Mechanic quantum

Quantum mechanics (QM) is the correct mathematical description of the behavior of electrons and thus of chemistry. In theory, QM can predict any property of an individual atom or molecule exactly. In practice, the QM equations have only been solved exactly for one electron systems. A myriad collection of methods has been developed for approximating the solution for multiple electron systems. These approximations can be very useful, but this requires an amount of sophistication on the part of the researcher to know when each approximation is valid and how accurate the results are likely to be. A significant portion of this book addresses these questions. 

Two equivalent formulations of QM were devised by Schrodinger and Heisenberg. Here, we will present only the Schrodinger form since it is the basis for nearly all computational chemistry methods. The Schrodinger equation is 

The wave function T is a function of the electron and nuclear positions. As the name implies, this is the description of an electron as a wave. This is a probabilistic description of electron behavior. As such, it can describe the probability of electrons being in certain locations, but it cannot predict exactly where electrons are located. The wave function is also called a probability amplitude because it is the square of the wave function that yields probabilities. This is the only rigorously correct meaning of a wave function. In order to obtain a physically relevant solution of the Schrodinger equation, the wave function must be continuous, single-valued, normalizable, and antisymmetric with respect to the interchange of electrons. 

In currently available software, the Hamiltonian above is nearly never used. The problem can be simplified by separating the nuclear and electron motions. This is called the Born-Oppenheimer approximation. The Hamiltonian for a molecule with stationary nuclei is 

the first term is the kinetic energy of the electrons only. The second term is the attraction of electrons to nuclei. The third term is the repulsion between electrons. The repulsion between nuclei is added onto the energy at the end of the calculation. The motion of nuclei can be described by considering this entire formulation to be a potential energy surface on which nuclei move. 

In quantum mechanics the state of a many-particle system is represented by a wavefunction T (r, Z), observables correspond to hermitian operators and results of measurements are represented by expectation values of these operators. 

When A is substituted with the unity operator, Eq. (1.108) shows that acceptable wavefunctions should be normalized to 1, that is, (/1 /) = I.A central problem is the calculation of the wavefunction, 4 (r, Z), that describes the time-dependent state of the system. This wavefunction is the solution of the time-dependent 

The hermitian conjugatre of an operator A is the operator that satisfies 

In the so-called coordinate representation the potential energy operator U amounts to a simple product, that is, (r- )4 (r-, Z) = t/(r ) h(r-, Z) where is 

Equation (1.116) is an eigenvalue equation, and andE are eigenfunctions and corresponding eigenvalues of the Hamiltonian. If at time t = Q the system is in a state which is one of these eigenfunctions, that is, 

We will discuss quantum mechanics extensively in Chapters 5 and 6. It provides the best description we have to date of the behavior of atoms and molecules. The Schrodinger equation, which is the fundamental defining equation of quantum mechanics (it is as central to quantum mechanics as Newton s laws are to the motions of particles), is a differential equation that involves a second derivative. In fact, while Newton s laws can be understood in some simple limits without calculus (for example, if a particle starts atx = 0 and moves with constant velocity vx,x = vxt at later times), it is very difficult to use quantum mechanics in any quantitative way without using derivatives. 

Unlike molecular mechanics, the quantum mechanical approach to molecular modelling does not require the use of parameters similar to those used in molecular mechanics. It is based on the realization that electrons and all material particles exhibit wavelike properties. This allows the well defined, parameter free, mathematics of wave motions to be applied to electrons, atomic and molecular structure. The basis of these calculations is the Schrodinger wave equation, which in its simplest form may be stated as  

Schrodinger equations for atoms and molecules use the the sum of the potential and kinetic energies of the electrons and nuclei in a structure as the basis of a description of the three dimensional arangements of electrons about the nucleus. Equations are normally obtained using the Born-Oppenheimer approximation, which considers the nucleus to be stationary with respect to the electrons. This approximation means that one need not consider the kinetic energy of the nuclei in a molecule, which considerably simplifies the calculations. Furthermore, the 

