Electronic .structure

Electronic Structure. - So far cluster geometry has been highlighted and the matter of electronic structure has been neglected. However, the electronic configurations of the surface atoms of a cluster will determine their reactivity, and the matter is of importance. 

The basic question to which we require an answer is how small may a metal particle become before it loses its metallic properties Or conversely, in the growth of a metallic cluster, at what point does the electronic structure closely approximate to that of the bulk metal Though the questions appear simple the answers are not. This is partly because the many properties that may be used to characterize the metallic state are differently related to cluster size, so that one may be achieved at a much earlier stage in cluster growth than another. Moreover, if one property is arbitrarily selected as a criterion, the various available theoretical methods for the calculation of that property may differ in their predictions as to when, during cluster growth, the metallic state is achieved. 

For present purposes we will deem that a cluster partakes substantially of the metallic state if the electronic density of states (DOS) curve calculated for it compares well with that of the bulk metal. To compare well, the valence bandwidth must be approximately correct, as must the density of states at the Fermi level, and the general features, particularly the major peaks, in the DOS curves for cluster and bulk metal, must correspond. This criterion has a point of contact with experiment, because DOS curves can be compared with photoelectron emission spectra. 

TTie three methods that have been most widely adopted to calculate DOS curves for clusters are (/) the semi-empirical extended-Huckel (EH) method, (n) the complete-neglect-of-differential-overlap (CNDO) method, and iii) the self-consistent-field Xa scattered-wave (Xa-SW) method. Calculations of electronic structures of many transition- and noble-metal clusters have been reported over the last decade. No attempt is made to summarize them all, because there appears to be a disparity of view among the expert practitioners as to the validities of the various methods of calculation. However, the story to date is as follows. 

Early calculations for copper, palladium, and silver clusters were carried out by various investigators using the EH and CNDO methods, and among these is an attempt by Baetzold to take into account the effect of a carbon support on the electronic structure of a palladium cluster.In 1976, Messmer etal. compared the efficacies of the three methods of calcu- 

In this chapter, we will discuss the electronic arrangement of atoms. We will also talk about quantum numbers, orbitals, various rules pertaining to electron-filling, and electronic configuration. 

In terms of the classical octet theory originated by G.N. Lewis [9] in 1916, the electronic configuration in pyramidal and tetrahedral phosphorus compounds is completed by an outer shell of eight electrons as indicated in (3.7). 

In phosphoryl compounds such as POCI3 the covalent bond to the oxygen atom is often regarded as formed by donation of the lone pair electrons from the P atom. Such donation confers semi-polar or part-ionic properties on the bond and it can be written as P+-0 , P O or as P=0, the double bond completing the formal pentavalency of the central phosphorus atom. There is, however, controversy over the nature of this bond (see below). 

By acquiring three extra electrons as in the stable orthophosphate anion, PO , the P atom can form donor-type P 0 linkages, while if an electron is lost from the P atom, four single covalent bonds are formed as in the tetrachlorophosphonium cation, PCIJ (3.7). 

In trigonal bipyramidal compounds such as PCI5, an outer shell of 10 electrons is involved (3.9a), while in octahedral configurations such as PFg, a negative charge is acquired and the outer shell probably contains 12 electrons (3.9b). The phosphide anion P probably exists in some metal phosphides and this will presumably be based on a completed octet of electrons (3.9c). A similar situation occurs in the phosphide PHj anion (3.9d). 

When trivalent phosphorus atoms link together as in diphosphine, P2H4 (3.10a), each P atom contributes an electron to form the single covalent bond. A similar situation exists with pentavalent derivatives such as hypophosphoric acid, H4P2O6 which can be represented as in (3.10b). The phosphoryl bonds in the compound are donor type as in POCI3 above, with two electrons being provided by each P atom to complete the formal octet around the O atoms. It is, however, usually more convenient to represent the electronic formulae as in (3.10c). Examples of donor-type P — P linkages are known but these are very rare. In MejP — PFj, both electrons for the bond are provided by the same P atom (3.10d). 

