Quantum Mechanical Many-Electron Problem

The material world of everyday experience, as studied by chemistry and condensed-matter physics, is built up from electrons and a few (or at most a few hundred) kinds of nuclei. The basic interaction is electrostatic or Coulom-bic An electron at position r is attracted to a nucleus of charge at R by the potential energy —Zj s — R, a pair of electrons at r and r repel one another by the potential energy l/ r — r, and two nuclei at R and R repel one another as Z Z/ R — R. The electrons must be described by quantum mechanics, while the more massive nuclei can sometimes be regarded as classical particles. All of the electrons in the lighter elements, and the chemically important valence electrons in most elements, move at speeds much less than the speed of light, and so are non-relativistic. 

In essence, that is the simple story of practically ever3dhing. But there is still a long path from these general principles to theoretical prediction of the structures and properties of atoms, molecules, and solids, and eventually to the design of new chemicals or materials. If we restrict our focus to the important class of ground-state properties, we can take a shortcut through density functional theory. 

These lectures present an introduction to density functionals for non-relativistic Coulomb systems. The reader is assumed to have a working knowledge of quantum mechanics at the level of one-particle wavefunctions (r) [1]. The many-electron wavefunction f (ri,r2. rjv) [2] is briefly introduced here, and then replaced as basic variable by the electron density n(r). Various terms of the total energy are defined as functionals of the electron density, and some formal properties of these functionals are discussed. The most widely-used density functionals - the local spin density and generalized gradient 

Fiolhais, F. Nogueira, M. Marques (Eds.) LNP 620, pp. 1-55, 2003. Springer-Verlag Berlin Heidelberg 2003 

These lectures are intended to teach at the introductory level, and not to serve as a comprehensive treatise. The reader who wants more can go to several excellent general sources [3,4,5] or to the original literature. Atomic units (in which all electromagnetic equations are written in cgs form, and 

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With the development of material science, fine chemistry, molecular biology and many branches of condensed-matter physics the question of how to deal with the quantum mechanics of many-particle systems formed by thousands of electrons and hundreds of nuclei has attained unusual relevance. The basic difficulty is that an exact solution to this problem by means of a straight-forward application of the Schrodinger equation, either in its numerical, variational or perturbation-theory versions is nowadays out of the reach of even the most advanced supercomputers. It is for this reason that alternative ways for handling the quantum-mechanical many-body problem have been vigorously pursued during the last few years by both quantum chemists and condensed matter physicists. As a consequence of... 

For physical systems with more than one electron, the problem of calculating the electronic properties is a quantum mechanical many-body problem. The prospect of an analytic solution is remote. For metals even the best approximate methods in use today must rely on over-simplications. Nevertheless, these methods have provided us with a great deal of insight, and will continue to be the major theoretical tool for many years to come. 

Density-functional theory has its conceptual roots in the Thomas-Fermi model of a uniform electron gas [325,326] and the Slater local exchange approximation [327]. A formalistic proof for the correctness of the Thomas-Fermi model was provided by Hohenberg-Kohn theorems, [328]. DFT has been very popular for calculations in sohd-state physics since the 1970s. In many cases DFT with the local-density approximation and plane waves as basis functions gives quite satisfactory results, for sohd-state calculations, in comparison to experimental data at relatively low computational costs when compared to other ways of solving the quantum-mechanical many-body problem. 

It seems that there is no A-representabihty problem since the conditions that ensure that a one-particle density comes from an A-particle wavefunction are well known [5]. Here, the obstacle is the constmction of the functional E [p] capable of describing a quantum mechanical A-electron system. This functional A-representabUity is still related to the A-representabUity problem of the 2-RDM. Many currently available functionals are not A-representable [6]. Consequently, the energies produced by these functionals can lie below the exact value. Even though these energy values may lie quite close to the exact ones, they do not guarantee, however, that the calculations are accurate. 

Let us start by considering the general many-electron problem of Ng valence electrons, which contribute to chemical bonding, and A ion ions, which contain the nuclei and the tightly bound core electrons. The positions of the electrons and ions are given by r, and Rj, respectively, referred to the same arbitrary origin. This problem can be described quantum-mechanically, in the absence of external fields, by the Hamilton operator Ho ... 