The precise mathematical form of E T for the Schrodinger equation will depend on the complexity of the structure being modelled. Its operator H will contain individual terms for all the possible electron-electron, electron-nucleus and nucleus-nucleus interactions between the electrons and nuclei in the structure needed to determine the energies of the components of that structure. Consider, for example, the structure of the hydrogen molecule with its four particles, namely two electrons at positions and r2 and two nuclei at positions R and R2. The Schrodinger Equation (5.5) may be rewritten for this molecule as  

It is not possible to obtain a direct solution of a Schrodinger equation for a structure containing more than two particles. Solutions are normally obtained by simplifying H by using the Hartree-Fock approximation. This approximation uses the concept of an effective field V to represent the interactions of an electron with all the other electrons in the structure. For example, the Hartree-Fock approximation converts the Hamiltonian operator (5.7) for each electron in the hydrogen molecule to the simpler form  

The postulates and theorems of quantum mechanics form the rigorous foundation for the prediction of observable chemical properties from first principles. Expressed somewhat loosely, the fundamental postulates of quantum mechanics assert dial microscopic systems are described by wave functions diat completely characterize all of die physical properties of the system. In particular, there aie quantum mechanical operators corresponding to each physical observable that, when applied to the wave function, allow one to predict the probability of finding the system to exhibit a particular value or range of values (scalar, vector. 

Atomic orbitals represent the locations of electrons in atoms, and are derived from quantum mechanical calculations. The consequence of being in considerable ignorance about the position of an electron in an atom is that calculations of the probability of finding an electron in a given position must be made. 

The solution of the Schrodinger equation gives mathematical form to the wave functions which describe the locations of electrons in atoms. The wave function is represented by /, which is such that its square, i /2, is the probability density of finding an electron. 

but the combination refers to 5g orbitals which are not used by elements in their ground states. 

The mathematical functions representing atomic orbitals are usually normalized, ie. the wave function is arranged so that the integral of its 

The solutions of the Schrodinger equation show how j/ is distributed in the space around the nucleus of the hydrogen atom. The solutions for v / are characterized by the values of three quantum numbers and every allowed set of values for the quantum numbers, together with the associated wave function, strictly defines that space which is termed an atomic orbital. Other representations are used for atomic orbitals, such as the boundary surface and orbital envelopes described later in the chapter. 

A promising simplification has been proposed by Bader (1990) who has shown that the electron density in a molecule can be uniquely partitioned into atomic fragments that behave as open quantum systems. Using a topological analysis of the electron density, he has been able to trace the paths of chemical bonds. This approach has recently been applied to the electron density in inorganic crystals by Pendas et al. (1997, 1998) and Luana et al. (1997). While this analysis holds great promise, the bond paths of the electron density in inorganic solids are not the same as the more traditional chemical bonds and, for reasons discussed in Section 14.8, the electron density model is difficult to compare with the traditional chemical bond models. 

Other simplified quantum treatments, such as the Lewis electron pair and orbital overlap models, have proved useful in teaching and they give qualitative predictions of the structures of molecular compounds, but they become unwieldy when applied to solids. They have not proved to be particularly helpful in the description of the complex structures found in inorganic chemistry and have therefore not been widely used in this field. 

Electronic structure theory, which is based on quantum mechanics, is the most accurate method to calculate the structure, energy, and properties of a molecule or an assembly of molecules. It helps in computing many of the thermochemical, physical, spectroscopic, and kinetic (rate of reactions) properties of the molecules. 

A brief description of the fundamental concepts underlying the ab initio molecular orbital computations is presented here. The reader is requested to refer to some of the textbooks providing comprehensive discussion of molecnlar orbital theory (Atkins 1991 Atkins and Friedman 1997 Cook 1998 Levine 1983 Simons and Nichols 1997). In electronic structure theory, given the position of atomic nuclei, R [under the Bom-Oppenheimer approximation (Bom and Oppenheimer 1927)], the Schrodinger equation for motion of electrons (r) is solved as  

For the purpose of our discussion, let us assume that only electrons have important quantum-mechanical behavior. Five assertions about quantum mechanics will enable us to discuss properties of electrons. Along with these assertions, we shall make one or two clarifying remarks and slate a few consequences. 

