27 research outputs found
Non-iterative computation of Gauss-Jacobi quadrature
Asymptotic approximations to the zeros of Jacobi polynomials are given, with methods to obtain the coefficients in the expansions. These approximations can be used as standalone methods for the noniterative computation of the nodes of Gauss--Jacobi quadratures of high degree (). We also provide asymptotic approximations for functions related to the first-order derivative of Jacobi polynomials which are used for computing the weights of the Gauss--Jacobi quadrature. The performance of the asymptotic approximations is illustrated with numerical examples, and it is shown that nearly double precision relative accuracy is obtained for both the nodes and the weights when and . For smaller degrees the approximations are also useful as they provide relative accuracy for the nodes when , and just one Newton step would be sufficient to guarantee double precision accuracy in that cases
Fast, reliable and unrestricted iterative computation of Gauss-Hermite and Gauss-Laguerre quadratures
Methods for the computation of classical Gaussian quadrature rules are described which are effective both for small and large degree. These methods are reliable because the iterative computation of the nodes has guaranteed convergence, and they are fast due to their fourth-order convergence and its asymptotic exactness for an appropriate selection of the variables. For Gauss?Hermite and Gauss?Laguerre quadratures, local Taylor series can be used for computing efficiently the orthogonal polynomials involved, with exact initial values for the Hermite case and first values computed with a continued fraction for the Laguerre case. The resulting algorithms have almost unrestricted validity with respect to the parameters. Full relative precision is reached for the Hermite nodes, without any accuracy loss and for any degree, and a mild accuracy loss occurs for the Hermite and Laguerre weights as well as for the Laguerre nodes. These fast methods are exclusively based on convergent processes, which, together with the high order of convergence of the underlying iterative method, makes them particularly useful for high accuracy computations. We show examples of very high accuracy computations (of up to 1000 digits of accuracy)
Uniform relations between the Gauss-Legendre nodes and weights
Three different relations between the Legendre nodes and weights are
presented which, unlike the circle and trapezoid theorems for Gauss-Legendre
quadrature, hold uniformly in the whole interval of orthogonality .
These properties are supported by strong asymptotic evidence.The study of these
results is motivated by the role they play in certain finite difference schemes
used in the discretization of the angular Fokker-Planck diffusion operator
Asymptotic expansions of Jacobi polynomials and of the nodes and weights of Gauss-Jacobi quadrature for large degree and parameters in terms of elementary functions
Asymptotic approximations of Jacobi polynomials are given in terms of elementary functions for large degree n and parameters α and ÎČ. From these new results, asymptotic expansions of the zeros are derived and methods are given to obtain the coefficients in the expansions. These approximations can be used as initial values in iterative methods for computing the nodes of Gauss--Jacobi quadrature for large degree and parameters. The performance of the asymptotic approximations for computing the nodes and weights of these Gaussian quadratures is illustrated with numerical examples
Global potential energy surface for the O2 + N2 interaction. Applications to the collisional, spectroscopic, and thermodynamic properties of the complex
A detailed characterization of the interaction between the most abundant
molecules in air is important for the understanding of a variety of phenomena
in atmospherical science. A completely {\em ab initio} global potential energy
surface (PES) for the O + N interaction is
reported for the first time. It has been obtained with the symmetry-adapted
perturbation theory utilizing a density functional description of monomers
[SAPT(DFT)] extended to treat the interaction involving high-spin open-shell
complexes. The computed interaction energies of the complex are in a good
agreement with those obtained by using the spin-restricted coupled cluster
methodology with singles, doubles and noniterative triple excitations
[RCCSD(T)]. A spherical harmonics expansion containing a large number of terms
due to the anisotropy of the interaction has been built from the {\em ab
initio} data. The radial coefficients of the expansion are matched in the long
range with the analytical functions based on the recent {\em ab initio}
calculations of the electric properties of the monomers [M. Bartolomei et al.,
