Band Structure Theory for Extended Systems

The electronic structure of infinite periodic systems (crystals) is treated with band structure theory, replacing molecular orbitals by crystal orbitals. The chapter starts out by introducing the electron gas and definitions of the Fermi momentum, the Fermi energy, and the density of states (DOS). A periodic linear combination of atomic orbitals (LCAO) type treatment of an infinite periodic system is facilitated by the construction of Bloch functions. The notions of energy band and band gap are discussed with band structure concepts, using the approximations made in Huckel theory (chapter 12). One, two, and three-dimensional crystal lattices and the associated reciprocal lattices are introduced. The band structures of sodium metal, boron nitride, silicon, and graphite, are discussed as examples of metals, insulators, semi-conductors, and semi-metals, respectively. The chapter concludes with a brief discussion of the projected DOS and measures to determine bonding or antibonding interactions between atoms in a crystal.

Vectors and Functions and Operators

This chapter introduces – briefly – vectors and functions and the similarities between them, some basic linear algebra concepts, operators (including the del and Laplace operators), eigenvalues and eigenvectors &eigenfunctions, the scalar (dot) and vector (cross) product between two vectors, the scalar product between two functions, the concepts of normalization, orthogonality, and orthonormality. The concept of an operator is first introduced by considering the rotation and stretching or compression of a vector. It is then generalized to a mathematical prescription that changes a function into another function.

Dynamic Fields and Response Properties

It is shown how electronic transitions can be induced by the interaction with an electromagnetic wave of a suitable frequency. The rate of a transition between two electronic states induced by a time-dependent field is derived. The transition rate expression is used to calculate the absorption coefficient due to electronic transitions. The differential absorption coefficient for left and right circular polarized light is specific to chiral molecules and has different signs for a pair of enantiomers. The discussion then shifts to general functions describing the response of an atom or molecule to an external. The ideas developed thus far are then applied to the dynamic polarizability, molecular linear response functions in general, and the optical rotation. Linear response theory is set up within time-dependent molecular orbital theory. The Chapter concludes with a discussion of non-linear response properties and two-photon absorption.

Approximate Molecular Orbital Theory: The Hückel/Tight-binding Model

Huckel molecular orbital (HMO) theory is a simple approximate parameterized molecular orbital (MO) theory that has been very successful in organic chemistry and other fields. This chapter introduces the approximations made in HMO theory, and then treats as examples ethane, hetratriene and other linear polyenes, and benzene and other cyclic polyenes. The pi binding energy of benzene is particularly large according to HMO theory, rationalizing the special ‘aromatic’ behaviour of benzene. But there is a lot more to benzene than that. It is shown that the pi bond framework of benzene would rather prefer a structure with alternating single and double C-C bonds, rather than the actually observed 6-fold symmetric structure where all C-C bonds are equivalent. The observed benzene structure is a result of a delicate balance between the tendencies of the pi framework to create bond length alternation, and the sigma framework to resist bond length alternation.

Orbital-based Descriptions of Electron Configurations, Ionization, Excitation, and Bonding

This chapter deals with quantitative aspects of molecular orbital (MO) theory: Construction of an orbital diagram, bonding and antibonding overlap, Koopmans’ theorem, orbital energies versus total energies, an explanation of the unintuitive ground state electron configurations seen for some neutral transition metals, and a discussion of orbital energy gaps versus electronic excitations and other observable energy gaps. Localized MOs show the chemical bonds expected from the Lewis structure more readily than the canonical orbitals obtained from solving the SCF equations. It is shown that the delocalization of localized, not the canonical, MOs shows whether a system is delocalized. Algorithms by which to obtain localized MOs are sketched.