The way we use a wave function of an electron and the operator representing an observable is stated in a third assertion  

In a similar way the denominator on the right side of Eq. (1-3) can be shortened to ij/11/ ). The angular brackets are also used separately. TheAra (l or means i/ i(r) the ket 2) or means (These terms come from splitting the 

(1-3) is the principal assertion of the quantum mechanics needed in this book. Assertions (a) and (b) simply define wave functions and operators, but assertion (c) makes a connection with experiment. It follows from Eq. (1-3), for example, that the probability of finding an electron in a small region of space, (fir, is i// (r)i//(r)d r TItus is the probability density for the electron. 

It follows also from Eq. (1-3) that there exist electron states having discrete or definite values for energy (or, states with discrete values for any other observable). This can be proved by construction. Since any measured quantity must be real, Eq. (1-3) suggests that the operator 0 is Hermitian. We know from mathematics that it is possible to construct eigenstates of any Hermitian operator. However, for the Hamiltonian operator, which is a Hermitian operator, eigenstates are obtained as solutions of a differential equation, the time-independent Schroedinger equation. 

A very simple problem which can be treated on the basis of the Schrodinger equation is that of an electron of mass m which is able to move in one dimension only, say along the X axis. The total energy of the electron is the sum of its potential and kinetic energies. The potential energy depends on the environment and may be written in general as V x). In quantum-mechanical theory, the familiar term for kinetic energy is replaced by the differential operator 

When this is attempted, solutions of the differential equation can only be obtained for certain values of the energy E. Such energy values are known as the eigenvalues of the equation. They are the permitted quantized energy levels for the system, and we see that the quantum restriction has appeared naturally, and not arbitrarily as in 

Usually, of course, we are concerned with three-dimensional rather than onedimensional problems. If we decide to work with Cartesian coordinates, the potential energy will be a function of x, y, and z, and can be written as U x,y,z). The Schrodinger equation now takes the form 

The three-dimensional-Schrodinger equation (1.13) could have been written in the form 

The operator in the brackets in equation (1,14) is known as the Hamiltonian operator, since it is related to an expression for energy given by the Irish mathematician Sir William Rowan Hamilton (1805- 1865) this operator is given the symbol H  

These limits can be pushed back by the extension of existing force fields and the development of new ones the refinement of generic force fields (see Section 3.3) quantum-mechanically driven molecular mechanics (e.g., for transition states see Section 3.3) the development of tools that refine parameter sets based on data banks, including genetic algorithms and neural networks or more conventional techniques (see Section 3.3 and 16.3). 

The advantage of ab-initio quantum-mechanical methods is their ability to handle any element of the Periodic Table, and ground states as well as excited and transition states. The cost is a heavy consumption of computing resources, and this limits the size of systems that can be treated. These limits can be overcome by using combined QM/MM methods (see Section 3.3), or by the thorough investigation of simplified models of the molecular systems of interest, and approximations to simplify ab-initio quantum mechanics, where certain quantities are neglected or replaced by parameters fitted to experimental data. 

An important modification of the general time-independent Schrodinger equation [Eq. (2.10)1 is that based on the Born-Oppenheimer approximation [23], which 

As the general multi-electron problem caimot be solved in closed form, the electron density for a given geometry is usually calculated until self-consistence is reached (SCF methods). 

An important modification of the general Schrodinger equation (Eq. 2.10) is that based on the Born-Oppenheimer approximation[l ], which assumes stationary nuclei. Further approximations include the neglect of relativistic effects, where they are less important, and the reduction of the many-electron problem to an effective one-electron problem, i. e., the determination of the energy and movement 

Using such an extremal principle, we can obtain also the equation of motion. Lagrange could show further that the use of the laws of Newton and the use of the principle of the least action are two equivalent formulations of the same problem. The expansion of these concepts on quantum mechanics is naturally possible. We become aware due to these considerations that symmetry obviously is related to extremal principles in the mechanical sense. On the other hand, obviously a system tries to achieve a symmetrical condition, when disturbances are missing, which tend to produce as3nmnetry. 

