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Advancing Theory for Kinetics and Dynamics of Complex, Many-Dimensional Systems

Physics meets art: The background of the cover represents a detail of a church window (© jorisvo/fotolia.com). The shining colors of church windows reflect their content in metallic particles, in modern words metallic nanoclusters.

Physics meets biology: illustrative example of gold nanoclusters attached to a DNA molecule. The presence of nobel metal nanoparticles in biological environments results in significant radiobiological and immunological effects the essential physical insights into which could be obtained through the molecular dynamics simulations, see http://www.mbnexplorer.com/.

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Cluster science developed as an independent branch of science only a few decades ago. Since then its remarkable achievements have turned it into a major branch of science bridging the gap between microscopic and macroscopic worlds. Cluster science joins efforts from physicists and chemists and has led to impressive technological developments, opening the door to the nanoworld. It is now an established field of research with an impressive network of connections to neighboring scientific domains such as material science, but also to more remote ones such as, in particular, biology. We aim in this short book at addressing both these aspects, first, presenting cluster science as such, and second, indicating the connections to other fields of research.

Cluster science: a young field with a long history and a promising future Clusters were recognized in technological applications well before they were identified as physical objects. It was well known to Roman craftsmen and into the Middle Ages that immersing small pieces of a noble metal into a glass allowed for beautiful colors to then reside within the glass. More recently, photography became possible when realizing that small aggregates, namely “small” pieces of matter, of AgBr had a remarkable sensitivity to light, which with a proper chemical treatment, allowed to record and print images. However, over these many centuries, the idea of clusters as a subject of scientific research remained absent. One major reason was that one did not know how to isolate and to identify such microscopically small objects from a carrier environment, for example a glass in the Middle Ages, or more recently a gel. It has only been during the last quarter of the twentieth century that researchers have succeeded in producing in a controlled manner aggregates of various (controlled) sizes, thus opening the door to dedicated studies of clusters, also known as nanoparticles. For example, studies of clusters of various sizes allowed for the first time to systematically track the transition between atom/molecule and a bulk material. Within a few decades, cluster physics has become a lively domain of research, and has “invaded” many other domains where it is now fully admitted that clusters do play a major role. As typical examples, let us cite the many potential applications of nanosystems in material science in the race for miniaturization, as well as in medicine for drug delivery and imaging. We could also mention astrophysics with the composition of the interstellar medium, or climate science with aerosols. Cluster science has thus a promising future which motivates us to present its many facets in this book. Thereby we try to stay at an introductory level to address a broad readership.

Cluster science: a merger between physics and chemistry Clusters are constituted from atoms and interpolate between small molecules and bulk materials. As such, they thus call for expertise from chemistry – chemistry of small molecules but also solid-state chemistry – as well as from physics – atomic and molecular physics and solid-state physics. Indeed, the cluster community formed as a merger between various fields of physics and chemistry, including researchers involved in the study of other finite systems, such as nuclei or helium droplets. From this somewhat heterogeneous background emerged an original and rich scientific field primarily dedicated to the study of clusters themselves. Cluster science indeed developed over the last few decades into a somewhat specific domain. The study of clusters themselves made tremendous progress, reaching now in some cases a remarkable degree of detail, for example, even a time-resolved account of the dynamical response to a dedicated excitation. This high degree of detail was made possible partly because of the growing versatility of lasers over the same period of time and to the fact that clusters may have a strong coupling to light, especially clusters made out of metallic material. We will see many examples along that line throughout this book, both in the study of clusters themselves as well as in applications to other fields.

Cluster science: an interface between many domains A fascinating aspect of cluster science is that clusters play a role in several somewhat unexpected scenarios. This holds, for example, in astrophysics where it was recently realized that the composition of cosmic dust is “full” of clusters, whose influence on light signals received on earth may be crucial. But this also holds in terms of applications, for example, in drug delivery on specific targets in the human body. But they may be essential building blocks of new materials as well. The range of “applications” is thus enormous, from the largest times and distances in the universe to nanosized devices and materials, with an excursion into mesoscopic constituents of living cells in the human body. These apparently remote domains of scientific knowledge happen to share common objects, namely clusters. It is thus certainly an important issue to understand the properties of these fascinating objects.

