Many-electron Quantum Mechanics

Motivation

The theory of interacting electrons is extremely rich and complex. This, along with recent advances in computing capabilities, means that more than 10,000 papers are published on electronic structure theory each year. This research is yielding novel understanding in a vast range of fields including physics, chemistry and materials science.

This is because, on the atomic level, electrons are the glue of matter and if we understand how electrons move in their environment and how they interact with each other, we can predict the electronic, optical and mechanical properties of materials.

Notation

We are going to use Hartree atomic units where \(e = \hbar = m_e = 4 \pi \varepsilon_0 = 1\) which saves a lot of clutter! This means the standard unit of length is the Bohr radius \(a_0 = 5.29 \times 10^{-11} \mathrm{m}\), and the unit of energy is the Hartree \(E_H = 2\mathrm{Ry} = 27.2 \mathrm{eV}\). Also be aware that capital \(\Psi\) refers to a many-body wave function whereas lower case \(\psi\) refers to single particle wave functions.

Below, we keep everything in 3 dimensions to keep things general, but be aware that iDEA works in 1D only.

Schrödinger equation

The name of the game is to solve the Schrödinger equation for both the ground state system, \(\hat{H} \Psi = E \Psi\) and for the time dependent one, \(\hat{H} \Psi = i \frac{\partial \Psi}{\partial t}\) (we’ll consider the time dependent case later).

In an ideal world, we would solve the many-body Schrödinger equation exactly and then armed with these wavefunctions, we’d have complete knowledge of the system and be able to make precise predictions. Unfortunately, however, the problem is much too hard to solve in the way. Let’s have a look at the Hamiltonian to see why.

\[\hat{H} = - \frac{1}{2} \sum_i \nabla_i^2 + V_{ext}(\textbf{r}_i) + V_{ee}(\mid \textbf{r}_i - \textbf{r}_j \mid ),\]

where we have used the Born-Oppenheimer approximation (treating the nuclei as so massive that they do not move on the timescale of electron motion), and the fact that the nuclei-nuclei interacting is constant and so we simply shift our zero of energy to absorb that term. \(V_{ext}\) is the external potential in which the electrons move, in the case of electrons moving in a potential set up by the nuclei, \(V_{ext} = -\sum_{i,I} \frac{Z_I}{\mid \textbf{r}_i - \textbf{R}_I\mid }\). \(V_{ee}\) is the electron-electron interaction which in 3 dimensions takes the form \(V_{ee} =\frac{1}{2} \sum_{i \neq j} \frac{1}{\mid \textbf{r$_i$} - \textbf{r$_j$} \mid}\), but of course has a different 1D form which is implemented in the iDEA code.

The difficulty arises with the third term, the electron-electron interaction. This term means that the Schrödinger equation isn’t seperable so the wavefunction isn’t simply a suitable anti-symmetric product of single particle wavefunctions. We now explore several approaches to solve this problem that are used by the iDEA code.

Exact solution

We are going to pretend from here that our electrons are spinless, so we can ignore any spin-orbit effects. This also has more profound impacts on the universe we’re considering. In a spinful universe, only electrons of equal spins would feel the exchange interaction, but in our spinless universe all electrons feel it. This means we get to see the effects of \(V_{xc}\) for systems with a smaller number of electrons so we get to maximise the amount of physics we can highlight for a given effort (on both your part and mine!).

As we mentioned earlier, the exact problem is, normally, much too hard to solve; the difficulty growing exponentially with the number of electrons in the system. However, for systems with 2 or 3 electrons, we are able to solve the problem exactly. In fact, this is one of the key concepts in iDEA - solving simple systems exactly to allow us to compare, and possibly improve, the approximate solutions. Armed with these improvements, we are in a better position to tackle the larger systems with many electrons.

Given we can’t solve the many-electron system exactly in general, we need to have a look at the various different approaches of tackling the problem in more complicated systems.

Complete neglect of interaction

This is by far the simplest approach - you just completely ignore the Coulomb interaction between the electrons. Clearly this massively simplifies the problem, giving a separable Hamiltonian for a start! This means that we are able to solve the Schrödinger equation by the method of separation of variables, solving it for each electron individually. Then the total wavefunction is just a product of these single particle wave functions, in a suitable anti symmetric arrangement. One often constructs this wavefunction with something known as a “Slater determinant”:

\[\begin{split} \Psi(\textbf{r}_1, \textbf{r}_2, \ ... , \textbf{r}_N) = \frac{1}{\sqrt{ N!}} \begin{vmatrix} \psi_1(\textbf{r}_1) & \psi_1(\textbf{r}_2) & \dots & \psi_1(\textbf{r}_N) \\ \psi_2(\textbf{r}_1) & \psi_2(\textbf{r}_2) & \dots & \psi_2(\textbf{r}_N) \\ \vdots & \vdots & \ddots & \vdots\\ \psi_N(\textbf{r}_1) & \psi_N(\textbf{r}_2) & \dots & \psi_N(\textbf{r}_N) \end{vmatrix} .\end{split}\]

Unfortunately, but unsurpringsingly, making this approximation means that you lose a lot of the physics in the problem (look at the double antisymmetric well notebook for a prime example). Generally, in this system electrons spend more time closer to each other than they would if the repulsion between them had been taken into account. This is often a useful thing to calculate, however, since the code runs quickly and can then provide a reasonable first guess for the density for a self-consistant field approach - more on this later.