We emphasize still another example from physical chemistry. In the derivation of the Kelvin equation, i.e., the vapor pressure of small droplets, we immediately and naturally accept that the shape of the droplets is spherical. Whoever spills mercury knows this. It is obviously not at all necessary to justify spherical droplets. Spherical droplets are in our common sense. But an extremal principle is behind the spherical droplets. A body with a given volume seeks to reach a minimum boundary surface, if no disturbance is aside. Particles that are bigger than a certain size exhibit an obvious deviation from the spherical shape under the effect of gravity. 

The evaluation of integrals in quantum mechanics is greatly facilitated by the symmetry properties of the integrals. Integrating from —oo to H-oo, in the case of an odd function the integral becomes zero. An odd function is a function with the property /(x) = -/(-x). An even function is a function with the property /(x) = /(—x). 

Examples for odd and even functions are sin(x) and cos(x). We can resolve each arbitrary function into an even part and an odd part. 

It is well known that the entropy is a measure for the disorder of a system. It is clear that the entropy can serve also as a measure for the order of a system. However, the fact that we can understand entropy as a measure of symmetry is rather emphasized rarely. 

Caution Comparing the shifted constant dielectric to a constant dielectric function without a cutoff shows that the shifted dielectric, unlike a switching function, perturbs the entire electrostatic energy curve, not only the region near the cutoff. 

This section provides an overview and review of quantum mechanics calculations. The information can help you use Hyper-Chem to solve practical problems. For quantitative details of quantum mechanics calculations and how HyperChem implements them, see the second part of this book. Theory and Methods. 

Ab initio quantum mechanics methods have evolved for many decades. The speed and accuracy oiab initio calculations have been greatly improved by developing new algorithms and introducing better basis functions. 

Semi-empirical quantum mechanics methods have evolved over the last three decades. Using today s microcomputers, they can produce meaningful, often quantitative, results for large molecular systems. The roots of the methods lie in the theory of % electrons, now largely superseded by all-valence electron theories. 

The British physicist and mathematician Sir Isaac Newton formulated his laws of motion in the 17th century. These laws predict the course of an object when subjected to various forces, such as a push, pull, or a collision with another object. In Newtonian physics, physicists can predict the motion of an object with any desired degree of accuracy if all the forces acting on it are precisely known. 

In the 20th century, physicists discovered to their surprise that small particles such as atoms and the components of atoms do not obey Newton s law of motion. Instead of being deterministic—following trajectories determined by the laws of physics—tiny bits of matter behave probabilistically, meaning that their state or trajectory is not precisely determined but can follow one of a number of different options. The German physicist Werner Heisenberg proposed his uncertainty principle in 1927, which states that there is generally some amount of uncertainty in measurements of a particle s state. 

Also in the 1920s, the Austrian physicist Erwin Schroding-er developed an equation that describes the motion of small particles. This equation, which continues to be used today. 

Quantum mechanics governs the behavior of tiny particles such as atoms. This theory has been confirmed in many different experiments 

This computer, Blue Gene/L at the Lawrence Livermore National Laboratory in California, is one of the world s fastest computers and can perform simulations of many physical processes. (Lawrence Livermore National Laboratory) 

Ernest Rutherford, first baron Rutherford of Nelson (1871-1937) 

Where the Schrodinger or Dirac equations apply, quantization will appear from sundry mathematical conditions on the existence of physically meaningful solutions to these differential equations. However, in nonrela-tivistic terms, spin does not come from a differential equation It comes from the assumptions of spin matrices, or from necessity (the Dirac equation does yield spin = 1/2 solutions, but not for higher spin). So we must posit quantum numbers (see Section 2.12) even when there are no differential equations in the back to comfort us. This is especially true for the weak and strong forces, where no distance-dependent potential energy functions have been developed. 

what is left Well, here are the requirements for quantum mechanics  

Particles and waves have a dual nature Particles with mass m and momentum p = mv can become waves, with an equivalent de Broglie5 wavelength X [4] given by 

The observation of a quantum-mechanical system involves the disturbance of the state being observed the Heisenberg6 uncertainty principle [5] dictates that the uncertainty Ax in position x and the uncertainty Apx in momentum px in the x direction (or in y or in z, or the uncertainty in any two canonically conjugate variables, e.g. energy E and time f, or angular momentum L and phase (f , i.e. variables whose 

The covalent bond is typical of the compounds of carbon it is the bond of chief tance in the study of organic chemistry. 