The aim of this book is to introduce the reader to these many aspects of cluster science. The domain is huge and cannot be covered in depth within the limited size of the present book which should be an introductory text. It nonetheless indicates the wide range of cluster physics. We thus have confined the presentation to the basic aspects of clusters, being well aware that some aspects and many details are missing. This book is not meant to be an exhaustive review but rather a survey to motivate the reader to go deeper into the material. We have thus tried to supply relevant citations, mostly to textbooks or review papers and, when found helpful, to the proper specific citation. A strong underlying idea was to precisely cover general characteristics, often on a schematic basis, as well as some actual recent scientific results in order to enlighten the ever-developing nature of the field. The book thus consists of two parts of about equal size. The first half of the book (Chapters 1 to 3) includes a general introduction and provides the basic notions and keywords in experiments and theory. These notions should suffice for further reading of papers in the field. The last three chapters (Chapters 4 to 6) gather a collection of illustrative results. These chapters cover both properties of clusters themselves and their applications in various domains of science from astrophysics to material science and biology.

A book is always the result of numerous interactions with many colleagues. It is obvious that our project would not have converged without these many interactions. We would thus like to acknowledge the help of all these colleagues and tell them how much they helped, long ago or more recently, both in terms of science and personal contacts. We would, in particular, like to mention here: E. Artacho, M. Bär, M. Belkacem, D. Berger, G.F. Bertsch, S. Bjornholm, C. Bordas, M. Brack, F. Calvayrac, B. and M. Farizon, F. Fehrer, T. Fennel, G. Gerber, E. Giglio, C. Guet, B. von Issendorf, H. Haberl, J.M. L’Hermite, P. Klüpfel, U. Kreibig, J. Kohanoff, C. Kohl, S. Kümmel, E. Krotscheck, P. Labastie, F. Lépine, F. Marquardt, K.-H. Meiwes-Broer, B. Montag, M. Moseler, J. Navarro, V. Nesterenko, A. Pohl, L. Sanche, L. Serra, R. Schmidt, A. Solov’yov, F. Spiegelman, F. Stienkemeier, J. Tiggesbäumker, C. Toepffer, C. Ullrich, R. Vuilleumier, Z.P. Wang, P. Wopperer, F.S. Zhang, and G. Zwicknagel. Finally we would like to mention that this book emerges from a long-standing collaboration between the authors. This would not have been possible without the help of funding from the French–German exchange program PROCOPE, the Institut Universitaire de France, and the Alexander-von-Humboldt Foundation. We are thankful to these institutions to have supported us in our common efforts.

We list here a few basic physical constants and units (data taken from [1]). We use the Gaussian system of units for electromagnetic properties (dielectric constant

In dynamics, one simultaneously treats energy, distance and time scales, so that one has to consider proper combinations of these three quantities. Some standard packages are:

Clusters in Nature

Clusters, also called nanoparticles, are special molecules. They are composed from the same building blocks, atoms or small molecules, stacked in any desired amount. This is similar to a bulk crystal. In fact, one may view clusters as small pieces of bulk material. It has only been within the past few decades that clusters have come into the focus of intense investigations. During these few decades, cluster science has developed into an extremely rich and promising field of research. As often in science, technological applications of clusters existed before they were identified and understood. One of the most famous and oldest examples of the application of clusters in technology is the coloring of glass by immersing small gold clusters into the glass itself. The process allowed for some tuning of colors depending on the inclusions’ size. This technology dates back to Roman times, where there is evidence craftmen had perfectly mastered this versatile technique. In scientific terms, such a phenomenon just reflects the size dependence of the optical response (that is, the color) of gold clusters in a glass matrix (although that prosaic formulation certainly does not give sufficient credit to the marvelous impressions attained that way). Another example of early applications is found in traditional photography which started about two centuries ago. The emulsion of a photographic film contains a dense distribution of AgCl (later AgBr) clusters whose special optical properties allowed to store information from light impulses and to visualize it later by chemical reduction. Progress in sensitivity and resolution was tightly bound in properly handling the cluster properties, where for a long time photographers did not even know that they were dealing, in fact, with clusters.