The bonding in HN3 is usually described in terms of equal contributions of the resonance structures HN =N =N and HN -N EN-y. The a framework of HN3 approximately results from an sp hybrid at Np which is bonded to a p5s orbital of N and a p or p6s orbital of N. A second p5s orbital of is used for the a bond to H. A supplemental, localized 71 bond connects Np and N. The bonding of the N3 group is completed by a delocalized 71 bond which links Np with N and N [4, 5]. The calculation of the bond orders at the MP2/6-31G(d, p) level predicts less than a single bond for H-N , an intermediate between a single and a double bond for N -Np, and nearly a triple bond for Np-N [6]. 

Six photoelectron bands, expected for nonlinear HN3 analogous to the spectrum of the isoelectronic linear N2O, were observed in the most recent studies of He I and He II photoelectron (PE) spectra. The first three bands exhibit vibrational splitting see p. 154 for details. 

The consequence of hydrogenation on the electronic structure can be broadly classified into four types of effects [32]. 

1) Modification of the symmetry of electronic states and reduction of the width of the bands due to expansion of the crystal lattice. 

2) Appearance of a metal-hydrogen bonding band below the metal d-band. Electrons are transferred from the s-d band to this new band and some metal states could be pulled down below the Fermi level. 

3) In hydrides that have more than one hydrogen atom per unit cell, the H-H interaction produces new attributes in the lower portion of the density of states. 

4) General upward shift of the Fermi level due to the inequality between the additional electrons brought by hydrogen and the number of new electron states. 

The lanthanide elements differ chemically according to the filling of seven orbitals from the 4f orbital. The electronic configurations span from lanthanum [Xe]5d 6s up to lutetium [Xe]4f 5d 6s. As you move across the series from left to right, the orbitals that experience the greatest addition of electrons are the f orbitals. Here it is broken down such that each line shows the changes in each orbital. 

Notice that there is no change of the d or s orbitals, only in the f orbitals. This filling of f orbitals ranges from 4f (for Ce) to 4f (for Lu). 

The 4f electrons play only a small role in bonding. These metals are highly electropositive with a +3 oxidation number being typical. Because of this, the 4f electrons are similar in terms of physical and chemical properties. Lanthanide chemistry changes gradually as you move across the series they are typically +3 oxidation state. 

The early lanthanides (on the left side of the periodic table) are more reactive than the late lanthanides because the basicity (the measure of ease in which an atom will lose an electron) decreases from left to right. In other words, lanthanum would be the most reactive in this series because it more readily gives up an electron, and lutetium ise the least reactive because it more likely holds on to its electrons. 

The 4f orbitals in general are much less reactive than the 5d orbitals (transition metals). The f orbitals do not span out as far into physical space as the d orbitals, so they are harder to reach and harder to do chemical reactions with. Additionally, the 4d and 5d elements are relatively inert in comparison to the 3d elements. Therefore, not only are the lanthanides less reactive because of the employment of 4f orbitals, but any d orbitals that they might employ are going to be less reactive than the d orbitals of the earlier d block elements. 

Matsumoto [98] tried to summarize the work done on the electronic structure of iron oxides and concluded that the about 2 eV in the bandgap of the hematite semiconductor measured in photoelectrochemistry indeed is based on the 3d band transition between the Fe3+ ions, which supports a Mott-Hubbard insulator. 

2 Single crystal Flux grown from an Li2M04 melt 9 

The bandgap (Eg) of a semiconductor is usually determined by means of optical absorption. The following equation gives a satisfactory description of the absorption behavior near the threshold [85]  

The bandgap of a semiconductor can also be determined photo-electrochemically [1, 2, 5, 7, 8], which is based on the fact that the wavelength corresponding to the onset of photocurrent agrees well with the optical absorption edge. For colloids and powders, diffuse reflectance spectroscopy method has been used to characterize the hematite bandgap [4]. 