J. Comp. Chem., {\bf 32}, 279 (2011)]. The PES is tested against the second
virial coefficient data and the integral cross sections measured with
rotationally hot effusive beams, leading in both cases to a very good
agreement. The first bound states of the complex have been computed and
relevant spectroscopic features of the interacting complex are reported. A
comparison with a previous experimentally derived PES is also provided
Asymptotic computation of classical orthogonal polynomials
The classical orthogonal polynomials (Hermite, Laguerre and Jacobi) are involved in a vast number of applications in physics and engineering. When large degrees n are needed, the use of recursion to compute the polynomials is not a good strategy for computation and a more efficient approach, such as the use of asymptotic expansions,is recommended. In this paper, we give an overview of the asymptotic expansions considered in [8] for computing Laguerre polynomials L(α)n(x) for bounded values of the parameter α. Additionally, we show examples of the computational performance of an asymptotic expansion for L(α)n(x) valid for large values of α and n. This expansion was used in [6] as starting point for obtaining asymptotic approximations to the zeros. Finally, we analyze the expansions considered in [9], [10] and [11] to compute the Jacobi polynomials for large degrees n
Pion-less effective field theory for atomic nuclei and lattice nuclei
We compute the medium-mass nuclei O and Ca using pionless
effective field theory (EFT) at next-to-leading order (NLO). The low-energy
coefficients of the EFT Hamiltonian are adjusted to experimantal data for
nuclei with mass numbers and , or alternatively to results from
lattice quantum chromodynamics (QCD) at an unphysical pion mass of 806 MeV. The
EFT is implemented through a discrete variable representation in the harmonic
oscillator basis. This approach ensures rapid convergence with respect to the
size of the model space and facilitates the computation of medium-mass nuclei.
At NLO the nuclei O and Ca are bound with respect to decay into
alpha particles. Binding energies per nucleon are 9-10 MeV and 30-40 MeV at
pion masses of 140 MeV and 806 MeV, respectively.Comment: 26 page
The Polarizable Continuum Model Goes Viral! Extensible, Modular and Sustainable Development of Quantum Mechanical Continuum Solvation Models
Synergistic theoretical and experimental approaches to challenging chemical problems have become more and more widespread, due to the availability of efficient and accurate ab initio quantum chemical models. Limitations to such an approach do, however, still exist. The vast majority of chemical phenomena happens in complex environments, where the molecule of interest can interact with a large number of other moieties, solvent molecules or residues in a protein. These systems represent an ongoing challenge to our modelling capabilities, especially when high accuracy is required for the prediction of exotic and novel molecular properties. How to achieve the insight needed to understand and predict the physics and chemistry of such complex systems is still an open question. I will present our efforts in answering this question based on the development of the polarizable continuum model for solvation. While the solute is described by a quantum mechanical method, the surrounding environment is replaced by a structureless continuum dielectric. The mutual polarization of the solute-environment system is described by classical electrostatics. Despite its inherent simplifications, the model contains the basic mathematical features of more refined explicit quantum/classical polarizable models. Leveraging this fundamental similarity, we show how the inclusion of environment effects for relativistic and nonrelativistic quantum mechanical Hamiltonians, arbitrary order response properties and high-level electron correlation methods can be transparently derived and implemented. The computer implementation of the polarizable continuum model is central to the work presented in this dissertation. The quantum chemistry software ecosystem suffers from a growing complexity. Modular programming offers an extensible, flexible and sustainable paradigm to implement new features with reduced effort. PCMSolver, our open-source application programming interface, can provide continuum solvation functionality to any quantum chemistry software: continuum solvation goes viral. Our strategy affords simpler programming workflows, more thorough testing and lower overall code complexity. As examples of the flexibility of our implementation approach, we present results for the continuum modelling of non homogeneous environments and how wavelet-based numerical methods greatly outperform the accuracy of traditional methods usually employed in continuum solvation models