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Introduction

Band diagrams are a very powerful concept to understand the properies of solids. Since they are obtained for the idealized situation of infinite and perfect lattices, they are not directly applicable as objects get small, for example nano-particles or polycrystalline material that consists of small crystallites. This is obviously a very vast topic and there is no generally applicable theory. Here, we limit ourselves to a few simplified scenarios.

The covalent binding

We start by recapitulating some ideas of how molecules are held together. Unfortunately, even for the easiest case, the hydrogen molecule, it is impossible to give an anylytic solution. Instead, the problem is approximated by separating the movement of the electrons from the one of the nuclei; generally the nuclei are supposed to be fixed at a given distance and the combined wavefunction of the electrons is approximated by superposing the wavefunctions of the isolated hydrogen atom.

Imagine we could get away with a one-dimensional treatment and forget about all the complications of three-dimensional orbitals. We could then represent the wavefunction of the hydrogen atoms by Gaussians around the nuclei. For simplicity, we fixed them at a distance of two units.

Ψb = exp{-(x+1)2}+ exp{-(x-1)2}

Obviously the wavefunction is symmetric with respect to exchange of the positions and there is a probability to find both electrons exactly in the middle between the atoms. We associate this function with a bonding state of reduced energy.

Likewise, we can define an anti-symmetric wavefunction where the probability of finding an electron exactly between the atoms is zero. This would probably make the atoms fly apart and is often called an "anti-bonding" state.

Ψa = exp{-(x+1)2}- exp{-(x-1)2}

The figure below plots the modulus of the two orbitals; in reality the atoms should be closer together, such that the wavefunction of the bonding state assumes a single maximum exactly between the atoms. This is not the case here, but just assume that we hold the atoms on a stick at this distance for illustration. The difference between the bonding and the anti-bonding states is still visible because the probabiltiy of finding an electron between the atoms is finite for the bonding state whereas it is zero for the anti-bonding state.

In molecular-orbital theory the states are normally ordered with respect to their energy. Starting from the energy level of the isolated atom, there is the bonding state at a lower energy and the anti-bonding state at a higher energy. The colored blobs to the right can serve as illustration which signs we actually used to construct our wavefunctions; positive and negative signs are denoted by red and blue, respectively.

Momentum representation

Whenever we plot the probability of finding an electron at a certain position, we use the "position-representation", a concept that we are more or less used to. In quantum mechanics we can also look at the Fourier-transform of the wavefunctions. This is the "momentum-representation". Using the simple 1D-wavefuctions of exponential form has the advantage that we can calculate the Fourier-transform analytically. The results are shown below:

The momentum-representation is totally equivalent, but it is maybe a little less intuitive. Essentially the figures show the probability to find an electron with a certain momentum. Clearly, the most probable momentum of the bonding state is zero. For the anti-bonding state, zero-momentum is actually excluded (maybe a sign that this state would fly apart immediately if we hadn't pierced the atoms on a stick).

The next section will make clearer why we want to use the momentum-representation.


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