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Bloch's Theorem

Thus far, the quantum mechanical approaches to solving the many-body problem have been discussed. However, the correlated nature of the electrons within a solid is not the only obstacle to solving the Schrödinger equation for a condensed matter system: for solids, one must also bear in mind the effectively infinite number of electrons within the solid.

One may appeal to Bloch's theorem in order to make headway in obviating this latter problem. Instead of being required to consider an infinite number of electrons, it is only necessary to consider the number of electrons within the unit cell (or half of this number if the electrons are spin degenerate).

Bloch's theorem [55] states that the wavefunction of an electron within a perfectly periodic potential may be written as


\begin{displaymath}
\psi_{j,\mathbf{k}}(\mathbf{r}) = u_{j}(\mathbf{r})e^{i \mathbf{k \cdot r}}
\end{displaymath} (2.37)

where $u_{i}(\mathbf{r})$ is a function that possesses the periodicity of the potential, i.e. $u_{i}(\mathbf{r} + \mathbf{l}) = u_{i}(\mathbf{r})$, where $\mathbf{l}$ is the length of the unit cell. In 2.37 i is the band index, and $\mathbf{k}$ is a wavevector confined to the first Brillouin Zone. Since $u_{i}(\mathbf{r})$ is a periodic function, we may expand it in terms of a Fourier series:


\begin{displaymath}
u_{j}(\mathbf{r}) = \sum_{\mathbf{G}} c_{j,\mathbf{G}}e^{i\mathbf{G \cdot r}}
\end{displaymath} (2.38)

where the $\mathbf{G}$ are reciprocal lattice vectors defined through $\mathbf{G}\cdot \mathbf{R} = 2\pi m$, where m is an integer, $\mathbf{R}$ is a real space lattice vector and the $c_{i,\mathbf{G}}$ are plane wave expansion coefficients. The electron wavefunctions may therefore be written as a linear combination of plane waves:


\begin{displaymath}
\psi_{j,\mathbf{k}}(\mathbf{r}) = \sum_{\mathbf{G}} c_{j,\mathbf{k}+\mathbf{G}}e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}}.
\end{displaymath} (2.39)

Given that each electron occupies a state of definite $\mathbf{k}$, the infinite number of electrons within the solid gives rise to an infinite number of k-points. At each k-point, only a finite number of the available energy levels will be occupied. Thus one only needs to consider a finite number of electrons at an infinite number of k-points. This may seem to be replacing one infinity (number of electrons) with another one (number of k-points) to little discernible advantage. However, one does not need to consider all of these k-points; rather, since the electron wavefunctions will be almost identical for values of $\mathbf{k}$ that are sufficiently close, one can represent the wavefunctions over a region of reciprocal space by considering the wavefunction at a single k-point. It is therefore sufficient to consider the electronic states at a finite number of k-points in order to determine the groundstate density of the solid. The net effect of Bloch's Theorem therefore has been to change the problem of an infinite number of electrons to one of considering only the number of electrons in the unit cell (or half that number, depending on whether the states are spin-degenerate or not) at a finite number of k-points chosen so as to appropriately sample the Brillouin Zone; this problem is returned to later.


next up previous contents
Next: Kohn-Sham Equations in Plane Up: The Many Body Problem Previous: The Exchange-Correlation Term   Contents
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