Table Of Content

P1: GKD Revised Pages Qu: 00, 00, 00, 00 Encyclopedia of Physical Science and Technology EN002D-71 May 17, 2001 20:25 Bonding and Structure in Solids J. C. Phillips Lucent Technologies I. Introduction: Molecules and Solids II. Molecular Crystals III. Ionic Crystals and Electronegativity IV. Covalent Crystals and Directed Valence Bonds V. Mixed Covalent and Ionic Bonding VI. Metallic Bonding VII. Quantum Structural Diagrams VIII. Complete Quantum Structure Analysis IX. Chemical Bonding in Solids in the Third Millennium GLOSSARY THE RELATIVE POSITIONS of atoms in molecules and solids are described and explained in terms of the ar- Atom Smallest unit of an element. rangements of their nearest neighbors. Together with the Bond Electronic configuration that binds atoms together chemical valences of the atoms as given by the periodic Covalent bond Chemical bond formed by electron table, these arrangements of the bonding determine the sharing. structure and physical properties of solids. Both structure Crystals Solids in which the atoms are arranged in a pe- and properties can be used to separate solids into various riodic fashion. classes where further quantitative trends can be systemi- Electronegativity Measure of the ability of an atom to cally described by structural diagrams. attract electrons. Glass Solid in which the atoms are not arranged in a periodic fashion and which melts into a supercooled liquid I. INTRODUCTION: MOLECULES when heated rapidly. AND SOLIDS Ionic bond Chemical bond caused by charge transfer. Metallic Material with high electrical conductivity at low The combinations of atoms found in the vapor phase are frequency. called molecules. Molecules containing a small number of Molecule Bonded atoms in a gas. atoms have been studied accurately and extensively. Most Valence Number of electrons used by an atom to form of our knowledge of chemical bonding between atoms chemical bonds. comes from these studies. When atoms are condensed to 281 P1: GKD Revised Pages Encyclopedia of Physical Science and Technology EN002D-71 May 17, 2001 20:25 282 Bonding and Structure in Solids form solids, the atomic density is much greater, as re- Certain general techniques are widely used for describ- flected by the number of atoms that are nearest neighbors ing bonding and structure in solids. Tables of atomic radii of any given atom. This number is called the coordination are available for ionic, covalent, and metallic bonding. De- number. An example is the molecule NaCl, in which each viations of bond lengths from values predicted by these atom has one nearest neighbor. In solid NaCl each atom radii of order 1–3% often reveal critical structural features has six nearest neighbors. of importance to material fabrication and properties. The Solids in their pure forms are nearly always crystalline. cohesion of solids can be connected to the cohesion of A crystal is a periodic arrangement of atoms along lines, the elements. A binary solid AmBn is said to have heat of which in turn is repeated periodically along planes. Fi- formation Hf, which is the difference between m times nally, the planes are repeated periodically to form the crys- the cohesive energy of A plus n times that of B minus the tal lattice. cohesive energy of AmBn. This heat of formation can be Most of our knowledge of crystal structures comes from estimated with often remarkable accuracy from Pauling’s the diffraction of waves of photons, electrons, or neu- table of elemental electronegativities X(A). This is proba- trons by lattice planes. Usually all the atomic positions bly the most widely used table in science apart from the pe- in the crystal can be determined this way. By comparing riodic table of the elements, and it is shown here as Table I. chemical trends in bond lengths in crystals with those in molecules, one can often infer the nature of the electronic charge distribution responsible for chemical bonding in II. MOLECULAR CRYSTALS the crystal. From this it may be possible to predict the nature of chemical bonding at crystalline defects or even in We now turn to the differently bonded main groups of noncrystalline solids, which are amorphous or glassy. solids. The molecular crystals are the simplest case, be- The structures of millions of solids are known by cause the intermolecular forces are typically much weaker diffraction. To understand these structures one begins by than the intramolecular ones. As a result the structure of studying the simplest cases and classifying them into the molecules, as reflected, for example, by bond lengths groups. The main groups are characterized as molecu- and