1991, 1993, for a discussion of this point), it seems to be
supported by the qualitative agreement of the two curves that I. be a multiple n of the de Broglie wavelength
in Fig. 12. In any case, this result demonstrates that the A,=2m. /k of the electron. Inserting in Eq. (5. 10) the
6nite temperature of the valence electrons can play an Thomas-Fermi expression for the Fermi momentum of
essential role in the mass yields. the valence electrons, 2'/kz=(32m /9)'~ r, =3.27r„
Rather direct evidence of the electronic nature of the and the average length responsible for the main-shell os-
observed supershells is also found from the radius incre- cillations from Eq. (5.4), L =5.42R =5.42r, N'~, one
ment AR =r, AXo between two clusters with neighbor- Ands that the Wigner-Seitz radius r, cancels, and one ob-
ing magic numbers Xo. As can be seen from Figs. 11 and tains immediately the increment of the magic "shell ra-
12, AXO is almost constant within each supershell. dius" Xo
This becomes even more evident if one plots the quantity
s =SX'" =
327 =0 603
versus the number of the magic shell, the "shell in- (5. 11)
dex" i, as done in Fig. 13 below: all points lie on portions 5. 42
of straight lines with a slope of s=hXO =0.61+0.01. This value agrees well with the experimental data. Actu-
This can be easily understood from the results of Balian ally, the result (5. 11) is valid only for an infinitely steep
and Bloch (1972) cited above, which lead to the form spherical potential well; in a realistic potential with a
(5.3) of the level-density oscillations. For a classical orbit diffuse surface, the "corners" of the classical trajectories
(with length L) that is planar, a simple one-dimensional will be rounded off and AXO will be somewhat larger
quantization can be used, than 0.603. This demonstrates that, indeed, the observed
main shells are the result of a quantization of the valence
gp dq =fikL =nh (n ))1) (5. 10) electrons in their mean field. [Finer details of the shell
which, after division by Ak, is equivalent to demanding
oscillations depend also on the quantization of the
motion perpendicular to the classical orbits, which is far
less trivial; see Gutzwiller (1971), Balian and Bloch
CLUSTER SIZE, N
(1972), and Strutinsky et al. (1977).]
100 500 1000 2000
10 I
I
I In Fig. 13, we have made a compilation of the "shell
ALL RUNS radius" No versus shell index i for various experimental
C9
CL
X
25
K
-2- N0
1/3
X ~o
ID O&O
20
8JpO
-1 0
&o (a)
-12
3 4 5 6 7 8 9 10 11 12 13 14 15 15
CUBE ROOT OF SIZE, N
10
o 1&9
N1/3
0
CD ~ Al {Lyon) — 10
CL + Al (LA)
OC
CU -20- I
10
I
15
I
20
X 600K
T=
—3p-
Ll
FICx. 13. "Shell radius" &0 vs "shell index" i for spherical
(b) clusters with "magic numbers" No. IONIC SHELLS corre-
I i I i I i I I i I i I i I i I i I i I i I
gp
&
6 8 'l0 'l2 14 spond to complete ionic icosahedral or cubo-octahedral

configurations. ELECTRONIC SHELLS correspond to filled
major spherical shells of the valence electrons. "?",see text for
FICx. 12. Supershell beats in large sodium clusters: (a) Loga- the interpretation of the Al data. The solid lines are theoretical
rithmic derivative of the experimental mass yield of sodium curves with their slopes s indicated in parentheses. The various
clusters from an adiabatic expansion source, by Pedersen et al. symbols correspond to the experimentally observed most abun-
(1991) {see this reference and the text for details); (b) second dant species: "
(Na "cold, MPI), Martin et al. (1990, 1991b);
differences of total free energy obtained in self-consistent spher- "
O (Na "hoi, MPI), Martin, Bj@rnholrn, et al. , 1991); X (Na,
ical jellium-Kohn-Sham-LDA calculations by Genzken and 4
NBI), Pedersen et al. {1991); (Li, Orsay), Brechignac et al.
Brack (1991). See the text for an explanation of the exponential (1992c); V (Al, Lyon), Lerme et al. (1992); + (Al, LA), Persson
scaling factors. et al. (1991).

and theoretical situations. The solid lines with slope smallest.

s=0. 61 represent both the results of Nishioka et al. As we have seen above, the average slope, s =0. 61, is
(1990), using a Woods-Saxon potential with constant sur- very close to that found in an infinitely steep spherical
face diffuseness, and the self-consistent jellium model re- potential well, Eq. (5. 11); it is slightly larger (although in-
sults of Genzken and Brack (1991) for Na clusters. Ex- side the theoretical and experimental uncertainties) due
actly the same slope is also found for Li clusters (Genzk- to the diffuseness of the realistic potential. We show in
en et al. , 1992), demonstrating its independence of the Fig. 13 the line corresponding to a pure spherical
Wigner-Seitz radius r, according to the above reasoning. harmonic-oscillator potential; its magic numbers vary as
The rupture of the lines around i = 14 —15, corresponding
to N-800 —900, rejects a phase shift of the main-shell Xo(i) = —i(i +1)(i +2) (i = 1, 2, 3, . . . ), (5. 12)
oscillations by 180' when passing from one supershell to
the next, in accordance with Eq. (5.3) above. For Li, this giving a slope s =0. 693. This is clearly outside the exper-
phase change occurs just above i =14 and for Na just imental error bars and also quite different from the self-
below i = 14; this is due to the slightly increased surface consistent jellium model results.
diffuseness for Li clusters compared to that of the Na A completely different slope of s =1.493 was found for
clusters. a series of magic numbers Xo & 1500 observed by Martin
The same phase shifts and slopes, within experimental et al. (1990, 199lb); it can be attributed to the ionic
uncertainties, have been measured by three independent structure, as will be discussed in the following subsection.
groups. In Fig. 13 the experimental shell radii, found by One might also want to compare the spherical "magic"
Martin, Bjdrnholm, et al. (1991) and Pedersen et al. shell-closure numbers Xo found in the Kohn-Sham calcu-
(1991) for Na clusters and by Brechignac et al. (1992c) lations directly with those observed in experiment. The
for Li clusters, are shown by different symbols. (Not all latter are typically given with an uncertainty of
symbols are indicated on those points where they all —1 —2%, depending somewhat on the kind of analysis
coincide. ) They nicely confirm the theoretical predic- done to the mass abundance data. Within these limits,
tions, particularly for the Li clusters. there is a rather good agreement found between experi-
At first glance, it appears that self-consistency is not ment and the jellium model predictions. There are a few
very important for the global effect shown in Fig. 13: the systematic differences, however. One example is that the
slopes s found for Na clusters with the phenomenological jellium model predicts a strong shell closure for % =186,
Woods-Saxon and the self-consistent Kohn-Sham poten- whereas the experiments point towards N = 196 or 198;
tials are identical. [This is not very surprising, since the similarly, N =254 is predicted and %=264 is seen experi-
Woods-Saxon potential of Nishioka et al. (1990) was ex- mentally. These shifts of the shell closures can partially
plicitly fitted to self-consistent Kohn-Sham potentials, be removed by the introduction of minor modifications to
though only for smaller clusters. ] However, the details of the jellium density distribution (see Sec. III.D. 1).
the phase change around i = 14 —15 do depend on the po- In some recent experiments on Al clusters, Lerme
tential and most sensitively on its surface steepness, et al. (1992) found a very regular shell structure in a size
which is not independent of X in the self-consistent region of 600&Z &2700 valence electrons; however, no
Kahn-Sham results (Genzken and Brack, 1991). Indeed, supershell beating could be observed in this region.
in the latter the phase change occurs at lower cluster Moreover, the slope s for these shells is smaller by about
sizes than for the Woods-Saxon potential, namely around a factor of 2 than that found for Na and Li clusters:
%-800, which compares favorably with the experimen- s =0. 318+0.004. Earlier Persson et al. (1991) had ob-
tal results of Martin, Bj6rnholm, et al. (1991) and served similar shells for Al clusters with 400&Z & 1300
Brechignac et al. (1992c). electrons; they fall on a slope s =0. 315+0.006. Both sets
It should be stressed that the counting of main shells, of data are included in the lower part of Fig. 13. If these
i.e., the attribution of the shell index i to the Inajor mini- shells are to be attributed to quantized electronic orbits,
ma of the oscillations, is not quite unique in the region of these cannot be single-turn trajectories but rather ones
interference minima, since the oscillations there are less that make two turns around the center before closing.
regular than in the middle of the supershells and exhibit Lerme et al. (1992) have proposed five-cornered starlike
minor subshells that are not easily distinguished from orbits that would lead to a calculated slope of s=0. 33.
what one should call main shells. (As mentioned above, Indeed, these authors point out that, for a sufficiently
this is due to the contributions of more complicated clas- diffuse surface of the potential, the triangular and
sical trajectories. ) Therefore, in establishing plots of the squared orbits cannot close any more — as was also ob-
type shown in Fig. 13, both from experimental and served by Nishioka et al. (1990). However, the surface
theoretical results, a certain bias cannot be excluded. diffuseness needed to obtain the desired star orbits is
The least one can say is that the experimental results are much larger than that of the fitted Woods-Saxon poten-
compatible with the above interpretation of an interfer- tial for Na clusters. Moreover, the potential of Lerme
ence between triangular and squared orbits with compa- et al. (1992), when used in a fully quantum-mechanical
rable amplitudes, and that a phase shift does occur in the calculation, does not give a shell structure that corre-
region where the amplitude of the shell oscillations is sponds either to the star orbits or to experiment (Genzk-

