Abstract This chapter begins by defining the electronic structure problem of matter. Following the introduction of the system of atomic units, the electron-nuclear many-body problem is then formulated as a Schrödinger equation of the coupled electron-nuclear many-body system. The electronic and nuclear degrees of freedom are then decoupled using the Born-Oppenheimer approximation, which defines the electronic many-body problem. An illustration of potential-energy curves of a diatomic molecule is given, as well as an example of an interacting and a noninteracting two-electron system such as the helium atom. The chapter concludes by discussing the N-electron wave function, arguing that it becomes meaningless when the number of electrons, N, becomes large, due to the exponential increase of complexity.
Abstract This chapter begins by defining the electronic structure problem of matter. Following the introduction of the system of atomic units, the electron-nuclear many-body problem is then formulated as a Schrödinger equation of the coupled electron-nuclear many-body system. The electronic and nuclear degrees of freedom are then decoupled using the Born-Oppenheimer approximation, which defines the electronic many-body problem. An illustration of potential-energy curves of a diatomic molecule is given, as well as an example of an interacting and a noninteracting two-electron system such as the helium atom. The chapter concludes by discussing the N-electron wave function, arguing that it becomes meaningless when the number of electrons, N, becomes large, due to the exponential increase of complexity.
The method of many-body Green's functions is developed for arbitrary systems of electrons and nuclei starting from the full (beyond Born-Oppenheimer) Hamiltonian of Coulomb interactions and kinetic energies. The theory presented here resolves the problems arising from the translational and rotational invariance of this Hamiltonian that afflict the existing many-body Green's function theories. We derive a coupled set of exact equations for the electronic and nuclear Green's functions and provide a systematic way to approximately compute the properties of arbitrary many-body systems of electrons and nuclei beyond the Born-Oppenheimer approximation. The case of crystalline solids is discussed in detail.
Abstract This chapter contains a review of quantum mechanics of one-electron systems, suitable for undergraduate students with a limited background of modern physics. It is first explained that matter consists of electrons and nuclei, and the concept of wave functions is introduced. The Schrödinger equation is then presented, and it is shown how the static Schrödinger equation emerges from the stationary limit of the time-dependent Schrödinger equation. Two examples are discussed: the particle in a box and plane waves. The chapter finishes by listing important concepts and properties of quantum mechanics, such as probability density, normalization, observables, operators, energy spectra, and orthonormal sets.
In this chapter we study the energy levels of two-electron atoms, principally helium. A two-electron atom is a system of three interacting particles, a problem extremely difficult to analyze within classical mechanics. Since the mass of the nucleus is much larger than that of the electrons, the nuclear motion may be ignored and the problem reduces to two interacting particles moving in an external potential. This is still difficult to solve within classical mechanics. The old Bohr-Sommerfeld quantum mechanics also had little success with this problem. Thus it was a critical triumph for wave mechanics to provide correct calculations of the energy levels of two-electron atoms. Indeed it is remarkable that very little effort is required, as we shall see, to give a good estimate for the ground state energy. Further calculations for two-electron atoms provided six-figure agreement between theory and experiment, a striking proof of the validity of wave mechanics.
The calculation of a wavefunction took about two afternoons, and five wavefunctions were calculated in the whole …. Wigner and Seitz, 1933 Summary In order to explain many important properties of materials and phenomena, it is necessary to go beyond independent-particle approximations and directly account for many-body effects that result from electronic interaction. The many-body problem is a major scientific challenge, but there has been great progress resulting from theoretical developments and advances in computation. This chapter is a short introduction to the interacting-electron problem, with some of the history that has led up to the concepts and methods described in this book. The many-body interacting-electron problem ranks among the most fascinating and fruitful areas of research in physics and chemistry. It has a rich history, starting from the early days of quantum mechanics and continuing with new intellectual challenges and opportunities. The vitality of electronic structure theory arises in large part from the close connection with experiment and applications. It is spurred on by new discoveries and advances in techniques that probe the behavior of electrons in depth. In turn, theoretical concepts and calculations can now make predictions that suggest new experiments, as well as provide quantitative information that is difficult or not yet possible to measure experimentally. This book is concerned with the effects of interactions between electrons beyond independent-particle approximations. Some phenomena cannot be explained by any independent-electron method, such as broadening and lifetime of excited states and two-particle bound states (excitons) that are crucial for optical properties of materials. There are many other examples, such as the van derWaals interaction between neutral molecules that arises from the dipole–induced dipole interaction. This force, which is entirely due to correlation between electrons, is an essential mechanism determining the functions of biological systems. Other properties, such as thermodynamically stable magnetic phases, would not exist if there were no interactions between electrons; even though mean-field approximations can describe average effects, they do not account for fluctuations around the average. Ground-state properties, such as the equilibrium structures of molecules and solids, can be described by density functional theory (DFT) and the Kohn–Sham independent-particle equations. However, present approximations are often not sufficient, and for many properties the equations, when used in a straightforward way, do not give a proper description, even in principle.
Abstract This chapter covers the rudiments of quantum theory, including the dual nature of light and matter, the Uncertainty Principle and probability concept, the electronic wavefunction, and the probability density function. Numerical examples are given to show that given the electronic wavefunction of a system, the probability of finding an electron in a volume element around a certain point in space can be readily calculated. Finally, the electronic wave equation, the Schrödinger equation, is introduced. This discussion is followed by the solution of a few particle-in-a-box problems, with the ‘box’ having the shape of a wire (one-dimensional), a cube, a ring, or a triangle. Where possible, the solutions of these problems are then applied to a chemical system.
Abstract Discusses with some rigour the properties of electrons, based on the Schrodinger equation. Introduces the concepts of wave function, quantum-mechanical operators, and wave packets. Examples cover the electron meeting an infinitely long potential barrier and the passage of electrons through a finite barrier (which leads to the phenomenon of tunnelling).The electron in a potential well is also discussed, solving the problem both for a finite and for an infinite well, and finding the permissible energy levels. The chapter is concluded with the philosophical implications that arise from the quantum-mechanical approach. Two limericks relevant to the subject are quoted.