Numerical methods for Kohn–Sham density functional theory

Lin, Lin; Lu, Jianfeng; Ying, Lexing

doi:10.1017/s0962492919000047

Cited by 40 publications

(24 citation statements)

References 274 publications

(478 reference statements)

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“…with A a symmetric tensor of order 4. For more details on these models or electronic structure in general, we refer to [12,40,56].…”

Section: Optimization On Grassmann Manifoldsmentioning

confidence: 99%

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Convergence Analysis of Direct Minimization and Self-Consistent Iterations

Cancès¹,

Kemlin²,

Levitt³

2021

SIAM J. Matrix Anal. Appl.

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This article is concerned with the numerical solution of subspace optimization problems, consisting of minimizing a smooth functional over the set of orthogonal projectors of fixed rank. Such problems are encountered in particular in electronic structure calculation (Hartree-Fock and Kohn-Sham Density Functional Theory-DFT-models). We compare from a numerical analysis perspective two simple representatives, the damped self-consistent field (SCF) iterations and the gradient descent algorithm, of the two classes of methods competing in the field: SCF and direct minimization methods. We derive asymptotic rates of convergence for these algorithms and analyze their dependence on the spectral gap and other properties of the problem. Our theoretical results are complemented by numerical simulations on a variety of examples, from toy models with tunable parameters to realistic Kohn-Sham computations. We also provide an example of chaotic behavior of the simple SCF iterations for a nonquadratic functional.

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“…with A a symmetric tensor of order 4. For more details on these models or electronic structure in general, we refer to [12,40,56].…”

Section: Optimization On Grassmann Manifoldsmentioning

confidence: 99%

“…using an orthonormal basis (φ i ) i=1,...,N for the subspace Ran(P ). This problem is of interest in a number of contexts, such as matrix approximation, computer vision [1], and electronic structure theory [12,28,39,40,45,56], the latter of which being the main motivation for this work.…”

Section: Introductionmentioning

confidence: 99%

Convergence Analysis of Direct Minimization and Self-Consistent Iterations

Cancès¹,

Kemlin²,

Levitt³

2021

SIAM J. Matrix Anal. Appl.

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“…Remark 2.2 Computing eigenvalues and eigenfunctions of the electronic Schrödinger equation is the primary concern of computational quantum chemistry; see, e.g. Szabo and Ostlund (1996) or Jensen (2016) and from a more mathematical viewpoint Cancès, Defranceschi, Kutzelnigg, Le Bris and Maday (2003), Cancès, Le Bris and Maday (2006), Le Bris (2005), and Lin, Lu and Ying (2019). Here we simply assume that this problem is solved in some satisfactory way.…”

Section: Electronic Schrödinger Equationmentioning

confidence: 99%

Computing quantum dynamics in the semiclassical regime

Lasser,

Lubich

2020

Preprint

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The semiclassically scaled time-dependent multi-particle Schrödinger equation describes, inter alia, quantum dynamics of nuclei in a molecule. It poses the combined computational challenges of high oscillations and high dimensions. This paper reviews and studies numerical approaches that are robust to the small semiclassical parameter. We present and analyse variationally evolving Gaussian wave packets, Hagedorn's semiclassical wave packets, continuous superpositions of both thawed and frozen Gaussians, and Wigner function approaches to the direct computation of expectation values of observables. Making good use of classical mechanics is essential for all these approaches. The arising aspects of time integration and high-dimensional quadrature are also discussed.

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“…Lu, Sogge and the author [12] gave a quantitative description of the smooth • smooth = smooth regime in the continuous setting (see also Jin [7] and Wyman [22]). The product φ λ • φ µ appear in a variety of different applications such as finding appropriate eigenfunctions for the dimensionality reduction of high-dimensional data (see Cloninger & Steinerberger [2], Kohli, Cloninger & Mishne [8]), in numerical aspects of the Kohn-Sham density functional theory (see Lin, Lu & Ying [10]) and in problems related to shape matching (see Litany, Rodola, Bronstein & Bronstein [11]). The largest eigenvectors (and the associated eigenvalues) are known to have great significance in combinatorics, specifically the chromatic number and size of independent set (i.e.…”

Section: Introduction and Resultsmentioning

confidence: 99%

The product of two high-frequency Graph Laplacian eigenfunctions is smooth

Steinerberger¹

2021

Preprint

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In the continuous setting, we expect the product of two oscillating functions to oscillate even more (generically). On a graph G = (V, E), there are only |V | eigenvectors of the Laplacian L = D − A, so one oscillates 'the most'. The purpose of this short note is to point out an interesting phenomenon: if φ 1 , φ 2 are delocalized eigenvectors of L corresponding to large eigenvalues, then their (pointwise) product φ 1 • φ 2 is smooth (in the sense of small Dirichlet energy): highly oscillatory functions have largely matching oscillation patterns.

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Numerical methods for Kohn–Sham density functional theory

Cited by 40 publications

References 274 publications

Convergence Analysis of Direct Minimization and Self-Consistent Iterations

Convergence Analysis of Direct Minimization and Self-Consistent Iterations

Computing quantum dynamics in the semiclassical regime

The product of two high-frequency Graph Laplacian eigenfunctions is smooth

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