2.2 The Hamiltonian

The time independent Schrödinger equation for a system of $N$ particles interacting via the Coulomb interaction is

$$\hat{H}\Psi = E\Psi$$ (2.1)

where

$$\hat{H} = \sum_{i=1}^{N}\left( -\frac{\hbar^2}{2m_{i}}\nabla... ...\frac{Z_{i}Z_{j}}{4\pi\epsilon_0\vert{\bf r}_i-{\bf r}_j\vert}$$ (2.2)

and $\Psi$ is an N-body wavefunction, ${\bf r}$ denotes spatial positions and $Z$ the charges of the individual particles. $E$ denotes the energy of either the ground or an excited state of the system.

Most physical problems of interest consist of a number of interacting electrons and ions. The total number of particles, $N$, is usually sufficiently large that an exact solution cannot be found. Controlled and well understood approximations are sought to reduce the complexity to a tractable level. Once the equations are solved, a large number of properties may be calculated from the wavefunction. Errors or approximations made in obtaining the wavefunction will be manifest in any property derived from the wavefunction. Where high accuracy is required, considerable attention must be paid to the derivation of the wavefunction and any approximations made.

2.2.1 The Born-Oppenheimer approximation

A common and very reasonable approximation used in the solution of equation 2.1 is the Born-Oppenheimer Approximation. [1] In a system of interacting electrons and nuclei there will usually be little momentum transfer between the two types of particles due to their greatly differing masses. The forces between the particles are of similar magnitude due to their similar charge. If one then assumes that the momenta of the particles are also similar, then the nuclei must have much smaller velocities than the electrons due to their far greater mass. On the time-scale of nuclear motion, one can therefore consider the electrons to relax to a ground-state given by the Hamiltonian equation 2.2 with the nuclei at fixed locations. This separation of the electronic and nuclear degrees of freedom is known as the Born-Oppenheimer approximation.

This approximation will be used for the remainder of this thesis. It is important to note that this approximation does not limit the techniques described to systems of fixed ions: in principle, once the electronic configuration is known, the nuclear degrees of freedom could also be solved for, giving rise to nuclear motion.

In practice Newtonian mechanics using forces calculated via quantum mechanics is often sufficient to solve for the motion of the nuclei, however, these aspects go beyond the scope of the thesis so that from now on a simpler version of the many-body Hamiltonian, equation 2.2, is used

$$\hat{H} = \sum_{i}-\frac{1}{2}\nabla^2_i + \sum_i\sum_{\alpha}\frac{Z_{\alpha}}{\vert{\bf r}_i-{\bf d}_\alpha\vert}$$

$$+ \frac{1}{2}\sum_i\sum_{j\neq i}\frac{1}{\vert{\bf r}_i-{\bf r}_j\... ...ha} \frac{Z_{\alpha}Z_{\beta}}{\vert{\bf d}_{\alpha}-{\bf d}_{\beta}\vert}\;\;.$$ (2.3)

The opportunity has been taken to separate the interacting particles into electrons and ions. The terms in the Hamiltonian are now expressed in terms of $N$ electrons of charge $-1$ at positions ${\bf r}_{i}$ and ions of charge $Z_\alpha$ at positions ${\bf d}_\alpha$. This simplified electronic Hamiltonian remains very difficult to solve. No analytic solutions exist for general systems with more than one electron.

Note that this equation has been written in atomic units ( $e=m_e=\hbar=4\pi\epsilon_0=1$) which are more convenient for quantum mechanical problems and will be used for the remainder of the equations in this thesis.