Eigenvalue lower bounds with Bazley’s special choice of an infinite-dimensional subspace

Marmorino, M. G.

doi:10.1007/s10910-011-9839-y

Cited by 6 publications

(6 citation statements)

References 7 publications

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“…The Temple lower bound for the ground state now takes the well-known form where ε 2 – , which must be greater than λ 1 , is defined as a lower bound to the first excited-state energy. This lower bound may be obtained through a variety of lower bound methods such as the Weinstein, 3 Bazley, 21 , 22 Miller, 23 and Marmorino 24 methods. Introduction of the residual energy made it possible to obtain a practical calculable form of Temple’s lower bound formula.…”

Section: Short Review Of Previous Lower Bound Theoriesmentioning

confidence: 99%

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Lower Bounds for Coulombic Systems

Pollak

Martinazzo

2021

J. Chem. Theory Comput.

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As of the writing of this paper, lower bounds are not a staple of quantum chemistry computations and for good reason. All previous attempts at applying lower bound theory to Coulombic systems led to lower bounds whose quality was inferior to the Ritz upper bounds so that their added value was minimal. Even our recent improvements upon Temple’s lower bound theory were limited to Lanczos basis sets and these are not available to atoms and molecules due to the Coulomb singularity. In the present paper, we overcome these problems by deriving a rather simple eigenvalue equation whose roots, under appropriate conditions, give lower bounds which are competitive with the Ritz upper bounds. The input for the theory is the Ritz eigenvalues and their variances; there is no need to compute the full matrix of the squared Hamiltonian. Along the way, we present a Cauchy–Schwartz inequality which underlies many aspects of lower bound theory. We also show that within the matrix Hamiltonian theory used here, the methods of Lehmann and our recent self-consistent lower bound theory ( J. Chem. Phys. 2020, 115, 244110) are identical. Examples include implementation to the hydrogen and helium atoms.

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Section: Short Review Of Previous Lower Bound Theoriesmentioning

confidence: 99%

“…This implies that the "complementary part" of the Hamiltonian takes the form Using the identity 24) one finds the needed relation Inserting this into eq 2. 20 gives a practical improved lower bound expression…”

Section: Short Review Of Previous Lower Boundmentioning

confidence: 99%

Lower Bounds for Coulombic Systems

Pollak

Martinazzo

2021

J. Chem. Theory Comput.

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“…Their energies bound the exact eigenvalues from below. Further analysis and improvements upon Bazley’s method as well as comparisons with Temple-based lower bounds have been presented by Marmorino. − However, here too the bottom line is not very encouraging. The complete basis set of the “base” separable Hamiltonian is not complete for a multielectron atom so that the method will not necessarily converge to the exact answer.…”

Section: Introductionmentioning

confidence: 99%

Lower Bounds for Nonrelativistic Atomic Energies

et al. 2021

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A recently developed lower bound theory for Coulombic problems (E. Pollak, R. Martinazzo, J. Chem. Theory Comput. 2021Comput. , 17, 1535 is further developed and applied to the highly accurate calculation of the ground-state energy of two-(He, Li + , and H − ) and three-(Li) electron atoms. The method has been implemented with explicitly correlated many-particle basis sets of Gaussian type, on the basis of the highly accurate (Ritz) upper bounds they can provide with relatively small numbers of functions. The use of explicitly correlated Gaussians is developed further for computing the variances, and the necessary modifications are here discussed. The computed lower bounds are of submilli-Hartree (parts per million relative) precision and for Li represent the best lower bounds ever obtained.Although not yet as accurate as the corresponding (Ritz) upper bounds, the computed bounds are orders of magnitude tighter than those obtained with other lower bound methods, thereby demonstrating that the proposed method is viable for lower bound calculations in quantum chemistry applications. Among several aspects, the optimization of the wave function is shown to play a key role for both the optimal solution of the lower bound problem and the internal check of the theory.

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“…31 Due to associated conceptual and numerical difficulties, lower bounds to true eigenvalues of the Hamiltonian are typically evaluated for simple quantum systems, e.g., the He atom. 5,6,9,11 In the present study we rest satisfied with the aim of bounding full CI eigenvalues but not those of the complete CI limit. In return, an algorithm computable for any quantum chemical system is devised.…”

Section: Evaluation Of Eq (1) Via Explicitly Expressing the Resolvenmentioning

confidence: 99%

“…Lower bound methods, based on the second moment, or variance of the Hamiltonian [1][2][3][4][5] are perhaps the most known. Some other approaches are based on operators bounding the Hamiltonian from below, [6][7][8][9] Padé approximants, 10 local energy, 11 spectral consideration, 12 or matrix representation of the Hamiltonian. 13 Eigenvalues of the two-electron reduced Hamiltonian also provide strict approximation to the exact eigenvalues from below, 14 and this property is usually conserved when Nrepresentability conditions are partially incorporated within a reduced density matrix approach.…”

Section: Introductionmentioning

confidence: 99%

Energy error bars in direct configuration interaction iteration sequence

Tóth

Szabados

2015

The Journal of Chemical Physics

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A computational scheme for approximate lower bound to eigenvalues of linear operators is elaborated, based on Löwdin's bracketing function. Implementation in direct full configuration interaction algorithm is presented, generating essentially just input-output increase. While strict lower bound property is lost due to approximations, test calculations result lower bounds of the same order of magnitude, as the usual upper bound, provided by the expectation value. Difference of upper and lower bounds gives an error bar, characterizing the wavefunction at the given iteration step.

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Eigenvalue lower bounds with Bazley’s special choice of an infinite-dimensional subspace

Cited by 6 publications

References 7 publications

Lower Bounds for Coulombic Systems

Lower Bounds for Coulombic Systems

Lower Bounds for Nonrelativistic Atomic Energies

Energy error bars in direct configuration interaction iteration sequence

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