Energy error bars in direct configuration interaction iteration sequence

Tóth, Zsuzsanna; Szabados, Ágnes

doi:10.1063/1.4928977

Cited by 7 publications

(5 citation statements)

References 40 publications

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“…But even for more ”standard” problems, knowledge of a tight lower bound would not only provide an estimate of the accuracy but could also be used to improve upon previous estimates by simple averaging of the upper and lower bounds. This has motivated many different efforts, ,,− but a real breakthrough did not come about.…”

Section: Introductionmentioning

confidence: 99%

A Tight Lower Bound to the Ground-State Energy

Pollak

2019

J. Chem. Theory Comput.

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Ninety years ago Temple (Proc. R. Soc. (London)1928A119276) derived a lower bound for the ground-state energy. The bound was tested and invariably found to be poor as compared to the upper bound obtained through the Rayleigh Ritz procedure due to the fact that it is based also on the second moment of the Hamiltonian. In this paper we (a) improve upon Temple’s lower bound estimate for the overlap squared of the true ground-state wave function with the approximate one and (b) describe in detail and generalize our recent improvement on the Temple lower bound based on utilization of higher-order basis functions derived by the Arnoldi algorithm. Both improvements combined lead to a lower bound on the ground-state energy whose accuracy is better than that of the Temple lower bound. This is exemplified by considering the ground-state energy of a quartic potential where one finds that the improvements lead to a lower bound whose quality is comparable to that of the upper bound. The applicability of the method to atoms and molecules is discussed.

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Section: Introductionmentioning

confidence: 99%

A Tight Lower Bound to the Ground-State Energy

Pollak

2019

J. Chem. Theory Comput.

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“…The optimal inclusion intervals introduced by Lehmann [19][20][21] were a significant development in relation to the original Temple bound. Further approaches of lower bound methods are based on bracketing functions [22][23][24][25][26][27] and on the method of intermediate operators [28,29]. Lower bounds are also of importance in the context of physical properties of few-electron atoms such as oscillator strengths [30][31][32][33][34].…”

Section: Introductionmentioning

confidence: 99%

Lower bounds on par with upper bounds for few-electron atomic energies

Ronto,

Jeszenszki,

Mátyus

et al. 2023

Preprint

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“…[28][29][30][31][32][33] Bracketing functions have also been successfully applied to lower bound problems. [34][35][36][37][38] Further lower bound calculation strategies are also available for He atoms. [39][40][41][42] A comprehensive discussion of these methods can be found in Ref.…”

Section: Introductionmentioning

confidence: 99%

Comparison of an Improved Self-consistent Lower Bound Theory with Lehmann’s Method for Low-lying Eigenvalues

Rontó

Pollak

Martinazzo

2021

Preprint

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Ritz eigenvalues only provide upper bounds for the energy levels, while obtaining lower bounds requires at least the calculation of the variances associated with these eigenvalues. The well-known Weinstein and Temple lower bounds based on the eigenvalues and variances converge very slowly and their quality is considerably worse than that of the Ritz upper bounds. Lehmann presented a method that in principle optimizes Temple’s lower bounds with significantly improved results. We have recently formulated a Self-Consistent Lower Bound Theory (SCLBT), which improves upon Temple’s results. In this paper we further improve the SCLBT and compare its quality with Lehmann’s theory. The Lánczos algorithm for constructing the Hamiltonian matrix simplifies Lehmann’s theory and is essential for the SCLBT. Using two lattice Hamiltonians, we compared the improved SCLBT with its previous implementation as well as with Lehmann’s lower bound theory. The novel SCLBT exhibits a significant improvement over the previous version. Both Lehmann’s theory and the SCLBT provide significantly better lower bounds than those obtained from Weinstein’s and Temple’s methods. Compared to each other, the Lehmann and SCLBT theories exhibit similar performance in terms of the quality and convergence of the lower bounds. By increasing the number of states included in the calculations, the lower bounds are tighter and their quality becomes comparable with that of the Ritz upper bounds. Both methods are suitable for providing lower bounds for low-lying excited states as well. Compared to Lehmann’s theory, one of the advantages of the SCLBT method is that it does not necessarily require the Weinstein lower bound for its initial input, but Ritz eigenvalue estimates can also be used. Especially owing to this property the SCLBT method sometimes exhibits improved convergence compared to that of Lehmann’s lower bounds.

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Energy error bars in direct configuration interaction iteration sequence

Cited by 7 publications

References 40 publications

A Tight Lower Bound to the Ground-State Energy

A Tight Lower Bound to the Ground-State Energy

Lower bounds on par with upper bounds for few-electron atomic energies

Comparison of an Improved Self-consistent Lower Bound Theory with Lehmann’s Method for Low-lying Eigenvalues

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