Figure 1.1 exemplifies clusters with an ancient and with a modern view. Figure 1.1a shows a church window from the St. Etienne cathedral in Bourges (France), whose impressive colors (not visible here, but which can be appreciated from the book cover) were fabricated to a large extent by Au clusters embedded in glass. Figure 1.1b was recorded with the most modern achievements of scanning tunneling microscopes (STM). It shows in detail Ag nanoparticles sitting on a highly oriented pyrolytic graphite (HOPG) surface. From the given length scale, we read off for this case cluster sizes in the range of a few nanometers, which corresponds to system sizes of about 100–10 000 Au atoms. As we will see, the combination of these quickly developing methods of nanoanalysis with nanoparticles, called clusters, constitutes a powerful tool for fundamental and applied physics.

Figure 1.1 (a) Glass window of the St. Etienne cathedral in Bourges, France. Colors of church windows reflect their content in metallic particles. (b) Topography of silver nanoparticles deposited on (HOPG), recorded with an in situ scanning tunneling microscope (STM), from [2].

A first theoretical study which these days plays a basic role in cluster physics goes back to G. Mie in the early twentieth century [3]. Mie considered the question of the response of small metal particles to light, and how this optical response might depend on the size of the considered particle. It is interesting to quote Mie who turned out to develop a remarkable intuition of the future of cluster science: “Because gold atoms surely differ in their optical properties from small gold spheres”, it would “probably be very interesting to study the absorption of solutions with the smallest submicroscopic particles; thus, in a way, one could investigate by optical means how gold particles are composed of atoms.” This apparently simple and “reasonable” statement actually covers a large fraction of today’s activities in cluster science as the interaction with light is a key tool for the investigation of cluster structure and dynamics.

In spite of Mie’s intuition, the study of clusters as physical individual objects remained rather limited for the subsequent decades. Most investigations concerned clusters in contact with an environment (embedded or deposited). This limitation was due to the difficulty to create isolated clusters in a controlled manner. During the last quarter of the twentieth century, the capability of producing free clusters from dedicated sources finally triggered the emergence of cluster science on a systematic basis. The identification of the remarkable C₆₀ clusters, the famous fullerenes [4], and the first systematic investigations of metal clusters [5] were impressive boosters. From then on, cluster science rapidly grew to an independent, although cross-disciplinary, field among the well-defined branches of physics and chemistry, ranging from fundamental research to applications in the context of nanotechnology.

Grossly speaking, clusters can be considered as large molecules or small pieces of bulk material. Their properties thus can be understood to some extent by methods from molecular and solid-state physics. Nonetheless, clusters represent a species of their own. One of the possibly most specific aspects of clusters is the fact that one can deliberately vary cluster size. Clusters are, so to say, “scalable” objects which bridge the gap between atoms/molecules and bulk material. This makes them useful testing grounds for the many-body problems, for example, to understand the path to bulk matter. They are true multidisciplinary objects staying in contact with many areas of science. This includes even such a remote field as astrophysics where clusters play a role in the formation mechanisms and the properties of cosmic dust. But the interest in clusters is not purely fundamental. We have already mentioned their early technological use in photography and artwork. Cluster research has meanwhile triggered a wide field of further applications. In chemistry, for example, scalability allows to play with the tunable surface to volume ratio which governs reactivity and may thus find key applications for catalysis. In material science, the discovery of fullerenes and nanotubes opened up new ways to design new materials [6] making carbon, and more recently “nano” science, an emerging field per se.

Cluster science with its many achievements now belongs to one of the most active fields in physics and chemistry, and offers, in particular through related fields, one of the fastest developing areas in applied as well as in fundamental science. One single and short book on cluster science can certainly not cover all aspects. We therefore confine ourselves to present key cluster properties and tools of investigation in a compact manner. The aim here is to give the reader a basic understanding and a motivation for further detailed readings. To reach this double goal (basics and motivation), we split our book in two major parts. The first half provides the essential concepts. The introductory Chapter 1 gives a first overview of the field. Chapter 2 introduces major experimental concepts, discussing cluster production and tools for the analysis of cluster properties. The subsequent Chapter 3 describes basic theoretical tools used in cluster science. These three first chapters, roughly representing half of the book, lay ground for all further reading. The second part tries to present selected examples superficially covering both, current cluster science and closely related fields. Chapter 4 presents in a compact manner some key properties of free clusters, both static and dynamic. It serves as a first application of the concepts introduced before. Chapter 5 concerns applications of clusters mostly in relation to material science. Finally Chapter 6 covers relations of cluster science to close domains such as astrophysics and biology. In the last two chapters, and to a lesser extent in Chapter 4, the topics to be covered are so vast that an extensive review is impossible in such a short book. We have thus chosen to select illustrative test cases which we discuss in some detail. We believe that such a strategy is more motivating and easier to grasp, though the selection may be somewhat subjective.