This section deals with the overall qualitative features of fullerene electronic structure rather than with highly accurate calculations on specific cages. Some ab initio calculations are reviewed in Refs. 45 and 46. 

Nuclear magnetic moments typically represent 10 of the electron magnetic moment and can be neglected. To a good approximation, magnetic moments in solids can therefore be described in terms of electronic structure alone. 

A key concept in the description of electrons in atoms is their wave-like nature, introduced by de Broglie in 1925. The momentum of a particle of mass m and velocity v can be represented by a wave with wavelength X X = hjmv, and kinetic energy = hv, where h is Planck s constant (6.626 X J s) and v is the frequency. 

Quantum mechanics is based on the wave nature of all atomic particles. In a H atom, an electron orbits around the nucleus (a proton) electron energies, or energy states, can be conveniently described in terms of a wave function, P(x , y, z, t), which depends on particle space coordinates, x, y, z, and time t. Stable states having well-defined (discrete) energies can be represented as the product of a sinusoidal time-dependent term of angular frequency co, and a time-independent wave function l/(x, y, z)  

The problem is now to find which time-independent wave functions describe the energy states of the electron in a H atom. In quantum mechanics, these wave functions are found as solutions to the Schrodinger equation  

The wave functions which are solutions to the Schrodinger equation and therefore correspond to stable energy states depend on three parameters, or quantum numbers  

How electrons are distributed about nuclear centers and how they participate in chemical bonds are crucial aspects of chemistry, one dictated by the laws of quantum mechanics. This is the problem of electronic structure, using the Schrodinger equation to find wave-functions for electrons in atoms and molecules. The atom with the fewest electrons, the hydrogen atom, is as an important model problem, and the quantum mechanical analysis of the hydrogen atom is carried out in detail in this chapter. Based on that discussion, we explore the qualitative features of the structure of more complicated atoms and molecules. 

BSCF orthorhombic unit cell (thick lines) and its reiation to the cubic lattice (thin lines), and (d) the relationship between the orthorhombic lattice (Oi ojOjojOjo) and the cubic lattice (o 2 c c)- Ba/3r and Co/Fe sites are split- 

However, it is also necessary to consider the Cl and F. Fluorine is the most electronegative atom known and as a result its compounds are 

Adoption of the oxidation state Fe( —I) for Fe(NO)2 fragments leads to the assignment of Co(O) in the binuclear heterometallic complex 17, [(ON)2Fe(SPh)2Co(NO)(PPh3)], while adoption of Fe(I) for the Fe(NO) fragments in the heterometallic cubane 19 leads to a formal oxidation state V(III) for vanadium. 

The diamagnetic behavior of the diiron and tetrairon complexes, despite the presence of formally d1 and/or d9 iron centers, indicates very strong coupling between the individual paramagnetic centers all theoretical treatments of polynuclear iron-sulfur-nitrosyl complexes to date have been based on the assumption of diamagnetism in even-electron species and have employed molecular orbital methods at various levels of approximation. 

The complexes [Fe2(SR)2(NO)4] contain a total of 34 valence electrons and are isoelectronic with [Fe2(SR)2(CO)6] and with [Fe2(CO)9] with a single Fe-Fe bond, each obeys the 18-electron rule. The cobalt complexes [Co2(SR)2(NO)4], on the other hand, contain 36 valence electrons and thus obey the 18-electron rule in the absence of any metal metal bond, while the heterometallic complex 17 is another 34-electron species containing a Co-Fe bond. 

In both [Fe4X4(N0)4](X = S or Se) and [Fe4S2(NO)4(NCMe3)2] the total valence electron count is 60. This is the number characteristic of tetrahedral tetranuclear metal clusters, such as [Ir4(CO)12], in the Wade and Mingos skeletal-electron counting schemes (76, 77) and, furthermore, each iron atom in these clusters obeys the 18-electron rule, provided that it forms single Fe-Fe bonds to each of the other iron atoms in the tetrahedron. 