From Atomic Orbitals to Molecular Orbitals and Chemical Bonds

It is shown how an aufbau principle for atoms arises from the Hartree-Fock (HF) treatment with increasing numbers of electrons. The Slater screening rules are introduced. The HF equations for general molecules are not separable in the spatial variables. This requires another approximation, such as the linear combination of atomic orbitals (LCAO) molecular orbital method. The orbitals of molecules are represented in a basis set of known functions, for example atomic orbital (AO)-like functions or plane waves. The HF equation then becomes a generalized matrix pseudo-eigenvalue problem. Solutions are obtained for the hydrogen molecule ion and H2 with a minimal AO basis. The Slater rule for 1s shells is rationalized via the optimal exponent in a minimal 1s basis. The nature of the chemical bond, and specifically the role of the kinetic energy in covalent bonding, are discussed in details with the example of the hydrogen molecule ion.

Particle in a Cylinder, in a Sphere, and on a Helix

The particle in a box from an earlier chapter is useful as a model to treat the electron motion in linear molecular ‘wires’, rectangular surfaces, and cuboid nano-particles. In this chapter, similar models are developed for spherical and cylindrical nano-particles, and helical nano-wires. Applications of these models that have been reported in the literature are discussed. For example, the particle in a cylinder model has been applied to treat the absorption spectra of silver nano-rods. The particle in a sphere model can be used tom rationalize the occurrence of sodium nano-clusters with certain ‘magic’ numbers of Na atoms. The model also explains the behaviour of potassium under high pressure, where the element starts to behave like a transition metal. The particle on a helix has been used to rationalize the optical activity of helical pi-conjugated molecules.

Many-electron Systems and the Pauli Principle

It is shown how the quantum Hamiltonian for a general molecule is set up, using the ‘quantum recipe’ of chapter 3. In the most restrictive Born Oppenheimer approximation, the nuclei are held fixed and the Schrodinger equation (SE) is set up for the electrons only. The wavefunction depends on the positions and spin projections of all electrons. The electron spin projection is introduced heuristically as another two-valued electron degree of freedom. The electronic SE cannot be solved exactly, and (spin-) orbitals are introduced to construct an approximate wavefunction. The Pauli principle demands that a many-electron wavefunction is antisymmetric upon the exchange of electron labels, which leads to the construction of the approximate orbital-model wavefunction as a Slater determinant rather than a simple Hartree product. The orbital model wavefunction does not describe the Coulomb electron correlation, but it incorporates the (Fermi) correlation leading to the Pauli exclusion.

A First Example: The “Particle in a Box“ and Quantized Translational Motion

The simple ‘particle in a box’ (Piab) is introduced in this chapter so that the reader can get familiar with applying the quantum recipe and atomic units. The PiaB is introduced in its one, two, and three dimensional variants, which demonstrates the use of the separation of variables technique as a strategy to solve the Schrodinger equation for a particle with two or three degrees of freedom. It is shown that the confinement of the particle causes the energy to be quantized. The one-dimensional PiaB is then applied to treat the electronic spectra of cyanine dyes and their absorption colors. The chapter then introduces more general setups with finite potential wells, in order to introduce the phenomenon of quantum tunnelling and to discuss more generally with the unintuitive ‘quantum behavior’ of particles such as electrons. Scanning tunnelling and atomic force microscopes are also discussed briefly.

Hydrogen-like Atoms

This chapter shows how the electronic Schrodinger equation (SE) is solved for a hydrogen-like atom, i.e. an electron moving in the field of a fixed point-like nucleus with charge number Z. The hydrogen atom corresponds to Z = 1. The potential in atomic units is –Z/r, with r being the distance of the electron from the nucleus. The SE is not separable in Cartesian coordinates, but in spherical polar coordinates it separates into a radial equation and an angular momentum equation. The bound states have a total energy of –Z2/(2n2), with n = nr + ℓ being the principal quantum number (q.n.), ℓ = 0,1,2,… the angular momentum q.n., and nr = 1,2,3,… being a radial q.n. Each state for a given ℓ is 2ℓ+1-fold degenerate, with the components labelled by the projection q.n. mℓ. The wavefunctions for mℓ ≠ 0 are complex, but real linear combinations can be formed. This gives the atomic orbitals known from general and organic chemistry. Different ways of visualizing the real wavefunctions are discussed, e.g. as iso-surfaces.