1.1 Atoms, Molecules and Bulk

Clusters have this remarkable and unique property to interpolate between individual atoms/molecules and bulk matter. Their size, in particular, can be deliberately chosen, so that this interpolation can be followed step by step if necessary. By being “in between” individuals and bulk, one could naively consider them as simple aggregates of individuals and/or finite pieces of bulk material. It turns out that they are neither the former nor the latter, and possess specific properties of their own. On the other side remain close relations with the extremes (atom/bulk). It is this subtle balance between specificity and connections with other systems that we first want to explore in this section. This provides a first overview of cluster science.

1.1.1 Scales of Matter Down to Atoms

We recall here a few basic scales of matter down to atoms and simple molecules, which in the following will constitute the building blocks of clusters. We focus on distances, energies and times. Figure 1.2 summarizes the following discussion in a compact and schematic manner indicating, in the time-distance plane, typical systems and observables of interest. The reader can also find a list at the beginning of this book summarizing the various units which are commonly used in atomic, molecular, cluster, and laser physics.

In bulk material, there is no clear macroscopic distance scale. We shall thus directly refer to the typical interconstituent distances, which lie in the nanometer and Å range. This also corresponds to the typical bond lengths in molecules. Atoms can be characterized by the typical radius of the electron cloud in the Å range. In the following, we shall thus use typical distances in the range 1 Å to 1 nm, and use as a standard unit Bohr’s radius (hydrogen atom “radius”) abbreviated as a₀ with

Figure 1.2 Schematic representation of typical time (abscissa) and length (ordinate) scales. We also indicate representative systems and components thereof, and associated energy scales for completeness. Today’s investigations allow to access a wide range of distances and times in electronic systems including clusters, both from an experimental and a theoretical point of view. From [7].

Energies of interest span a much wider range with variations of six orders of magnitude. We shall take as a reference unit the electronvolt (1 eV = 1.6 × 10⁻¹⁹ J). It provides an appropriate unit for characterizing the binding of a valence electron (least bound electrons) to the system, often called the ionization potential (IP). The IP is typically a few eV varying, of course, from one system to the other, but within at most a factor of 10. The eV also roughly corresponds to the typical dissociation energy of a molecule or cohesive energy of a solid. Much lower energies, in the meV range, are associated to vibrational motion (atomic vibrations) in bulk and molecules. The meV is the typical energy range of rotational motion in molecules as well. This range of energies, though, is directly connected to atomic masses and nature of bonding. It thus shows larger variations reflecting the large span of atomic masses and electronic binding. On the side of higher energies, the binding of deeply bound electrons in atoms (and thus in molecules and bulk) ranges up to the keV domain. In most low-energy situations, such electrons remain safely bound to their parent nucleus. We shall therefore denote by “ion” the nucleus and the core electrons, to distinguish the latter ones from valence electrons. There however exist high-energy processes where core electrons can be excited, such as irradiation by high intensity laser fields or high-frequency photons (X-rays).

The above range of energies fixes a range of associated time scales on the basis of the energy-time uncertainty relation. The eV is closely related to the femtosecond (10⁻¹⁵ s) which appears as the natural time scale of valence electron motion in atoms, molecules and bulk. The meV range in turn is associated to the picosecond domain, characteristics of atomic motion. On the other extreme, keV electrons are typically associated to attoseconds (10⁻¹⁸ s). This very short time scale was not accessible until very recently. The advent of new light sources now allows to produce light pulses with durations down to a few tens of attoseconds. Such short pulses suffice to directly excite low lying, deeply bound electrons. In a way similar to energies, accessible time scales today thus span several orders of magnitude in the domain of interest of cluster science.

