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Next: 9.3 Non-selfconsistent Harris-Foulkes functional Up: 9 Theoretical Background Previous: 9.1.6 Residual minimization scheme

9.2 Wrap-around errors - convolutions

 

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 G vectors up to tex2html_wrap_inline4323 .

It can be shown that the charge density contains components up to tex2html_wrap_inline4323 , where tex2html_wrap_inline4323 is the 'longest plane' wave in the basis set:

The wavefunction is defined as

displaymath1701

in real space it is given by

displaymath1711

Using Fast Fourier transformations one can define

 

Therefore the wavefunction can be written in real space as

 

The charge density is simply given by

displaymath1761

in the reciprocal mesh it can be written as

 

Inserting tex2html_wrap_inline4897 from equation (9.4) and tex2html_wrap_inline4899 from (9.3) it is very easy to show that tex2html_wrap_inline4901 contains Fourier-components up to tex2html_wrap_inline4323 .

Generally it can be shown that a the convolution tex2html_wrap_inline4905 of two 'functions' tex2html_wrap_inline4907 with Fourier-components up to tex2html_wrap_inline4909 and tex2html_wrap_inline4911 with Fourier-components up to tex2html_wrap_inline4913 contains Fourier-components up to tex2html_wrap_inline4915 .

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

displaymath4887

Only the components tex2html_wrap_inline4329 with tex2html_wrap_inline5030 are taken into account (see section 9.1: tex2html_wrap_inline4329 is added to the wavefunction during the iterative refinement of the wavefunctions tex2html_wrap_inline4923 , and tex2html_wrap_inline4923 contains only components up to tex2html_wrap_inline4927 ). From the previous theorem we see that tex2html_wrap_inline4929 contains components up to tex2html_wrap_inline4327 ( tex2html_wrap_inline4933 contains components up to tex2html_wrap_inline4323 ).

  figure1505
Figure 2:   The small sphere contains all plane waves included in the basis set tex2html_wrap_inline4321 . The charge density contains components up to tex2html_wrap_inline4323 (second sphere), and the acceleration a components up to tex2html_wrap_inline4327 , which are reflected in (third sphere) because of the finite size of the FFT-mesh. Nevertheless the components tex2html_wrap_inline4329 with tex2html_wrap_inline5030 are correct i.e. the small sphere does not intersect with the third large sphere

If the FFT-mesh contains all components up to tex2html_wrap_inline4323 the resulting wrap-around error is once again 0. This can be easily seen in Fig. 2. \


next up previous contents
Next: 9.3 Non-selfconsistent Harris-Foulkes functional Up: 9 Theoretical Background Previous: 9.1.6 Residual minimization scheme

MASTER USER VASP
Mon Mar 29 10:38:29 MEST 1999