Introduction

A significant part of condensed matter physics and chemistry would be solved if the electronic structure of atoms, molecules and solids could be determined exactly. This however is a formidable task for two main reasons. Firstly, electrons in matter must be treated using the laws of quantum mechanics rather than classical physics - the quantum length scale is set by Planck's constant $h$, and the onset of quantum effects occurs when the de Broglie wavelength of a particle, $\lambda$, given by

$$\lambda = \frac{h}{p}$$ (1.1)

is comparable to the average inter-particle separation. Rearranging the energy-momentum equation $E = p^2/2m_{\hbox{{e}}}$, and the electronic thermal energy relationship $E \sim k_{\hbox{{B}}} T$, leads to a relation for the de Broglie wavelength of an electron given in terms of the electron mass $m_{\hbox{{e}}}$ and temperature $T$,

$$\lambda = \frac{h}{\sqrt{2m_{\hbox{{e}}}k_{\hbox{{B}}} T}} \, .$$ (1.2)

For solid-state systems the average inter-electron separation is usually represented by the Seitz radius, $r_s$, which is the radius of a sphere whose volume encloses a single electron in the system, and is normally written in terms of the Bohr radius, $a_0~\sim~0.529~\times~10^{-10}{\hbox{m}}$,

$$\frac{r_s}{a_0} = \bigg(\frac{3}{4 \pi n_0} \bigg)^{1/3} \, ,$$ (1.3)

where $n_0$ is the average electron density. For most systems of interest $r_s$ typically ranges from $0.1$ up to $10$, which means that the electron de Broglie wavelength is larger or comparable to the average separation up to $T \sim 10^4 K$, according to relation (1.2). Therefore within this temperature range the de Broglie wavelength of the electrons overlap and the interactions between the electrons become quantum-mechanically correlated.

The second problematic issue concerns the number of electrons that are involved - the coupling of the electron interactions due to de Broglie wavelength overlap renders an analytic solution impossible for systems with more than one electron, and the complexity grows dramatically with increasing electron number. It is for these reasons that the electronic structure of matter is known as the quantum many-body problem.

The quantum many-body problem is unusual within the realm of theoretical physics because the equations required for an exact solution are known. The properties of any (non-relativistic) time-independent quantum system can be determined by solving the Schrödinger equation [1],

$$\hat{H} \Psi({\bf r}_1,{\bf r}_2 \ldots {\bf r}_N) = E \Psi({\bf r}_1,{\bf r}_2 \ldots {\bf r}_N) \, ,$$ (1.4)

where $\hat{H}$, $\Psi({\bf r}_1,{\bf r}_2 \ldots {\bf r}_N)$ and $E$ are the Hamiltonian, many-body wavefunction and total energy of the system. Matter consists of electrons and nuclei interacting with each other Coulombically, consequently the Hamiltonian for any such system is given by,

$${\hat{H} = - \sum_{i=1}^{M} \frac{\hbar^2}{2 m_{\hbox{{Z}}_i}} \n... ...sum_{i}^M \sum_{j > i}^{M} \frac{Z_i Z_j}{\vert\,{\bf R}_i - {\bf R}_j\,\vert}}$$

$$\hspace{2.0cm} - \; \frac{1}{4 \pi \epsilon_0} \sum_{i=1}^{N} \su... ...um_{i=1}^{N} \sum_{j > i}^{N} \frac{e^2}{\vert\,{\bf r}_i-{\bf r}_j\,\vert} \,,$$ (1.5)

where $M$ and $N$ are the number of nuclei and electrons in the system, $m_{{\hbox{{Z}}}}$, $Z$ and ${\bf R}$ are the mass, charge and position of the nuclei, $m_e$ and $e$ are the mass and charge of an electron, and ${\bf r}$ represents the position of the electrons. The first two terms in (1.5) are the kinetic energy contributions from the nuclei and the electrons respectively, and the rest are Coulombic potential energy terms arising from the ion-ion repulsion, ion-electron attraction and the electron-electron repulsion respectively. Although in principle everything is known exactly, the Schrödinger equation (1.4) with this Hamiltonian is simply too difficult to solve directly. Hence, the quantum many-body problem is centred upon finding intelligent approximations to the Hamiltonian (1.5) and the many body wavefunction $\Psi$, that retain the correct physics and are computationally tractable to solve.