A simple estimate of the computational difficulties involved with the customary quantum mechanical approach to the many-electron problem illustrates vividly the point [255]. Consider a real-space representation of ( ii 2, , at) on a mesh in which each coordinate is discretized by using 20 mesh points (which is not very much). For N electrons, becomes a variable of 3N coordinates (ignoring spin), and 20 values are required to describe on the mesh. The density n(r) is a function of three coordinates and requires only 20 values on the same mesh. Cl and the Kohn-Sham formulation of DFT (see below) additionally employ sets of single-particle orbitals. N such orbitals, used to build the density, require 20 values on the same mesh. (A Cl calculation employs in addition unoccupied orbitals and requires more values.) For = 10 electrons, the many-body wave function thus requires 20 °/20 10 times more storage space than the density and sets of single-particle orbitals 20 °/10x 20 10 times more. Clever use of symmetries can reduce these ratios, but the full many-body wave function remains inaccessible for real systems with more than a few electrons. 

In the following three sections we shall discuss four applications of quantum mechanics to miscellaneous problems, selected from the very large number of applications which have been made. These are the van der Waals attraction between molecules (Sec. 47), the symmetry properties of molecular wave functions (Sec. 48), statistical quantum mechanics, including the theory of the dielectric constant of a diatomic dipole gas (Sec. 49), and the energy of activation of chemical reactions (Sec. 50). With reluctance we omit mention of many other important applications, such as to the theories of the radioactive decomposition of nuclei, the structure of metals, the diffraction of electrons by gas molecules and crystals, electrode reactions in electrolysis, and heterogeneous catalysis. 

A number of recent calculations have compared the classical result with quantum mechanical calculations. In many cases, the results from the latter techniques confirm those from classical calculations with a gratifying accuracy. However, one topic on which there is continuing controversy is the nature of the polarons in transition metal oxides. Since the classical method subsumes all the quantum mechanics of the problem into the potential function, it can only tackle problems of electronic structure in a few specific cases, the most common example of which is in non-stoichiometric oxides. Here the question is the location of the electronic hole when the system is metal deficient. The only way such a problem can be tackled by classical methods is to use the small polaron approximation and assume that the hole resides on an ion to produce a new (in effect substitutional) ion with an extra positive charge. This can be successful and the use of the small polaron approximation in crystals is discussed in detail by Shluger and Stoneham (1993). However, all calculations on the first-row transition metal oxides have assumed that the extra charge resides on the metal ion. Recent quantum calculations (Towler et al., 1994) have thrown doubt on this assumption, suggesting that the hole is on the oxide ion. Moreover, the question of whether the hole is a small polaron for all these oxides is, at present, quite uncertain. Further discussion is given in Chapter 8. 

Almost all applications of quantum mechanics to the problem of quantitatively describing molecular electronic structure begin with the choice of a suitable basis set in terms of which the electronic wavefunction is then parametrized. This choice of basis set is crucial, since it ultimately determines the accuracy of the calculation, whether it be a matrix Hartree-Fock calculation, a conflguration-interaction study or a many-body perturbation theory expansion, whether it is a calculation of the total energy of a system, the energy of interaction between two subsystems, or the determination of some electric or magnetic property. 

To understand more complex atoms containing many electrons, we must solve the many-electron Schrodinger equation. Even in classical mechanics, many-body problems are difficult, so it is not surprising that many-electron quantum mechanics, usually called quantum chemistry,... 

Finding and describing approximate solutions to the electronic Schrodinger equation has been a major preoccupation of quantum chemists since the birth of quantum mechanics. Except for the very simplest cases like H2, quantum chemists are faced with many-electron problems. Central to attempts at solving such problems, and central to this book, is the Hartree-Fock approximation. It has played an important role in elucidating modern chemistry. In addition, it usually constitutes the first step towards more accurate approximations. We are now in a position to consider some of the basic ideas which underlie this approximation. A detailed description of the Hartree-Fock method is given in Chapter 3. 

In MOT the goal is compute the electronic wavefnnction by solving the Schrodinger equation (i.e., non-relativistic quantum mechanics). Many textbooks are available on the topic. The classic text by Hehre et al. [61] emphasizes understanding and planning actual calculations. This is complemented by the theoretical approach taken by Szabo and Ostlund [62]. Unlike SEMOT, no empirical parameters are used. However, approximations are necessary to make the problem computationally tractable. The approximations can be classihed as either physical or numerical. The most serious physical approximations involve the treatment of the electrostatic repulsion among the electrons. The principal numerical approximation is the choice of mathematical functions for describing the molecular orbitals. Popular approximations are described in this section. 

One of the fundamental problems in condensed-matter physics and quantum chemistry is the theoretical study of electronic properties. This is essential to understand the behaviour of systems ranging from atoms, molecules, and nanostructures to complex materials. Since electrons are governed by the laws of quantum mechanics, the many-electron problem is, in principle, fully described by a Schrodinger equation (supposing the nuclei to be fixed). However, the electrostatic repulsion between the electrons makes its numerical resolution an impossible task in practice, even for a relatively small number of particles. 

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