vibration frequencies, is almost the same in the solid lar, metallic, ionic, and covalent. In most solids the actual as in the gas phase. Some examples of materials that form bonding is a mixture to some degree of these different molecular solids are the inert gases, diatomic halogens, kinds of chemical interaction. While most solids are com- closed-shell molecules such as methane, and many planar plex, the inorganic solids, which are best understood be- aromatic molecules such as benzene. Typically, in molec- cause they have had the widest technological applications, ular crystals the heat of fusion per molecule per bond is at are usually either simple examples from a main group or least 10 times smaller than the bond dissociation energy. are closely related to them. In contrast, organic and bio- The binding forces that hold molecular crystals together logically important molecules may be quite complex. The may arise from electric dipoles if the molecules carry per- chemical and structural simplicity of technologically im- manent dipole moments (e.g., HCl). When the molecules portant inorganic solids stems from the requirement of have no permanent moment, binding arises from mutually availability of techniques for production in bulk. induced dipole moments (van der Waals interactions). TABLE I Electronegativity Table of the Elements According to Pauling Li Be B C N O F 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Na Mg Al Si P S Cl 0.9 1.2 1.5 1.8 2.1 2.5 3.0 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br 0.8 1.0 1.3 1.5 1.6 1.6 1.5 1.8 1.8 1.8 1.9 1.6 1.6 1.8 2.0 2.4 2.8 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I 0.8 1.0 1.2 1.4 1.6 1.8 1.9 2.2 2.2 2.2 1.9 1.7 1.7 1.8 1.9 2.1 2.5 Cs Ba La–Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At 0.7 0.9 1.1–1.2 1.3 1.5 1.7 1.9 2.2 2.2 2.2 2.4 1.9 1.8 1.8 1.9 2.0 2.2 Fr Ra Ac Th Pa U Np–No 0.7 0.9 1.1 1.3 1.5 1.7 1.3 P1: GKD Revised Pages Encyclopedia of Physical Science and Technology EN002D-71 May 17, 2001 20:25 Bonding and Structure in Solids 283 The structures of molecular crystals are determined pri- These correspond to packing cations and anions of nearly marily by packing considerations and thus vary from ma- equal size (CsCl structure), and then successively larger terial to material according to molecular shape. Molec- anion/cation size ratios lead to increasing anion–anion ular solids are generally poor conductors of electric- contacts, thus reducing coordination numbers. These “ra- ity, and even the photoconductivity is generally small dius ratio” rules do not actually describe the crystal struc- unless metallic impurities are added to “sensitize” the tures, as shown in Fig. 1. What this means is that the material. ions should not be regarded as hard spheres, but rather as centers of quantum mechanically determined electronic charge distributions. Additional evidence for the break- III. IONIC CRYSTALS AND down of classical electrostatic models is contained in the ELECTRONEGATIVITY elastic constants of the alkali halides. If these models were correct, the elastic constants would satisfy certain rela- Before discussing the structure of ionic crystals in detail, tions (the Cauchy relations) valid for central force inter- we shall familiarize ourselves with the concept of elec- actions. These relations are not satisfied for most of the tronegativity, defined by Pauling as “the ability of atoms alkali halides, another indication of quantum mechani- in the bonded state to attract electrons to themselves.” cal interactions. Some simplified modern treatments of Atoms in solids are in a variety of bonded states, and it these and related problems are discussed in the following is due to Pauling’s insight that we have come to realize sections. that the atomic electronegativity that he defined in terms of heat of formation (Section I) is indeed nearly constant for each element. His idea is that in solids charge flows from cations with smaller electronegativity to anions with greater electronegativity and that the heat of formation re- 2 sulting from this charge flow is proportional to (Xc− Xa) v, where Xc and Xa are the cation and anion electronegativities, respectively. Ionic crystals are composed of cations and anions with very large electronegativity differences, such as alkali metals and halides, columns I and VII of the periodic table, respectively. In this case the charge transfer of valence electrons is almost complete, so that the core configu- rations become isoelectronic to those of inert-gas atoms + − (e.g., Na to Ne, Cl to Ar). While some energy is required to ionize the cations and transfer