ProWem Lt Which of the fj lowing would you expect to be ionic, and which non-ionic C c a simple electronic struciure for each, showing only valence sheii elections. (a) KBr. (c) NFj (e)CaS04 (g) PH  

Problem 1.2 Give a likely simple electronic structure for each of the foliowiof, uming them to be completely covalent. Assume that every atom (except hydro n, ot course) has a complete octet, and that two atoms may shard more than one pair of electrons..  

In 1926 there emerged the theory known as quantum mechanics, developed, in e form most useful to chemists, by Erwin Schrodinger (of the University of r. . h). He worked out mathematical expressions to describe the motion of an - ron in terms of its energy. These mathematical expressions are called wave. - 15, since they are based upon the concept that electrons show properties not 

It is possible to use the concept of partide-wave duality to provide a simple explanation of the behavior of carriers in a semiconductor nanocrystal. In a bulk inorganic semiconductor, conduction band electrons (and valence band holes) are free to move throughout the crystal, and their motion can be described satisfactorily by a linear combination of plane waves, the wavelength of which is generally of the order of nanometers. This means that, whenever the size of a semiconductor solid 

The spectacular success of Bohr s theory was followed by a series of disappointments. Bohr s approach did not account for the emission spectra of atoms containing more than one electron, such as atoms of helium and lithium. Nor did it explain why extra lines appear in the hydrogen emission spectrum when a magnetic field is applied. Another problem arose with the discovery that electrons are wavelike How can the position of a wave be specified We cannot define the precise location of a wave because a wave extends in space. 

To describe the problem of trying to locate a subatomic particle that behaves like a wave, Werner Heisenberg formulated what is now known as the Heisenberg uncertainty principle it is impossible to know simultaneously both the momentum p (defined as mass times velocity) and the position of a particle with certainty. Stated mathematically. 

In reality, Bohr s theory accounted for the observed emission spectra of He and ions, as well as that of hydrogen. However, all three systems have one feature in common—each contains a single electron. Thus, the Bohr model worked successfully only for the hydrogen atom and for hydrogenlike ions.  

Applying the Heisenberg uncertainty principle to the hydrogen atom, we see that in reality the electron does not orbit the nucleus in a well-defined path, as Bohr thought. If it did, we could determine precisely both the position of the electron (from its location on a particular orbit) and its momentum (from its kinetic energy) at the same time, a violation of the uncertainty principle. 

The wave function itself has no direct physical meaning. However, the probability of finding the electron in a certain region in space is proportional to the square of 

In the general case, the Schrodinger equation takes the form 

Atomic systems consist of electrons and nuclei, hence Eq. 4.2 becomes 

The reader probably remembers from chemistry courses that the electrons orbiting around in the outer shells of an atom can only have certain energies, but not others. Nature allows these energy levels to be whole numbers of the lowest energy units, such as 2.0 times the lowest level, or 3.0 times, but not 2.1 or 2.2. 

It matters little whether we can fully understand a new scientific principle, or whether we like it and feel comfortable with it. What matters most is whether we can use it to make valuable things (without doing any harm, of course). Programs that will operate in a desktop computer can now use quantum mechanics calculations to design new medicines and new microwave semiconductors for cellular phones, things which are obviously of great value. 

As far as understanding quantum mechanics is concerned, the math has now advanced to the uncomfortable point where some of its principles say certain conclusions have to be true, and others say they can t be true, simultaneously. In other words, human beings don t seem to be able to understand this relatively new science. However, we certainly are making good use of it, without a full understanding. 

Left X-ray diffraction pattern of aluminum foil. Right Electron diffraction of aluminum foil. The similarity of these two patterns shows that electrons can behave like X rays and display wave properties. 

To be sure, Bohr made a significant contribution to our understanding of atoms, and his suggestion that the energy of an electron in an atom is quantized remains unchallenged. But his theory did not provide a complete description of electronic behavior in atoms. In 1926 the Austrian physicist Erwin Schrodinger, using a complicated mathematical technique, formulated an equation that describes the behavior 

A representation of the electron density distribution surrounding the nucleus in the hydrogen atom. It shows a high probability of finding the electron closer to the nucleus. 