en et al. , 1992). The jellium model does predict an in- giving the slope s=1. 493. This atomic shell structure
crease in the surface diffuseness with decreasing ~„but has also been observed by the same authors for K and Ca
the effect is much smaller (see also Lang and Kohn, clusters with 1980~ % ~ 8170 and in Mg clusters with
1970). On the other hand, Persson et al. (1991) found a 146 & N & 2870 (Martin et al. , 1991a, 1991b; Martin,
reasonable agreement of their observed magic numbers Naher et al. , 1991).
400 & Z & 1300 with jellium model predictions. The situ- The interpretation is that, at lower temperatures, there
ation is therefore not quite clear. More investigations on might be a transition around N-1500 —1800 from liquid
Al clusters, for which the jellium model is definitely less to crystalline clusters. This is consistent with the results
justified than for the alkalis, will be necessary to under- on "warm" clusters (Martin, Bjdrnholm et al. , 1991),
stand these very interesting results. Very recently, Mar- where the average initial temperature of the cluster beam
tin et al. (1992) have also observed a similar structure in is estimated to be -500 K.
the mass spectrum of cold Al clusters with It constitutes a considerable challenge to verify the
250&N & 10000 and interpreted them in terms of ionic above interpretation of a phase transition by theoretical
shells. We shall discuss this in the next section. calculations. Of course, the ionic shell structures cannot
be described by the jellium model. On the other hand,
3. From electronic to ionic shells the large clusters discussed here are far beyond the reach
of purely microscopic treatments such as quantum-
It is easy to show that magic numbers for h, F in very chemical or molecular-dynamics theories. There is there-
large cold clusters must be determined by crystal struc- fore a definite need to develop simplified models that are
ture rather than electronic shells. When an atom is add- able to describe the interplay between ionic geometry and
ed to a crystalline cluster, it may complete a layer or electronic shell effects in large clusters. Some interesting
start a new layer, and this gives a contribution to h, F in- steps in this direction have been taken recently by Maiti
dependent of the size of the cluster. On the other hand, and Falicov (1991, 1992) using perturbative pseudopoten-
according to Strutinsky theory the Auctuation in the elec- tial calculations. The spherically averaged pseudopoten-
tronic energy contribution obeys the proportionality tial (SAPS) model (see the end of Sec. II.C) might also be
a promising tool for such studies.
56,F-5eI; —eF 5g (5. 13) The regular shell structure in large Al clusters dis-
cussed above in connection with Fig. 13 has recently been
observed by Martin et al. (1992). They interpreted it in
From Eq. (5.2) and the relation go-R it follows that
terms of subshells of close-packed octahedral ionic
this contribution decreases as R ' or X ' and is thus
shapes, correlating the maxima in the mass spectra with
small for very large clusters.
the filling numbers corresponding to the addition of suc-
Indeed, some of the data on magic numbers in very
cessive triangular facets.
large clusters favor an interpretation as crystal faceting
effects. Martin et ai. (1990, 1991b) found an interesting
transition of the shell spacings above %=1500, using B. Semiclassical theory and large-N
photoionization time-of-Aight mass spectroscopy for rela- expansions: links to the macroscopic world
tively cold Na clusters. Up to this size, they observed
magic numbers falling exactly on the lower part of the
line with slope s=0. 61 in Fig. 13, extrapolated up to The present section is devoted to a discussion of
i =17 corresponding to NO=1430+20. (These results are density-variational calculations in the strict sense, i.e.,
indicated by square boxes in the figure, but shown only where the density p(r) of the valence electrons is the
for i ~ 14 in order not to overload the lower part of that direct variational quantity in contrast to the single-
line. ) These magic numbers up to i=13 were later particle wave functions y,. (r) varied in the Kohn-Sham
confirmed in experiments in which the clusters were laser or Hartree-Fock methods. This becomes possible
warmed before ionization (Martin, Bjprnholm, et al. , through the use of explicit semiclassical approximations
1991). However, in the region 1980 & N & 21 300, a total- to the kinetic-energy functional T, [p] in terms of p(r)
ly different spacing between the magic-shell radii was and its gradients, instead of Eq. (3.5), which involves the
found, corresponding to a slope s =1.49, as shown in the y;(r). The attribute "semiclassical" is used to indicate
upper left of Fig. 13. These shells were identified by that expansions in powers of A are involved in deriving
Martin et al. (1990, 1991b) as atomic shells correspond- the explicit functionals used for T, [p(r)]. The two most
ing to icosahedral or cubo-octahedral close-packed ionic common functionals, that of the Thomas-Fermi (TF)
configurations, as they are well known for van der Waals theory and its extensions, are presented in Appendix
clusters. Indeed, both these configurations lead to the A. 2.a.
magic numbers With such an explicit functional T, [p(r)] for the kinet-
ic energy, the variation principle for the total energy can
No(i)= —(10i —15i + lli —3) (i=1,2, 3, . . . ), be applied, according to the Hohenberg-Kohn theorem
(cf. Appendix A. 2.a), by a direct variation of the density
(5. 14) p(r):

714 Matthias EIrack: The physics of simple metal clusters
1989), who demonstrated how the expressions for ioniza-

tion energies and electron affinities obtained by such
methods (cf. Sec. V. B.3 below) could be used to interpo-
where the Lagrange multiplier p fixes the number of elec- late from the bulk all the way to single atoms.
trons and physically is understood as the chemical poten-
tial. This leads to a Euler-Lagrange-type differential
equation for the density which is nonlinear and whose or- 3. Semiclassical density-variational calculations
der depends on the number of gradient terms included.
The main advantage of this method is that one only The solution of the variational equation (5. 15) for the
has to vary one density function p(r) (or, if spin degrees density p(r) can, in principle, be obtained directly in
of freedom play a role, two spin densities) instead of coordinate space. This may, however, be numerically
many single-particle wave functions. This often gives very dificult due to the high degree of nonlinearity, in
more physical insight than microscopic methods, since particular if higher gradient corrections are included in
the observables can be connected directly to the density the energy functional. Reasonable approximations are
p(r) and other local functions and many mechanisms often obtained in a restricted basis of trial density func-
may become more transparent. tions by minimizing the total energy with respect to some
The price one pays is that the functional T, [p] forbids variational parameters.
the inclusion of shell effects. One can thus obtain only In the remainder of this section, we shall briefly men-
average properties (total energy, density and its motion density-variation al calculations of metal cluster
ments, ionization potential, electron affinity, polarizabili- properties using the Thomas-Fermi (TF) functional
ty, plasmon energies, etc. ) but in a parameter-free and TTF[p] and its extensions (TFW, TFWD, ETF, etc. ; see
self-consistent way even for very large systems. Appendix A. 2.a). Some of their results have already been
Due to the missing shell effects, the semiclassical re- quoted in the earlier chapters of this review; some of the
sults cannot usually be directly compared with experi- papers are mentioned merely for historical reasons. Re-
ment. However, it is possible to treat shell effects pertur- sults of large-X expansions and liquid-drop parameters
batively at relatively low cost, using the ideas developed will be discussed in the following two sections. In all
by Strutinsky (1968) in nuclear physics and sketched in these calculations, exchange and correlation energies
Sec. V.A. 1 above. In fact, the semiclassical variational were included in the local-density approximation. Only
results for the average energy E and for the average po- spherical clusters have been treated so far.
tentials, from which the shell-corrections 5E can be ex- The first jellium model calculations for metal clusters
tracted, represent the ideal input into a Strutinsky calcu- altogether, of which we are aware, were done in 1975 by
lation in which the total energy is written in the form of Cini. He used the Thomas-Fermi-Weizsacker kinetic-
Eq. (5. 1). Alternatively, the shell efFects may be added at energy density and a variational space of spherical
the end of a semiclassical density-variational calculation double-exponential trial densities. He discussed ioniza-
simply by solving once the Kohn-Sham equations using tion potentials (IP) and electron affinities (EA) and their
the variational average potential V. With the latter X dependence. Snider and Sorbello (1983a) used a very
method one obtains not only the total energy, but also a similar model, varying the Weizsacker coefficient (dis-
good approximation to the self-consistent Kohn-Sham cussed in Appendix A. 2.a), obtained IPs and EAs, and
orbitals from which other observables can be calculated. found the slope parameter a in the IP to be different
Both methods have proven useful in nuclear physics as from 3/8 (see Sec. V.B.3 below for discussion). They
economical substitutes for fully microscopic Hartree- then applied the same model to the calculation of static
Fock calculations (see, for example, Brack et aI. , 1985). dipole polarizabilities by applying an external electrical
They have not been used in cluster physics so far, but field (Snider and Sorbello, 1983b; Sorbello, 1983) and dis-
might prove useful for systematic calculations of very cussed a dipole force sum rule (see Sec. IV). Snider and
large clusters in which the fully self-consistent micro- Sorbello (1984) extended their earlier model to the spin-
scopic Kohn-Sham method becomes too time consuming. density formalism in order to study odd-even effects in
Finally, the semiclassical density-variational method the IPs of microclusters.
gives access to a self-consistent determination of the Iniguez et al. (1986) did variational Thomas-Fermi-
coeKcients in liquid-drop-type asymptotic expansions of Weizsacker-Dirac (TFWD) calculations both for the jelli-
the average energy and other variables in powers of um model and using pseudopotentials and obtained IPs,
N ' . This provides links between the finite system and EAs, and cohesive energies of small sodium clusters.
properties of the semi-infinite system (i.e. , an infinite Kresin (1988—1991) used the Thomas-Fermi theory for
plane surface), such as the surface energy and the bulk small metal clusters and developed an approximate
work function. ' We refer the reader to Perdew (1988, analytical solution of the Thomas-Fermi equation. He
discussed diamagnetism and later applied his model to
surface plasmons and static dipole polarizabilities of
spherical clusters (cf. Sec. IV).
~7The inclusion of ionic structure eftects can here be rather Brack (1989) used the full extended Thomas-Fermi (4)
crucial; see Sec. V.B.2 below. kinetic-energy functional, Eq. (A21), with three-

parameter variational densities and found a rather accu- trated in Fig. 14. It shows the density profile of Na2654
rate reproduction of the average Kohn-Sham densities obtained in self-consistent jellium model Kohn-Sham cal-
and potentials by Ekardt (1984b), particularly in the sur- culations by Genzken and Brack (1991), overlayed with
face region. He studied dipole polarizabilities and mul- the profile of an infinite plane surface taken from Lang
tipole vibrations via sum rules and the "local RPA" ap- and Kohn (1970). The oscillations near the surface of the
proach discussed in Sec. IV.C.2 and Appendix B.2.c. spherical cluster reproduce rather accurately the Friedel
Spina et al. (1990) showed that these variational densities oscillations of the semi-infinite profile that will be
reproduce very accurately the bulk work functions and
surface energies found by Tarazona and Chacon (1989),
reached asymptotically for N ~~. The oscillations near
the interior of the finite cluster are due to the filled spher-
who solved the full extended Thomas-Fermi (4) Euler ical shells near the Fermi energy. '
equation (see Appendix A. 2.a) in the semi-infinite For a neutral spherical system with X particles and a
geometry. reference radius R =roN', the above technique leads to
Spina and Brack (1990) used the same parametrized the liquid-drop model expansion of the total binding en-
trial densities in a semiclassical jellium model that in- ergy:
cludes schematic ionic structure through spherically
averaged pseudopotentials, with results similar to the mi-
E(N) =a„N+a, N +a, N' +ao+ (5. 16)
croscopic spherically averaged pseudopotential model by Here a, is the volume or bulk energy, which is defined as
Inigues et al. (1989, 1990; see Sec. II.C). the energy per particle of the infinite system with con-
Serra et al. (1989a, 1989b, 1990) and Balbas and Rubio stant density po:
(1990) also studied the multipole response of spherical
clusters in density variational calculations with an ap- a. =eh @lS'o~~po. (5. 17)
proximate TFWD functional (see Sec. IV).
The surface energy a, in Eq. (5. 16) is given by
Engel and Perdew (1991) were the first to solve the full
extended Thomas-Fermi (4) Euler equation for spherical a, =4~roo (5. 18)
clusters directly in r space. They discussed the asymptot-
ic behavior of ionization potentials and electron affinities in terms of the surface tension o. , i.e. , the energy per unit
(see Sec. V.B.3 below). area of an infinite plane surface. Not only a, but also the
curvature energy a„and the higher-order coe%cients in
Eq. (5. 16) can be obtained uniquely in terms of the densi-
2. Liquid-drop model expansion of the energy
ty profile perpendicular to the surface of the semi-infinite
Density-variational calculations give a natural starting 1985). "