electrons to the anions, this energy is more than recovered thanks to the larger electronegativity of the anions and the mutual at- traction of cations by their anion neighbors. In the case of the alkali halides, the cohesive energies can be estimated within a few percentage points by assuming complete charge transfer and evaluating the electrostatic energies (including ion polarization energies). A core–core repulsive energy, required by the exclusion principle, com- pletes the calculation, which was first sketched around 1910. As one might expect, the overall features of the crystal structures of ionic crystals are given quite well by pack- + − FIGURE 1 The structures of the alkali halides M X as func- ing spherical cations and anions in the appropriate propor- tions of classical ionic radii r + and r−, respectively. (Coordination tions indicated by their chemical formulas. However, the numbers in parentheses are those predicted by the classical ionic + − ions are not quite the incompressible spheres suggested model.) In the upper left corner, for example, Li I is predicted by the classical model to have coordination number 4, but the by their isoelectronic analogy to inert-gas atoms. If they symbol indicates it is actually sixfold coordinated. Key: ■, sixfold were, one could use simple geometrical arguments (origi- ❤ coordinated; ⋄, eightfold coordinated; , six- or eightfold coor- nating around 1930) to predict a coordination number of 8 dinated. [From Phillips, J. C. (1974). In “Solid State Chemistry” (CsCl structure), 6 (NaCl structure), or 4 (ZnS structure). (N. B. Hannay, ed.), Vol. 1, Plenum, New York.] P1: GKD Revised Pages Encyclopedia of Physical Science and Technology EN002D-71 May 17, 2001 20:25 284 Bonding and Structure in Solids 2 2 configuration of the atom is ns np , with n = 3 for silicon. 3 In the crystal this becomes ns np so that (counting electron spin twofold degeneracy) both the ns and np levels are half-filled. These four states can be combined to form four directed valence orbitals with tetrahedral geometry. The wave functions on nearest neighbors can be combined in phase to form bonding states or out of phase to form antibonding states. Then wave-function overlap produces an energy gap between these states (Fig. 3). This energy gap is the basis of the technologically important electronic and optical properties of semiconductors. The covalent energy gained by wave-function overlap or interference is much more sensitive to structural per- FIGURE 2 The tetrahedrally coordinated diamond structure, fection than is the energy associated with classical ionic which describes many technologically important semiconductors interactions. A very important consequence of this sensi- such as silicon. [From Phillips, J. C. (1970). Phys. Today 23 (Feb.), 23.] tivity is that it is possible to produce semiconductor crystals such as silicon in far purer and far more structurally perfect states than has been possible with any other solid. IV. COVALENT CRYSTALS AND It is possible to add impurities with designed concentra- DIRECTED VALENCE BONDS tions and locations to tailor the chemical and mechanical design of the solid with far greater precision and ease than Whereas ionic crystals can be (at least roughly) described for any other solid. Thus, the quantum mechanical nature in classical terms, structure and bonding in covalent crys- of structure and bonding in covalent silicon is the key to tals can be understood only in terms of quantum mechan- its technological significance. ical electron orbital wave functions. Prototypical covalent crystals have the diamond structure. Many technologically important semiconductors such as silicon and germanium V. MIXED COVALENT AND have this structure or a closely related one, the zinc blend IONIC BONDING or wurtzite structure. In these structures each atom is tetrahedrally coordinated (Fig. 2). Most semiconductors and insulators have neither purely The structure shown in Fig. 2 can be explained simply covalent nor purely ionic bonding, but their bonding is in terms of directed valence electron orbitals. The valence described as a mixture of covalent and ionic effects. The FIGURE 3 Sketch of electronic interactions between directed valence orbitals that produce an energy gap between bonding states (electrons shared between nearest neighbors) and antibonding states (no electron sharing). [From Phillips, J. C. (1970). Phys. Today 23 (Feb.), 23.] P1: GKD Revised Pages Encyclopedia of Physical Science and Technology EN002D-71 May 17, 2001 20:25 Bonding and Structure in Solids 285 way in which the mixture occurs is of great importance The last