Bohr s theory firmly established the concept of atomic energy levels. It was unsuccessful, however, in accounting for the details of atomic structure and in predicting energy levels for atoms other than hydrogen. Further understanding of atomic structure required other theoretical developments. 

Current ideas about atomic structure depend on the principles of quantum mechanics, a theory that applies to submicroscopic (that is, extremely small) particles of matter, such as electrons. The development of this theory was stimulated by the discovery of the de Broglie relation. 

A photon has a rest mass of zero but a relativistic mass m as a result of its motion. Einstein s equation E = mc relates this relativistic mass to the energy of the photon, which also equals hv. Therefore, mc = hv,or me = hv/c = h/k. 

This microscope can resolve details down to 3 nm in a sample.The operator places the sample inside the chamber at the left and views the image on the video screens. 

Scanning electron microscope image This image is of a wasp s head.Color has been added by computer for contrast in discerning different parts of the image. 

The discovery that waves could have matterlike properties and that matter could have wavelike properties was revolutionary. Although scientists had long believed that energy and matter were distinct entities, the distinction between them, at lea.st at the atomic level, was no longer clear. Bohr s theory was tremendously successful in explaining the line spectrum of hydrogen, but it failed to explain the spectra of atoms with more than one electron. The electron appeared to behave as a particle in some circumstances and as a wave in others. Neither description could completely 

Clinton Joseph Davisson (I88I-I958). American physicist. He and G. P. Thomson shared the Nobel Prize in Physics in 1937 for demonstrating the wave properties of electrons. 

Before developing the occupation number representation outlined in Sects. 

we begin with the more familiar Schrbdinger equation formulation. 

In Sect.2.2.2 we shall demonstrate that each term of the Hamilton functions (2.63) or (2.69) can be quantized as a simple harmonic oscillator. We therefore start with the wave mechanics of a simple oscillator. 

In this section we summarize the wave mechanics of the harmonic oscillator. Excellent treatments are given in [2.8,9]. 

We consider a harmonic oscillator with mass m, displacement coordinate u and momentum p = mu. The classical Hamilton function is 

The domination of science through the 19th century by the Newtonian model came to an end with the discovery of the subatomic particles in the turn of this century. Physicists soon realized that, in this fine scale, nature portrays a dual behavior sometimes particle-like, other times wavelike, the latter demonstrated in electron and proton diffraction. 

Conversely, it was also realized that waves behave some times like particles as in the case of the photoelectric effect. Here, they were forced to create the photon as the hypothetical particle corresponding to a certain amount, a quantum, of electromagnetic energy. The very existence, of course, of these quanta of energy contrasted the Newtonian notion of continuity of energy. 

Modern physics, where the concept of quantum plays a pivotal role, was thus born. Here is a small sample of the difficulties that arose in describing this dual-nature of the very small particles  

As for the conceptual difficulties that arose from this dual character of nature in the fine atomic scale, here is what two of the pioneers of the field said, as quoted by Capra, p.42  

The apparent contradiction between the particle and the wave picture was solved in a completely unexpected way which called in question the very foundation of the mechanistic world-view - the concept of the reality of matter. At the subatomic level, matter does not exist with certainty in definite places, but rather shows tendencies to exist , and atomic events do not occur with certainty at definite times and in definite ways, but rather show tendencies to occur . In the formalism of quantum theory, these tendencies are expressed as probabilities and are associated with mathematical quantities which take theform of waves. This is why particles can be waves at the same time. They are not real three dimensional waves like sound or water waves. They are probability-waves, abstract mathematical quantities with all the characteristic properties of waves which are related to the probabilities of finding the particles at particular points in space and at particular times. All the laws of atomic physics are expressed in terms of these probabilities. We can never predict an atomic event with certainty we can say only how likely it is to happen. (Capra, p.77) 

Consider a single particle with a constant energy E = p lm, constrained on a two-dimensional surface (Fig. 3.9). The phase space is now four-dimensional. Note that two-dimensional projections are shown in Fig. 3.9. 

At this point, we need to bring quantum mechanics into the discussion. The ultimate goal of this section is to discuss how quantum mechanical principles discretize the phase space of systems. 

