Plane Waves

From Bloch's theorem we know that

$$\phi_i({\bf r}) = e^{i{\bf k}\cdot {\bf r}} u({\bf r})$$ (205)

where the function $u({\bf r})$ is a periodic function with the periodicity of the lattice. This means that we can always expand it in a Fourier series as

$$u({\bf r}) = \frac{1}{\sqrt{\Omega}} \sum_{\bf K=0}^\infty C_{\bf k}({\bf K}) e^{i{\bf K}\cdot {\bf r}}$$ (206)

where ${\bf K}$ is a vector of the recirpocal lattice.

Let us define the PW basis as

$$\phi_{\bf K}({\bf r}) = \frac{1}{\sqrt{\Omega}} e^{i{\bf K}\cdot {\bf r}}$$ (207)

We can see that this is an orthonormal basis

$$\langle \phi_{\bf K} \vert \phi_{\bf K'} \rangle = 0$$ (208)

We can now write the wave function $\phi_{\bf k}$ in this basis as:

$$\phi_{\bf k}({\bf r}) = e^{i{\bf k}\cdot {\bf r}} \sum_{\bf K=0}^\infty C_{\bf k} \phi_{\bf K}({\bf r}).$$ (209)

We can redefine the basis by including the phase in the exponential

$$\phi_{\bf K+k}({\bf r}) = \frac{1}{\sqrt{\Omega}} e^{i({\bf K+k})\cdot {\bf r}}$$ (210)

to obtain

$$\phi_{\bf k}({\bf r}) = \sum_{\bf K=0}^\infty C_{\bf k} \phi_{\bf K+k}({\bf r}).$$ (211)