case is primarily ionic materials with a covalent both scientifically and technologically. We shall discuss component. Many oxides in which the oxygen atoms are several important examples. three- to sixfold coordinated fall in this class, and this in- The simplest case occurs for the tetrahedrally coordi- cludes many ceramic materials. These materials can have nated covalent structure shown in Fig. 2. This structure high melting points and good chemical stability, but they contains two kinds of atomic sites: site A with only B are brittle and for this reason their range of technological neighbors, and vice versa. In silicon and gemanium both applications is limited. sites are occupied by the same atom, which has four valence electrons. However, one can occupy the two sites with different atoms, such that the total number of valence VI. METALLIC BONDING electrons is eight per atom pair (formally represented by N 8−N A B ). Many compounds of this type with N = 3 and Broadly speaking, three kinds of elements are found in N = 2 are known. Two examples with N = 3 are GaAs metals. They are the simple s–p metallic elements from and InP and ternary and quaternary alloys (Ga, In) (As, P). the left-hand side of the periodic table, such as lithium, alu- These materials are transparent in the near infrared, and minum, and lead; the rare-earth and transition elements their optical properties can be adjusted by “band gap engi- with f and d valence electrons, such as titanium, iron, neering.” They are important as high intensity, low power and nickel; and metalloid elements, such as carbon, sil- monochromatic light sources or as light amplifiers (lasers). icon, and phosphorus, which may also in certain combi- The next interesting case is the triatomic material SiO2 nations form covalent solids. In metals the coordination (silica). The electronegativity difference between silicon number (or number of nearest neighbors) is much larger and oxygen is large, so the bonding here contains a large (usually twice as large or more) than the number of va- ionic component. At the same time each oxygen atom lence electrons. This means that the directed valence bonds contains six valence electrons while silicon has four, so found in molecules or in covalent crystals are much weaker the total number of valence electrons per molecular unit (although not completely absent) in metals. is 16 = 8 + 8. This favors covalent bonding. In the solid The high electrical and thermal conductivity of metals each silicon atom has tetrahedral oxygen neighbors, while is a result of the absence of a gap in the energy spec- each oxygen atom has two silicon neighbors, which is trum between filled and empty electronic states. This high again the coordination characteristic of covalent bonding. electrical conductivity in turn reduces the contribution to Silica is chemically stable and can be made very pure, for cohesion associated with charge transfer because the inter- much the same reasons that silicon can. This high purity nal electric fields are limited by electronic redistribution is essential to technological applications in the context of or charge flow on an atomic scale. Thus, ionic interactions optical fibers for communications. are reduced in metals compared with ionic crystals. Another feature of silica is that it can easily be cooled The reduction of covalent or molecular bonding as well into a solid state that is not crystalline but more like a as ionic bonding in metals presents a paradox. If nei- frozen liquid. The state is called a glass. The ductility of ther of these bonding mechanisms is fully effective, to glasses at high temperatures is essential to the manufac- what forces do metals owe their cohesion? Modern quan- ture of optical fibers. However, glasses are also ductile on tum theory shows a complex correlation of the motion of a molecular scale and so do not form molecular “cracks,” metallic valence electrons, which reduces the Coulomb which would be arrays of broken bonds that might be elec- repulsive energy between these like charges while leaving trically active and destructive to the electronic capabilities almost unchanged the attractive Coulomb interaction be- of semiconductor devices such as transistors. It is one of tween negatively charged electrons and positively charged nature’s most felicitous accidents that silicon electronic atom cores. It is this correlation energy that is primarily devices can be packaged by simply oxidizing the surfaces responsible for metallic cohesion. of solid silicon to form a protective coating of silica, SiO2. From studies of the structure and cohesion of metals The