Ab initio molecular dynamics: Theory and Implementation

CALL SGEMM(’T’,’N’,M,N b ,2*N PW ,2,&& scr(1,0),2*N PW ,cr(1,1),2*N PW ,0,fnl(i,1),N p )ENDFourier transform routines are assumed to work on complex data and returnalso arrays with complex numbers. The transform of data with the density cutoff isshown in the next two pseudo code sections. It is assumed that a three dimensionalfast Fourier transform routine exists. This is in fact the case on most computerswhere optimized scientific libraries are available. The next two pseudo code segmentsshow the transform of the charge density from Fourier space to real spaceand back.MODULE INVFFTscr(1:N x ,1:N y ,1:N z ) = 0FOR i=1:N Dscr(ipg(1,i),ipg(2,i),ipg(3,i)) = rhog(i)scr(img(1,i),img(2,i),img(3,i)) = CONJG[rhog(i)]ENDCALL FFT3D("INV",scr)n(1:N x ,1:N y ,1:N z ) = REAL[scr(1:N x ,1:N y ,1:N z )]MODULE FWFFTscr(1:N x ,1:N y ,1:N z ) = n(1:N x ,1:N y ,1:N z )CALL FFT3D("FW",scr)FOR i=1:N Drhog(i) = scr(ipg(1,i),ipg(2,i),ipg(3,i))ENDSpecial kernels are presented for the calculation of the density and the applicationof the local potential. These are the implementation of the flow charts shown inFig. 8. The operation count of these routines is N b N log[N]. In most applicationsthese routines take most of the computer time. Only for the biggest applicationspossible on todays computers the cubic scaling of the orthogonalization and thenonlocal pseudopotential become dominant. A small prefactor and the optimizedimplementation of the overlap are the reasons for this.In the Fourier transforms of the wavefunction two properties are used to speedup the calculation. First, because the wavefunctions are real two transforms can bedone at the same time, and second, the smaller cutoff of the wavefunctions can beused to avoid some parts of the transforms. The use of the sparsity in the Fouriertransforms is not shown in the following modules. In an actual implementation amask will be generated and only transforms allowed by this mask will be executed.Under optimal circumstances a gain of almost a factor of two can be achieved.81