Density-functional expansion methods: Generalization of the auxiliary basis

Abstract

The formulation of density-functional expansion methods is extended to treat the second and higherorder terms involving the response density and spin densities with an arbitrary single-center auxiliary basis. The two-center atomic orbital products are represented by the auxiliary functions centered about those two atoms, and the mapping coefficients are determined from a local constrained variational procedure. This two-center variational procedure allows the mapping coefficients to be pretabulated and splined as a function of internuclear separation for efficient look up. The splines of mapping coefficients have a range no longer than that of the overlap integrals, and the auxiliary density appears as a single point-multipole expansion to all nonoverlapping atoms, thus allowing for the trivial implementation of a linear-scaling algorithm. The method is tested using Gaussian multipole expansions, and the effect of angular and radial completeness is explored. Several auxiliary basis sets are parametrized and compared to an auxiliary basis analogous to that used in the selfconsistent-charge density-functional tight-binding model, and the method is demonstrated to greatly improve the representation of the density response with respect to a reference expansion model that does not use an auxiliary basis.

Density-functional expansion methods: grand challenges

“…When reading those works [ 56 -58 ], a reader may misconstrue the use of a third-order expansion to mean that the underlying problem giving rise to the quadratic behavior of the hardness is a premature truncation of the Taylor series to second-order. Our previous works suggest that this is not the case [26,32]. The pure-quadratic response self-energy of an isolated atom, as modeled by DFTB2, results only from having limited the auxiliary basis so completely that both the diffuse and tightlybound electron densities are represented by the same spatial function, and thus contribute equally to the self-energy.…”

“…Table 1 compares the geometry optimized bond, angle, dipole moment, and relative energy errors of various PBE/6-31G*-based models to standard PBE/6-31G*. The statistics include results from 52 molecules [ 81 ] taken from the G2/97 neutral small molecule test set [ 82 ], which were also used in our previous VE studies [26,32]. For the purpose of discussing the generalization of the auxiliary basis, the reader should compare VEJ/1S(M) and VEJ/2P,3D,4F to VEJ, and note how well VEJ compares to VE.…”

“…For the purpose of discussing the generalization of the auxiliary basis, the reader should compare VEJ/1S(M) and VEJ/2P,3D,4F to VEJ, and note how well VEJ compares to VE. The parametrization of the VEJ/2P,3D,4F auxiliary basis Slater exponents is described elsewhere [32]. Table 1 is used to compare the errors of the VE1, VE1S, and VE1W models, all of which are variations of the VE0 model.…”

“…The VEJ model, where the AO products are represented with an auxiliary basis of Gaussian multipole expansions (GME) (32) whose primitive contraction coefficients c γ , and exponents ζ γ (ξ), have been chosen to mimic a Slater multipole-like expansion with exponent ξ. The notation 2P,3D,4F indicates that period 1 elements use two GMEs of l max = 1, each corresponding to one of two Slater exponents; period 2 elements use three GMEs of l max = 2; and period 3 elements use four GMEs of l max = 3.…”

Attractive electron–electron interactions within robust local fitting approximations

“…Essentially, the same idea was used more recently by Giese and York. [52] In the PARI scheme, we exploit locality in a manner similar to that of Baerends et al except that we approximate the electron integrals ðabjcdÞ individually for each jcdÞ rather than its total Coulomb interaction ðabjqÞ. Also, we apply the robust Dunlap correction.…”