silica coating is not only chemically stable (because of it appears that d valence electrons (as in the transition its covalent-ionic bonding), but is also mechanically stable metals) contribute almost as effectively to metallic cohe- because of the ductility of vitreous silica at the molecular sion as s and p electrons. The f electrons in rare-earth level. Thus, the interface between the crystalline silicon metals, on the other hand, play a minor role in metallic electronic device and the silica coating is itself almost per- cohesion but occasionally have magnetic properties. Tran- fect. It does not store fixed charge, even when the thickness sition metals are notable for their strong magnetic proper- of the silica is only a few molecular layers. This greatly ties (iron, cobalt, and nickel), as well as their high melting enhances the performance of microelectronic devices (in- points and refractory properties, which result from the tegrated circuits on silicon “chips.” large number of combined s, p, and d valence electrons. P1: GKD Revised Pages Encyclopedia of Physical Science and Technology EN002D-71 May 17, 2001 20:25 286 Bonding and Structure in Solids The compound with the highest known melting point is This has been done in several ways, which are substan- N M tungsten carbide (WC), an interesting combination of a tially equivalent. The simplest case is A B compounds transition element whose d levels are half-full with a met- where A and B have only s and p valence electrons and alloid element whose s and p valence levels are half-full. N + M = P = 8, which means that the s and p valence Also, here tungsten is very large and carbon is very small, levels are half-full. In this case one can separate ionic and which makes possible an ionic contribution to the cohesive covalent crystal structures by separating the average en- and refractory properties. ergy gap between occupied and empty electronic states into ionic and covalent components, represented by C and Eh, respectively. Both NaCl (ionic) and diamond, silicon, VII. QUANTUM STRUCTURAL DIAGRAMS and germanium (covalent) crystals (Fig. 2) belong in this group, with C/Eh = 0 in the latter and C /Eh largely in N 8−N The description of structure and bonding in solids given the former. The quantum structural diagram for A B in the preceding sections is largely qualitative, but it is a nontransition-metal compounds shown in Fig. 4 not only fair (although abbreviated) account of most of what was is a huge improvement on the classical structure diagram generally known as a result of quantum mechanical anal- shown in Fig. 1 but also is an exact separation of covalent ysis in the period from 1930 to 1960. Starting in 1960 a and ionic crystal structures. more quantitative description was developed that enables us to inspect systematic trends in structure and bonding with the aid of quantum structural diagrams. With a structural diagram one assigns to each element certain characteristics and then treats these characteristics as configuration coordinates, which are used to construct structural maps. The natural classical configuration coordinates are atomic size and electronegativity, as defined by Pauling (see Table I). To these we may add the number of valence electrons per atom. One then takes a N M class of binary compounds, say A B , with the same value of P = N + M and uses size differences (or ratios) as well as electronegativity differences as Cartesian coordinates. If the characteristics or configuration coordinates have genuine value for describing structure and bonding, compounds composed of different elements A and B but with similar values of their Cartesian coordinates, should have the same crystal structure. Put somewhat differently, the structural map should separate into simple regions, with each region containing compounds with the same crystal structure. Early attempts to construct structural maps of this kind using classical coordinates were only partially successful; as many as 10 or 20% or more of the compounds were misplaced. From this failure most workers concluded that the problem of structure and bonding in solids, and espe- cially in metals where the number of known compounds 4 exceeds 10 , was simply too complex to solve in any simple way. Finding a solution was left to the indefinite future, when computers became large enough and quantum mechanical methods accurate enough to predict structures on a case-by-case basis. FIGURE 4 The separation of the energy gap shown in Fig. 3 into Recent research has shown that the idea of structural di- covalent and ionic components (Eh and C, respectively) generates agrams is