1.1.2 More on Time Scales

Figure 1.2 provides a very general overview of typical time scales of cluster properties, cluster dynamics, and excitation mechanisms. Clusters are built of atoms and thus naturally cover the various time scales associated both to individual atoms and assemblies thereof. Furthermore, as already mentioned in Figure 1.1, the possible contact with an environment (surface, matrix, solvent, …) may introduce more time scales to the picture. Finally, it is interesting to detail a bit the typical times scales associated to lasers, which, as we shall see at several places throughout the book, constitute a basic tool of investigation of cluster properties. We thus present in Figure 1.3 a schematic but more detailed overview of times related to electronic and ionic motion, lifetimes for relaxation processes, and laser characteristics. Numbers are given in the case of metal clusters because specifying the cluster type allows to be more specific on time scales. Changing the nature of the cluster would probably alter the detailed numbers, but not the qualitative relations between the various time scales. The fastest cluster time scales are related to electronic motion. Core electrons have cycle times of 0.1 fs and faster (depending on the element). The Mie plasmon period, see Eq. (2.6), most important for metal clusters, is on the order of femtosecond, see Section 2.2.3. In the same range, but with wider span from subfemtosecond to several femtosecond, are cycle times for other single-particle excitations and direct electron escape, that is single-particle excitation directly into the continuum. Somewhat slower is the plasmon decay due to Landau damping, a mechanism which is well known from plasma physics [8], see also the discussion of Figure 4.7. The most widely varying electronic times are related to damping from electron–electron collisions and thermal electron evaporation. Both strongly depend on the internal excitation energy of the cluster. They are long for low and moderate excitations, as indicated in Figure 1.3. Ionic motion spans a wide range of slow time scales. Vibrations, which may be measured by Raman scattering, typically in the meV regime, have cycle times of order 100 fs to 1 ps. Strong laser irradiation can lead to Coulomb explosion of the cluster where ionic motion becomes somewhat faster. Besides the ionization effects, the thermal relaxation between electrons and ions takes much longer, up to the nanosecond range. Ionic relaxation processes are even slower, for example thermal emission of a monomer can easily last µs. As indicated in Figure 1.3, the pulse duration of optical lasers may be varied over a wide range and extends in principle from fs (even some hundreds of attoseconds) to ps or even ns.

Figure 1.3 Typical time scales for the various dynamical processes in a cluster, here for the example of free sodium clusters, and clusters in contact with an inert, insulating substrate.

The above discussion reveals that cluster dynamics comprises a huge span of time scales which are extremely hard to treat at the same level of refinement. A most basic and widely used approximation is to treat nuclei, or ionic cores, respectively, as classical particles while the electrons remain quantum mechanical particles. We will come back to that aspect in more detail in Chapter 3.

1.1.3 Binding in Atoms, Molecules and Bulk

Atoms, molecules and solids are all made from the same constituents, namely nuclei and electrons. And yet, matter provides a great variety of properties. The reason is that electron binding shows up in very different manners, depending on the elements involved and the composition thereof. In this section, we will briefly discuss the different mechanisms of electron binding.

1.1.3.1 Atoms

Atoms consist, as is well known, of a nucleus at the center surrounded by a cloud of electrons. They interact with each other and with the central nucleus mainly through Coulomb interaction. The immense attraction provided by the positively charged nucleus suffices to counterweight the repulsion between electrons themselves. The basic theory is known as the central field approach [9]. Each electron independently moves in a common mean field, the central field, which consists of the Coulomb field of the central nucleus augmented by the screening field from the other electrons. The predominantly spherical shape of the central field leads to a pronounced bunching of the single-electron spectrum into shells, each one containing a couple of degenerated states. The Pauli principle determines the filling of these shells with increasing electron number. This simple shell model picture provides the key to a first sorting of atomic structure. Particularly important are shell closures which arise for those electron numbers where the occupied shells are all exactly filled. In the Mendeleev classification, this situation corresponds to the rare gas atoms He, Ne, Ar, Kr, Xe, and Rn. These are particularly inert in chemical reactions. In general, the amount of shell filling determines the chemical properties. Extreme cases are those next to shell closure. The alkalines Li, Na, K, Rb, Cs, and Fr, have one single electron on top of a closed shell and are considered as simple metals. At the other side of shell filling are the halogens F, Cl, Br, I, and At which have an almost complete shell, missing just one electron. Both these groups are particularly reactive. One can go stepwise further, two additional electrons or two missing electrons, and so forth. The properties of some transitional atoms may require to watch not only the shell of least bound electrons but also the next lower shell. This is, for example, the case for the transition metals Cu, Ag, and Au. All in all, the ensemble of these “active” electrons constitute what one usually calls the valence electrons, namely those which determine binding structure and low-energy dynamical properties.

