A Tight Lower Bound to the Ground-State Energy

Pollak, Eli

doi:10.1021/acs.jctc.9b00344

Cited by 12 publications

(10 citation statements)

References 22 publications

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“…It is necessary to obtain accurate lower bounds and it is the combination of the upper and lower bounds on the eigenvalues that gives upper and lower bounds on the gap between the eigenvalues. In a recent series of publications, [40][41][42][43] we have developed a highly accurate Self-Consistent Lower Bound Theory (SCLBT), which can provide lower bounds to eigenvalues that are of similar and sometimes even have greater accuracy than those of the upper bounds obtained with the Ritz-MacDonald variational method. 44,45 The theory generalizes and improves upon Temple's lower bound expression.…”

Section: Introductionmentioning

confidence: 99%

Upper and lower bounds for tunneling splittings in a symmetric double-well potential

Rontó

Pollak

2020

RSC Adv.

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Section: Introductionmentioning

confidence: 99%

Upper and lower bounds for tunneling splittings in a symmetric double-well potential

Rontó

Pollak

2020

RSC Adv.

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“…5-12. In the past year a different approach has been suggested and shown to improve upon Temple's method (13,14). It improved upon Temple's original bound; however, to ensure rapid convergence it was necessary to use an approximate estimate for overlap matrix elements (14).…”

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confidence: 99%

Lower bounds to eigenvalues of the Schrödinger equation by solution of a 90-y challenge

Martinazzo

Pollak

2020

Proc. Natl. Acad. Sci. U.S.A.

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The Ritz upper bound to eigenvalues of Hermitian operators is essential for many applications in science. It is a staple of quantum chemistry and physics computations. The lower bound devised by Temple in 1928 [G. Temple,Proc. R. Soc. A Math. Phys. Eng. Sci.119, 276–293 (1928)] is not, since it converges too slowly. The need for a good lower-bound theorem and algorithm cannot be overstated, since an upper bound alone is not sufficient for determining differences between eigenvalues such as tunneling splittings and spectral features. In this paper, after 90 y, we derive a generalization and improvement of Temple’s lower bound. Numerical examples based on implementation of the Lanczos tridiagonalization are provided for nontrivial lattice model Hamiltonians, exemplifying convergence over a range of 13 orders of magnitude. This lower bound is typically at least one order of magnitude better than Temple’s result. Its rate of convergence is comparable to that of the Ritz upper bound. It is not limited to ground states. These results complement Ritz’s upper bound and may turn the computation of lower bounds into a staple of eigenvalue and spectral problems in physics and chemistry.

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“…Many other generalisations and variations of Temple's bound have been made, including methods involving the minimisation of the variance σ E 2 by H. Kleindienst and W. Altmann [25] and by combining Temple's bound with the the inner projection method [26,27]. More recently, E. Pollak has made modifications to Temple's bound which have been applied to various model systems, in particular the quartic oscillator [19,21,22,28].…”

Section: Further Notes On Lower Boundsmentioning

confidence: 99%

Integrals for lower bounds to the exact energy

Ireland

2021

Preprint

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Methods for calculating lower bounds to the exact energy using the variance of the upper bound energy are discussed and explored. All the matrix elements of the Hamiltonian squared are collected and considered, and those for which no known solutions could be found in the literature are derived for an explicitly correlated Gaussian (ECG) basis set. Analytical Solutions are determined for twoelectron, mono-nuclear systems, in addition to a one-dimensional integral expression which has use in polyatomic calculations. The newly derived integral expressions have been implemented in the integral library of the QUANTEN computer program. I. INTRODUCTIONSince the successful application of Schrödinger equation to the hydrogen atom in the mid 1920's, few-body systems have been of great interest to physicists and chemists. The ability to describe few-particle systems such as small atoms and molecules however becomes problematic due to a lack of analytical solutions. As such, quantum chemistry requires the use of approximations in order to simplify quantum problems, allowing for approximate solutions to be obtained for the physical properties of the systems in question.In atomic and molecular physics, one property of great interest is the energy of different eigenstates. The energy of a particular state can be calculated by approximating the wave function through linear and non-linear parameterisation. This approximate wave function is called the trial wave function for the system, and it can be shown that the energy expectation value associated with this trial wave function serves as an upper bound to the exact energy [1,2]. Calculating 'tight' upper bounds to the exact energy is well known and occurs frequently in modern quantum chemistry. However, the upper-bound value itself does not give information on how close it is to the exact energy. If a lower bound could also be calculated then this, along with the upper-bound value, would give an interval within which the exact energy can be found. Of course, the lower-bound value is only meaningful if the lower bound is of the same quality as the upper bound.As is true with upper bounds, improving the basis set with respect to the variational principle improves the lowerbound value, though the convergence of lower bounds is slower. Nevertheless, convergence of both upper and lower bounds leads to ever-tighter intervals, within which the exact energy lies. Two such methods for calculating lower bounds are the Weinstein criterion [3] and Temple's lower bound [4], presented by D. Weinstein and G. Temple respectively. Weinstein's bound tends to give lower bounds that are of too low a quality compared to the upper bounds, whereas Temple's bound is regarded to give better quality lower bounds [5,6]. Both the Weinstein criterion and Temple's bound make use of the variance of the upper bound energy, σ E 2 , which requires the matrix elements of the Hamiltonian operator squared, Ĥ2 .The trial wave function can be written as a linear combination of basis functions in the many-particle sta...

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A Tight Lower Bound to the Ground-State Energy

Cited by 12 publications

References 22 publications

Upper and lower bounds for tunneling splittings in a symmetric double-well potential

Upper and lower bounds for tunneling splittings in a symmetric double-well potential

Lower bounds to eigenvalues of the Schrödinger equation by solution of a 90-y challenge

Integrals for lower bounds to the exact energy

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