A brief comparison between grid based real space algorithms andspectrum algorithms for electronic structure calculations Page: 2 of 11
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here the occupied charge density is
P(r) = V i, (r)2 (2)
and Vion(r) is the ionic potential, and ZR is the nuclei charge at R, and the function Fxc(x)
is the LDA exchange correlation function. The wavefunctions Wi satisfies the following
f V/*(r)V,-(r)d3r = 5 (3)
Finding the minimum of the energy in Eq.(1) is equivalent to find the solution for the
following Kohn-Sham equation  (sometime it is also called Schrodinger's equation):
[--IV2+V(r)+YNLVr = ,,r 4
here the total potential V(r) has the following expression:
V(r)=VdO) p(r)' dr+pxc(p(r)) (5)
1r -r J
here the second term is the Coulomb potential due to charge p(r), and the third term is the
LDA exchange correlation potential coming from the derivative of p~xc(p). In Eq.(4), we
have also introduced a nonlocal potential VNL, which is a nonlocal operator acting on the
wavefunction. This nonlocal potential is needed for pseudopotential calculations .
Given the above formalism, the different numerical methods distinguish themselves by
their ways to represent the wavefunction Wi. In the plane wave (PW) method, the
wavefunctions are expanded by plane wave basis set as:
V/, (r) = C, (q)e*'r (6)
Usually a periodic box (supercell) is chosen. Then the reciprocal lattice of the supercell
defines a grid of q in the Fourier space. As a convention, all the q points within a sphere
defined by a kinetic energy cutoff E, - q' is chosen in the summation of Eq.(6). In the
PW method, the wavefunctions are kept in reciprocal space (q-space), represented by the
coefficients Ci(q). Major operations, like the enforcement of the orthonormal conditions
of Eq.(3) are carried out in this reciprocal space representation. However, to carry out
operations like the V(r)i(r), the wavefunction is transformed via FFT into the real space
on a regular real space grid. After the V(r) and 'i(r) multiplication, the result is then
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Wang, Lin-Wang. A brief comparison between grid based real space algorithms andspectrum algorithms for electronic structure calculations, report, December 1, 2006; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc897800/m1/2/: accessed July 22, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.