In tackling problems of molecular and electronic structure such as those posed by these sulfur-bridged tetrametallic clusters, two approaches are possible. One is the construction, often heavily reliant upon symmetry arguments, of a general, essentially qualitative model intended to provide a broad description of a series of molecules the other is a detailed quantum mechanical study, at an appropriate level of theory, of individual molecules, followed by a search for generalization or patterns. A combination of both approaches is usually the most valuable for chemical understanding, and much effort along these lines has been expanded on cubane-type clusters and their derivatives. 

A common feature of metal atoms is that they are generally larger in size in comparison with nonmetal atoms. A characteristic of nonmetals is that their atoms have the ability to attach electrons to themselves, leading to the formation of anions. The opposite is true for the metals and as told they alter to cationic forms when their removable electrons leave them. 

The higher the valency of a metal, the greater will be the number of electrons in the outermost shell. Now, since the positive charge residing in the nucleus remains unaltered by the removal of electrons, its attractive influence will progressively increase with the removal of each successive electron. It follows that when there are a number of electrons in the outermost shell, the removal of electrons will progressively tend to be more and more difficult as each electron is taken out. 

The larger the atomic volume of an atom, the less strongly will the positively charged nucleus be able to hold the electrons in the outermost or the valency shell. In other words, 

Although it is possible, by the loss of several electrons, for certain metal atoms to form polyvalent cations upto a maximum valency of four (e.g., tin forms the tetravalent stannic ion, Sn4+), the formation of polyvalent anions is extremely difficult since for the acquisition of each additional electron the attractive force exerted by the nucleus on each individual electron becomes progressively smaller. It is for this reason that the maximum valency for a simple anion is found to be two. 

The metallic structure essentially consists of atomic nuclei and associated core electrons, surrounded by a sea of free electrons. The high electrical conductivity of metals is derived from the presence of these free electrons. In addition to high electrical conductivity, the free electrons provide the metals with good thermal conductivity as well. The electrical resistivity of a metal increases with temperature. 

These results were most clearly stated by Weaire and Thorpe (Weaire 1971, Thorpe and Weaire 1971), who described the bonding by a tight binding Hamiltonian of the form 

At first sight, it appears that the developers of quantum chemistry (QC) codes, such as Gaussian,SPARTAN,ACES, °i and TURBOMOLE,i°2 made only modest progress in moving to parallel platforms partly because of the size of these programs. Despite an explosion in the number of people using 

A secondary concern, but one that definitely impacts the chemistry community, is the increasing role played by the user interface in the scientific exploitation of a particular high performance machine. Given that the dependency of users to a particular code is often linked to the convenience of the associated user interfaces, it is clear that efforts to develop new codes for MPP hardware must be accompanied by work to bring flexible and friendly interfaces into these codes. 

Few of the published efforts in parallelizing QC codes appear to be directed at the most widely used packages the exceptions include the development of parallel versions of GAMESS (at Iowa State University 3,94,io6 at the EPSRC Daresbury Laboratory 96.u 4,io8) FIONDO (at IBM Kingston ), and TURBOMOLE (at Karlsruhe ). Though not as widely used. 

In addition to ab initio methods, developments for semiempirical methods are being investigated on parallel machines. Notable in this regard are MOPAC 20 (at the San Diego Supercomputer Center i EPSRC Dares-bury Laboratory ) and AMSOL.122 expected, many of the problems encountered with ab initio codes are also found with these programs. 

