Rapid evaluation of two-center two-electron integrals

Avoiding Negligible Shell Pairs and Quartets in Electronic Structure Calculations

We define a significant shell pair in an electronic structure calculation as one that generates at least one two-electron integral larger than a preset threshold. We define a significant shell quartet similarly. We then explore several methods for identifying nonsignificant pairs and quartets so that they can be avoided and computational efficiency improved. We find that the widely used Cauchy-Schwarz bound identifies most nonsignificant quartets but that the Hölder bound is slightly more powerful for identifying nonsignificant pairs.

Molecular integrals from Fast Fourier Transforms (FFT) instead of recurrences: The McMurchie-Davidson case

We report a closed formula expressing the McMurchie-Davidson (MD) key intermediates {[r]⁽⁰⁾; r_x + r_y + r_z ≤ L} directly in terms of the set of basic integrals {[0]^(m); m ≤ L}, without any recurrences. This formula can be evaluated at O(L) cost per output [r]⁽⁰⁾ with dense matrix multiplications and Fast Fourier Transforms (FFT). Key to this is the fact that the transformation that builds Cartesian angular momentum from the basic integrals, {[0_κ]^(m+m')}↦{[l_κ]^(m)} (κ ∈ {x, y, z}), can be phrased as a circulant-matrix/vector product, which is susceptible to FFTs. After simplification, a simple formula yields the final [r]⁽⁰⁾ in one step, as contraction of four auxiliary vectors over a common Fourier index k-one vector for the [0]^(m) and one for each Cartesian axis. Similar transformations occur in many integral approaches beside MD, making this idea potentially broadly applicable. The simple resulting code and data structures may make it attractive for novel hardware platforms.

Fast Evaluation of Two-Center Integrals over Gaussian Charge Distributions and Gaussian Orbitals with General Interaction Kernels

We present efficient algorithms for computing two-center integrals and integral derivatives, with general interaction kernels K(r₁₂), over Gaussian charge distributions of general angular momenta l. While formulated in terms of traditional ab initio integration techniques, full derivations and required secondary information, as well as a reference implementation, are provided to make the content accessible to other fields. Concretely, the presented algorithms are based on an adaption of the McMurchie-Davidson Recurrence Relation (MDRR) combined with analytical properties of the solid harmonic transformation; this obviates all intermediate recurrences except the adapted MDRR itself, and allows it to be applied to fully contracted auxiliary kernel integrals. The technique is particularly well-suited for semiempirical molecular orbital methods, where it can serve as a more general and efficient replacement of Slater-Koster tables, and for first-principles quantum chemistry methods employing density fitting. But the formalism's high efficiency and ability of handling general interaction kernels K(r₁₂) and multipolar Gaussian charge distributions may also be of interest for modeling electrostatic interactions and short-range exchange and charge penetration effects in classical force fields and model potentials. With the presented technique, a 4894 × 4894 univ-JKFIT Coulomb matrix J_AB = (A|1/r₁₂|B) (183 MiB) can be computed in 50 ms on a Q2'2018 notebook CPU, without any screening or approximations.

An examination of the nature of localized molecular orbitals and their value in understanding various phenomena that occur in organic chemistry

While canonical molecular orbitals have been used in computational chemistry for almost a century, the use of localized molecular orbitals is relatively new, and generating them has been difficult until recently. This has impeded their routine use in modeling chemical systems and reactions so that, even though localized molecular orbitals can now be generated easily, their usefulness in interpreting chemical phenomena has not been properly appreciated. Localized molecular orbitals can provide useful insights into chemical phenomena such as two-electron bonds, π delocalization, and lone pairs. A potentially important application would be interpreting the phenomena that occur in chemical reactions, in particular those reactions which can be described using the Lewis curly-arrow electron pushing convention. This paper considers how canonical and localized molecular orbitals are generated, their usefulness and limitations, and some issues that could be considered controversial regarding their nature, and it presents examples of the usefulness of LMOs in describing six chemical systems and one reaction.

Contracted auxiliary Gaussian basis integral and derivative evaluation

The rapid evaluation of two-center Coulomb and overlap integrals between contracted auxiliary solid harmonic Gaussian functions is examined. Integral expressions are derived from the application of Hobson's theorem and Dunlap's product and differentiation rules of the spherical tensor gradient operator. It is shown that inclusion of the primitive normalization constants greatly simplifies the calculation of contracted functions corresponding to a Gaussian multipole expansion of a diffuse charge density. Derivative expressions are presented and it is shown that chain rules are avoided by expressing the derivatives as a linear combination of auxiliary integrals involving no more than five terms. Calculation of integrals and derivatives requires the contraction of a single vector corresponding to the monopolar result and its scalar derivatives. Implementation of the method is discussed and comparison is made with a Cartesian Gaussian-based method. The current method is superior for the evaluation of both integrals and derivatives using either primitive or contracted functions.