Singular analysis and coupled cluster theory

Flad, Heinz-Jürgen; Harutyunyan, Gohar; Schulze, Bert-Wolfgang

doi:10.1039/c5cp01183c

Cited by 7 publications

(6 citation statements)

References 70 publications

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“…[113][114][115][116][117][118][119][120][121][122] Many real-space methods are wavelet based, 123 take sparsity into account, 124 provide controls for numerical errors, 125,126 are robust and efficient, 127 allow for basis set truncation errors below chemical accuracy and with predetermined numerical accuracy, Fig. 4 (d), 128 and they are easily adapted to massively parallel simulations. [129][130][131] Some of them scale linearly with system size 132 and they have a wide range of potential applications, including optimal control of quantum systems or simulations of plasmonic systems 133 and spectra.…”

Section: Reliable and Predictivementioning

confidence: 99%

From Prescriptive to Predictive: an Interdisciplinary Perspective on the Future of Computational Chemistry

Rommel

2021

Preprint

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Reliable predictions of the behaviour of chemical systems are essential across many industries, from nanoscale engineering over validation of advanced materials to nanotoxicity assessment in health and medicine. For the future we therefore envision a paradigm shift for the design of chemical simulations across all length scales from a prescriptive to a predictive and quantitative science. This paper presents an integrative perspective about the state-of-the-art of modelling in computational and theoretical chemistry with examples from data-and equation-based models. Extension to include reliable risk assessments and quality control are discussed. To specify and broaden the concept of chemical accuracy in the design cycle of reliable and robust molecular simulations the fields of computational chemistry, physics, mathematics, visualisation science, and engineering are bridged. Methods from electronic structure calculations serve as examples to explain how uncertainties arise: through assumed mechanisms in form of equations, model parameters, algorithms, and numerical implementations. We

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Section: Reliable and Predictivementioning

confidence: 99%

From Prescriptive to Predictive: an Interdisciplinary Perspective on the Future of Computational Chemistry

Rommel

2021

Preprint

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“…Let us just mention that the results of the present work can be applied to these models with minor modifications. For further details and first applications we refer to [10]. There are other approaches in singular analysis which have been applied to electronic structure theory as well, see e.g.…”

Section: Singular Analysis Meets Quantum Chemistrymentioning

confidence: 99%

Explicit Green operators for quantum mechanical Hamiltonians. II. Edge-type singularities of the helium atom

Flad

Flad-Harutyunyan

Schulze

2019

Asian-European J. Math.

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We extend our approach of asymptotic parametrix construction for Hamiltonian operators from conical to edge-type singularities which is applicable to coalescence points of two particles of the helium atom and related two electron systems including the hydrogen molecule. Up to second order we have calculated the symbols of an asymptotic parametrix of the nonrelativisic Hamiltonian of the helium atom within the Born-Oppenheimer approximation and provide explicit formulas for the corresponding Green operators which encode the asymptotic behaviour of the eigenfunctons near an edge. 1 We do not consider spin degrees of freedom or equivalent permutational symmetries of the electron coordinates in our discussion.2 The manifold M \ M0 actually correponds to the mathematical notion of a configuration space of N ordered particles in R 3 . more generally, as the inner part of an open smooth manifold with boundary. Next, let us consider the subset M 1 ⊂ M 0 of all coalescence points of more than two particles. The stratum M 0 \ M 1 is an open smooth manifold representing edges of M. Correspondingly, we denote M \ M 1 as a singular manifold with edges. Higher order strata can be constructed along the same lines, e.g., let M 2 ⊂ M 1 denote the set of coalescence points of more than three particles. Again the stratum M 1 \M 2 is an open smooth manifold representing the lowest order type of corners in M. Therefore, M \ M 2 is a singular manifold with edges and corners. In this way the configuration space can be decomposed into its strata, i.e.,The singular operator calculus associates classes of degenerate differential operators to the singular manifolds M \ M i , i = 0, 1, . . . , and a corresponding hierarchy of symbols to the strata.Within the present work we want to study edge singularities corresponding to coalescence points of two particles, i.e., two electrons or an electron and a nucleus. Higher order corner sigularities are subject of our future work.Near an edge, M \ M 1 is identified with a wedgewith smooth base X, homeomorphic to S 2 , the unit sphere 3 , and edge Y . The class Diff µ deg (W) of edge-degenerate Fuchs-type differential operators of order µ ∈ Nis defined on the associated open stretched wedgeHere r ∈ R + denotes the distance to the open edge Y ⊂ R q and y is a q-dimensional variable, varying on Y (for a Coulomb system consisting of N electrons the dimension q of Y equals 3N − 3). The coefficients a jα (r, y) take values in differential operators of order µ − (j + |α|) on the base X of the cone and are smooth in the respective variables up to r = 0. On an open stretched wedge it is the distance variable r ∈ R + which carries the asymptotic information. In order to incorporate asymptotics into Sobolev spaces let us proceed in a recursive manner. Weighted Sobolev spaces K s,γ (X ∧ ) on an open stretched cone with base X are defined with respect to the corresponding polar coordinatesx → (r, x) viafor a cut-off function ω, i.e., ω ∈ C ∞ 0 (R + ) such that ω(r) = 1 near r = 0. Here H s,γ (X ∧ ) = r γ H s,0 (...

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“…Wolfgang Hackbusch wh@mis.mpg.de (for instance, f and g describe the pair amplitude and the pair interaction; cf. Flad-Flad-Harutyunyan [5]). A discretisation by a uniform grid {ih = (i 1 h, i 2 h, i 3 h) : 0 ≤ i 1 , i 2 , i 3 ≤ n − 1} (h: grid size) in a cube leads to the discrete problem…”

Section: Introductionmentioning

confidence: 99%

Numerical Tensor Techniques for Multidimensional Convolution Products

Hackbusch

2018

Vietnam J. Math.

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In order to treat high-dimensional problems, one has to find data-sparse representations. Starting with a six-dimensional problem, we first introduce the low-rank approximation of matrices. One purpose is the reduction of memory requirements, another advantage is that now vector operations instead of matrix operations can be applied. In the considered problem, the vectors correspond to grid functions defined on a three-dimensional grid. This leads to the next separation: these grid functions are tensors in R n ⊗ R n ⊗ R n and can be represented by the hierarchical tensor format. Typical operations as the Hadamard product and the convolution are now reduced to operations between R n vectors. Standard algorithms for operations with vectors from R n are of order O(n) or larger. The tensorisation method is a representation method introducing additional data-sparsity. In many cases, the data size can be reduced from O(n) to O(log n). Even more important, operations as the convolution can be performed with a cost corresponding to these data sizes.

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Singular analysis and coupled cluster theory

Cited by 7 publications

References 70 publications

From Prescriptive to Predictive: an Interdisciplinary Perspective on the Future of Computational Chemistry

From Prescriptive to Predictive: an Interdisciplinary Perspective on the Future of Computational Chemistry

Explicit Green operators for quantum mechanical Hamiltonians. II. Edge-type singularities of the helium atom

Numerical Tensor Techniques for Multidimensional Convolution Products

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