itself valid but that previous failures arose from a structural map that separates fourfold- and sixfold-coordinated N 8−N A B crystals perfectly (no transition or rare-earth elements). the use of largely classical coordinates. In addition to the The structures and coordination numbers (in parentheses) are as number of valence electrons per atom (a quantum con- follows: ♦, diamond, zinc blend (4); , wurtzite (4); ■, rock salt cept), one must also use other quantum variables to replace (6); ❤, rock salt/wurtzite (6, 4). [From Cohen, M. L., Heine, V., the classical variables of atomic size and electronegativity. and Phillips, J. C. (1982), Sci. Am. 246 (6), 82.] P1: GKD Revised Pages Encyclopedia of Physical Science and Technology EN002D-71 May 17, 2001 20:25 Bonding and Structure in Solids 287 VIII. COMPLETE QUANTUM STRUCTURE ANALYSIS On a case-by-case basis a full discussion of structure and bonding in a given solid can be achieved using the most ad- vanced computational techniques combined with the most sophisticated computers. Work with sufficient precision and flexibility to describe the structure of solid surfaces, point defects, and solid transitions under high pressures became available in selected cases in the 1980s. An ex- cellent example is shown in Fig. 6, which gives the total energy of crystalline silicon in different crystal structures as a function of volume. From these curves transition pressures and volumes can be obtained from the tie-line (common tangent) construction due (∼100 years ago) to Gibbs. It is interesting that all the results shown in Figs. 4, 5, and 6 are based on a particular approach to the quantum structure of solids that is known as the pseudopotential method. N 8−N FIGURE 5 A general separation of A B crystal structures utilizes quantum coordinates defined for all elements including rare-earth and transition metals. Compounds containing the latter are indicated by open symbols. [From Villars, P. (1983). J. Less- Common Met. 92, 215.] To extend this analysis to transition and rareearth metals as well as compounds in which the valence shell is not exactly half-full is a monumental taks that includes ∼1000 AB compounds, ∼1000 AB2 compounds, and more than 1000 AB3 and A3B5 compounds, as well as more than 7000 ternary compounds. The correct quantum coordinates for these 10,000 compounds have been identified from a field of 182 candidate coordinates, some classical and some quantum coordinates. All the best coordinates are found to be quantum coordinates, and these turn out to be the atomic ionization potential and a suitably N 8−N defined quantum core size. The result for A B compounds (where A or B or both may be transition or rare- earth elements) is shown in Fig. 5. It is representative of the best global analysis of structure and bonding in solids available in 1992. This structural map is 97% successful. In addition to binary compounds one can use diagrams to analyze ternary compounds. Ternary ionic compounds usually contain two kinds of cations, and their structures are determined by cation radius ratios. Ternary metallic compounds are more complex, and their structures FIGURE 6 A plot of the total energy of silicon crystals in different are determined by valence electron numbers, size differ- crystal structures as a function of atomic volume. At atmospheric ences, and electronegativity differences, much as for the pressure the diamond structure has the lowest energy, but at pressure of hundreds of thousands of atmospheres silicon is more binary compounds in Fig. 5. Many structure–property re- stable in other structures. Such high pressures can be produced lationships can be recognized with these diagrams which in the laboratory, and they are also found at great depths below conveniently display general trends in both binaries and the earth’s surface. [From Chang, K. J., and Cohen, M. L. (1984). ternaries. Phys. Rev. B 30, 5376.] P1: GKD Revised Pages Encyclopedia of Physical Science and Technology EN002D-71 May 17, 2001 20:25 288 Bonding and Structure in Solids IX. CHEMICAL BONDING IN SOLIDS • SOLID-STATE CHEMISTRY • SOLID-STATE ELECTRO- IN THE THIRD MILLENNIUM CHEMISTRY • SUPERCONDUCTIVITY • VALENCE-BOND THEORY • X-RAY ANALYSIS The evolution of microelectronic devices towards smaller and smaller dimensions will soon reach the level of 2.5 nm BIBLIOGRAPHY (25A) or less, which is basically the molecular level. At this level the concepts of chemical bonding discussed here Adams, D. M. (1974). “Inorganic Solids,” Wiley, New York. cease to be only theoretical abstractions and become valu- Chang, K. J., and Cohen, M. L. (1984). Phys. Rev. B 30, 5376. able