iVI: An iterative vector interaction method for large eigenvalue problems

Huang, Chao; Liu, Wenjian; Xiao, Yunlong; Hoffmann, Mark R.

doi:10.1002/jcc.24907

Cited by 32 publications

(62 citation statements)

References 40 publications

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“…The iVI method for standard eigenvalue problems is extended here to generalized eigenvalue problems with positive‐definite metrics, viz.,

boldHC = boldSCE, C^{†} boldSC = boldI .

…”

Section: Ivi For Generalized Eigenvalue Problemsmentioning

confidence: 99%

“…Therefore, eq. had better be revised to

{[Δ (ζ)]}_{qq} = - \frac{1}{\max (()||,, ζ S_{qq} - H_{qq}, ɛ)}

for exterior roots or

{[Δ (ζ)]}_{qq} = ({, \begin{array}{l} - \frac{1}{\max ((),, H_{qq} - ζ S_{qq}, ɛ)}, & H_{qq} > ζ S_{qq} \\ 0, & H_{qq} \leq ζ S_{qq} \end{array}),

{[Δ (ζ)]}_{qq} = ({, \begin{array}{l} \frac{1}{\max ((),, ζ S_{qq} - H_{qq}, ɛ)}, & H_{qq} \leq ζ S_{qq} \\ 0, & H_{qq} > ζ S_{qq} \end{array})

for interior roots. Equation amounts to treating the perturbers { q } with H qq ≤ ζ S qq and those with H qq > ζ S qq in the same manner (which is in the spirit of absolute value preconditioning), whereas eq.…”

Section: Ivi For Generalized Eigenvalue Problemsmentioning

confidence: 99%

“…If this can be done, different energy windows can be embarrassingly parallelized. The recently proposed iterative vector interaction (iVI) method for Hermitian eigenvalue problems is very promising in this regard. The underlying idea is rooted in the “static‐dynamic‐static” framework for solving the many‐electron Schrödinger equation.…”

Section: Introductionmentioning

confidence: 99%

“…The underlying idea is rooted in the “static‐dynamic‐static” framework for solving the many‐electron Schrödinger equation. Although it works with a fixed‐dimensional search space, it has been shown both theoretically and numerically that iVI can converge quickly and monotonically from above to the exact exterior/interior roots. Encouraged by this observation, we here extend iVI to generalized eigenvalue problems, including linear‐response time‐dependent density functional theory (TD‐DFT) as a special case.…”

Section: Introductionmentioning

confidence: 99%

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iVI‐TD‐DFT: An iterative vector interaction method for exterior/interior roots of TD‐DFT

Huang

Liu

2018

J Comput Chem

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The recently proposed iterative vector interaction (iVI) method for large Hermitian eigenvalue problems (Huang et al., J. Comput. Chem. 2017, 38, 2481 is extended to generalized eigenvalue problems, HC = SCE, with the metric S being either positive definite or not. Although, it works with a fixeddimensional search subspace, iVI can converge quickly and monotonically from above to the exact exterior/interior roots. The algorithms are further specialized to nonrelativistic and relativistic time-dependent density functional theories (TD-DFT) by taking the orbital Hessian as the metric (i.e., the inverse TD-DFT eigenvalue problem) and incorporating explicitly the paired structure into the trial vectors. The efficacy of iVI-TD-DFT is demonstrated by various examples.

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“…The iVI method for standard eigenvalue problems is extended here to generalized eigenvalue problems with positive‐definite metrics, viz.,

boldHC = boldSCE, C^{†} boldSC = boldI .