Certainly, experimental evidence and empirical phenomena are irrefutable and although even the best human minds, like Einstein s, find it difficult to accept quantum mechanical concepts, there is no question as to their truth. Thankfully, rich and intellectually satisfying insight can result from mathematical descriptions and interpretations of quantum mechanical phenomena. 

In this chapter we rely on mathematical descriptions and interpretations to discuss quantum mechanical concepts only to the extent they can serve our purpose of presenting the statistical thermodynamics theory needed to connect microscopic to macroscopic states of matter. Specifically, we will briefly discuss three quantum mechanical dictums  

Matter exhibits a dual particle-wave character. 

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The second model is a quantum mechanical one where free electrons are contained in a box whose sides correspond to the surfaces of the metal. The wave functions for the standing waves inside the box yield permissible states essentially independent of the lattice type. The kinetic energy corresponding to the rejected states leads to the surface energy in fair agreement with experimental estimates [86, 87],... 

Actual crystal planes tend to be incomplete and imperfect in many ways. Nonequilibrium surface stresses may be relieved by surface imperfections such as overgrowths, incomplete planes, steps, and dislocations (see below) as illustrated in Fig. VII-5 [98, 99]. The distribution of such features depends on the past history of the material, including the presence of adsorbing impurities [100]. Finally, for sufficiently small crystals (1-10 nm in dimension), quantum-mechanical effects may alter various physical (e.g., optical) properties [101]. 

While field ion microscopy has provided an effective means to visualize surface atoms and adsorbates, field emission is the preferred technique for measurement of the energetic properties of the surface. The effect of an applied field on the rate of electron emission was described by Fowler and Nordheim [65] and is shown schematically in Fig. Vlll 5. In the absence of a field, a barrier corresponding to the thermionic work function, prevents electrons from escaping from the Fermi level. An applied field, reduces this barrier to 4> - F, where the potential V decreases linearly with distance according to V = xF. Quantum-mechanical tunneling is now possible through this finite barrier, and the solufion for an electron in a finite potential box gives... 

MSS Molecule surface scattering [159-161] Translational and rotational energy distribution of a scattered molecular beam Quantum mechanics of scattering processes... 

It is now necessary to examine the partition function in more detail. The energy states for translation are assumed to be given by the quantum-mechanical picture of a particle in a box. For a one-dimensional box of length a. 

Chemisoq)tion bonding to metal and metal oxide surfaces has been treated extensively by quantum-mechanical methods. Somoijai and Bent [153] give a general discussion of the surface chemical bond, and some specific theoretical treatments are found in Refs. 154-157 see also a review by Hoffman [158]. One approach uses the variation method (see physical chemistry textbooks) ... 

The purpose of this chapter is to provide an introduction to tlie basic framework of quantum mechanics, with an emphasis on aspects that are most relevant for the study of atoms and molecules. After siumnarizing the basic principles of the subject that represent required knowledge for all students of physical chemistry, the independent-particle approximation so important in molecular quantum mechanics is introduced. A significant effort is made to describe this approach in detail and to coimnunicate how it is used as a foundation for qualitative understanding and as a basis for more accurate treatments. Following this, the basic teclmiques used in accurate calculations that go beyond the independent-particle picture (variational method and perturbation theory) are described, with some attention given to how they are actually used in practical calculations. 

While not unique, the Scluodinger picture of quantum mechanics is the most familiar to chemists principally because it has proven to be the simplest to use in practical calculations. Hence, the remainder of this section will focus on the Schrodinger fomuilation and its associated wavefiinctions, operators and eigenvalues. Moreover, effects associated with the special theory of relativity (which include spin) will be ignored in this subsection. Treatments of alternative fomuilations of quantum mechanics and discussions of relativistic effects can be found in the reading list that accompanies this chapter. 

Like the geometry of Euclid and the mechanics of Newton, quantum mechanics is an axiomatic subject. By making several assertions, or postulates, about the mathematical properties of and physical interpretation associated with solutions to the Scluodinger equation, the subject of quantum mechanics can be applied to understand behaviour in atomic and molecular systems. The fust of these postulates is ... 