system (see Myers and Swiatecki, 1969; Brack et al. ,
point for the self-consistent determination of liquid-drop Although extended Thomas-Fermi variational densi-
parameters by means of a "leptodermous" expansion of ties serve as a natura1 starting point, the leptodermous
the total binding energy of a saturated ferrnion system. expansion is not restricted to semiclassical theory and
This idea lies behind the famous mass formula for nu- can also be applied to microscopic densities obtained
clear binding energies developed by von Weizsacker within the LDA-Kohn-Sham approach. The Friedel os-
(1935) and Bethe (1937); it was successfully further cillations in the semi-infinite density profile (Lang and
developed from the basis of Thomas-Fermi theory by Kohn, 1970, see Fig. 14) do not disturb the principle of
Strutinsky and Tyapin (1964) and by Myers and the leptodermous expansion discussed here. Numerical-
Swiatecki (1969). We shall only sketch here the principal ly, however, they lead to convergence problems in the
ideas and refer the reader to the literature for details. As evaluation of the curvature energy a, (Stocker and Fa-
an application, we discuss the asymptotic behavior of rine, 1985). Their relative contribution to the surface en-
ionization potentials and electron amenities of metal clus- ergy a, of metals is found to be only a few percent (Seidl,
ters in Sec. V.B.3 below. Spina, and Brack, 1991; Engel and Perdew, 1991;
Assume that the density is going to a constant value po Fiolhais and Perdew, 1992).
in the interior of the system and that there exists a well-
defined surface region where p(r) drops from po to zero.
Introduce o. as a measure for the surface thickness and a i80nly oscillations due to shells with low angular momenta l
reference radius R (e.g. , the average location of the sur- can be distinguished near the center. See also Thorpe and
face, measured from the center). When a
tern is "thin-skinned" or "leptodermous. " «R,
the sys-
The basic idea
Thouless (1970), who have discussed both types of density oscil-
lations in the nuclear physics context.
then is to perform a so-called leptodermous expansion in Note that the direct determination of the coefficients a„a„
powers of the small variable x =a/R around the leading etc. by a least-squares fit of Eq. (5. 16) to microscopically calcu-
volume term, which is given in terms of a steplike density lated energies is hampered by shell effects (see, for example,
profile. This expansion is asymptotic in nature and a Utreras-Diaz and Shore, 1989). This may be done with serni-
priori valid only for large enough systems. classically obtained average energies, if sufficiently large parti-
That the electronic densities of large metal clusters cle numbers are used (up to X ~ 10; see Seidl, Spina, and Brack,
fulfill the assumption of leptodermicity very well is illus- 1991, and Fiolhais and Perdew, 1992).

1.2 I I I I I I I I I I I
)
wrong sign for r, 4 a. u. (i.e. , where the simple jellium
model works best). Curvature energies a, have been ex-
1.0 tracted recently from experimental vacancy formation
energies by Perdew et al. (1991) and agree with the
values calculated by Fiolhais and Perdew (1992) within a
0.8
factor of less than two for r, ~ 5 a.u. and three for r, (5
a.u. ; no difference was found here between jellium and
'0.6 stabilized jellium.
———infinite plane surface
04- 3. Asymptotic behavior of ionization potentials

and electron affinities
0.2-
Since much experimental information is available on
I I I I I I I I I I I
ionization potentials IP and electron amenities EA of met-
0 I
0 5 10 15 20 25 30 35 40 45 50 55 60 65 10 al clusters, and their large-X behavior has received con-

r (a.u. ) siderable attention in the literature, we shall discuss here
their asymptotic expressions derived by the techniques
FIG. 14. Asymptotic behavior of electronic density: solid line,
discussed above. They are defined by
electron density of a spherical sodium cluster with %=2654
atoms, from spherical jellium-Kohn-Sham-LDA calculations by IP=E(N, —1) —E(N, O), EA=E(N, O) E(N, +—1)
Genzken and Brack (1991).Dashed line, electron density profile
perpendicular to an infinite plane surface of sodium metal, ob- (5. 19)
tained in jellium-Kohn-Sham-LDA calculations by Lang and
Kohn (1970). The two curves are adjusted so that the jellium in terms of the total energy E(N, q) of a cluster with N
edge is at the same location along the r axis. atoms and q excess electrons. When the above leptoder-
Inous expansion was generalized for a charged system the
following expansion of E(N, q) was derived by Seidl,
For Inetal clusters, the leptodermous expansion of the Meiwes-Broer, and Brack (1991) and Seidl, Spina, and
total energy has been studied by Seidl, Spina, and Brack, Brack (1991) within the spherical jellium model, to the
(1991) with extended Thomas-Fermi (4) results using the leading orders in q «&:
jellium model, and by Fiolhais and Perdew (1992), who
used both semiclassical and Kohn-Sham calculations for E(N, q) = —qb, y'"'+ (qe)
2R
the jellium and the stabilized jellium model (see Sec.
III.D.2). Pseudopotential corrections to the surface ener- +(N+q)eb+a, N i +. (5.20)
gy have also been studied in extended Thomas-Fermi (4)
calculations by Spina et al. (1990). Here Acp'"' is the outer part of the Coulomb barrier of an
The agreement of the calculated liquid-drop model pa- infinite plane metal surface, i.e. , the work required to
rameters with experimental quantities depends on the bring a test charge from the jellium edge to infinity:
quality of the model used. The simple jellium model can-
not yield the correct cohesive energy of the bulk metal;
the volume energy eb here is just the energy per electron =4~e f z[p(z) —poe( —z)]dz . (5.21)
of a structureless infinite gas with the r, value of the bulk
Inetal. Surface energies a, obtained in the jellium model The second term in Eq. (5.20) is just the classical electro-
)
for metals with r, 4 a.u. (i.e. , K, Rb, and Cs) agree
static energy of a surface-charged metal sphere with ra-
dius Rr.
within 10—20% with experimental values, as shown by
With q =+1 one finds from Eq. (5.20) the following
Lang and Kohn (1970) in Kohn-Sham calculations with
semi-infinite geometry. For metals with smaller r, the
asymptotic expressions for IP and EA which are valid for
large iV:
agreement becomes worse; for r, & 2. 3 a. u. (e.g. , for
2
aluminum), one even obtains unphysical negative values
IP(N) = 8'„+a +O(RI ), (5.22)
of a, . This is greatly improved when the ionic structure I
is accounted for. Lang and Kohn (1970, 1971) showed
that a perturbative inclusion of pseudopotentials allows EA(N) = Wb —13 +O(RI ), (5.23)
one to reproduce the experimental surface energies I
within 10—30% for most metals. (See Monnier et al. , where 8'b is the bulk work function given by
1978, for a nonperturbative inclusion of pseudopoten-
tials. ) The stabilized jellium model, which yields the (5.24)
correct cohesive and bulk energies (cf. Sec. III.D.2),
reproduces these results more or less (Fiolhais and Per- The "slope parameters" a and /3 in Eqs. (5.22) and (5.23),
dew, 1992), although the ionic correction to a, has the which dominate the size dependence in large clusters, re-

ceive their leading contribution '

—, from the classical talists par ametrized their data by truncating these
charging energy, i.e., the second term in Eq. (5.20). asymptotic expressions after the terms linear in 3f
Quantum-mechanical corrections due to the diffuseness fitting their results by straight lines in a plot versus 1/RI
of p and to explicit kinetic, exchange, and correlation en- going through the measured bulk work function 8'b at
ergy contributions, which are all contained in the 1/RI =O. This is dangerous, and the resulting slope pa-
higher-order terms (indicated by dots) in the expansion rameters cannot be trusted for the following reasons: (i)
(5.20), lead to deviations of a and p from their common many experimental data for small clusters fall into a re-
classical value —, '. gion where the higher-order terms in 1/EI cannot be
Perdew (1989) and Engel and Perdew (1991) derived neglected, so that the curves are no longer straight lines,
Eqs. (5.22) and (5.23) starting from the variational equa- particularly for EA; (ii) the shell oscillations are rather
tion (5. 15) and expanding the chemical potential p as a strong and make the fits ambiguous; and (iii) there is
function of the cluster radius Rl, leading to the same sometimes a rather large uncertainty in the measured
I/Rz corrections for IP and EA (see also Balba, s and Ru- values of 8'b, which affects the values of the slope param-
bio, 1990). The numerical values obtained for a and P eters. Furthermore, a spillout correction has often been
by Seidl, Meiwes-Broer, and Brack (1991) and Seidl, Spi- included in the definition of the radius by setting
na, and Brack (1991) in variational extended Thomas- RI=r, X'~ +a in the denominators of the 1/RI terms of
Fermi calculations with parametrized trial densities were Eqs. (5.22) and (5.23), using a more or less ad hoc chosen
well confirmed by Engel and Perdew (1991) with fully
variational solutions of the extended Thomas-Fermi (4)
Euler-Lagrange equation.
',
value for a. This effectively includes higher-order terms
in X albeit with a biased coefficient; it may improve
the local fits in a limited size range, but it also affects the
The calculated values of a and p for various simple apparent values of the slope parameters.
metals are close to, but not exactly equal to, —, ' and —,', re- The work of Engel and Perdew (1991) and Seidl, Spina,
spectively; they depend slightly, but systematically, on and Brack (1991) shows that very large particle numbers
the VAgner-Seitz radius r, . In the analysis of experimen- X ~ 10 are needed in order to determine the asymptotic
tal results on IP and EA, similar values have been found, slopes a and P uniquely. This can only be done, of
although their correct values often cannot be extracted course, in semiclassical calculations. This also shows
from small clusters (see Secs. VI and VII of de Heer, that their determination from experimental data is not
1993, and the discussion further below). Their approxi- easy, particularly in view of the shell effects.
mate agreement with the values a= — 'and
, p= —,', obtained Seidl, Meiwes-Broer, and Brack (1991) have shown
from a classical image-charge argument (Smith, 1965; that the variational semiclassical results for IP and EA fit
Wood, 1981), has unfortunately led to a great deal of con- the size dependence of the experimental data of' simple
fusion in the literature (see, for example, Haberland, metal clusters surprisingly well on the average, although
1992). As pointed out by Makov et al. (1988), Perdew the bulk 11m1t 8 b 1s off by some 5 10 %~ which 1s a
(1989), and de Heer and Milani (1990), this argument is well-known defect of the jellium model without pseudo-
not physically justified: the classical image potential can- potential corrections (Lang and Kohn, 1971). It is also
not be applied to a point charge at distances of atomic di- clear from their results that the quadratic and higher-
mensions from a metal surface. Therefore the approxi- order terms in N ' of the asymptotic expansions (5.22)
mate equality of the correct slope parameters with the and (5.23) cannot be neglected when fitting data of clus-
values — ', and —
', , respectively, must be taken to be acciden- (
ters with X 100, in particular for the electron affinities.
tal. As clearly shown within density-functional theory, In view of these results and in order to avoid the ambi-
the correct classical limit for a charge continuously dis- guities outlined above, we strongly advocate the use of
tributed over a sphere leads to their common value —,' the particular forms (5.22) and (5.23) in fitting the X
(Perdew, 1989; Seidl, Meiwes-Broer, and Brack, 1991). dependence of IP and EA data (cf. also the analysis of the
The deviations from this value can be accounted for by experimental data in Secs. VI and VII of de Heer, 1993).
quantum-mechanical corrections, as mentioned above Equation (5.24) for the bulk work function is not very
(see also Makov and Nitzan, 1991). Actually, this should frequently used in the literature; it was derived by Mahan
not be so surprising, since the leading term of IP and EA, and Schaich (1974) employing a theorem by Budd and
namely the bulk work function Wb, is also a purely Vannimenus (1973) (see also Monnier et al. 1978). For
quantum-mechanical entity. There is no reason why the densities p(z) which solve the variational Euler equation
next-order corrections to IP and EA should be explicable for the semi-infinite problem exactly, Wb (5.24) is identi-
by purely classica1 arguments. cal to the more widely used expression introduced by
Another confusion in the literature concerns the appli- Lang and Kohn (1970):
cation of the expansions (5.22) and (5.23) to measured Rb=ky —Pb . (5.25)
values of IP and EA of small clusters. Many experimen-
Here Ay is the full Coulomb barrier of the plane metal
surface
In this derivation, the bulk work function 8'& is given by the
equivalent expression {5.25) discussed below. Ap =y( oo ) —p( —oo ), (5.26)