…”

Section: Ivi For Generalized Eigenvalue Problemsmentioning

confidence: 99%

“…Therefore, eq. had better be revised to

{[Δ (ζ)]}_{qq} = - \frac{1}{\max (()||,, ζ S_{qq} - H_{qq}, ɛ)}

for exterior roots or

{[Δ (ζ)]}_{qq} = ({, \begin{array}{l} - \frac{1}{\max ((),, H_{qq} - ζ S_{qq}, ɛ)}, & H_{qq} > ζ S_{qq} \\ 0, & H_{qq} \leq ζ S_{qq} \end{array}),

{[Δ (ζ)]}_{qq} = ({, \begin{array}{l} \frac{1}{\max ((),, ζ S_{qq} - H_{qq}, ɛ)}, & H_{qq} \leq ζ S_{qq} \\ 0, & H_{qq} > ζ S_{qq} \end{array})

Section: Ivi For Generalized Eigenvalue Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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iVI‐TD‐DFT: An iterative vector interaction method for exterior/interior roots of TD‐DFT

Huang

Liu

2018

J Comput Chem

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“…The popularity of this approach for a wide range of systems is also due to efficient algorithms available for solving this equation. [12][13][14][15][16][17] However, since eigenvalue calculations normally proceed from the lowest excitation energy and the computational cost increases with the number of eigenvalues, its applications in high-frequency spectral regions and regions with high density-of-states remain challenging and require a development of special techniques. 15,16,18,19 Moreover, due to its perturbative nature, the response eigenvalue equation requires the evaluation of the derivatives of DFT exchangecorrelation potentials (so-called kernels) that must be formulated carefully, particularly in relativistic multi-component theories with spin-orbit coupling.…”

Section: Introductionmentioning

confidence: 99%

Relativistic four-component linear damped response TDDFT for electronic absorption and circular dichroism calculations

Konecny

Repiský

Ruud

et al. 2019

The Journal of Chemical Physics

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We present a detailed theory, implementation, and a benchmark study of a linear damped response time-dependent density functional theory (TDDFT) based on the relativistic four-component (4c) Dirac-Kohn-Sham formalism using the restricted kinetic balance condition for the small-component basis and a non-collinear exchangecorrelation kernel. The damped response equations are solved by means of a multifrequency iterative subspace solver utilizing decomposition of the equations according to hermitian and time-reversal symmetry. This partitioning leads to robust convergence and the detailed algorithm of the solver for relativistic multicomponent wavefunctions is also presented. The solutions are then used to calculate the linear electricand magnetic-dipole responses of molecular systems to an electric perturbation, leading to frequency-dependent dipole polarizabilities, electronic absorption and circular dichroism (ECD), and optical rotatory dispersion (ORD) spectra. The methodology has been implemented in the relativistic spectroscopy DFT program ReSpect, and its performance assessed on a model series of dimethylchalcogeniranes, C 4 H 8 X (X = O, S, Se, Te, Po, Lv) and on larger transition metal complexes that have been studied experimentally, [M(phen) 3 ] 3+ (M = Fe, Ru, Os). These are the first 4c damped linear response TDDFT calculations of ECD and ORD presented in the literature.

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Libkrylov: A modular open‐source software library for extremely large on‐the‐fly matrix computations

et al. 2023

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We present the design and implementation of libkrylov, an open-source library for solving matrix-free eigenvalue, linear, and shifted linear equations using Krylov subspace methods. The primary objectives of libkrylov are flexible API design and modular structure, which enables integration with specialized matrix-vector evaluation "engines". Libkrylov features pluggable preconditioning, orthonormalization, and tunable convergence control. Diagonal (conjugate gradient, CG), Davidson, and Jacobi-Davidson preconditioners are available, along with orthonormal and nonorthonormal (nKs) schemes. All functionality of libkrylov is exposed via Fortran and C application programming interfaces (APIs). We illustrate the performance of libkrylov for eigenvalue calculations arising in time-dependent density functional theory (TDDFT) in the Tamm-Dancoff approximation (TDA) and discuss the convergence behavior as a function of preconditioning and orthonormalization methods.

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iVI: An iterative vector interaction method for large eigenvalue problems

Cited by 32 publications

References 40 publications

iVI‐TD‐DFT: An iterative vector interaction method for exterior/interior roots of TD‐DFT

iVI‐TD‐DFT: An iterative vector interaction method for exterior/interior roots of TD‐DFT

Relativistic four-component linear damped response TDDFT for electronic absorption and circular dichroism calculations

Libkrylov: A modular open‐source software library for extremely large on‐the‐fly matrix computations

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