Einally, we draw attention to several review articles in this area. In 1986 Lowdin 23 considered various aspects of the historical development of computational QC in view of the development of both conventional supercomputers and large-scale parallel computers. More recently, Weineri24 presented a discussion on the programming of parallel computers and their use in molecular dynamics simulations, free energy perturbation, and large scale ab initio calculations, as well as the use of very elaborate graphical display programs in chemistry research. We also note a review on the use of parallel processors in 

We adopt here an axis convention in which the ground state of the phenolate anion (PhO ) is described by the following basic orbital configuration . .. (13ai). .. (8 2) (3i i) The reference configurations for the A2, 5i and 

FIGURE 39. Schematic potential energy profiles showing the interconversion between phenol and cyclohexa-2,4-dienone in free and water-assisted systems (a) in the neutral state and (b) in the ionized state. Values given in kJ moU were obtained from B3LYP/6-31G(d,p)-hZPE computations 

FIGURE 40. A representation of four different singly-occupied orbitals (SOMO) of the phenoxyl radical in the corresponding electronic states 

When comparing all the available observed absorption bands and the energies calculated using the multi-reference CASSCF methods with large active space , the following assignments of the observed transitions can be proposed (i) the band at 1200 nm is due to the 1 52 X Bi transition, (ii) 611 nm to l Ai X B, (hi) 395 nm to 2 B  

A possible problem concerns the transition f 2 Bi, which is symmetry forbidden under C2v symmetry and might cast doubt on the assignment of the 292 nm band. Experimentally, this band was observed to be weak and the relevant peak is almost completely obscured by the strong peak centred at 240 nm . The CASSCF excitation energies were found to be overestimated by up to 0.5 eV, indicating the importance of dynamic electron correlation for a reasonable description of the excited states. Calculations on PhO using small atomic basis sets turned out to give incorrect results. 

Furthermore phase separation of Ceo and CuPe is deseribed at elevated temperatures [55] whieh affeets the moleeular arrangement, but ean not be de-teeted by the teehniques used here. Altogether, film morphology and strueture of blends of flat CuPe and spherieal Ceo moleeules are still not very well understood and need further investigation. This will beeome partieularly important in photovoltaic cells, where this material combination is a potentially promising candidate for so-called buUc-heterojunction cells [21, 56], 

It is remarkable that the seeondary electron cut-offs used to determine the work function of all mixtures were sharp and did not show a double step that would refleet two different loeal surface potentials [59], although the films have a nanogranular morphology as determined by atomic force microscopy [27]. Thus, the blends show a eommon vacuum level which shifts linearly with the concentration between the work functions of the neat materials given by d cuPc 3.8 eV and = 4.3 eV. The linear change of the work function in our studies suggests that CuPe/Cgo mixtures are electronically non-interacting (unless they are optieally exeited). 

If we limit ourselves to octahedral complexes with six identical ligands that only have a interactions with the metal (the model 

We shall now study other geometrical arrangements (ligand fields) that are frequently met in transition metal complexes. We shall generally limit ourselves to the characterization of the structure of the d block, that is, the shape and the relative energy of the five orbitals 

Higher excited states that correlate with H+ and the ground and first excited states of the neutral P atom, i.e., (P( S) + H+) and 211, 2A, 22 (P(2D) + H+), have been mentioned (cf. p. 38) 

FSGO [35] (the abbreviations are explained on pp. 3/4), deal with various ground-state properties of PH and, in some cases [14,18,19, 29, 31], with the lowest excited states. The so far best theoretical value for the ground-state total molecular energy at the equilibrium internu-clear distance, Et=-341.05765 E, was obtained by a recent G2 calculation using a method that treats the electron correlation by Moller-Plesset perturbation theory (MP4) and quadratic configuration interaction (QCI) [25]. 

The total energy Ej = -174.19502 a.u. was calculated [4], Et = -173.91011 and -171.80257 a.u. come from ab initio SCF-MO calculations with a double zeta and a minimal basis set, 

The Linnett [12,13] treatment of the Lewis-Langmuir octet as a double quartet of electrons leads to an OF structure with three bonding electrons, see also [14] and the preliminary considerations in [15, 16]. 

A radical structure with a single bond plus a (weaker) three-electron bond in the Pauling sense [17] was suggested [18]. The three-electron bond in OF has also been discussed [19]. 

