Abstract
In this chapter, we will discuss O(N) methods for first-principles calculations based on density functional theory (DFT) which provides a first-principles framework enabling us to efficiently calculate electronic structures of materials with quantitative accuracy, where N is the number of atoms.
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Notes
- 1.
Note that the second and third terms in the parenthesis are calculated using the output charge density \(n_\mathrm{out}\), while the fourth term is calculated using the input charge density \(n_\mathrm{in}\).
- 2.
Even if the overlap between the associated localized orbitals is zero, the two-electron exchange integral is not zero. Thus, strictly speaking all the elements are nonzero for hybrid functionals. Even in the case of hybrid functionals with a screened Coulomb interaction, the number of nonzero elements largely increases compared to semi-local functionals, while the resultant matrix will be sparse due to the limited interaction.
- 3.
The formulation does not rely on the band gap of a systems.
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Ozaki, T. (2019). O(N) Methods. In: Geshi, M. (eds) The Art of High Performance Computing for Computational Science, Vol. 2. Springer, Singapore. https://doi.org/10.1007/978-981-13-9802-5_4
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