which can be given in terms of simple integrals over the

density profile p(z), and pb is the bulk chemical potential
defined by
I b= @ipol .
dpp
For approximate density profiles (e.g. , parametrized trial 3—
densities used in restricted variational calculations), it I
turns out that the expression Wb (5.24) is much less sensi-

AIN
tive to numerical errors or approximations than the stan-
dard expression Wb (5.25); this has been investigated in
detail by Monnier et al. (1978) and by Perdew and Sahni
(1979).
In Fig. 15, taken from Spina et al. (1990), we illustrate
the asymptotic behavior of IP and EA. The crosses con- 0. 2
I
0.4
i I
0. 6
i I
0. 8
(
nected by the solid hnes represent the variational semi- -1(3

g
classical results obtained in the extended Thomas-Fermi
(4) approximation for spherical Na clusters with
8&%&125000. The nonlinearity of EA in X ' is
clearly visible. Note also the correct limiting value of the
tron a%nity (A) of Al clusters vs X ';
FIG. 16. Difference between ionization potential (I) and elec-
, experimental results;
solid line, results of variational extended Thomas-Fermi(4) cal-
theoretical bulk work function 8'b, which is diA'erent culations in the spherical jellium model. (From Seidl, Meiwes-
here from 8'b due to the use of a restricted set of trial Broer, and Brack, 1991; see this reference for the experimental
density functions. data. )
A surprising result is the curve labeled 8'* in Fig. 15.
The quantity 8'* is calculated with the same expressions tion of the inverse cluster radius. The dots are experi-
(5.21) and (5.24) which defined W&, but using the finite- mental results and the solid line is the result of a semi-
cluster density profile p(r) of the actual cluster with elec- classical density-variational calculation with the full ex-
tron number N, extrapolated to r = —oo. The curve 8" tended Thomas-Fermi (4) kinetic-energy functional in the
is practically constant with the value O'I, . This means spherical jellium model. Note that in the difference IP-
that we can obtain the correct theoretical bulk work EA, the bulk work function (which is not correctly ren-
function, to within a few percent, from a simple semiclas- dered in the jellium model) cancels. This difference thus
sical variational calculation (which, by the way, can be focuses on the finite-size eAects. The good agreement is
done on a simple personal computer) for a microcluster another example of the fact that the jellium model can
with as few as eight atoms. correctly reproduce average trends of finite-size
In Fig. 16, taken from Seidl, Meiwes-Broer, and Brack eA'ects— in the present case even down to the dimer.
(1991), we show the difFerence between the ionization po-
tential and the electron affinity of Al clusters as a func- VI. SUMMARY AND CONCLUSIONS
In this review article we have given a survey of

theoretical approaches for the description of simple met-
4
(( VI al clusters. We have focused on mean-field theory ap-
propriate to finite fermion systems using the Hartree-
Fock (HF) and the density-functional methods, the latter
mainly in the local-density approximation (LDA). We
have extensively discussed the electric response proper-
ties and their description in the time-dependent LDA
and the random-phase approximation (RPA).
In many respects, the metal clusters appear as droplets
of a quantum Fermi liquid in which the valence electrons
1.0
are the dominant degrees of freedom and the ionic struc-
0 0, 2 0, 3 Ot 0. 5 ture seems to have little inhuence. This is particularly so
Z
for the observed magic numbers in alkali clusters with up
potentials I (Z) and electron aftinities
to %=3000 atoms, in which the picture of valence elec-
FIG. 15. Ionization
3 (Z) trons confined in a smooth —
self-consistent or suitably
of sodium clusters with Z atoms, obtained in semiclassi-
cal ETF(4) density-variational calculations by Spina et aI. parametrized —potential also can account for the super-
(1990), vs Z ' . The quantity 8'*(Z) and the two theoretical shell structure.
values 8'and 8" for the bulk work function are defined and ex- In theoretical investigations of electronic shell struc-
plained in the text. From Spina et al. (1990). ture, the jellium model is playing an important role due

to its computational simplicity, which permits us to con- packing of the atoms can be observed if the clusters are
nect small and very large cluster sizes in a unified picture. sufticiently cold. These ionic structures have, however, a
It is very hard, if not impossible, to assess its limitations different symmetry from that of the ionic lattice in the
in a quantitative manner from within the model itself. bulk metal. Therefore the old question: " How many
For microclusters with up to X-20 atoms, where ab ini- atoms are needed to make a piece of bulk material?" is
tio quantum-chemistry or molecular-dynamics calcula- still not answered. Apparently 20000 alkali atoms are
tions are technically feasible, it has become evident that not enough.
details of the experimental ionization potentials, electron The relative importance of electronic and ionic shell
affinities, polarizabilities, and fine structure of the photo- effects in these large metal clusters depends crucially on
absorption cross section do depend on the ionic geometry their temperature. This subject deserves more experi-
and that such calculations give a better quantitative mental and theoretical attention in future research. A
description than the jellium model. But still, the average description by first-principles ab initio methods is impos-
trends of these observables can often be described sible for such sizes. Approximate models, such as the
surprisingly well by the jellium model even for small clus- tight-binding or the Hueckel model, perturbative or
ters. If deformations of the positive charge distribution spherically averaged pseudopotential calculations, or the
are included, the jellium model can also account for the effective-medium theory, must be applied and further
averaged ionic geometry, leading, for example, to a shape developed.
isomerism similar to that found in the structural models. We have given some emphasis to semiclassical varia-
For the static electric dipole polarizabilities and the tional methods and a local-current approximation to the
positions of the collective electronic dipole resonances, RPA built on sum-rule relations, which allow one to
the jellium model with LDA misses some 10—20% of the evaluate average static and dynamic response properties
average experimental results. Two competing explana- for very large systems in which fully microscopic calcula-
tions for this failure have been given. One of them in- tions are no longer possible. We think that these
vokes the missing ionic contributions, the other points to methods might be helpful in future investigations, partic-
the failure of the LDA in yielding the correct 1/r falloff ularly in the mesoscopic domain. The large-X expansion
of the total potential, which can be partially overcome by of the semiclassical results also yields direct contact to
self-interaction corrections or extensions of the LDA (the volume and surface properties of the bulk metal. In this
so-called weighted density approximation, WDA). The connection, we have found that the jellium model is a
LDA, which has been used with considerable success in useful mediator between the microcosm and the macro-
many branches of physics, faces a rather crucial test in cosm.
calculations for metal clusters —
as well as for plane met-
al surfaces — due to the steep surface of the electronic
density. We have discussed some recent calculations, ACKNOWLEDGMENTS
performed within the jellium model, using a HF basis
plus perturbation expansion and explicit evaluations of This review would not have been initiated without the
RPA correlations. The results for ionization potentials stimulation of Walt de Heer, and it would never have
and polarizabilities are very close to Kohn-Sham-LDA been completed without the continuous support and en-
results and thus seem to confirm the validity of the local- couragement of Sven Bjprnholm. I am deeply indebted
density approximation even in small alkali clusters. We to them, as well, for their careful reading of the
therefore tend to believe that the missing electronic manuscript and many valuable suggestions for improve-
response is due to the missing ionic structure. Indeed, ments. I am grateful to P. Ballone, G. F. Bertsch, O.
explicit structural pseudopotential calculations for clus- Genzken, C. Guet, H. Nishioka, and P.-G. Reinhard for
ters up to X-10 tend to give significantly improved re- clarifying discussions and for their reading parts of the
sults, even within the LDA. When the ionic structure manuscript, and to V. Bonacic-Koutecky, J. Borggreen,
effects are partially simulated in the jellium model by in- W. Ekardt, K. Hansen, M. S. Hansen, B. Mottelson, J.
troducing a diffuse surface of the positive charge distribu- Pacheco, and J. Pedersen for further stimulating discus-
tion, the discrepancy is also removed. In any case fur- sions. Thanks are due to O. Genzken, M. S. Hansen, G.
ther theoretical investigations, both extending the LDA Lauritsch, H. Nishioka, and T. Hirschmann for the com-
and including ionic structure, will be necessary to settle munication of unpublished results, which have been used
this question. in some figures. Last but not least, I want to express my
The interplay of electronic and ionic binding effects gratitude to my family for their patience and their indul-
contributes in an essential way to the richness of the gence.
structural forms of matter, in both the inorganic and the This work was partially supported by the Danish Nat-
organic worlds. It dominates the smallest micro- ural Science Foundation, the Deutsche Forschungs-
molecules, where a distinction between metals and non- gemeinschaft, and the Commission of the European
metals is hardly possible. But it is also important in very Communities. The hospitality extended to me during
large metal clusters with up to %=20000 and more several visits to the Niels Bohr Institute, Copenhagen, is
atoms, in which ionic shells corresponding to a dense gratefully acknowledged.

