Temple's Formula Research Articles

Comparison and union of the Temple and Bazley lower bounds

The Lehmann-Maehly approach and Bazley’s method of special choice are matrix eigenvalue problems that allow the calculation of lower bounds to energies of atomic and molecular systems. We introduce a common derivation of their scalar versions using the overlap of a trial function with the unknown ground-state wave function. In the scalar setting, the Lehmann-Maehly approach reduces to the Temple formula. The common derivation allows us to easily unite and improve both methods in several stages within this restricted application. Finally we offer a different union that allows generalization to arbitrary dimension matrix methods. Calculations on the helium atom ground state illustrate the improvements and mergers.

Optimization of the temple lower bound

The Temple formula is perhaps the most common method used in the uncommon endeavor of calculating a lower bound to the ground-state energy of an atomic or molecular system. We generalize the Temple formula by introducing a parameter that can be varied to optimize the lower bound. This generalization does not require any information that is not already used for the traditional Temple lower bound. Examples with the helium cation and neutral atom show that improvement is greatest when the approximate wave function poorly approximates the true ground-state wave function. The examples also show that in some cases the traditional Temple lower bound may already be optimal so that our generalization gives no improvement.

An extension of the rayleigh-ritz method for finding upper and lower bounds of eigenvalues

A simple variational method is described for the determination of upper and lower bounds for the eigenvalues of an Hermitian operator M. For a selected vector subspace of n dimensions it involves the diagonalization of a 2n × 2n matrix, the elements of which depend on the matrix elements of M and M2 as well as on upper and lower bounds of M in the complementary subspace. The method is equivalent to one given by H. F. Weinberger; it includes the Rayleigh-Ritz procedure and a modified form of Temple's formula as limiting cases.

Computing upper and lower bounds using Monte Carlo methods

AbstractWe computed lower bounds for some ground and excited states of the helium atom using variational Monte Carlo and the Weinstein, Temple, and Stevenson formulas. For these systems, the Temple and Stevenson bounds are approximately the same and have similar standard deviations. The Weinstein bounds are much further from the actual energies and have much larger standard deviations. We also investigated the reliability of these lower bound formulas when nonorthogonal wave functions are used. The Temple formula has been shown to produce an accurate lower bound even under such circumstances, while the Weinstein and Stevenson formulas are shown to yield incorrect results. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

Equivalence of Two Lower Bound Methods

Two lower bound methods are compared: the inverse method and a generalized variance method. As the generalized variance method is a generalization of the simple method of variance, the inverse method is shown to be a generalization of the Temple formula. We show that for the same a priori information and basis set, the generalized variance and inverse methods are equivalent.

Accurate upper and lower bounds to the 2S states of the lithium atom

AbstractThe first nonrelativistic lower bound to the ground state of the lithium atom is give with E0 > −7.47816 au using the method of variance minimization and an extension of Temple's formula. With large Hylleraas–CI basis sets, high‐precision upper bounds and isotope shifts are calculated for the three lowest 2S states of the lithium atom, which are best to date. © 1994 John Wiley & Sons, Inc.

Atomic integrals containing rrr with λ, μ, ν ≥ −2

AbstractIn this paper, the efficient evaluation of the atomic integrals I =∫rrrrrre−αr1−βr2−γr3dτ with one or two factors r is described. These integrals are necessary for a lower‐bound calculation for Li‐like systems using the method of variance minimization or Temple's formula. © 1993 John Wiley & Sons, Inc.

A lower bound for the two‐dimensional Helmholtz equation

AbstractA variational expression (Temple's formula), requiring C1 continuity, is used to transform the two‐dimensional Helmholtz equation into a matrix equation. Finite element techniques are used and homogeneous Neumann or Dirichlet boundary conditions are assumed. It is demonstrated that under certain conditions lower bounds to the eigenvalues are obtained.

Variance minimization. A variational principle for accurate lower and upper bounds of the eigenvalues of a selfadjoint operator, bounded below

In connection with Temple's formula variance minimization yields accurate values especially for groundstate energies of Schrodinger operators with a discrete spectrum. The result in a.u. for the groundstate E 0 of the He-atom in the infinite nuclear mass approximation is −2.90372438655≤E 0≤−2.90372437696 i.e. E 0 is determined with an absolute error smaller than 0.0022 cm−1.

A lower bound for the one-dimensional Helmholtz equation

A variational expression (Temple's formula), requiring CI continuity, is used to transform the one-dimensional Hclmholtz equation into a matrix equation. Finite-element techniques are used and homogeneous Neumann and/or Dirichlet boundary conditions are assumed. It is shown that lower bounds to the eigenvalues are obtained.

Error analysis of variationally optimized upper‐ and lower‐bound wave functions for H

AbstractWave functions which are a linear combination of H‐type elliptical orbitals are optimized to provide either an upper bound or a lower bound to the H ground state. For the latter, Temple's formula is used. Three criteria are considered to determine the relative accuracy of these wave functions: (i) energy (calculated versus exact eigenvalue); (ii) average error; and (iii) local energy. Although the lower‐bound optimized wave functions obtained are the most accurate available for H from approximate wave functions, they are still inferior to the corresponding upper‐bound wave functions by criteria (i) and (ii). In particular, using criterion (ii), it is shown numerically that the upper‐bound functions are “correct to second order,” while the lower‐bound functions are almost, but not quite, “correct to second order.” Despite this, the local energy analysis, criterion (iii), reveals that the lower‐bound wave functions can be more accurate than the upper‐bound functions in some regions of space, and hence give more accurate values for physical properties sensitive to these regions. Examples considered are the dipole–dipole and Fermi contact interactions.

Error Bounds for the Energy and Overlap of Approximate Wavefunctions

A sequence of variational principles for upper and lower bounds to the energy of the ground state of a quantum-mechanical system is constructed. The first approximations in the sequence are simply the Ritz upper bound and the Temple lower bound. Higher approximations give improved upper and lower bounds, at the cost of computing additional integrals of the form 〈φ | Hm | φ〉 over the approximate wavefunction. As in the Temple formula, the lower bounds to the ground-state energy also depend on a knowledge of the energy of the first excited state. If in addition one knows an accurate energy for the ground state (say from experiment), then it is shown that one can construct a sequence of upper and lower bounds to the overlap between the approximate function and the true (but unknown) ground-state wavefunction. The first approximation in this sequence is the Eckart lower bound to this overlap, and the higher approximations provide improved upper and lower bounds to the overlap.