New index functions for storing Gaunt coefficients

Pinchon, Didier; Hoggan, Philip E.

doi:10.1002/qua.21337

Cited by 22 publications

(8 citation statements)

References 13 publications

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“…A change of basis set requires relatively few simple new evaluations. A modular or object oriented program is being designed to do this more efficiently 11, 17, 18…”

Section: Atomic Basis Functionsmentioning

confidence: 99%

Quantum Monte Carlo simulation of carbon monoxide reactivity when adsorbed at metal and oxide catalyst surfaces: Trial wave‐functions with exponential type basis and quasi‐exact three‐body correlation

Hoggan

2012

Int J of Quantum Chemistry

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Review publications show the advantages of using quantum Monte Carlo (QMC) methods when bonding interactions are modified and consequently, where the electron correlation energy varies and needs to be evaluated accurately. This article considers a model efficient metal catalyst for reactions of carbon monoxide. The ultimate aim is to help design industrially useful catalysts, since the reaction with water produces hydrogen gas selectively, that is, a clean fuel and a sustainable energy source. To narrow the gap between the industrial process and calculation, a model of platinum/oxide is studied. It is mimicked here by copper with and without adsorbed oxygen. Periodicity as well as a validated pseudo-potential are used to limit the number of active electrons considered. A suitable code to carry out this task is CASINO. Furthermore, this code scales linearly to 100,000 processors and possesses a shared memory facility so that it is well-suited to runs on the Blugene/P. It has been in production on such computers since early in 2011. A mixed plane-wave/exponential type orbitals (ETO) basis is used for the systems studied. Plane waves are well-suited to periodic solid substrates and a linear combination of ETOs for molecules. The atomic orbitals have direct physical interpretation, i.e., Coulomb Sturmians and hydrogen-like orbitals. Their radial nodes are shown to be essential in obtaining the vitally important accurate QMC trial wave-functions. Until 2008,ETO products on different atoms were difficult to manipulate for the evaluation of two-electron integrals. Coulomb resolutions provide an excellent approximation that reduces these integrals to a sum of one-electron overlap-like integral products that each involve orbitals on at most two centers. They are thus readily evaluated. Only these integrals need to be re-evaluated to change basis functions. In this article, QMC-VMC variational optimisation is used with a quasi analytic two-electron and nucleus correlation factor. The QMC diffusion Monte Carlo methodology is applied to adsorbed carbon monoxide and some of its reactions.

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“…A change of basis set requires relatively few simple new evaluations. A modular or object oriented program is being designed to do this more efficiently 11, 17, 18…”

Section: Atomic Basis Functionsmentioning

confidence: 99%

Quantum Monte Carlo simulation of carbon monoxide reactivity when adsorbed at metal and oxide catalyst surfaces: Trial wave‐functions with exponential type basis and quasi‐exact three‐body correlation

Hoggan

2012

Int J of Quantum Chemistry

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“…Note that unlike the atomic case, in which the angular expansion is always limited, the angular momentum l may attain large values in the partial‐wave expansion for diatomic molecules: for instance, the calculations on Cu 2 and CuLi in the present work used expansions up to l = 46. Although elegant schemes for the sparse storage of Gaunt coefficient tables have been discussed in the literature, in the present case only a small subset of m values is needed—from m = 0 for σ orbitals to m = ±3 for ϕ orbitals—and so a simple dense cubic array storage scheme [( l 1 , m 1 ), ( l 2 , m 2 ); ( L , M )] is sufficient for our work.…”

Section: Theorymentioning

confidence: 99%

“…The examination of equation (42) shows that the kinetic energy density diverges for µ → 0 for m = 0. This means that non-σ states must vanish at µ = 0…”

Section: Kinetic Energymentioning

confidence: 99%

Fully numerical Hartree‐Fock and density functional calculations. II. Diatomic molecules

Lehtola

2019

Int J of Quantum Chemistry

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We present the implementation of a variational finite element solver in the HELFEM program for benchmark calculations on diatomic systems. A basis set of the formÞ is used, where (μ, ν, φ) are transformed prolate spheroidal coordinates, B n (μ) are finite element shape functions, and Y m l are spherical harmonics. The basis set allows for an arbitrary level of accuracy in calculations on diatomic molecules, which can be performed at present with either nonrelativistic Hartree-Fock (HF) or density functional (DF) theory. Hundreds of DFs at the local spin density approximation (LDA), generalized gradient approximation (GGA), and the meta-GGA level can be used through an interface with the LIBXC library; meta-GGA and hybrid DFs are not available in other fully numerical diatomic program packages. Finite electric fields are also supported in HELFEM, enabling access to electric properties. We introduce a powerful tool for adaptively choosing the basis set by using the core Hamiltonian as a proxy for its completeness. HELFEM and the novel basis set procedure are demonstrated by reproducing the restricted open-shell HF limit energies of 68 diatomic molecules from the first to the fourth period with excellent agreement with literature values, despite requiring orders of magnitude fewer parameters for the wave function. Then, the electric properties of the BH and N 2 molecules under finite field are studied, again yielding excellent agreement with previous HF limit values for energies, dipole moments, and dipole polarizabilities, again with much more compact wave functions than what were needed for the literature references. Finally, HF, LDA, GGA, and meta-GGA calculations of the atomization energy of N 2 are performed, demonstrating the superb accuracy of the present approach. K E Y W O R D S density functional, diatomic molecule, finite element, Hartree-Fock, partial-wave expansion 1 | INTRODUCTIONIn the first part of this series, [1] we presented fully numerical Hartree-Fock (HF) and density functional calculations on atoms. The present article focuses on diatomic molecules, which may serve as even more stringent tests of ab initio as well as density functional methods than atoms, as the methods' accuracy can be probed for multiple kinds of covalent bonds, as well as for ionic bonds and dispersion effects. As reviewed at length in Ref.[2], the history of fully numerical calculations on diatomic molecules is almost as long as that on atoms, starting with the partial-wave approach of McCullough. [3,4] The partial-wave approach appears to have since fallen out of use, with the well-known programs by Heinemann, Laaksonen, Sundholm, Kobus, and coworkers [5][6][7][8] relying on grid-based approaches for the angular expansion.

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“…Luckily, these need to be done only once. These optimizations were inspired by Pinchon & Hoggan (2006).…”

Section: An Example Applicationmentioning

confidence: 99%

Evolution of non‐linear dynamics using Graphical Processing Units

2013

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A new Graphical Processing Unit (GPU) accelerated general lattice PDE solver and real-time visualisation framework, called Gripper, is presented. Besides its scientific goals, the purpose of this project is to provide a publicly available, general purpose GPU enabled framework with visualisation capabilities. One of the application of this newly developed software, is aimed to provide valuable insight into non-linear evolution of relativistic dynamical systems, such, as for example, neutron stars or black holes.

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New index functions for storing Gaunt coefficients

Cited by 22 publications

References 13 publications

Quantum Monte Carlo simulation of carbon monoxide reactivity when adsorbed at metal and oxide catalyst surfaces: Trial wave‐functions with exponential type basis and quasi‐exact three‐body correlation

Quantum Monte Carlo simulation of carbon monoxide reactivity when adsorbed at metal and oxide catalyst surfaces: Trial wave‐functions with exponential type basis and quasi‐exact three‐body correlation

Fully numerical Hartree‐Fock and density functional calculations. II. Diatomic molecules

Evolution of non‐linear dynamics using Graphical Processing Units

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