APPENDIX A: MEAN-FIELD THEORIES Pi

2 N e2
+ V,„,(r;)+—
1
(A 1)
We start from the Hamiltonian of a system of X elec- The exact wave function %(r&, r2, . . . , r&) belonging to
trons moving in an external potential V, „,(r) and in- this Hamiltonian generally cannot be calculated. From it
teracting through the Coulomb two-body potential: we define the one-body density matrix ' p, (r', r):
p, (r', r)= fd fd r2 r3 .
f d'r„q*(r', r„.. . , r~)e(r, rz, . . . , rz) . (A2)
Its diagonal part is the density p(r) which will be normal- T+ V,„,(r)+ VH(r) Ip;(r)+ V~p, (r) =c,; y, (r), . (AS)
I
ized to the number N of electrons:
f,
the so-called HF equations. Here VH(r) is the Hartree
p(r)=p, (r, r), f p(r)d r=X . (A3) (or direct, or classical) Coulomb potential
In both Hartree-Fock (HF) theory and density- (r')
VH(r)=e d r' (A9)
functional theory, to be sketched in the next two subsec- r —r'l
tions, the density io( r ) is written in terms of single- and VF is the nonlocal Pock (or exchange) potential,
particle wave functions ip, (r ): which is an integral operator and originates from the an-
N tisymmetrization of the wave function It is defined 4.
p(r)=g ly;(r)l'. (A4) (apart from spin complications) by
HF(ri ' r)
VFy, (r) = ——e' f y, (r')d'p' . (A 10)
1. Hartree-Fock theory
Since both VII and VI; depend on the wave functions, the
In HF theory the ground-state wave function of an X- HF equations (A8) are nonlinear and must be solved self-
body system is approximated by a Slater determinant @ consistently; this is usually done iteratively. The biggest
built from a complete orthogonal set Iy (r)I of single- complication in this procedure is the integral operator
particle wave functions: VF for the exchange.
The lowest energy obtained after convergence is usual-
@(rl r2 ~ riv)=detlv'(rJ)l, J=1,2, . . . , N . ly called the HF energy EHF, and the corresponding
The density matrix (A2) then takes the form Sinter determinant is denoted by HF ): l
N (Al 1)
p, "(r', r) = g y,'(r')y, (r),
The sum of Hartree and Fock potentials in Eq. (A8) is
from which Eq. (A4) follows. The choice of the single- usually referred to as the "HF potential, "
particle wave functions y; is made by a variational prin- V»= VH+ V„. Naturally, the HF energy may be bro-
ciple: One makes the expectation value of the total Ham- ken up into its difFerent contributions by writing
iltonian (Al) between the Slater determinants (A5) sta-
tionary with respect to the wave functions y, , subject to
the condition of their orthogonalization
E„F— ~r + V. . r p. r + ' V„r p
—, r +E. ,
by means of
(A12)
Lagrange multipliers c;:
where the kinetic-energy density r(r) is given by
(elHl@& —E; f q;(r)l'd'r =0.
&q,*(r ) g2 N
r(r)= g l&q;(r)l' (A13)
2 fPl;
The variation (A7) leads to a set of coupled integro-
difFerential equations of Schrodinger form: and E is the exchange Coulomb energy corresponding
to Eq. (A10):
p, "(r', r)p& "(r, r'}

For the sake of simplicity, and since they will not really be E = ——e d r'd r. (A14)
4
needed here, we do not exhibit the spin degrees of freedom.
They would, in fact, render the expressions for the exchange It is a well-known feature of the self-consistent mean-
(Fock) terms given in Appendix A. 1 below somewhat more field theory that the total energy is not equal to the sum
complicated; for that we refer the reader to any standard text- of occupied single-particle energies c;. Indeed, from Eqs.
book on many-body theory. (A8) and (A12) one easily verifies that

independent of ground-state degeneracy and of the so-

E„„=g , ——
2
(HFI VH+ VFIHF)
E, . (A15) called V representability assumed by Hohenberg and
Kohn, was given by Levy (1979). The Hohenberg-Kohn
Strictly speaking, the above expressions for VH, VF, theorem states that the exact ground-state energy of a
and E, contain unphysical contributions due to the in- correlated electron system is a functional of the density
teraction of the electron in the ith state with itself, which p(r) and that this functional has its variational minimum
should have been omitted [see the condition i%j in Eq. when evaluated for the exact ground-state density. This
(Al) above]. However, when one takes the sum of direct means that, ideally, the variational equation
and exchange terms in EH„, these contributions cancel 5
exactly. Leaving them out of both potentials would E[p(r)] —A, f p(r)d r =0, (A17)
5p(r)
make the latter state dependent —
as in simple Hartree
theory — and render the HF equations still more compli- using the Lagrange multiplier k to fix the number of par-
ticles according to (A3), would lead to a knowledge of the
cated to solve. It is therefore standard praxis to keep
them in both potentials. [As is known from classical exact ground-state energy and density —
if the exact func-
physics, the inclusion of the self-interaction in the Har- tional E[p] were known (which, alas, it is not).
%'e do not need to go into further details about this
tree potential (or the corresponding classical Coulomb
energy) does not cause any harm for a continuous density basic theorem and the general formalism of density-
distribution p(r). ] This point, however, becomes of cru- functional theory, since this is the subject of many excel-
cial importance as soon as different approximations are lent reviews. For further reference, let us just sketch the
made for the direct and the exchange terms of the Inain steps and give the most important formulae needed
Coulomb energy, as is the case in most applications of in the main text. The usual way to break up the energy
the density-functional theory. functional (A16) for the Hamiltonian given by Eq. (Al) is
E[p] = T, [p]+ f I V, „,(r)p(r)+

'
—, VH [p(r)]p(r)]d'r
2. Density-functional theory
+E„,[p] . (A18)
Density-functional theory goes beyond the HF ap-
proach in that correlations are taken into account which Here T, [p] contains that part of the kinetic energy that
are not contained in the HF energy (A12). In principle, corresponds to a system of independent particles with
this theory maps the full many-body problem for the density p, the external potential energy and the Hartree-
ground state of a correlated fermion system onto simple Coulomb energy are clear from the above. The last term
mean-field equations. Practically, however, the exchange in Eq. (A18) is the so-called exchange-correlation energy;
and correlation contributions can only be evaluated ap- it contains the exchange part of the Coulomb energy, i.e.,
proximately. Still, density-functional theory has had E in Eq. (A14) above, plus all the contributions due to
considerable success in many branches of physics. For other correlations related to the fact that the exact wave
recent reviews on density-functional theory and its appli- function is not a Slater determinant, including the corre-
cations in atomic, molecular, and solid-state physics, we lation part of the kinetic energy.
refer the reader to Jones and Gunnarsson (1989) and to E„,[p] is not known exactly for any finite interacting
Dreizler and Gross (1990). fermion system, and it is a matter of state-of-the-art
density-functional theory to use more or less fancy ap-
proximations to it. The simplest but very successful ap-
proximation is the local-density approximation (LDA) to
a. Hohenberg-Kohn theorem be discussed in the Sec. A. 2.c below. The same holds for
and density-variational equations the kinetic-energy functional T, [p], which is not known
explicitly for many-fermion systems.
The basic idea of density-functional theory is almost as The famous Thomas-Fermi (TF) model of the atom
old as quantum mechanics and was used by Thomas (Thomas, 1927; Fermi, 1928) represents a textbook exam-
(1927) and Fermi (1928) in their famous work: to calcu- ple of density-functional theory, in which the density is
late the total energy of a system by an integral over an varied directly according to Eq. (A17). Here one exploits
expression depending only on the local ground-state den- the fact that for a Fermi gas with constant density p, the
sity p(r): kinetic-energy density is proportional to
local-density approximation one therefore
' in the
has the
p,
E„„=f A'[p(r) ]d r =E [p] . (A16) kinetic-energy functional
Mathematically speaking, the energy is assumed to be a

functional of p(r), denoted by E[p]. The formal basis of TTF[p]= s fp ~
(r)d r, a= —(3' ) (A19)
the ensuing theory was laid by Hohenberg and Kohn
(1964) in their famous theorem, which they proved for a Using this functional for T, [p] in Eq. (A18), omitting the
nondegenerate electronic system. A more general proof, exchange-correlation energy, and performing the varia-

Matthias Brack: The phYsics of simple metal clusters
tion (A17), one arrives at the following equation for the derived an inhomogeneity correction to the kinetic-
density: energy functional leading to an additional term
(V'p) /(4p) in the integrand of Eq. (A19). The corre-
5 sponding approaches are usually denoted by the letters
i~p
/
(r)+ V,„,(r)+ Vtt(r)=A, ,
3 2m TFD, TFW, and TFWD, depending on the number of
terms included. In the so-called extended Thomas-Fermi
which is equivalent to the well-known Thomas-Fermi model (see, for example, Kirzhnits, 1957), a systematic
equation. [The latter is usually derived for the total po- expansion of the kinetic-energy functional in terms of
tential with V, „,(r) = —Xe /r, after eliminating the den- gradients and higher derivatives of the density is derived
sity p(r) with the help of the Poisson equation. ] with semiclassical methods. Either by an expansion of
Many improvements to the Thomas-Fermi theory have the density matrix in powers of A (Wigner, 1932; Kirk-
been proposed over the time. Dirac (1930) introduced wood, 1933), or by a commutator algebra (Kirzhnits,
the exchange-energy correction (A31) in the local-density 1957) which is fully equivalent, one arrives at the follow-
approximation discussed below. Von Weizsacker (1935) ing functional (Hodges, 1973):
4 2
$2
[p ]
2m I imp
5/ 3 1
+ 36
(/'
P
p
+ 6480 ( 3 2
)
—2 / 3 i/3
8
p
P Vp
P
~P +24
P P
+ + ~ +
~d p
(A21)
The coefficient of the second term is nine times smaller densities (see Murphy and Wang, 1980, and references
than that of the original Weizsacker term. Equation quoted therein). The finite-temperature extension of the
(A21) has the correct coefficient in the limit of slowly functional TETF [p] up to fourth order has been derived
varying densities, whereas the Weizsacker coefficient ( —, by Bartel et al. (1985); from its T~O limit one obtains a
instead of — is correct in the limit of rapid density oscil-
„) rigorous proof (see also Brack, 1984) of the correctness of
lations with small amplitude (see Jones and Gunnarson, the functional (A21) in the classically forbidden region,
for which the above-mentioned A expansions are
1989, for a detailed discussion). The terms in square
brackets in Eq. (A21) come from the A' terms of the semi- mathematically not well founded at T=0.
classical expansion, and the dots stand for contributions Using the extended Thomas-Fermi kinetic-energy
from the higher orders (6,8, . . . ) in fi. (Note that up to functional (A21), one can still perform the variation
nth derivatives of the density appear originally under the (A17) directly. This leads to a nonlinear fourth-order
integral when expanding to order A"; the two highest diA'erential equation for the density. In the main text, we
ones can, however, be removed by partial integration if refer to it as the extended Thomas-Fermi (4) equation, if
the density is assumed to be analytical and to vanish at all explicitly shown gradient terms are included. The
infinity. ) Some of the gradient terms in Eq. (A21) have asymptotic decrease of the solution for p(r) at large dis-
also been derived in linear-response theory (Kohn and tances depends solely on the highest derivative term in-
Sham, 1965). Similar gradient corrections leading cluded in TET„[p]. In Thomas-Fermi theory, the density
beyond the local-density approximation have also been of an atom is well known to fall off as r . Including the
derived for the exchange-correlation energy functional second-order Weizsacker term (i.e., in the TFW or
E„,[p] (see Sec. III.C. 5 for a brief discussion). TFWD approximation), one finds an exponential de-
The series (A21) represents an asymptotic expansion of crease, which, however, is too fast if the coefficient —is
used. Therefore in many Thomas-Fermi-Weizsacker (or
„
the noninteracting kinetic-energy functional T, [p]. It is
semiclassical in the sense that it does not correctly repro- TFWD) calculations the coefficient of the Weizsacker
duce sheH eAects but converges towards an average part term has been treated as a fIt parameter. Going up to or-
of the kinetic energy, which varies smoothly with the der 2m with m ~2 in the expansion (A21), one finds
number of particles and with the deformation of the sys- asymptotically [Guet and Brack, 1980 (note added in
tem, if a correspondingly averaged density is used. Guet proof)]
and Brack (1980) analyzed the convergence of Eq. (A21) —3m /(m —i)
using smoothed densities obtained by the Strutinsky (A22)
averaging method (Strutinsky, 1968; Brack et al. , 1972}
and found that including terms up to order A' [i.e. , the for a spherical system. The highest derivative terms in
terms shown in (A21)] it reproduces very accurately the the integrand of Eq. (A21), which have the slowest falloff,
average kinetic energy of a system of fermions in vary as the density (A22) and therefore lead to finite con-
harmonic-oscillator or %'oods-Saxon potentials, indepen- tributions to the kinetic energy at all orders 2m, contrary
dently of particle number and deformation. The same to a rather widespread belief. Even though the asymptot-
functional has also been tested for atoms in terms of HF ic decrease (A22} of p(r) is not realistic, the extended