The dipole moment i (in D) was measured directly for the ground state from the saturated absorption signals of the LMR spectrum (see above) in the presence of an electric field [i=( )0.0043 0.0004 for v = 0, i=( )0.0267 0.0009 for v=1, with the same sign in both vibrational states [29, 30]. Similarly low values were theoretically calculated (up to v = 6) fi= -0.0089 (v = 0), -0.0318 (v=1). The negative sign implies the polarity 0 F . The effect of rotation was rather small. A theoretical dipole moment function adjusted so as to exactly reproduce the experimental dipole moments for v = 0 and v=1 yielded the polarity O F for the equilibrium internuclear distance [31]. [i=-0.361 was calculated by an ab initio SCF-MO method [4], fi = ( )0.29 [9] by the semiempirical SINDO method [10]. 

The quadrupole moment Q = 0.134x10 esu cm was calculated by an ab initio SCF-MO method [4]. 

General Features of the Band Models. Calculation Methods 

All these calculations and also the APW calculations by Farberovich, Vlasov [5] (see below) show that the d band X3 point is at lower energy than the 6s band point. However, Kasuya [6], as a consequence of the concept of the magnetic exciton and the magnetic impurity state (see p. 232) sets the 6s state lower than the 5dt2g. His calculations are based on atomic spectra and the pure ionic crystal model [6]. For a simple ionic model proposed by Dagys, Anisimov [7] for EuSe, SmSe, and YbSe, see p. 150. 

The optical absorption edge corresponds to the onset of the 4f ( Sr ) 4f ( Fj)5dt2g transition of Eu2+. Its energy Eg (-1.80 eVat300 K) as a function of temperature, magnetic field, and pressure is discussed on p. 252. In the APW calculations. Eg (=1.78 eV) corresponds to the distance from the top of the f bandsto the d band X3I point [1]. In the band scheme of Kasuya [6] with the 6s band as the bottom of the conduction band, the distance between the 4f bands to the lowest 6s band is 2.1 eV. 

The controversial position of the 6s or 5d bands at the bottom of the conduction band has been discussed in the C 7-review. The conduction mechanism indicates that the 5d band is lower than the 6s band, because it is the exchange parameter (-0.1 eV) and not (which is an order of magnitude lower) that can explain the observed transport phenomena, see p. 231. 

For the temperature dependence of the conduction band edge in the AF-I, ferrimagnetic, and ferromagnetic states, as obtained from the bound magnetic polaron model, see p. 234. 

169 and continues to 218. Lack of detail precludes any close criticism of this scheme. 

Returning to more domestic territory, we note that Carlson et al. have calculated electron binding energies, and in particular K X-xdiy energies for elements up to Z = 120. The Z-ray energies are thought to be known to high accuracy and could be most useful for experimental identification. 

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Although a separation of electronic and nuclear motion provides an important simplification and appealing qualitative model for chemistry, the electronic Sclirodinger equation is still fomiidable. Efforts to solve it approximately and apply these solutions to the study of spectroscopy, stmcture and chemical reactions fonn the subject of what is usually called electronic structure theory or quantum chemistry. The starting point for most calculations and the foundation of molecular orbital theory is the independent-particle approximation. 

The result is that, to a very good approxunation, as treated elsewhere in this Encyclopedia, the nuclei move in a mechanical potential created by the much more rapid motion of the electrons. The electron cloud itself is described by the quantum mechanical theory of electronic structure. Since the electronic and nuclear motion are approximately separable, the electron cloud can be described mathematically by the quantum mechanical theory of electronic structure, in a framework where the nuclei are fixed. The resulting Bom-Oppenlieimer potential energy surface (PES) created by the electrons is the mechanical potential in which the nuclei move. Wlien we speak of the internal motion of molecules, we therefore mean essentially the motion of the nuclei, which contain most of the mass, on the molecular potential energy surface, with the electron cloud rapidly adjusting to the relatively slow nuclear motion. 

Herzberg G 1966 Molecular Spectra and Molecular Structure III Electronic Spectra and Electronic Structure of Polyatomic Molecules (New York Van Nostrand-Reinhold)... 

