Wrap-around errors: Difference between revisions
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Wrap around errors arise if the FFT meshes are not | Wrap around errors arise if the FFT meshes are not | ||
sufficiently large. It can be shown that no errors exist | sufficiently large. It can be shown that no errors exist | ||
if the FFT meshes contain all | if the FFT meshes contain all <math>\bold{G}</math> vectors up to <math>2 G_{\rm cut}</math>. | ||
[[File:Wrap errors spheres.png|350px|thumb|Fig. 1: Sphere intersections for <math>G_{\mathrm{cut}}</math>]] | [[File:Wrap errors spheres.png|350px|thumb|Fig. 1: Sphere intersections for <math>G_{\mathrm{cut}}</math>]] | ||
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---- | ---- | ||
[[Category:Electronic | [[Category:Electronic minimization]][[Category:Charge density]][[Category:Theory]] | ||
Latest revision as of 16:07, 6 April 2022
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\mathbf {G}} vectors up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 2G_{{{\rm {cut}}}} .
It can be shown that the charge density contains components up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 2G_{{{\mathrm {cut}}}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 2G_{{{\mathrm {cut}}}} is the "longest" plane wave in the basis set:
The wavefunction is defined as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): |\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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \rho _{{{\mathbf {r}}}}^{{{\mathrm {ps}}}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): C_{{{\mathbf {r}}n{\mathbf {k}}}} it is very easy to show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \rho _{{{\mathbf {r}}}}^{{{\mathrm {ps}}}} contains Fourier-components up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 2G_{{{\mathrm {cut}}}} .
Generally it can be shown that a the convolution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): f_{r}=f_{r}^{1}f_{r}^{2} of two functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): f_{r}^{1} with Fourier-components up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): G_{1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): f_{r}^{2} with Fourier-components up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): G_{2} contains Fourier-components up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 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
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): a_{{{\mathbf {r}}}}=V_{{{\mathbf {r}}}}C_{{{\mathbf {r}}n{\mathbf {k}}}}.
Only the components Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): a_{{{\mathbf {G}}}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): |{\mathbf {G}}|<G_{{{\mathrm {cut}}}} are taken into account (see section ALGO: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): a_{{{\mathbf {G}}}} is added to the wavefunction during the iterative refinement of the wavefunctions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): C_{{{\mathbf {G}}n{\mathbf {k}}}} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): C_{{{\mathbf {G}}n{\mathbf {k}}}} contains only components up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): G_{{{\mathrm {cut}}}} ). From the previous theorem we see that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): a_{{{\mathbf {r}}}} contains components up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 3G_{{{\mathrm {cut}}}} (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): V_{{{\mathbf {r}}}} contains components up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 2G_{{{\mathrm {cut}}}} ).
If the FFT-mesh contains all components up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): G<G_{{{\mathrm {cut}}}} . The charge density contains components up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 2G_{{{\mathrm {cut}}}} (second sphere), and the acceleration Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): a components up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 3G_{{{\mathrm {cut}}}} , which are reflected in (third sphere) because of the finite size of the FFT-mesh. Nevertheless the components Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): a_{{{\mathbf {G}}}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): |{\mathbf {G}}|<G_{{{\mathrm {cut}}}} are correct i.e. the small sphere does not intersect with the third large sphere}