Thomas-Fermi density-variational method has been quite Sham level (EHo) and the lowest unoccupied Kohn-Sham
successful at obtaining average energies, densities, and level (e„U) on either side of the Fermi energy. They can
other properties of finite fermion systems. [Obviously, be used to estimate ionization potentials and electron
the asymptotic behavior (A22) of p(r) is only reached far affinities, respectively (see, for example, Levy and Per-
outside the physically important surface region, so that, dew, 1985, and references quoted therein). The physics
in practice, it affects the interesting observables very little of this is very similar to that of the so-called Koopmans
if at all (see also Engel and Perdew, 1991).] We refer the theorem, which is usually derived within HF theory but
reader to Jones and Gunnarsson (1989) and to Dreizler applies also in density-functional theory. Apart from
and Gross (1990) for applications to electronic systems, some small rearrangement corrections due to the self-
and to Brack et al. (1985) and Treiner and Krivine (1986) consistent change of the mean field upon taking out the
for applications to nuclei. last electron of an atom, the ionization potential is given
by
b. Kohn-Sham equations
IP=E(N —1) —E(X) = —sHo= —E~ . (A26)
In order to avoid the difficulty of finding an explicit
Similarly, the electron affinity is approximately found as
density functional for the kinetic energy, Kohn and Sham
(1965) proposed to write the density p(r) in the form of EA=E(X) E(X + 1 ) — FLU= (A27)
Eq. (A4) in terms of some trial single-particle wave func-
tions y;(r). This is, in fact, possible for any non-negative [In exact density-functional theory, — EHo can be shown
normalizable density (Gilbert, 1975). The noninteracting to be identical to the ionization potential for atoms, or
part of the kinetic-energy density can then be given in the the work function for bulk metal (see, for example, Alm-
form ~(r) (A13) in terms of the same y;(r). The variation bladh and von Barth, 1985). These "ideal" statements
(A17) of the energy functional can now be done through are, however, violated in practice by the use of approxi-
a variation of the trial functions y;(r) with a constraint mate energy functionals using, e.g. , the LDA (see the
on their norms, as in the HF variation (A7), except that next subsection) or the generalized gradient approxima-
(NIHIL&) here is replaced by E[p] (A16). This leads to tion (see Sec. III.C. 5). If these approaches are combined
the widely used Kohn-Sham equations, with the jellium model, a correction e /2RI must be add-
ed to — cHo in order to obtain the IP of metal clusters
[V'+ V~s(r) I q, (r) =E, q, (r), (A23)
(Perdew, 1989). We refer the interested reader to an ex-
in which the local potential V&s(r) is a sum of three tensive discussion of the Koopmans theorem for solids
terms: and atoms by Perdew (1985).]
The density-functional theory can easily be extended to
VKs(r) = Vxs[p(r)] = V,„,(r)+ VH[p(r)]+ V„,[p(r)] . take the electron spin explicitly into account by introduc-
ing a spin-up density and a spin-down density. This
(A24)
leads, instead of Eq. (A23), to two coupled equations for
The first two terms are the same as above, and the third the two spin densities. In metal clusters, there is so far
term is just the variational derivative of the exchange- no evidence for any spin-orbit splitting effects. Therefore
correlation energy: the only place where the spin densities are needed here is
the case of an odd number of valence electrons, in which
.
V. [p(r)]=
5
$5p r
E..[p] . (A25) one orbit is only occupied by a single electron. Since we
shall only discuss clusters with even N in this article, we
Like the HF equations, the Kohn-Sham equations (A23) do not go into the details of the spin-dependent density-
are nonlinear due to the density dependence of Vzs functional theory and instead refer the reader to the
(A24). The important difference, however, is that the po- literature (Jones and Gunnarsson, 1989; Dreizler and
tential VKs(r) is local and the Kohn-Sham equations Gross, 1990).
therefore are much easier to solve. Another extension of the density-functional theory
A remark is necessary concerning the interpretation of concerns the inclusion of a finite temperature T & 0 of the
the wave functions y,. (r) and the energies E; obtained electrons. Mermin (1965) derived the Hohenberg-Kohn
from the Kohn-Sham equations: they do not have the theorem and the Kohn-Sham formalism at T & 0 for a
same physical meaning as in HF theory. The ansatz (A4) grand canonical system of electrons. Later Evans (1979)
for the density does not imply that the total wave func- showed that the density-functional theory also applies to
tion of the system here is taken to be a Slater deter- canonical systems. In essence, one goes over from the
minant. In fact, one does not know the total wave func- (internal) energy E [p] (A18) of the system to the free en-
tion in density-functional theory; the functions y;(r) are ergy E[p],
[pl —7'~, [p l,
just a variational tool to obtain the approximate ground-
+I p]=E (A28)
state density. Likewise, the c; do not, in general, have
the meaning of single-particle energies. An exception is where S, is the noninteracting part of the entropy. The
made for the energies of the highest occupied Kohn- exchange-correlation energy E„[p] will, in general, de-

pend on T explicitly (i.e. , not only through the density). imate functionals for the exchange-correlation part of the
The Kohn-Sham formalism then is obtained by including energy are at hand. The simplest and most frequently ap-
in the definition of the densities (A4) and (A13) the plied functionals for E„,[p] make use of the local den-sity
finite-temperature occupation numbers n;, approximation (LDA). One performs more or less sophis-
ticated many-body calculations for a hypothetical infinite
p(r)=& ly, (r)l n, , r(r)= & l&g;(r)l n;, system of electrons with constant density p, whereby the
diverging Hartree energy is canceled by embedding the
(A29)
n;=N, electrons in a jelliumlike background of opposite charge
density. The resulting energy per electron is used to ex-
tract the corresponding exchange-correlation part e„,(p),
and by minimizing F[p] with respect to both the y, and
which is a function of the variable p. The LDA for a
the n;. Since S, does not depend explicitly on the wave finite system with variable density p(r) then consists in
functions cp;, the variation of the latter gives exactly the
assuming the local exchange-correlation energy density
same form (A23) of the Kohn-Sham equations, the only to be that of the corresponding system with density
difference being that the potential VKs becomes ternpera- p=p(r):
ture dependent. Variation of the n, gives their explicit
form in terms of the c;; the result depends on whether
one treats the system as a canonical or a grand canonical
I
[p]= p(r)e„, (p(r))d'r . (A30)
ensemble. (In the latter case, in which the chemical po- The extension to the spin-density formalism is straight-
tential p is used to constrain the average particle number forward; it is usually termed "local-spin-density" (LSD
N, one obtains the familiar Fermi occupation numbers. ) or LSDA) formalism.
For an extensive discussion of the finite-temperature The exchange energy part of the local-density approxi-
density-functional theory and calculations for T &0, see mation was derived by Dirac (1930),
the review article by Gupta and Rajagopal (1982). Its ap- 1/3
plication to metallic clusters is discussed in Secs. III.B.3
and V.A. 2.
LDA[p]ez I [p(r)]4/3d3r (A31)
and is also often referred to as the Slater approximation.

c. Local-density approximation The most commonly used correlation energy function-
als in cluster physics are those of Wigner (1934), with
The Kohn-Sham approach is very appealing since, 0. 88
ideally, it allows one to reduce the correlated many-body (A32)
r, (p)+7. 8
problem to the solution of a self-consistent one-body
problem of Hartree type. The reality is that only approx- and of Gunnarsson and Lundqvist (1976), with
CxL
(p)= —0. 0666 (1+x 3 )log 1+ — + —x —x2 ——'
r, (p)
(A33)
x 2 3 114
both are in atomic energy units (Ry) and are written in ing with increasing number of atoms. The variation of
terms of the electronic %'igner-Seitz radius the positions of all atoms and a simultaneous, fully self-
r, (p) =(3/4vrp)'/ . consistent treatment of all electrons in systems with more
A lot of research has been done in going beyond the than 10—20 atoms exceeds the capacities even of modern
LDA and LSDA schemes. Both density gradient expan- computers. To restrict the number of degrees of free-
sions and explicitly nonlocal forms of E„,[p] have been dom, one often exploits the approximate separability of
developed and extensively studied (see, for example, an atom into one or a few valence electrons and an ionic
Dreizler and Gross, 1990, Chap. 7). Some of them are core. The idea is to treat only the valence electrons ex-
briefly reviewed in Sec. III.C, although not Inuch work plicitly by density-functional theory as interacting parti-
has been done with them for metal clusters so far. Bal- cles in the field created by the ions. The effects of the
lone et al. (1992) have performed variational quantum core electrons (screening and the Pauli exclusion princi-
Monte Carlo calculations for energies, densities, and ple) are taken into account for each atom by introducing
pair-correlation functions of electrons confined by a a so-called effective core potential or pseudopotential
spherical jellium potential, in order to test the LDA. seen by the valence electron(s). Pseudopotential theory
thus makes the assumption "atom =ion+ valence
"
electron(s), which generally works very well; it has been
d. Pseudopotentials
successfully used in atomic and molecular physics (see,
The application of the Kohn-Sham method to complex for example, Szasz, 1985, for an extensive review). For
atomic molecules or clusters becomes very time consum- metaIs like K and Cs, the assumption of a single valence