Using the Hamiltonian in equation Al.3.1. the quantum mechanical equation known as the Scln-ddinger equation for the electronic structure of the system can be written as... 

Applying Flartree-Fock wavefiinctions to condensed matter systems is not routine. The resulting Flartree-Fock equations are usually too complex to be solved for extended systems. It has been argried drat many-body wavefunction approaches to the condensed matter or large molecular systems do not represent a reasonable approach to the electronic structure problem of extended systems. 

The periodic nature of crystalline matter can be utilized to construct wavefunctions which reflect the translational synnnetry. Wavefiinctions so constructed are called Bloch functions [1]. These fiinctions greatly simplify the electronic structure problem and are applicable to any periodic system. 

There are a variety of other approaches to understanding the electronic structure of crystals. Most of them rely on a density functional approach, with or without the pseudopotential, and use different bases. For example, instead of a plane wave basis, one might write a basis composed of atomic-like orbitals ... 

A1.3.6 EXAMPLES FOR THE ELECTRONIC STRUCTURE AND ENERGY BANDS OF CRYSTALS... 

Many phenomena in solid-state physics can be understood by resort to energy band calculations. Conductivity trends, photoemission spectra, and optical properties can all be understood by examining the quantum states or energy bands of solids. In addition, electronic structure methods can be used to extract a wide variety of properties such as structural energies, mechanical properties and thennodynamic properties. 

Another usefiil quantity is defining the electronic structure of a solid is the electronic density of states. In general the density of states can be defined as... 

Cohen M L and Cheiikowsky J R 1989 Electronic Structure and Optical Properties of Semiconductors 2nd edn (Springer)... 

Cox P A 1987 The Electronic Structure and Chemistry of Solids (Oxford Oxford University Press)... 

Harrison W A 1999 Elementary Electronic Structure (River Edge World Scientific)... 

Harrison W A 1989 Electronic Structure and the Properties of Solids The Physics of the Chemical Bond (New York Dover)... 

Surfaces are found to exliibit properties that are different from those of the bulk material. In the bulk, each atom is bonded to other atoms m all tliree dimensions. In fact, it is this infinite periodicity in tliree dimensions that gives rise to the power of condensed matter physics. At a surface, however, the tliree-dimensional periodicity is broken. This causes the surface atoms to respond to this change in their local enviromnent by adjusting tiieir geometric and electronic structures. The physics and chemistry of clean surfaces is discussed in section Al.7.2. 

The study of clean surfaces encompassed a lot of interest in the early days of surface science. From this, we now have a reasonable idea of the geometric and electronic structure of many clean surfaces, and the tools are readily available for obtaining this infonnation from other systems, as needed. 

Adsorbates can physisorb onto a surface into a shallow potential well, typically 0.25 eV or less [25]. In physisorption, or physical adsorption, the electronic structure of the system is barely perturbed by the interaction, and the physisorbed species are held onto a surface by weak van der Waals forces. This attractive force is due to charge fiuctuations in the surface and adsorbed molecules, such as mutually induced dipole moments. Because of the weak nature of this interaction, the equilibrium distance at which physisorbed molecules reside above a surface is relatively large, of the order of 3 A or so. Physisorbed species can be induced to remain adsorbed for a long period of time if the sample temperature is held sufficiently low. Thus, most studies of physisorption are carried out with the sample cooled by liquid nitrogen or helium. 

The surface work fiincdon is fonnally defined as the minimum energy needed m order to remove an electron from a solid. It is often described as being the difference in energy between the Fenni level and the vacuum level of a solid. The work ftmction is a sensitive measure of the surface electronic structure, and can be measured in a number of ways, as described in section B 1.26.4. Many processes, such as catalytic surface reactions or resonant charge transfer between ions and surfaces, are critically dependent on the work ftmction. 

