Ab initio molecular dynamics: Theory and Implementation

with system size.eigr(N D ,N at ) structure factorsfnl(N p ,N b ) overlap of projectors and bandsdfnl(N p ,N b ,3) derivative of fnlsmat(N b ,N b ) overlap matrices between bandscr(N PW ,N b ) bands in Fourier spacecv(N PW ,N b ) velocity of bands in Fourier spacecf(N PW ,N b ) forces of bands in Fourier spaceIn order to save memory it is possible to store the structure factors only for theG vectors of the wave function basis or even not to store them at all. However,this requires that the missing structure factors are recomputed whenever needed.The structure factors eigr and the wavefunction related quantities cr, cv, cf arecomplex numbers. Other quantities, like the local pseudopotential vps, the corecharges rpc, and the projectors pro can be stored as real numbers if the factor(−i) l is excluded.3.8.2 CPMD Program: Computational KernelsMost of the calculations in a plane wave code are done in only a few kernel routines.These routines are given in this section using a pseudo code language. Wherepossible an implementation using basic linear algebra (blas) routines is given. Thefirst kernel is the calculation of the structure factors. The exponential function ofthe structure factor separates in three parts along the directions s x , s y , s z .MODULE StructureFactorFOR i=1:N ats(1:3) = 2 * PI * MATMUL[htm1(1:3,1:3),r(1:3,i)]dp(1:3) = CMPLX[COS[s(1:3)],SIN[s(1:3)]]dm(1:3) = CONJG[dp(1:3)]e(0,1:3,i) = 1FOR k=1:g maxe(k,1:3,i) = e(k-1,1:3,i) * dpe(-k,1:3,i) = e(-k+1,1:3,i) * dmENDFOR j=0:N Deigr(j,i) = e(g(1,j),1,i) * e(g(2,j),2,i) * e(g(3,j),3,i)ENDENDIn the module above htm1 is the matrix (h t ) −1 . One of the most important calculationis the inner product of two vectors in Fourier space. This kernel appears forexample in the calculation of energiese = ∑ GA ⋆ (G)B(G) . (262)Making use of the fact that both functions are real the sum can be restricted to half78