In classical mechanics, the state of the system may be completely specified by the set of Cartesian particle coordinates r. and velocities dr./dt at any given time. These evolve according to Newton s equations of motion. In principle, one can write down equations involving the state variables and forces acting on the particles which can be solved to give the location and velocity of each particle at any later (or earlier) time t, provided one knows the precise state of the classical system at time t. In quantum mechanics, the state of the system at time t is instead described by a well behaved mathematical fiinction of the particle coordinates q- rather than a simple list of positions and velocities. 

To extract infomiation from the wavefimction about properties other than the probability density, additional postulates are needed. All of these rely upon the mathematical concepts of operators, eigenvalues and eigenfiinctions. An extensive discussion of these important elements of the fomialism of quantum mechanics is precluded by space limitations. For fiirther details, the reader is referred to the reading list supplied at the end of this chapter. In quantum mechanics, the classical notions of position, momentum, energy etc are replaced by mathematical operators that act upon the wavefunction to provide infomiation about the system. The third postulate relates to certain properties of these operators ... 

Although not a unique prescription, the quantum-mechanical operators A can be obtained from their classical counterparts A by making the substitutions x x (coordinates) t t (time) p -Uid/dq (component of... 

Dynamical variable A Classical quantity Quantum-mechanical operator A... 

The relationship between tire abstract quantum-mechanical operators /4and the corresponding physical quantities A is the subject of the fourth postulate, which states ... 

If the system property is measured, the only values that can possibly be observed are those that correspond to eigenvalues of the quantum-mechanical operator 4. 

Wliat does this have to do with quantum mechanics To establish a coimection, it is necessary to first expand the wavefiinction in tenns of the eigenfiinctions of a quantum-mechanical operator A,... 

This provides a recipe for calculating the average value of the system property associated with the quantum-mechanical operator A, for a specific but arbitrary choice of the wavefiinction T, notably those choices which are not eigenfunctions of A. 

The fifth postulate and its corollary are extremely important concepts. Unlike classical mechanics, where everything can in principle be known with precision, one can generally talk only about the probabilities associated with each member of a set of possible outcomes in quantum mechanics. By making a measurement of the quantity A, all that can be said with certainty is that one of the eigenvalues of /4 will be observed, and its probability can be calculated precisely. However, if it happens that the wavefiinction corresponds to one of the eigenfunctions of the operator A, then and only then is the outcome of the experiment certain the measured value of A will be the corresponding eigenvalue. 

Up until now, little has been said about time. In classical mechanics, complete knowledge about the system at any time t suffices to predict with absolute certainty the properties of the system at any other time t. The situation is quite different in quantum mechanics, however, as it is not possible to know everything about the system at any time t. Nevertheless, the temporal behavior of a quantum-mechanical system evolves in a well defined way drat depends on the Hamiltonian operator and the wavefiinction T" according to the last postulate... 

Suppose that the system property A is of interest, and that it corresponds to the quantum-mechanical operator A. The average value of A obtained m a series of measurements can be calculated by exploiting the corollary to the fifth postulate... 

Starting with the quantum-mechanical postulate regarding a one-to-one correspondence between system properties and Hemiitian operators, and the mathematical result that only operators which conmuite have a connnon set of eigenfiinctions, a rather remarkable property of nature can be demonstrated. Suppose that one desires to detennine the values of the two quantities A and B, and that tire corresponding quantum-mechanical operators do not commute. In addition, the properties are to be measured simultaneously so that both reflect the same quantum-mechanical state of the system. If the wavefiinction is neither an eigenfiinction of dnor W, then there is necessarily some uncertainty associated with the measurement. To see this, simply expand the wavefiinction i in temis of the eigenfiinctions of the relevant operators... 

The one-dimensional cases discussed above illustrate many of die qualitative features of quantum mechanics, and their relative simplicity makes them quite easy to study. Motion in more than one dimension and (especially) that of more than one particle is considerably more complicated, but many of the general features of these systems can be understood from simple considerations. Wliile one relatively connnon feature of multidimensional problems in quantum mechanics is degeneracy, it turns out that the ground state must be non-degenerate. To prove this, simply assume the opposite to be true, i.e. 

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