Linear-scaling DFT LCAO Methods for Solids

The increase in computer power and the progress in the methodology have allowed the ab-initio calculations of increasingly more complex and larger systems with an increasing number of atoms N (for periodic qrstems N is the number of atoms in the primitive unit cell). In the HF LCAO and DFT PW methods the computer time and memory scales with N like [377,399]. inner only recently that two linear 

The Siesta (Spanish Initiative for Electronic Simulations with Thousands of Atoms) method [383,384,400] achieves linear scaling by the exphcit use of locahzed Wannier-like functions and numerical pseudoatomic orbitals confined by a spherical infinite-potential wall [401]. As the restriction of the Siesta method we mention the difficulty of the all-electron calculations and use of only LDA/GGA exchange-correlation functionals. 

The second DFT LCAO linear-scaling method by Scuseria and Kudin (SK method) [379] uses Gaussian atomic orbitals and a fast multipole method, which achieves not only linear-scaling with system size, but also very high accuracy in aU infinite summations [397]. This approach allows both all-electron and pseudopotential calculations and can be applied also with hybrid HF-DFT exchange-correlation functionals. 

Within the pseudopotential approximation, the standard KS one-electron Hamiltonian is written as 

The matrix elements of the first two terms involve only two-center integrals that are calculated in reciprocal space and tabulated as a function of interatomic distance. The remaining terms involve potentials that are calculated on a three-dimensional real-space grid. 

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