Plane waves basis functions

For a periodic system such as a crystal, plane-waves (PW) of the type:

$$\phi_{\bf g}({\bf r}) = \frac{1}{\sqrt{V}}e^{i {\bf g}\cdot {\bf r}},$$ (8.49)

where ${\bf g}$ is a reciprocal lattice vector and $V$ is the volume of the repeating unit cell, are particularly useful, because they naturally encode the periodicity of the crystal. Any function can be represented with arbitrary accuracy using enough PW's, and so they form an unbiased and systematically improvable basis set. One starts by including PW's with the lowest $g$'s ( $g = \vert{\bf g}\vert$), which are usually the most important, and works its way up by including PW's with progressively larger $g$'s. The systematic way of doing this is to define a maximum value $g_{max}$, and include PW's with $g \le g_{cut}$. For convenience, rather than specifying $g_{cut}$, one defines a PW energy cutoff $E_{cut} = \frac{\hbar^2}{2 m_e} g_{cut}^2$, and then use $E_{cut}$ as a knob to drive the system to the desired convergence. A Bloch function expanded in PW's therefore assumes the simple form:

$$\psi_{{\bf k},n}({\bf r}) = \frac{1}{\sqrt{V}}\sum_{\bf g} c_{\bf g}({\bf k},n) e^{i ({\bf g + k})\cdot {\bf r}}.$$ (8.50)

If the $KS$ hamiltonian is real then $\psi_{{\bf -k},n} = \psi^*_{{\bf k},n}$, and so at ${\bf k}= \Gamma$ the wavefunctions can be chosen to be real. The plane wave expansion reads:

$$\psi_{{\bf\Gamma},n}({\bf r}) = \frac{1}{\sqrt{V}}\sum_{\bf g} c_{\bf g}(\Gamma,n) e^{i {\bf g}\cdot {\bf r}}.$$ (8.51)

The wavevectors in the sum in  run inside a sphere of radius $g_{cut}$ which can be split in the sum over two hemispheres, one of which includes the wavevectors with $g_z>0$, indicated with ${\bf g} > 0$ below:

$$\psi_{{\bf\Gamma},n}({\bf r}) = \frac{1}{\sqrt{V}}\left [ \sum_{{... ... + \sum_{{\bf g} < 0} c_{\bf g}(\Gamma,n) e^{i {\bf g}\cdot {\bf r}} \right ] =$$

$$\frac{1}{\sqrt{V}}\left [ \sum_{{\bf g} > 0} c_{\bf g}(\Gamma,n) ... ...+ \sum_{{\bf g} > 0} c_{\bf -g}(\Gamma,n) e^{-i {\bf g}\cdot {\bf r}} \right ].$$ (8.52)

Consider now the complex conjugate of $\psi_{{\bf\Gamma},n}$:

$$\psi^*_{{\bf\Gamma},n}({\bf r}) = \frac{1}{\sqrt{V}}\left [ \sum_... ... \sum_{{\bf g} > 0} c^*_{\bf -g}(\Gamma,n) e^{i {\bf g}\cdot {\bf r}} \right ].$$ (8.53)

Since this can be chosen to be equal to $\psi_{{\bf\Gamma},n}$, we have $c_{\bf -g}(\Gamma,n) = c^*_{\bf g}(\Gamma,n)$. As a result, only the coefficients at ${\bf g}$ vectors in one hemisphere are needed, and both storage and number of operations are reduced by a factor of two.