The calculated shell structure of two typical metals is depicted in Figure 1.4. The states are labeled in standard atomic physics notation. The letters denote the orbital angular momentum l with the series s, p, d, f, g,… corresponding to l = 0, 1, 2, 3, 4,… The numbers count the occurrence of an angular momentum in increasing order of energies. For example, the 4p states denotes the fourth l = 1 state and 3s the third l = 0 states. Figure 1.4a shows the atomic level sequence for the noble metal Cu, and Figure 1.4b shows that for the simple metal Na. Note that there are several lower states (1s for Na and 1s, 2s, 2p for Cu) which are so deeply bound that they fall far below the bounds of the plot. For the remaining states, one still sees a clear distinction. The 3s, 3p states for Cu and the 2s, 2p states for Na are far away from the least bound states (which constitute the Fermi surface of the system). These are rather inert to bonding and low-frequency dynamics, and are thus considered as core states building together with the nucleus the ionic core. There remains one occupied valence state, the 3s for Na and the group 3d, 4s for Cu. The distinction between core and valence electrons is, of course, a matter of decision and may differ under different circumstances. For example, one may put the 3d state in Cu into the core states for low-energy situations where the 4s–3d gap of about 10 eV is sufficiently large. On the other hand, highly dynamical processes (involving high energies and/or frequencies) may require one to include the 3d state in Cu amongst the active electrons.

Figure 1.4 Comparison of the calculated level sequence in a noble metal atom, Cu (a), with a simple metal atom, Na (b). Single-particle (s.p.) energies are indicated by horizontal bars. The naming of the states (atomic convention) is given to the right near each level line. Occupied states are indicated by filled circles and unoccupied ones by open circles. Deeply lying states are not shown. All results have been computed by density functional theory (see Section 3.2.4).

Quantitative indicators of the chemical properties of an atom are the binding energies of the lowest unoccupied molecular orbital (LUMO) and of least bound electrons known as highest occupied molecular orbitals (HOMO). The latter energy is directly measurable as the ionization potential (IP), that is the energy which is needed to singly ionize the atom. For example, rare gas atoms are associated with particularly large IP which indicate a high resistance against ionization (and chemical reactions in general). The binding energy of the LUMO is related to the electron affinity which determines whether an atom is an electron donor (e.g., alkalines) or acceptor (e.g., halogens).

Figure 1.5 Schematic view of potentials and valence levels in dimers with the four basic types of binding as indicated (a–d). The potential seen by an electron in the molecule is drawn with solid lines, while those originating from single atoms are represented by dashes. The valence levels are indicated by horizontal straight lines.

1.1.3.2 Simple Molecules

Because chemical binding is by nature a low-energy phenomenon, involving at most a few eV energy, the properties of the valence electrons mainly determine how atoms bind together. The Mendeleev classification thus provides a gross map of which atom may possibly bind with another one. We first discuss the dimer molecule which is the simplest conceivable case. The question is how electrons will rearrange themselves between both atoms. The answer is to a large extent related to electronic energies. Strongly bound electrons will tend to remain on their parent atom, while loosely bound ones will have a tendency to leave it. Depending on the partner atom, this leads to different situations, as sketched in Figure 1.5.

In the case of ionic binding (Figure 1.5a), one loosely bound electron is transferred from its parent atom to a host atom offering an empty, more deeply bound level. The transfer is all the easier the smaller the IP of the parent atom and the higher the electron affinity of the host. The electron then predominantly localizes on the host atom: the donor atom becomes positively charged and the host one negatively, leading to a robust binding between the two such formed ions. Typical examples of ionic binding are realized between alkaline (electron donor) and halogen (electron acceptor) atoms such as, for example, in NaCl, NaI, or LiF.