tools for guiding microelectronic device design and Cohen, M. L., Heine, V., and Phillips, J. C. (1982). Sci. Am. 246(6), 82. manufacture. A remarkable aspect of much recent research Pauling, L. (1960). “Nature of the Chemical Bond,” Cornell Univ. Press, is that it demonstrates that both macroscopic and quantum Ithaca. ideas of materials can be implemented at the molecular Phillips, J. C. (1970). The chemical bond and solid state physics, Phys. Today 23 (February), 23. level when the processes involved are well controlled. Phillips, J. C. (1974). In “Solid State Chemistry” (N. B. Hannay. ed.), Vol. 1: The Chemical Structure of Solids, Plenum, New York. Tosi, M. P. (1964). Solid-State Phys. 16, 1. SEE ALSO THE FOLLOWING ARTICLES Villars, P. (1983). J. Less-Common Met. 92, 215. Villars, P. (1985). J. Less-Common Met. 109, 93. CRYSTALLOGRAPHY • EXCITONS, SEMICONDUCTOR • Villars, P., and Phillips, J. C. (1988). Phys. Rev. B 37, 2345. FERROMAGNETISM • GLASS • QUANTUM MECHANICS Wigner, E. P., and Seitz, F. (1955). Solid-State Phys. 1, 1. P1: GNH Qu: 00, 00, 00, 00 Encyclopedia of Physical Science and Technology EN002J-99 May 17, 2001 20:50 Chemical Physics Richard Bersohn Bruce J. Berne Columbia University I. Properties of Individual and Pairs of Molecules II. Collective Properties GLOSSARY chanics and quantum statistical mechanics developed by Feynman. Born–Oppenheimer approximation A quantum me- Radial distribution function The average density of chanical explanation for the approximate separation of fluid atoms as a function of distance from a given fluid molecular energy into electronic, vibrational, and rota- atom. tional energies. Raman scattering An inelastic scattering of a photon by Electric multipole moment If the charge density of a a molecule; the difference in energy between the inci- system is ρ(r, θ, φ) where r, θ, φ are spherical polar dent and scattered photon is a difference of molecular ocofoarvdeirnaagtes,∫thρen(rt,hθe,lφth)rm1Yu1lmti(pθo,lφe)mdoVm.eTnhtseamreotmheensetst SpecnterrogsycolepvyelsT.he measurement of energy levels. of a spherically symmetric charge distribution are Statistical mechanics A general theory of many parti- zero. cle systems that relates bulk properties to microscopic Green–Kubo relations Expressions for transport coeffi- properties. cients such as viscosity, thermal conductivity, and rate Time correlation functions A function that describes the constants in terms of time correlation functions. correlation between properties of a system at different Molecular dynamics method A method for simulating times. the properties of many-body systems based on solving classical equations of motion. Monte Carlo method A method for simulating the equi- CHEMICAL PHYSICS is the physics of the individual librium properties of many-body systems based on ran- and collective properties of molecules. However, the dis- dom walks. tinction between chemistry and chemical physics is largely Normal coordinates The coordinates of a vibrating sys- a matter of emphasis. The approach of the chemical physi- tem that oscillate with a single frequency. cist is theoretical. He searches for underlying theoreti- Partition function A sum over quantum states used to de- cal principles, and the molecules that he uses are often a termine thermodynamic properties from the quantum means to an end, whereas the synthetic chemist usually mechanical energy levels. considers the molecules that he synthesizes and their re- Path integral methods A formulation of quantum me- actions as ends in themselves. This article on chemical 739 P1: GNH Encyclopedia of Physical Science and Technology EN002J-99 May 17, 2001 20:50 740 Chemical Physics physics is divided into two sections, one on phenomena units of energy used to describe spectroscopic phenom- 7 which depend primarily on the properties of individual ena. The SI unit of energy is the joule (1 J = 10 ergs), and pairs of molecules and the other on phenomena which but this unit is rarely used by spectroscopists. Just as are primarily collective. the nuclear spectroscopist uses one million electron volts 10 (1 MeV = 9.65 × 10 J) as a unit of energy, the X-ray 4 spectroscopist uses electron volts (1 eV = 9.65 × 10 J). I. PROPERTIES OF INDIVIDUAL The ultraviolet, visible, and infrared spectroscopists use −1 AND PAIRS OF MOLECULES the cm unit, which is not an energy unit but a reciprocal −1 of a wavelength (1 cm × h/c = 11.96 J). The microwave Studies in chemical physics can be loosely classified as and radiofrequency spectroscopists use the megahertz −4 spectroscopic, structural, and dynamic. Spectroscopy is (1 MHz × h = 3.99 × 10 J), and