electron and a structureless ion is less justified, since the positions R of the ions (the latter entering the Kohn-
ionic cores of these atoms are highly polarizable. Sham equations through the pseudopotentials). Treating
There exist various pseudopotentials of different de- both y; and R as independent degrees of freedom, one
grees of phenomenology in the literature —
the most so- finds that the variational principle applied to the total
phisticated ones, built on ab initio quantum-chemical cal- Lagrangian of the system leads to the following coupled
culations, being nonlocal or semirelativistic and free of equations of motion (Car and Parrinello, 1985):
adjustable parameters but also difficult to use in complex
pij;(r, t)= —
5
molecules or clusters. At short distances from the atom-
5@,*(r, t)
E[p, R ]++A;krak(r, t),
ic nucleus, they are repulsive due to the Pauli principle,
(A35)
which excludes the valence electrons froID the states oc- MR= —VR E[p, R],
cupied by the core electrons. At large distances, they fall
off like the spherical Coulomb potential of the un- where M are the masses of the ions, p is a fictitious
screened effective charge of the ion. A peculiarity of the "mass" of the electronic degrees of freedom (but not the
pseudopotentials for simple metals like the alkalis is that electron mass!), and the Lagrange multipliers A;k ensures
they are rather weak; this gives a qualitative understand- the orthonormalization of the Kohn-Sham orbits. In the
ing of the relative success of the jellium model for these limit p «M no energy will be transferred from the elec-
metals. tronic to the ionic degrees of freedom, in consistency
An extremely simple but effective pseudopotential has with the Born-Oppenheimer approximation. Solving
been introduced by Ashcroft (1966). It simulates the Eqs. (A35) allows one to follow the time evolution of the
main requirements by a simple Coulomb potential that is ionic coordinates and thus to describe their dynamics (or
cut off at the so-called empty-core radius r, and is set thermodynamics); the valence electrons hereby follow
equal to zero inside: self-consistently and adiabatically the time-dependent
mean field. For p =0, the upper equation in (A35)
Vps (r)=- we for r &r, reduces with A;k=5;kE; to the stationary Kohn-Sham
equation (A23).
=0 for r&r, . (A34)
APPENDIX B: LINEAR-RESPONSE THEORY
Here w is the number of valence electrons of the atom.
With an empirical parameter r, for each atom, the Ash- 1. RPA and TDLDA
croft pseudopotentia1 has successfu11y been applied to
bulk and surface properties of many solids (see, for exam- The random-phase approximation (RPA) formulated
ple, Ashcroft and Mermin, 1976) and it is therefore very by Bohm and Pines (1953) can be derived as the small-
popular in cluster calculations. amplitude limit of the time-dependent Hartree-Fock
Another simple local pseudopotential has been pro- (HF) theory by linearizing the quantal equations of
posed and used for metal clusters by Manninen (1986b). motion. It is most successfully used to describe collective
However, it should be mentioned that the pseudopoten- small-amplitude excitations in many-body systems. The
tials used in quantum chemistry and in condensed-matter essence of the RPA is to construct excited states as su-
physics are considerably more sophisticated. When in- perpositions of particle-hole excitations. We shall first
tended for self-consistent mean-field calculations, an im- present it in the framework of HF theory (Appendix A. l)
portant criterion for a pseudopotential is that of and then discuss its application to density-functional
"transferability, "
i.e., the requirement that the valence theory (Appendix A. 2). We give only a few basic formu-
electron charge in the vicinity of the ionic core agree lae here; for a detailed presentation of the RPA formal-
with what would be calculated in an ab initio approach ism, see, for example, Thouless (1961) or Rowe (1968),
(see, for example, Hamann, Schliiter, and Chiang, 1979). Let HF ) be a Slater determinant that describes the
~
ground state (A5) of the system in the HF approxima-

e. Car-ParrinelIo equations tion, where all "hole" states below the Fermi energy
(E„(EF) are filled and the "particle" states above the
We shall finally sketch the equations derived by Car Fermi energy (E )
Ez) are empty. To define a correlated
and Parrinello (1985) for the so-called molecular- RPA ground state ~RPA), one adds to HF) a superpo-
dynamics (MD) method. One starts from the fact that, in sition of 2p-2h excitations:
density-functional theory, the total energy of a cluster is = 1+
a functional E[p, R ] of both the electronic density p,
~RPA) g
pp'hh '
yi'i'" a a~ahah, ~HF) .
and thus the Kohn-Sham orbital functions y, , and the
Excited RPA states ~n ) are defined as linear combina-
tions of lp-1h excitations from the ground state ~RPA):
In some applications, the pseudopotential is put to a nonzero ~n ) =g (x~"atai, y~"ahta )~RPA) . — (B2)
ph
constant value for r & r, (see, for example, Maiti and Falicov,
1991). In the above definitions, a and a are creation and annihi-

lation operators, respectively, for particle and hole states. slow collective motion (Kohl and Dreizler, 1986). We
In principle, the sum in Eq. (82) runs over all possible refer the reader to Gross and Kohn (1990) for a recent re-
1p-1h excitations; practically, one limits oneself to a view on time-dependent density-functional theory, in
configuration space that must be large enough to give which an explicitly frequency-dependent exchange-
convergence of the final results. The RPA amplitudes correlation energy functional is also discussed.
x~" and y~", from which the coefficients y~~ in (Bl) can Second, there is a danger of double-counting correla-
be computed, are found from diagonalization of the total tions when using the standard exchange-correlation LDA
Hamiltonian (including the two-body interaction V) in functionals in an RPA or TDLDA calculation, since
the restricted space of 1p-1h states. Retaining only the RPA correlations are usually already built into the
terms of first order in the x„" and y„, which is tan- ground-state energy E[p] via these functionals. Strictly,
tamount to linearizing the equations of motion, one ar- one should take these contributions out of the functional
rives at the RPA equation that determines the excitation E„,[p] before using it via Eqs. (83) —(85) to obtain the
energy spectrum Ac@„: RPA excitation energies fico„. This, however, is not easy,
since they are usually lumped together with other corre-
X„
—AG)~ (83) lations in a parametrized way. Numerically, their contri-
Y„ Yn bution to the calculated RPA excitation energies Ace„ is
not very large, so that this problem is not a very serious
Here X„=x„and P„=y„are the RPA amplitudes ap-
one. But it would be definitely wrong to use the correlat-
pearing in Eq. (82), and A and B are the following com- ed RPA ground state (Bl) to evaluate the ground-state
binations of antisymmetrized two-body matrix elements
energy. A partial remedy to this problem consists in
of the interaction V: making use only of certain moments of the RPA strength
A = A i'" ~ " =5 5hh (.E eh ) + ( h—
'p l
I lp'h ), (84) function whose values do not depend on the inclusion of
the RPA correlations in the ground state (see the follow-
B =B&"'&" = (pp'l I lhh') . ing subsection).
(If one starts from a fully self-consistent HF basis for the
p and hstates, the mean-field part of the interaction V
does not contribute, and only the residual interaction is 2. Sum rules and relations to classical hydrodynamics
needed in the above matrix elements. )
Once one has solved the above RPA equations, the a. Sum-rule expressions
spectrum I fico„, n ) ] can be used
l
to calculate the
response to an external excitation operator, as discussed It is often useful to discuss the global properties of a
in Appendix B.2 below and in Sec. IV. spectral distribution in terms of its moments. Starting
The use of the RPA within the framework of density- from an RPA spectrum taco„, ln )], one defines the
functional theory is straightforward. Although the strength function S&(E) for the linear response of a sys-
ground-state wave function here is not explicitly taken as tern to an external excitation operator Q:
a Slater determinant, it is still possible to use the Kohn-
Sham orbitals to create particle-hole excitations. This S&(E) = g l ( n Q RPA ) l'5(E —
l l
g~„), (86)
n&0
method, usually called the time-dependent local-density
approximation (TDLDA), was developed for the calcula- where we have put the energy of the RPA ground state
tion of atomic polarizabilities by Zangwill and Soven lRPA) (81) equal to zero. The kth energy-weighted mo-
(1980) and by Stott and Zaremba (1980). The TDLDA is ment of the strength function is given by
equivalent to the RPA if the residual interaction used in
the matrix elements (84) is obtained from the energy den- m„(g)= f0
E'S~(E)dE= y
n&0
(g~„)'l(nip lRPA) l'.
sity functional E [p] (A16) by a double variational deriva-
tive:
V„,[p(r) ] =, $2
5p(r)
E [p] .
Two caveats must be given here. First, there is a for-

(8&)
Many useful quantities and relations can be derived from
these moments. For example, the centroid (i.e. , the mean
energy) E and the variance o. of the distribution (86) are
(87)
given by
mal difficulty in using the LDA in a time-dependent
theory. The Hohenberg-Kohn theorem and the density- E=milmo q
0' =mplmp (piillno) (88)
functional theory built upon it, as we have presented it in
Upper and lower bounds for these two quantities can be
Appendix A. 2, is strictly limited to the static ground
state. The general formulation of a time-dependent given (Bohigas et al. , 1979) by
density-functional theory is a problem fraught with
difficulties, and in general the static functional E[p] can-
(E(E o (o. (89)
not be used by just inserting the time variable as a param-
eter of the density, p(r, t), except in the adiabatic limit of in terms of the two energies E3, F. , defined by
Rev. Mod. Phys. , Vol. 65, No. 3, Juty 1993

