Approximating a wavefunction as an unconstrained sum of Slater determinants

Beylkin, Gregory; Mohlenkamp, Martin J.; Pérez, Fernando

doi:10.1063/1.2873123

Cited by 39 publications

(67 citation statements)

References 54 publications

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“…where the constant C 0 > 0 does not depend on M or on n. Notice that a similar result was proven in [4,Theorem 22]. The following lemma proves the spectral equivalence estimates:…”

Section: In the Case Of Variable Coefficients The Related Cost Is Boumentioning

confidence: 51%

“…Here the parameter z 2 ≥ 0 can be specified via some optimization criteria; however, z = 0 will be the standard choice (convolution with the Poisson kernel). Under certain assumptions, G z can be proven to be a bounded operator in L 2 (R d ) (see [4,27]) that permits simple and robust iterative solution methods provided that the convolution transform can be computed in an efficient way (cf. [18]).…”

Section: The Green Function Formulationmentioning

confidence: 99%

“…The Green function (fixed point) formulation applied to the spectral problems in quantum chemistry (the Lippmann-Schwinger equation) leads to the efficient numerical methods based on data-sparse representation of operators involved (see [4,16] and references therein). This concept can also be applied as a preconditioning scheme for solving the elliptic boundary value problems.…”

Section: The Green Function Formulationmentioning

confidence: 99%

“…The idea of tensor-structured approximation of the elliptic inverse was first addressed in [3,4,10]. Particular constructions based on the sinc-approximation [29] applied to the operators (− + λI ) −α , λ ∈ R + , α > 0 on a hypercube were considered in [9,10,19].…”

Section: Introductionmentioning

confidence: 99%

“…[3,4,13,15,19,24]). In this way the construction of spectrally close tensor-structured preconditioners for the class of elliptic integral-differential operators in R d is one of the building blocks in development of efficient numerical methods in the case d ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

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Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝ d

Khoromskij

2009

Constr Approx

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In the present paper we analyze a class of tensor-structured preconditioners for the multidimensional second-order elliptic operators in R d , d ≥ 2. For equations in a bounded domain, the construction is based on the rank-R tensor-product approximation of the elliptic resolvent B R ≈ (L − λI ) −1 , where L is the sum of univariate elliptic operators. We prove the explicit estimate on the tensor rank R that ensures the spectral equivalence. For equations in an unbounded domain, one can utilize the tensor-structured approximation of Green's kernel for the shifted Laplacian in R d , which is well developed in the case of nonoscillatory potentials. For the oscillating kernels e −iκ x / x , x ∈ R d , κ ∈ R + , we give constructive proof of the rank-O(κ) separable approximation. This leads to the tensor representation for the discretized 3D Helmholtz kernel on an n × n × n grid that requires only O(κ | log ε| 2 n) reals for storage. Such representations can be applied to both the 3D volume and boundary calculations with sublinear cost O(n 2 ), even in the case κ = O(n).Numerical illustrations demonstrate the efficiency of low tensor-rank approximation for Green's kernels e −λ x / x , x ∈ R 3 , in the case of Newton (λ = 0), Yukawa (λ ∈ R + ), and Helmholtz (λ = iκ, κ ∈ R + ) potentials, as well as for the kernel functions 1/ x and 1/ x d−2 , x ∈ R d , in higher dimensions d > 3. We present numerical results on the iterative calculation of the minimal eigenvalue for the d-dimensional finite difference Laplacian by the power method with the rank truncation and based on the approximate inverse B R ≈ (− ) −1 , with 3 ≤ d ≤ 50.

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“…where the constant C 0 > 0 does not depend on M or on n. Notice that a similar result was proven in [4,Theorem 22]. The following lemma proves the spectral equivalence estimates:…”

Section: In the Case Of Variable Coefficients The Related Cost Is Boumentioning

confidence: 51%

Section: The Green Function Formulationmentioning

confidence: 99%

Section: The Green Function Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝ d

Khoromskij

2009

Constr Approx

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Preliminary results on approximating a wavefunction as an unconstrained sum of Slater determinants

Beylkin

Mohlenkamp

Pérez

2007

Proc Appl Math and Mech

Self Cite

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A multiparticle wavefunction, which is a solution of the multiparticle Schrödinger equation, satisfies the antisymmetry condition, thus making it natural to approximate it as a sum of Slater determinants. Many current methods do so but, in addition, they impose structural constraints on the Slater determinants, such as orthogonality between orbitals or a particular excitation pattern. By removing these constraints, we hope to obtain much more efficient expansions.We use an integral formulation of the problem, a Green's function iteration, and a fitting procedure based on the computational paradigm of separated representations. For constructing and solving a matrix-integral system of equations derived from antisymmetric inner products, we develop new algorithms with computational complexity competitive with current methods.We describe preliminary numerical results and make some observations.Given the difficulties of solving the multiparticle Schrödinger equation, current numerical methods in quantum chemistry/physics are remarkably successful. Part of their success comes from efficiencies gained by imposing structural constraints on the wavefunction to match physical intuition. However, such methods scale poorly to high accuracy, and are biased to only reveal structures that were part of their own construction. In [3] we develop a method that allows better scaling to high accuracy and an unbiased exploration of the structure of the wavefunction by approximating it as an unconstrained sum of Slater determinants.Motivated by the physical intuition that electrons may be excited into higher energy states, the Configuration Interaction (CI) family of methods choose a set of determinants with predetermined orbitals, and then optimize the coefficients used to combine them. When it is found insufficient, methods to optimize the orbitals, work with multiple reference states, etc., are introduced. A common feature of all these methods is that they impose some structural constraints on the Slater determinants, such as orthogonality of orbitals or an excitation pattern. As the requested accuracy increases, these structural constraints trigger an explosion in the number of determinants used, making the computation intractable for high accuracy. The a priori structural constraints present in CI-like methods also force the wavefunction to comply with such structure, whether or not it really is the case. For example, if you use a method that approximates the wavefunction as a linear combination of a reference state and excited states, you could not learn that the wavefunction is better approximated as a linear combination of several non-orthogonal, near-reference states. Thus, the choice of numerical method is not just a computational issue; it can help or hinder our understanding of the wavefunction.Our goal is to construct an adaptive numerical method without imposing a priori structural constraints besides that of antisymmetry. In [3] we derive and present an algorithm for approximating a wavefunction with an unconstrained su...

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Pauli’s Principle in Probe Microscopy

Jarvis

Sweetman

Kantorovich

et al. 2015

Advances in Atom and Single Molecule Machines

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It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation. The explanation is deep down in relativistic quantum mechanics. This probably means that we do not have a complete understanding of the fundamental principle involved.RP Feynman, The Feynman Lectures on Physics, Vol III, Chapter 4 (1964) AbstractExceptionally clear images of intramolecular structure can be attained in dynamic force microscopy through the combination of a passivated tip apex and operation in what has become known as the "Pauli exclusion regime" of the tip-sample interaction. We discuss, from an experimentalist's perspective, a number of aspects of the exclusion principle which underpin this ability to achieve submolecular resolution. Our particular focus is on the origins, history, and interpretation of Pauli's principle in the context of interatomic and intermolecular interactions.

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Approximating a wavefunction as an unconstrained sum of Slater determinants

Cited by 39 publications

References 54 publications

Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝ d

Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝ d

Preliminary results on approximating a wavefunction as an unconstrained sum of Slater determinants

Pauli’s Principle in Probe Microscopy

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