Resolutions of the Coulomb operator

Reexamination of the calculation of two-center, two-electron integrals over Slater-type orbitals. I. Coulomb and hybrid integrals

In this paper, which constitutes the first part of the series, we consider calculation of two-center Coulomb and hybrid integrals over Slater-type orbitals. General formulas for these integrals are derived with no restrictions on the values of the quantum numbers and nonlinear parameters. Direct integration over the coordinates of one of the electrons leaves us with the set of overlaplike integrals which are evaluated by using two distinct methods. The first one is based on the transformation to the ellipsoidal coordinates system and the second utilizes a recursive scheme for consecutive increase of the angular momenta in the integrand. In both methods simple one-dimensional numerical integrations are used in order to avoid severe digital erosion connected with the straightforward use of the alternative analytical formulas. It is discussed that the numerical integration does not introduce a large computational overhead since the integrands are well-behaved functions, calculated recursively with decent speed. Special attention is paid to the numerical stability of the algorithms. Applicability of the resulting scheme over a large range of the nonlinear parameters is tested on examples of the most difficult integrals appearing in the actual calculations including, at most, 7i-type functions (l=6).

Resolutions of the Coulomb operator: VIII. Parallel implementation using the modern programming language X10

Use of the modern parallel programming language X10 for computing long-range Coulomb and exchange interactions is presented. By using X10, a partitioned global address space language with support for task parallelism and the explicit representation of data locality, the resolution of the Ewald operator can be parallelized in a straightforward manner including use of both intranode and internode parallelism. We evaluate four different schemes for dynamic load balancing of integral calculation using X10's work stealing runtime, and report performance results for long-range HF energy calculation of large molecule/high quality basis running on up to 1024 cores of a high performance cluster machine.

Resolutions of the Coulomb Operator: VII. Evaluation of Long-Range Coulomb and Exchange Matrices

Use of the resolution of Ewald operator method for computing long-range Coulomb and exchange interactions is presented. We show that the accuracy of this method can be controlled by a single parameter in a manner similar to that used by conventional algorithms that compute two-electron integrals. Significant performance advantages over conventional algorithms are observed, particularly for high quality basis sets and globular systems. The approach is directly applicable to hybrid density functional theory.

A remarkable identity involving Bessel functions

We consider a new identity involving integrals and sums of Bessel functions. The identity provides new ways to evaluate integrals of products of two Bessel functions. The identity is remarkably simple and powerful since the summand and the integrand are of exactly the same form and the sum converges to the integral relatively fast for most cases. A proof and numerical examples of the identity are discussed.

Resolutions of the Coulomb operator. VI. Computation of auxiliary integrals

We discuss the efficient computation of the auxiliary integrals that arise when resolutions of two-electron operators (specifically, the Coulomb operator [T. Limpanuparb, A. T. B. Gilbert, and P. M. W. Gill, J. Chem. Theory Comput. 7, 830 (2011)] and the long-range Ewald operator [T. Limpanuparb and P. M. W. Gill, J. Chem. Theory Comput. 7, 2353 (2011)]) are employed in quantum chemical calculations. We derive a recurrence relation that facilitates the generation of auxiliary integrals for Gaussian basis functions of arbitrary angular momentum and propose a near-optimal algorithm for its use.

Resolutions of the Coulomb Operator: V. The Long-Range Ewald Operator

We show that the long-range Ewald operator can be resolved as erf(ωr12)/r12 = ∑k ϕk*(r1) ϕk(r2), where ϕk is proportional to the product of a spherical Bessel function and a spherical harmonic. We demonstrate the use of this new resolution by calculating the long-range Coulomb energy of the nanodiamond crystallite C84H64 and the long-range exchange energy of the graphene C96H24. The resolution appears particularly effective for long-range exchange calculations.

Diagnostics of molecular orbital quality

We discuss several measures of the quality of a molecular orbital. Each requires only that the orbital be associated with a well-defined Fock operator and is thus applicable to both Hartree–Fock and density functional orbitals. One of the measures, the γ diagnostic, ranges from γ = 0 (perfect) to γ = π/2 (poor) and is conceptually simple. We illustrate its usefulness by applying it to a number of small atoms and ions.