' I/2 with

(Q~ ) m, (Q)
E3(Q)= Ei(g)=
mi(Q) m, (Q) u(r) = —$2 Vg(r) . (815)
(810)
The moment m, (g) is then easily shown by partial in-
These energies are particularly easily accessible for the tegration to equal
following reason. Under the assumption that iiico„, In )
are eigenenergies and eigenstates of the total Hamiltoni-
an H of the system (which is the assumption of the RPA),
mi(g)=
2fi I u(r) u(r)p(r)d r, (816)
the moments I
3 and I
can be written as
&
which is proportional to a hydrodynamical mass parame-
ter if u(r) is interpreted as a displacement (or static ve-
m, (Q) = —,'(RPAI[Q, [H, Q)]IRPA), (811) locity) field [see Eq. (823) below]. On the other hand, the
moment m3(Q) can be expressed as
m, (g)=-,'«PAI[[H, g], [H, [g, H]]]IRPA) . (812)
According to a theorem proved by Thouless (1961), the m, (g) = —(HFI [S,[S,H ]]IHF)
expectation values (811) and (812) can be evaluated
without loss of accuracy replacing the correlated RPA 1 A'e
(HFIe IHF) (817)
ground state RPA ) by the uncorrelated HF ground
I de 'a=O
state IHF) belonging to the same Hamiltonian. [This
holds exactly for a density-independent Hamiltonian. and is thus proportional to the restoring force parameter
Applied to the Kohn-Sham formalism, the theorem must related to a collective "deformation" variable a(t).
be generalized to the case of a density-dependent Hamil- The energy E3 in Eq. (810) is therefore identified with
tonian (due to the form of the exchange and correlation the harmonic-oscillator energy fico,
energy functional in the LDA); it can, indeed, be shown
E3=+m~lm, =An) =&C/B (818)
(Bohigas et al. , 1979) to hold within the quasiboson ap-
proximation that is used in all practical RPA calcula- corresponding to the lowest excitation of a collective
tions. ] Thus m3 and mi can be evaluated as HF (or Hamiltonian H„ii(a ),
Kohn-Sham) ground-state expectation values without ex-
plicit calculation of the RPA spectrum. H„ii(a)= —,'Ba + V„ii(a) (819)
Similarly, the RPA moment m,
can be shown (Thou-
less, 1961; Marshalek and da Providencia, 1973) to be in the harmonic approximation. The collective potential
proportional to the static polarizability of the HF n, &
energy is the "scaled" HF energy
V„ii(a) = (HFle 8e
(or Kohn-Sham) ground state with respect to the external
field Q,
IHF) (820)
obtained by a unitary transformation of the ground state
m, (Q ) = 'ap, i(Q ) .
—, (813)
IHF) through the scaling operator e . Since S, as well
Exploiting the above relations, it is thus possible to find as Q, is a single-particle operator, all wave functions
upper and lower bounds for the centroid and an upper q&; ( r ) are transformed independently in the same way:
limit for the variance of an RPA excitation spectrum
merely from static ground-state wave functions.
e y;(r)=y;(r, a) . (821)
Explicit expressions of m and m3 for the electric di- The collective mass parameter 8 and the restoring
pole operator Q = r Yz 0(8) are given in the main text (Sec.
&
force parameter C in (818) are thus given by

IV.C); sum rules for the momentum-dependent excitation
operators of the form jr(qr) YI o(8) have been discussed B =2iri m, (Q), C=2m3(Q) . (822)
by Serra et al. (1990). The interpretation of u(r) as the displacement field be-
longing to the collective fIow pattern generated by the
b. Scaling model interpretation of moments m3 and m, scaling transformation (821) is verified by defining the to-
tal velocity field v (r, t )
The energy E3 in Eq. (810) has a simple and transpar-
ent physical interpretation in terms of a scaling transfor- v (r, t)=a(t)u(r) . (823)
mation (Bohigas et al. , 1979), if the external excitation Together with the scaled ground-state density
operator Q commutes with the potential-energy part of
the Hamiltonian. This is the case for any /ocal operator p (r, t)= (rp, a(t))=ply;(r, a(t))l (824)
Q=Q(r) in connection with a Coulombic system. One
may then define an anti-Hermitian "scaling operator" S v is, indeed, found to fulfill the continuity equation
by
(r, t)+V [p (r, t)v (r, t)]=0 . (825)
S=[H, Q]=[T, Q]= '(V u)+u. V, —, (814) Bt p

Note that the above interpretation of the energy E3 is are given by

exact within the HF+RPA approach; the term "scaling
model" should therefore not be misunderstood as indicat-
ing a further approximation beyond those inherent al-
4, =&HFltQ„[H, Q, ]]lHF&, (827)
ready in an RPA calculation built on top of the HF
ground state. As already mentioned above, all these ar- m„=&HF~t [a, Q, ], [II, [Q, , H]]]~HF& . (828)
guments carry over directly to the case in which the HF
calculation for the ground state is replaced by a Kohn-
Sham calculation.
They are nondiagonal generalizations of the moments
and m3 in Eqs. (811) and (812), so that the secular equa-
I &
The physical meaning of the estimate E3 for the collec-

tion (826) represents an extension of the simple sum-rule
tive vibrational energy now is clear: It corresponds to a
expression E3 (810) to the case of several coupled modes.
diabatic oscillation of the single-particle states (i.e., here
For sufficiently simple operators Q;, the expressions (827)
of the valence electrons) around their equilibrium
and (828) may be evaluated analytically and lead to in-
configuration; this oscillation is rapid, so that the mean
tegrals involving only HF (or Kohn-Sham) ground-state
field (i.e., the HF or Kohn-Sham potential) is not
densities.
changed during the vibration. All wave functions scale
Solving the secular equation (826) then gives a spec-
coherently according to Eq. (821); no change of their no-
trum of eigenenergies %co„ that represents an approxima-
dal structure occurs.
tion to the RPA spectrum. The corresponding one-
In contrast to this, the energy E, in Eq. (810) contains
phonon states n & exhaust the m, and m 3 sum rules for
the static polarizability [see Eq. (813)] in its denominator
~
any external operator that lies in the space spanned by

and thus corresponds to a slow, adiabatic motion of the
particles, which adjust their wave functions at any mo-
the trial set Q;. Evaluating the moment m, also gives,
by virtue of Eq. (813), the static polarizability. The only
ment to the static external field Q.
restriction of this approach with respect to the full mi-
croscopic RPA is the choice of a (finite) set of local
operators Q;(r), i.e. , the assumption of the local nature of
c. Local-current RPA, fluid dynamics,
the associated velocity fields or currents.
and normal hydrodynamics
In fIuid dynamics, emphasis has been put on the
dynamical distortions of the Fermi sphere in momentum
Reinhard et al. (1990) have shown that the exact RPA
space, which for uniform systems leads to zero sound
equations (83) are obtained if one makes the energy
efFects. These efFects are fully included in the local-
E3(Q) stationary by a variation of the operator Q in full current RPA if the kinetic-energy contribution to the re-
particle-hole space. Taking Q to be a local function
storing force tensor A;~ (828) is evaluated microscopical-
Q(r), one is led to a nonlinear fourth-order differential ly in terms of the scaled single-particle wave functions y;
equation for the velocity field, i.e., the gradient of Q(r)
(821) through Eq. (A13) (cf. Brack, 1983). If, however,
via Eqs. (815) and (823) above, which is identical in
the kinetic-energy density is evaluated in the Thomas-
structure to that of the so-called fIuid dynamics ap-
Fermi or ETF approximation (see Appendix A. 2.a), the
proach. This latter approach was initialized by Bertsch
zero-sound efFects are lost, since the momentum distribu-
(1975) and by Sagawa and Holzwarth (1978) for the
tion is always spherical in the (extended) Thomas-Fermi
description of giant resonances in nuclei, and put in a
model, and one obtains standard classical hydrodynam-
variational form by Krivine et al. (1980) and by da
ics. For pure electric dipole vibrations, or for monopole
Providencia and Holzwarth (1983, 1985). The fluid-
vibrations described by the operator Qo =r (leading to a
dynamical equations have usually been solved for
"breathing mode" ), there is no difference in the restoring
simplified liquid-drop model densities with sharp surfaces
forces obtained by Quid dynamics or ordinary hydro-
(see Lipparini and Stringari, 1989, for a recent review).
An alternative approach, which avoids the numerically
dynamics. For most other modes, however, the
difFerences can become important. A textbook example
difficult solution of the full Auid-dynamical equations and
is the nuclear giant quadrupole mode, in which hydro-
makes use of exact variational (HF, Kohn-Sham, or semi-
dynamics gives wrong values and the wrong dependence
classical extended Thomas-Fermi) ground-state densities,
on the nucleon number 3, whereas Auid dynamics leads
was recently proposed by Brack (1989) and Reinhard
et al. (1990) (see also Reinhard and Gambhir, 1992, for
to an excellent description of the peak energies even with
the simple E3 sum-rule expression (Bohigas et al. , 1979;
an exhaustive presentation of the formalism). Here the
variation of E3(Q) is done on a set of local trial operators Lipparini and Stringari, 1989; Gleissl et al. , 1990).
I Q;(r)] and leads to a secular equation for coupled har-
monic vibrations generated by these operators:
detP(, —(irico„) X,, =0 .

~
(826) See Brack (1989) and Reinhard et al. (1990) for expressions
valid for multipole operators of the form Q;lr)=r ' Yco(0) in
Here the mass tensor X, and the restoring force tensor connection with the spherical jellium model.

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1993-RevModPhys-metal Clusters Self-Consistent Jellium Model

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The physics of simple metal clusters: self-consistent jellium model

and semiclassical approaches

Rev. Mod. Phys. , Vol. 65, No. 3, July 1 993

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

atom is first studied microscopically in density-functional D. Jellium model

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

E[p]=T, [p]+E„[p]+ f VI(r)p(r)+ —

Here T, [p] is the (noninteracting) kinetic-energy density trons per atom).

—which is always possible for physical (i.e. , non-

electrostatic energy associated with the jellium back-

where w is the valence factor (number of valence elec-

Rev. Mod. Phys. , Vol. 65, No. 3, July 1 993

Usually, one takes the bulk value for r, corresponding to

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

reduction of the 2p lf -gap.

which are too large in the jellium model, as is well known

jellium model; the experimentally observed odd-even

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

volume and half-axes zo, po given in terms of a single de-

each deformation 6, and the ground state is found by

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

FIG. 3. Total energy per electron F. /X vs X for sodium clusters 05

Rev. Mod. Phys. , Vol. 65, No. 3, Juiy 1993

3. Finite-temperature effects II A f de Heer, 1993). The manifesta-

8 10 12 14 16 18 20 temperature a ffec t s th e magnitude of the electron'nic shell

10 12 14 16 18 20 '-D' distribution. This will also smoo th outou shell

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

correlation energy functional. This approach retains all (4.2)

IV. ELECTRIC DIPOLE RESPONSE (4. 3)

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

calculations that 5 in Eq. (4.7) is proportional to the elec-

O'ITco As discussed by de Heer (1993), the experimental po-

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

which, with R =r, Z'~, is equal to 1/&3 times the bulk

Re~. Mod. Phys. , VoI. 65, Np. 3, Juiy 1993

Rev. IVlod. Phys. , Vol. 65, No. 3, July 1993

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

discussed a "dipole force sum rule" that is closely related

where b, Z is the electronic spillout defined in Eq. (4.8)

Rev. Mod. Phys. , Vol. 65, No. 3, July 1 993

equations might give some information about the overlap

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

1/2 2. Coupling of surface and volume plasmons

Rev. Mod. Phys. , Vol. 65, No. 3, July 1993

(i) The compressional volume-type modes are no longer 100-

degenerate; the volume plasmon is fragmented into a fo m) I

bunch of scattered eigenfrequencies. That part of the

damped resonance, as shown by Ekardt (1985a, 1985b), 5 l6 ~(ev)