A brief comparison between grid based real space algorithms andspectrum algorithms for electronic structure calculations Page: 3 of 11
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Fourier transformed back to the reciprocal space. This due space representation is
illustrated in Fig. 1.
ReaI space grid G-space grid
V(' V(r) I)(r) 19 (G) V(G) P (G)
Fig.1, the due space representation of the PW method. The reciprocal space (right
box) and real space (left box). The wavefunction q vectors (G=q in this figure) inside
the cutoff Gel are chosen in the summation of Eq.(6), while the potential V and
charge density A are represented by the planewave q inside a larger cutoff Gc2. Gc 1
is half of Gc2. FFT is used to transform the wavefunction from reciprocal space
representation to a real space grid.
The PW formalism has the following advantage. Given the wavefunction expression in
Eq.(6), the local minimum of Eq.(1) can be evaluated almost numerically exactly without
further approximations. The only numerical approximation for the evaluation of Eq.(1) is
the calculation of the LDA exchange correlation term (the fourth term) in Eq(1), but the
error is found to be very small . Thus, the solution is numerically exact, and it is truly
variational. The only approximation is the use of limited planewave basis function in
Eq.(6). This numerical exactness is only rivaled by the Gaussian method in quantum
chemistry calculation, where the numerical integrals have analytical expressions.
The numerically variational feature provide the following two properties: (1) the total
energy found from Eq.(1) is always an upper bound of the exact solution; (2) the error in
the energy AE is proportional to the square of the error in the wavefunction A'. Besides,
a variational solution allows one to calculate the atomic forces using the Hellman-
Feynman theory . This is very important for atomic relaxation and molecular dynamics
The computational disadvantage of the PW method for large scale parallel computation is
the requirement for the FFT. While the number of floating point operation is moderate,
the global communication for the FFT can be a serious bottleneck. Efficient FFT have
been demonstrated using a few thousand processors in electronic structure calculations
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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/m1/3/: accessed January 20, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.