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# Wrap-around errors

In this section we will discuss wrap around errors. Wrap around errors arise if the FFT meshes are not sufficiently large. It can be shown that no errors exist if the FFT meshes contain all ${\displaystyle {\mathbf {G}}}$ vectors up to ${\displaystyle 2G_{\rm {cut}}}$.

Fig. 1: Sphere intersections for ${\displaystyle G_{\mathrm {cut} }}$

It can be shown that the charge density contains components up to ${\displaystyle 2G_{\mathrm {cut} }}$, where ${\displaystyle 2G_{\mathrm {cut} }}$ is the "longest" plane wave in the basis set:

The wavefunction is defined as

${\displaystyle |\phi _{n{\mathbf {k}}}\rangle =\sum _{\mathbf {G}}C_{{\mathbf {G}}n{\mathbf {k}}}|{\mathbf {k}}+{\mathbf {G}}\rangle ,}$

and in real space it is given by

${\displaystyle \langle {\mathbf {r}}|\phi _{n{\mathbf {k}}}\rangle =\sum _{\mathbf {G}}\langle {\mathbf {r}}|{\mathbf {k}}+{\mathbf {G}}\rangle \langle {\mathbf {k}}+{\mathbf {G}}|\phi _{n{\mathbf {k}}}\rangle ={\frac {1}{\Omega ^{1/2}}}\sum _{\mathbf {G}}e^{i({\mathbf {k}}+{\mathbf {G}}){\mathbf {r}}}C_{{\mathbf {G}}n{\mathbf {k}}}.}$

Using Fast Fourier transformations one can define

${\displaystyle C_{{\mathbf {r}}n{\mathbf {k}}}=\sum _{\mathbf {G}}C_{{\mathbf {G}}n{\mathbf {k}}}e^{i{\mathbf {G}}{\mathbf {r}}}\qquad \qquad \qquad C_{{\mathbf {G}}n{\mathbf {k}}}={\frac {1}{N_{\mathrm {FFT} }}}\sum _{\mathbf {r}}C_{{\mathbf {r}}n{\mathbf {k}}}e^{-i{\mathbf {G}}{\mathbf {r}}}.}$

Therefore the wavefunction can be written in real space as

${\displaystyle \langle {\mathbf {r}}|\phi _{n{\mathbf {k}}}\rangle =\phi _{n{\mathbf {k}}}(r)={\frac {1}{\Omega ^{1/2}}}C_{{\mathbf {r}}n{\mathbf {k}}}e^{i{\mathbf {k}}{\mathbf {r}}}.}$

The charge density is simply given by

${\displaystyle \rho _{\mathbf {r}}^{\mathrm {ps} }\equiv \langle {\mathbf {r}}|\rho ^{\mathrm {ps} }|{\mathbf {r}}\rangle =\sum _{\mathbf {k}}w_{\mathbf {k}}\sum _{n}f_{n{\mathbf {k}}}\phi _{n{\mathbf {k}}}(r)\phi _{n{\mathbf {k}}}^{*}(r),}$

and in the reciprocal mesh it can be written as

${\displaystyle \rho _{\mathbf {G}}^{\mathrm {ps} }\equiv {\frac {1}{\Omega }}\int \langle {\mathbf {r}}|\rho ^{\mathrm {ps} }|{\mathbf {r}}\rangle e^{-i{\mathbf {G}}{\mathbf {r}}}\,d{\mathbf {r}}\to {\frac {1}{N_{\mathrm {FFT} }}}\sum _{\mathbf {r}}\rho _{\mathbf {r}}^{\mathrm {ps} }e^{-i{\mathbf {G}}{\mathbf {r}}}.}$

Using the above equations for ${\displaystyle \rho _{\mathbf {r}}^{\mathrm {ps} }}$ and ${\displaystyle C_{{\mathbf {r}}n{\mathbf {k}}}}$ it is very easy to show that ${\displaystyle \rho _{\mathbf {r}}^{\mathrm {ps} }}$ contains Fourier-components up to ${\displaystyle 2G_{\mathrm {cut} }}$.

Generally it can be shown that a the convolution ${\displaystyle f_{r}=f_{r}^{1}f_{r}^{2}}$ of two functions ${\displaystyle f_{r}^{1}}$ with Fourier-components up to ${\displaystyle G_{1}}$ and ${\displaystyle f_{r}^{2}}$ with Fourier-components up to ${\displaystyle G_{2}}$ contains Fourier-components up to ${\displaystyle G_{1}+G_{2}}$.

The property of the convolution comes once again into play, when the action of the Hamiltonian onto a wavefunction is calculated. The action of the local-potential is given by

${\displaystyle a_{\mathbf {r}}=V_{\mathbf {r}}C_{{\mathbf {r}}n{\mathbf {k}}}.}$

Only the components ${\displaystyle a_{\mathbf {G}}}$ with ${\displaystyle |{\mathbf {G}}| are taken into account (see section ALGO: ${\displaystyle a_{\mathbf {G}}}$ is added to the wavefunction during the iterative refinement of the wavefunctions ${\displaystyle C_{{\mathbf {G}}n{\mathbf {k}}}}$, and ${\displaystyle C_{{\mathbf {G}}n{\mathbf {k}}}}$ contains only components up to ${\displaystyle G_{\mathrm {cut} }}$). From the previous theorem we see that ${\displaystyle a_{\mathbf {r}}}$ contains components up to ${\displaystyle 3G_{\mathrm {cut} }}$ (${\displaystyle V_{\mathbf {r}}}$ contains components up to ${\displaystyle 2G_{\mathrm {cut} }}$).

If the FFT-mesh contains all components up to ${\displaystyle 2G_{\mathrm {c} ut}}$ the resulting wrap-around error is once again 0. This can be easily seen in Fig. 1. Here we see that the small sphere contains all plane waves included in the basis set ${\displaystyle G. The charge density contains components up to ${\displaystyle 2G_{\mathrm {cut} }}$ (second sphere), and the acceleration ${\displaystyle a}$ components up to ${\displaystyle 3G_{\mathrm {cut} }}$, which are reflected in (third sphere) because of the finite size of the FFT-mesh. Nevertheless the components ${\displaystyle a_{\mathbf {G}}}$ with ${\displaystyle |{\mathbf {G}}| are correct i.e. the small sphere does not intersect with the third large sphere}