nuclear magnetic res- concerned with the determination of molecular energy lev- onance (NMR) spectroscopists often use a dimensionless els. Structural studies are aimed at finding the distribution quantity, the relative frequency shift in parts per million. of particles within a molecule and molecules within a liq- Some spectroscopists use units of energy, others units of uid or solid. The location of the nuclei defines the structure reciprocal wave length, and still others frequency units. of the molecule, that is, the distances between nuclei and Indeed the reference to spectral features by their wave the angles between internuclear vectors. The distribution lengths is still a very common practice, although it is to of electrons is intimately connected with the forces that be deplored. There are natural reasons for these differ- hold the atoms together. Dynamics involves the relation of ent choices of energy units but they are bewildering to the the rate of molecular transformations and changes of state beginner. In a way, these units are like the light year, a non- caused by collision to the intra- and intermolecular forces. standard unit convenient to the astronomer but to nobody else. Molecular spectroscopy is generally divided into four A. Molecular Spectroscopy branches, each corresponding to a different type of motion Spectroscopy is the measurement of energy level differ- and, in general, to a different frequency range. Electronic ences. This is most usually accomplished by measuring the spectroscopy is the study of the differences in electronic frequencies of light absorbed or emitted by a molecule, but energy levels which occur in atoms and molecules; the 14 17 it is sometimes done by measurements of the energy of an corresponding frequencies are 10 −10 Hz. Vibrational incident photon or particle together with a measurement spectroscopy is the study of molecular vibrations, whose 12 14 of a scattered photon or particle. For example, the fre- frequencies are typically of the order of 10 −10 Hz. Ro- quency of scattered light may differ from that of the inci- tational spectroscopy is the study of the rotational energy dent light. The absolute value of the frequency difference levels of molecules; corresponding rotational frequencies 10 12 is a difference of energy levels of the molecule divided are typically in the range of microwaves, 10 −10 Hz. by Planck’s constant. This phenomenon, called Raman Finally, NMR spectroscopy is the study of the magnetic scattering, has many analogs. Electron loss spectroscopy fields acting on a nucleus. is extensively used to measure vibrational frequencies of The given divisions are not quite as sharp as represented. surfaces. The difference in energy between incident and Frequency ranges overlap, and electronic spectroscopy is scattered electrons is, in general, a quantized energy left in often a source of information of vibrations and rotations the solid. When very slow (“cold”) neutrons are scattered as well as electronic motion. Nevertheless, this classifi- by a warm liquid or solid, the scattered neutrons move cation of spectra is of fundamental importance. The the- faster than those in the incident beam. oretical justification for this classification is the Born– In some spectroscopies the scattered particle whose en- Oppenheimer approximation. This is an approximation 4 5 ergy is measured is not the same as the incident particle. that exploits the fact that the electron is 10 −10 times For example, in photoelectron spectroscopy an incident lighter than a typical nucleus. Classically speaking, the photon with known energy whose wavelength is in the electron revolves around the molecule so rapidly that dur- XUV (<100 nm) or X-ray region produces a photoelec- ing an electron period the nuclei do not have time to react tron. Careful measurements of the kinetic energy of the to the different positions of the electron and barely move. electron yield energy level spacings in the positive ion. In Thus, electronic energies can be calculated with the as- photodissociation spectroscopy, when a molecule is dis- sumption that the nuclei are stationary. They are not sta- sociated into fragments by light, measurement of the ki- tionary, of course, but the electronic energy levels can (in netic energy of a fragment yields the internal energy of the principle) be calculated for any given arrangement of the fragments. nuclei. The arrangement or configuration that produces Before discussing spectroscopy, it might be appropri- the least electronic energy is the equilibrium structure of ate to warn and console the reader about the multitude of the molecule.