The first simplification of this problem is attributed to Born and Oppenheimer [2] who recognised that in most cases the nuclear and electronic degrees of freedom can be decoupled since they exhibit vastly different dynamics - the nuclei are of order $\sim10^3$ times heavier than the electrons and so are considered to be stationary with respect to the electrons. The electrons therefore move within a fixed external potential due to the nuclei. Within the Born-Oppenheimer approximation the complexity of the full many-body Hamiltonian (1.5) reduces to that of an electronic Hamiltonian,

$$\hat{H} = - \, \sum_{i=1}^{N} \frac{\hbar^2}{2 m_e} \nabla^2_i \,... ...m_{i=1}^{N} \sum_{j > i}^{N} \frac{e^2}{\vert\,{\bf r}_i-{\bf r}_j\,\vert} \, .$$ (1.6)

Solving the Schrödinger equation with the above Hamiltonian is however still too complex for most cases since the many-electron wavefunction contains $3N$ variables, which for a solid containing $N \sim 10^{26}$ electrons, is simply an intractable number of degrees of freedom.

Devising accurate schemes to approximate the many-electron problem has been an important goal since the founding of quantum mechanics in the early $1900$s. Several notable advances have been made, starting from Thomas-Fermi theory in the late $1920$s [3,4] which made a significant conceptual presumption by having the electron density, $n({\bf r})$, as the central unknown variable, rather than the many-electron wavefunction. This approach simplified the problem considerably since the density contains just three degrees of freedom, namely the $x, y, z$ coordinates of the system. In $1930$ came Hartree-Fock theory [5,6] which builds upon the single-particle approximation proposed earlier by Hartree [7], but in addition correctly accounts for the exchange interactions between electrons that are a consequence of the Pauli principle, by antisymmetrising the single-particle functions $\psi_i({\bf r}_i s_i)$,

$$\Psi({\bf r}_1 s_1,{\bf r}_2 s_2 \ldots {\bf r}_N s_N) \approx \f... ...bf r}_1 s_2) \, \psi_2({\bf r}_2 s_2) \ldots \psi_N({\bf r}_N s_N) \right] \, .$$ (1.7)

The symbol $\mathcal{A}$ represents the antisymmetric nature of the single-particle products, and $s_i$ gives the spin dependence. This has the desired effect of decoupling the $3N$ degrees of freedom in the many-electron wavefunction, and so allows each degree of freedom to be solved independently.

A significant leap in electronic structure theory was made in $1964$ with the remarkable theorems of density functional theory (DFT), proved by Hohenberg and Kohn [8]. DFT allows the ground-state properties of a many-electron system to be determined exactly through the electron density $n({\bf r})$, and therefore in a computationally tractable manner, however DFT is only a proof of existence, it does not give details of how this can be achieved in practice. In $1965$ Kohn and Sham [9] devised an ingeniously practical single-particle scheme for performing DFT calculations, which is still exact, in principle. The price to be paid for the benefits of Kohn-Sham DFT is that the single-particle Hamiltonian is only partly known in practice - approximations must be made for a single unknown component that accounts for electron many-body effects, known as exchange and correlation. Improving the exchange-correlation approximation in DFT is the object of this thesis.

The many-body methods just introduced will be discussed in more detail in the following sections. Unless otherwise stated, all equations, figures and tables in the remainder of this thesis will use atomic units, whereby $\hbar = e = m_e = 4 \pi \epsilon_0 = 1$.