A DIET process involves tliree steps (1) an initial electronic excitation, (2) an electronic rearrangement to fonn a repulsive state and (3) emission of a particle from the surface. The first step can be a direct excitation to an antibondmg state, but more frequently it is simply the removal of a bound electron. In the second step, the surface electronic structure rearranges itself to fonn a repulsive state. This rearrangement could be, for example, the decay of a valence band electron to fill a hole created in step (1). The repulsive state must have a sufficiently long lifetime that the products can desorb from the surface before the state decays. Finally, during the emission step, the particle can interact with the surface in ways that perturb its trajectory. 

Flamers R J, Tromp R M and Demuth J M 1986 Surface electronic structure of Si(111)-7 7 resolved in real space Phys. Rev. Lett. 56 1972... 

How are fiindamental aspects of surface reactions studied The surface science approach uses a simplified system to model the more complicated real-world systems. At the heart of this simplified system is the use of well defined surfaces, typically in the fonn of oriented single crystals. A thorough description of these surfaces should include composition, electronic structure and geometric structure measurements, as well as an evaluation of reactivity towards different adsorbates. Furthemiore, the system should be constructed such that it can be made increasingly more complex to more closely mimic macroscopic systems. However, relating surface science results to the corresponding real-world problems often proves to be a stumbling block because of the sheer complexity of these real-world systems. 

In order to describe the second-order nonlinear response from the interface of two centrosynnnetric media, the material system may be divided into tlnee regions the interface and the two bulk media. The interface is defined to be the transitional zone where the material properties—such as the electronic structure or molecular orientation of adsorbates—or the electromagnetic fields differ appreciably from the two bulk media. For most systems, this region occurs over a length scale of only a few Angstroms. With respect to the optical radiation, we can thus treat the nonlinearity of the interface as localized to a sheet of polarization. Fonnally, we can describe this sheet by a nonlinear dipole moment per unit area, -P ", which is related to a second-order bulk polarization by hy P - lx, y,r) = y. Flere z is the surface nonnal direction, and the... 

Luce T A and Bennemann K H 1998 Nonlinear optical response of noble metals determined from first-principles electronic structures and wave functions calculation of transition matrix elements P/rys. Rev. B 58 15 821-6... 

Venanzi T J 1982 Nuclear magnetic resonance coupling constants and electronic structure in molecules J. Chem. Educ. 59 144-8... 

STM found one of its earliest applications as a tool for probing the atomic-level structure of semiconductors. In 1983, the 7x7 reconstructed surface of Si(l 11) was observed for the first time [17] in real space all previous observations had been carried out using diffraction methods, the 7x7 structure having, in fact, only been hypothesized. By capitalizing on the spectroscopic capabilities of the technique it was also proven [18] that STM could be used to probe the electronic structure of this surface (figure B1.19.3). 

Hamers R J and Kohler U K 1989 Determination of the local electronic structure of atomic-sized defects on Si(OOI) by tunnelling spectroscopy J. Vac. Sc/. Technol. A 7 2854... 

Electronic structure theory describes the motions of the electrons and produces energy surfaces and wavefiinctions. The shapes and geometries of molecules, their electronic, vibrational and rotational energy levels, as well as the interactions of these states with electromagnetic fields lie within the realm of quantum stnicture theory. 

B3.1.1.3 WHAT IS LEARNED FROM AN ELECTRONIC STRUCTURE CALCULATION ... 

Also produced in electronic structure sunulations are the electronic waveftmctions and energies F ] of each of the electronic states. The separation m energies can be used to make predictions on the spectroscopy of the system. The waveftmctions can be used to evaluate the properties of the system that depend on the spatial distribution of the electrons. For example, the z component of the dipole moment [10] of a molecule can be computed by integrating... 

Returning now to the issue of the accuracy of various electronic structure predictions, it is natural to ask why... 

C) All mean-field models of electronic. structure require large corrections. Essentially all ab initio quantum chemistry approaches introduce a mean field potential F that embodies the average interactions among the electrons. The difference between the mean-field potential and the true Coulombic potential is temied [20] the "fluctuationpotentiar. The solutions Ef, to the true electronic... 

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