The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices

Davidson, Ernest R.

doi:10.1016/0021-9991(75)90065-0

Cited by 2,298 publications

(814 citation statements)

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“…The working equations may then be solved using non-Hermitian variants of the Davidson algorithm. [23][24][25][26] When excitation energies are the subject of interest, the EOM-CC equation is equivalent to the Jacobian in the LR-CC formalism 12 for excitation energy calculations. The mathematical difference between LR and EOM formalisms arises when they are used to compute excitation properties, such as transition dipole and oscillator strengths.…”

Section: Theorymentioning

confidence: 99%

Assessment of low-scaling approximations to the equation of motion coupled-cluster singles and doubles equations

Goings

Caricato

Frisch

et al. 2014

The Journal of Chemical Physics

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Articles you may be interested inReuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Methods for fast and reliable computation of electronic excitation energies are in short supply, and little is known about their systematic performance. This work reports a comparison of several low-scaling approximations to the equation of motion coupled cluster singles and doubles (EOM-CCSD) and linear-response coupled cluster singles and doubles (LR-CCSD) equations with other single reference methods for computing the vertical electronic transition energies of 11 small organic molecules. The methods, including second order equation-of-motion many-body perturbation theory (EOM-MBPT2) and its partitioned variant, are compared to several valence and Rydberg singlet states. We find that the EOM-MBPT2 method was rarely more than a tenth of an eV from EOM-CCSD calculated energies, yet demonstrates a performance gain of nearly 30%. The partitioned equation-of-motion approach, P-EOM-MBPT2, which is an order of magnitude faster than EOM-CCSD, outperforms the CIS(D) and CC2 in the description of Rydberg states. CC2, on the other hand, excels at describing valence states where P-EOM-MBPT2 does not. The difference between the CC2 and P-EOM-MBPT2 can ultimately be traced back to how each method approximates EOM-CCSD and LR-CCSD. The results suggest that CC2 and P-EOM-MBPT2 are complementary: CC2 is best suited for the description of valence states while P-EOM-MBPT2 proves to be a superior O(N 5 ) method for the description of Rydberg states. © 2014 AIP Publishing LLC.

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Section: Theorymentioning

confidence: 99%

Assessment of low-scaling approximations to the equation of motion coupled-cluster singles and doubles equations

Goings

Caricato

Frisch

et al. 2014

The Journal of Chemical Physics

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“…Since solving for a subset of eigenvalues of large matrices is a standard problem in computational science, a variety of different algorithms exist that vary in convergence properties and memory requirements [51]. Among the most popular in electronic structure theory are the Davidson [52] and Conjugate Gradients algorithms [53], which are based on minimising the Rayleigh quotient…”

Section: Iterative Eigensolversmentioning

confidence: 99%

Computing the Optical Properties of Large Systems

Zuehlsdorff

2015

Springer Theses

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“…For each iteration the step to the minimum at the boundary of the trust-region ensures that we are moving downhill. When solving the level-shifted Newton equations, explicit computation and storage of the Hessian is avoided by solving the equations using iterative algorithms 107 where only linear transformations of the Hessian on trial vectors are required. The trust-region method was introduced by Fletcher 106 and was adapted by Høyvik and Jørgensen to include a line search step to make it an efficient and reliable algorithm for optimizing localization functions for both the occupied and the virtual orbitals, for large molecular systems containing diffuse basis functions, 108 and also when large negative Hessian eigenvalues are encountered in the initial iterations.…”

Section: Localization Function Optimizationmentioning

confidence: 99%

Characterization and Generation of Local Occupied and Virtual Hartree–Fock Orbitals

2016

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The scope of this review article is to discuss the locality of occupied and virtual orthogonal Hartree-Fock orbitals generated by localization function optimization. Locality is discussed from the stand that an orbital is local if it is confined to a small region in space. Focusing on locality measures that reflects the spatial extent of the bulk of an orbital and the thickness of orbital tails, we discuss, with numerical illustrations, how the locality may be reported for individual orbitals as well as for sets of orbitals. Traditional and more recent orbital localization functions are reviewed, and the locality measures are used to compare the locality of the orbitals generated by the different localization functions, both for occupied and virtual orbitals. Numerical illustrations are given also for large molecular systems and for cases where diffuse functions are included in the atomic orbital basis. In addition, we have included a discussion on the physical and mathematical limitations on orbital locality.

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The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices

Cited by 2,298 publications

References 9 publications

Assessment of low-scaling approximations to the equation of motion coupled-cluster singles and doubles equations

Assessment of low-scaling approximations to the equation of motion coupled-cluster singles and doubles equations

Computing the Optical Properties of Large Systems

Characterization and Generation of Local Occupied and Virtual Hartree–Fock Orbitals

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