Basis functions for linear-scaling first-principles calculations

Abstract

In the framework of a recently reported linear-scaling method for density-functionalpseudopotential calculations, we investigate the use of localized basis functions for such work. We propose a basis set in which each local orbital is represented in terms of an array of "blip functions" on the points of a grid. We analyze the relation between blip-function basis sets and the plane-wave basis used in standard pseudopotential methods, derive criteria for the approximate equivalence of the two, and describe practical tests of these criteria. Techniques are presented for using blipfunction basis sets in linear-scaling calculations, and numerical tests of these techniques are reported for Si crystal using both local and non-local pseudopotentials. We find rapid convergence of the total energy to the values given by standard plane-wave calculations as the radius of the linear-scaling localized orbitals is increased.

Elimination of basis set superposition error in linear-scaling density-functional calculations with local orbitals optimised in situ

“…Linear scaling is obtained by truncating the kernel and localizing the NGWFs according to variational spatial cut-offs. Most linear-scaling methods fall into two categories: the first equivalent to optimizing the kernel only [17][18][19] for a fixed (but potentially large) set of local orbitals and the second involving both kernel and NGWF optimisation [16,15,20]. In the first case, the local orbitals are optimised beforehand for isolated atoms of each species and act as the basis set.…”

“…In all of these methods, local orbitals play a key part. For example, in the onetep method [14] the density-matrix is expressed in separable form [15] as…”

Linear-scaling density-functional theory with tens of thousands of atoms: Expanding the scope and scale of calculations with ONETEP

“…In other methods real-space grids with finite-difference [22,24] or multigrid [25,26] techniques, B-splines or blip functions [27], localised spherical-waves [28], numerical atomic orbitals [29,30,31,32] and Gaussians [33,34] have been proposed. In onetep a set of periodic cardinal sine or PSINC functions [23,35] are used which are equivalent to a set of plane-waves and which therefore inherit their accuracy (particularly with regard to the kinetic energy [36,37]) and the ability to improve the basis set completeness systematically via a single parameter, the energy cut-off.…”

Preconditioned iterative minimization for linear-scaling electronic structure calculations

“…Some of these approaches use a relatively small basis set of numerical atomic orbitals 9 or Gaussian atomic orbitals 10,11 that have been preoptimized for other environments and transferred to the system under consideration; other approaches [12][13][14][15][16] use much larger localized basis sets of simple functions such as polynomials, 17,18 spherical waves, 19 or bandwidth limited delta functions. 20 Each of these philosophies has its advantages and drawbacks: The former can suffer from transferability problems but is capable of providing good accuracy with modest effort; the latter is computationally more intensive but is capable of giving an accuracy that is systematically tunable with a parameter that controls the completeness of the basis set that is being used, akin to the kinetic energy cut-off in plane-wave methods.…”