A divide and conquer real-space approach for all-electron molecular electrostatic potentials and interaction energies

Abstract

A computational scheme to perform accurate numerical calculations of electrostatic potentials and interaction energies for molecular systems has been developed and implemented. Molecular electron and energy densities are divided into overlapping atom-centered atomic contributions and a three-dimensional molecular remainder. The steep nuclear cusps are included in the atom-centered functions making the three-dimensional remainder smooth enough to be accurately represented with a tractable amount of grid points. The one-dimensional radial functions of the atom-centered contributions as well as the three-dimensional remainder are expanded using finite element functions. The electrostatic potential is calculated by integrating the Coulomb potential for each separate density contribution, using our tensorial finite element method for the three-dimensional remainder. We also provide algorithms to compute accurate electron-electron and electron-nuclear interactions numerically using the proposed partitioning. The methods have been tested on all-electron densities of 18 reasonable large molecules containing elements up to Zn. The accuracy of the calculated Coulomb interaction energies is in the range of 10(-3) to 10(-6) E(h) when using an equidistant grid with a step length of 0.05 a(0).

The Elephant in the Room of Density Functional Theory Calculations

“…31 Our MW results provide quasi-exact reference values that can be employed to quantify the accuracy of standard basis sets, such as GTO, NAO, and APW methods, as well as of novel approaches based for instance on finite element methods [32][33][34][35] or discontinuous Galerkin methods.…”

The accuracy of the Gaussian-and-finite-element-Coulomb (GFC) method for the calculation of Coulomb integrals

“…(7), the coefficients in the Gaussian expansion of the atomic potential were determined by solving a system of linear equations similar to that in Eqs. (10)- (12). In the optimized sets, the ratios between two consecutive exponents (sorted in decreasing order) are always monotonically increasing (but with a less regular behavior for the ratios involving the largest exponents).…”

“…In particular, a linear combination of auxiliary Gaussians, with coefficients computed by solving the GFC equations, Eqs. (10)- (12), in the whole mixed GF basis set, takes at all points on some nonzero values that cannot be predicted exactly in advance. The existence of these artificial boundary values violates the assumption that the solution to the problem in Eq.…”

“…As the system size increases and d is sufficiently small, ε J eventually becomes negative. With d = 1.5a 0 , negative errors appear for alanine (18); with d = 1.2a 0 , they appear already with alanine (12). It means that the Coulomb energy obtained from the GFC approach with point-charge boundary conditions cannot any longer be considered as a lower bound to the correct energy and, therefore, the GFC method loses its variational character.…”

“…The GFC method can be viewed as belonging to a larger class of methods that decompose molecular functions such as the density and the potential into a sum of spherical components (of analytical or numerical form) centered on the atoms and a remaining part, represented as an expansion in plane waves or local finite-element basis functions. [4][5][6][7][8][9][10][11][12] In these methods, the atom-centered functions are utilized to describe, in an efficient manner, the complicated behavior of the considered molecular function that occurs near the nuclei.…”