Associated Legendre Polynomials and Spherical Harmonics Computation for Chemistry Applications

Associated Legendre polynomials and spherical harmonics are central to calculations in many fields of science and mathematics - not only chemistry but computer graphics, magnetic, seismology and geodesy. There are a number of algorithms for these functions published since 1960 but none of them satisfy our requirements. In this paper, we present a comprehensive review of algorithms in the literature and, based on them, propose an efficient and accurate code for quantum chemistry. Our requirements are to efficiently calculate these functions for all non-negative integer degrees and orders up to a given number (<=1000) and the absolute or the relative error of each calculated value should not exceed 10E-10. We achieve this by normalizing the polynomials, employing efficient and stable recurrence relations, and precomputing coefficients. The algorithm presented here is straightforward and may be used in other areas of science.Comment: The 40th Congress on Science and Technology of Thailand (STT40

Applications of Resolutions of the Coulomb Operator in Quantum Chemistry

This dissertation shows that the Coulomb operator and the long-range Coulomb operators can be resolved as a sum of products of one-particle functions. These resolutions provide a potent new route to tackle quantum chemical problems. Replacing electron repulsion terms in Schrodinger equations by the truncated resolutions yields the reduced-rank Schrodinger equations (RRSE). RRSEs are simpler than the original equations but yield energies with chemical accuracy even for low-rank approximations. Resolutions of the Coulomb operator factorize Coulomb matrix elements to Cholesky-like sums of products of auxiliary integrals. This factorization is the key to the reduction of computational cost of quantum chemical methods

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 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.Comment: 10 pages, 2 figure

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 and long-range Ewald operators) 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

Quantum Chemical Investigation of Polychlorinated Dibenzodioxins, Dibenzofurans and Biphenyls: Relative Stability and Planarity Analysis

All the possible polychlorinated aromatic compounds in the classes of dibenzodioxins (PCDDs), dibenzofurans (PCDFs), and biphenyls (PCBs) were studied by the quantum chemical methods of HF/6-311++G(d,p), B3LYP/6-311++G(d,p), and MP2/cc-pVTZ. The calculated stabilities and structures of these compounds were compared with the available data on their abundance and toxicity. Prediction models for trends in energy and planarity among these congeners were proposed. The results discussed here can help contribute to the understanding of the role of dioxin-like compounds (DLCs) in the environment

Z-Matrix template-based substitution approach for enumeration of 3D molecular structures

The exhaustive enumeration of 3D chemical structures based on Z-matrix templates has recently been used in the quantum chemical investigation of constitutional isomers, diastereomers and rotamers. This simple yet powerful initial structure generation approach can apply beyond the investigation of compounds of identical formula by quantum chemical methods. This paper aims to provide a short description of the overall concept followed by a practical tutorial to the approach. · - The four steps required for Z-matrix template-based substitution are template construction, generation of tuples for substitution sites, removal of duplicate tuples and substitution on the template. · -The generated tuples can be used to create chemical identifiers to query compound properties from chemical databases. · - All of these steps are demonstrated in this paper by common model compounds and are very straightforward for an undergraduate audience to reproduce. A comparison of the approach in this tutorial and other options is also discussed.<br /

Resolutions of the Coulomb operator Part III.Reduced-rank Schroedinger equations

We consider a modified Schrödinger equation wherein the electron-electron repulsion terms rij-1 are approximated by truncated one-particle resolutions. Numerical results for the He atom and H2 molecule at the Hartree-Fock, second-order Møller-Plesset,

Exposure of Plants to Static Electromagnetic Fields: The Early Growth of Basil and Waxy Corn

The effects of a static low-intensity electromagnetic field (EMF) on higher plants were investigated. Thai basil (Ocimum basilicum var. thyrsiflora) and waxy corn (Zea mays var. ceratina) were chosen as samples for the experiments. There were 2 methods of electromagnetic exposure, indirect exposure (applying magnetically treated water to plants) and direct exposure (plants are grown under the electromagnetic field). For the indirect exposure experiment, magnetically treated water (0.34-4.54 mT, 24 h) was applied to Thai basil seeds daily for 1 week and the experiment was repeated once. No significant difference in the heights was found between the control groups and the treated groups whether deionized or tap water was used. For the direct exposure experiment (0.28-3.83 mT), waxy corn seeds were grown between a pair of electromagnets. Two controls were grown in the absence of an extra magnetic field; 1 control group (no electromagnets (EM)) was grown at room temperature while the other control group (EM-0) was grown between a pair of electromagnets with a similar temperature to the other magnetically treated groups. The experiment was completed in 4 days and was repeated once. The results indicated that there is no height difference between the control (EM-0) and the groups grown between electromagnets; however, the controls (no EM), which were grown at room temperature were shorter compared to those grown between electromagnets. Our results suggest that the level of the applied magnetic field has no effect on the heights of the plants in this study