Ionic binding is all the more likely when the energy difference between the valence shells of the two atoms is large. In other cases, electron transfer becomes unfavorable and electrons tend to remain closer to their parent atoms. Binding then stems from charge sharing rather than charge transfer: the valence electrons tend to form a common electronic cloud establishing the binding of the two atoms together. Such a picture traditionally leads to two kinds of binding, depending on the actual strength of binding of each electron on its parent atom. If the valence electrons are weakly bound and if the atoms become sufficiently close to each other, the electronic wave functions become delocalized and fill the space all around the two atoms leading to a so-called metallic bond (Figure 1.5c). Alkalines are the typical elements which establish metallic bonds between each other. Metallic binding is especially simple in alkalines as it involves only one loosely bound electron per atom, hence the notion of “simple” metals to characterize these systems. Metallic binding is also observed in more complicated metals such as, for example, Cu, Ag, Au, or Pt. The shell below the HOMO is taking part in the binding. In this case, it is the d shell energetically close to the valence s shell (see Figure 1.4a). The energetic neighborhood of the d shell adds strong polarization effects with impact on binding and dynamical properties.

When electrons are initially more strongly bound to their parent atom, electrons rather gather in the region of smallest potential energy, that is between the two partner atoms, instead of fully delocalizing themselves. This leads to the so-called covalent bond realized between atoms like C or Si (Figure 1.5b). However, it should be noted that covalent and metallic bonds as described above are idealized situations. There is a smooth transition between these extremes. In addition, it is usually hard to disentangle these two types in small molecules. The distinction becomes better defined in bulk material.

In the extreme case where valence electrons are too deeply bound in the atom, as in rare gas atoms, neither charge transfer nor charge sharing are possible between the two atoms. Electrons remain basically localized to their parent atom. Still, the electronic cloud of each atom is influenced by the partner atom by mutual polarization. This polarization of the electronic clouds has to be understood as a dynamical correlation effect, with constantly fluctuating dipoles, rather than as a mere static polarization. It results in mutual dipole–dipole interactions between the two atoms, which finally establishes a binding of the system. Although the resulting binding is much weaker than the previous ionic, metallic or covalent bonds, it suffices for the binding of rare gas molecules up to possibly large compounds. This type of binding is known as van der Waals (or molecular) binding, in reference to the interactions between two neutral atoms or molecules (Figure 1.5d). It primarily concerns binding between rare gas atoms but will also be useful to describe binding between small molecules such as CO₂, because well-bound molecules as such represent also closed shell situations.

Simple energetic considerations for HOMO and LUMO thus allow one to identify four major types of bonding between atoms, as schematically represented in Figure 1.5. These energetic considerations are directly linked to the degree of localization of the electrons binding the two atoms: the more bound the electrons, the more localized their wave functions. As already pointed out, there is no clear separation between the various binding mechanisms, but rather a continuous path between them. Still, the sorting in four classes provides useful guidelines for understanding the binding of most molecular systems. As a final remark, one should note that some other “classes” of binding are sometimes introduced such as the separation between van der Waals (for rare gas) and molecular (for molecules), and that between covalent (generic one) and hydrogen binding (specific to covalent binding involving hydrogen atoms). Such fine distinctions do not basically alter our global classification scheme and we shall thus ignore such details in the following.

1.1.3.3 Bulk

We have classified binding on the basis of binding energies of the valence electrons. Although bulk spectra differ by nature from the discrete atomic spectra, it turns out that binding proceeds in a similar way and finally allows a similar classification.

In a solid, the infinity of connected atoms smears discrete energy levels into bands of continuous energy levels. These bands are separated from each other by energy gaps. Band structure is related to the width of the gaps, that of the bands, and where they are located in the energy spectrum. One distinguishes between two extremes, conductors and insulators. In the former ones, the valence bands containing the least occupied electronic states overlaps with the conduction band of empty electronic states so that electrons can switch empty conductance levels at no energetic cost. Such a system is ideally represented by simple metals such as alkaline metals in which, again, electronic wave functions delocalize over the whole system. The other extreme are insulators in which valence and conduction bands are separated from each other by a finite, mostly large, energy gap. There is, of course, a continuous path between these two extremes depending on the size of the energy gap. The transitional cases (small gaps) are known as semiconductors where conductivity can be easily induced by thermal agitation or dedicated doping [10].