Variational bounds to the overlap

Starting from a closed expression for the overlap, variational upper and lower bounds to the overlap are derived by means of operator inequalities.     

Semirigorous bounds for the overlap of approximate wavefunctions

Alexander's probable upper and lower bounds to the overlap S between an approximate and the true wavefunctions are based on second-order perturbation theory and a special ordering of configurations. These weak points are removed in the present treatment and replaced by an exact expression for the energy lowering, plus a very reasonable postulate. The resulting lower bound to the overlap S is illustrated with examples taken from the literature.     

A study of the weinhold-wang bounds to overlap integrals

The Weinhold-Wang extension of the Braun-Rebane formula for bounds to overlap integrals has been investigated numerically for the first and second excited s states of the hydrogen atom. The effect of the choice of basis sets is demonstrated with particular emphasis on the difference between a complete set and an incomplete set of expansion functions     

Error bounds for the overlap of approximate wave functions

Variational upper and lower bounds for the overlap between an approximate and the true wave function are proposed, and it is shown that the error bounds introduced recently by Gordon are special cases of the variational formulas.     

On a variational approach to the Soret coefficient

In this paper, a variational approach to the Soret coefficient initiated by Kempers [J. Chem. Phys. 90, 6541 (1989)] is critically revisited. We show that the physical coherence of the whole procedure leads to a very peculiar choice of the type of constraint one can select to mimic the nonequilibrium stationary state. However, we demonstrate that its precise definition would require a statistical evaluation of the heat of transfer, or a variational approach based on more microscopic ingredients.     

On a variational method for determining excited state wave functions

A variational method is proposed which results in an approximate wave function for an excited state which has the maximum overlap with the true excited state eigenfunction. The method involves the calculation of the quantities E=¦H¦ and =¦(H-E)²¦, but is free of the constraint that the trial function must remain orthogonal to all states of the same symmetry which lie beneath it. One must know, however, an approximation to the true eigenvalue. A discussion is given on how one might gain the latter information, lacking knowledge of the spectrum, from the repeated application of the method.

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Zusammenfassung Es wird eine Variationsmethode vorgeschlagen, die auf eins genäherte Wellenfunktion für einen angeregten Zustand führt, welche sich maximal mit der wahren Eigenfunktion des angeregten Zustandes überlappt. Die Methode ermöglicht die Berechnung der Größen: E=¦H¦ und =¦(H-E)²¦, ist aber frei von der Nebenbedingung, daß die Versuchsfunktion orthogonal zu allen benachbarten Zuständen mit derselben Symmetrie bleiben muß. — Man will jedoch eine Näherung des wahren Eigenwertes wissen. — Eine Diskussion darüber, wie man durch wiederholte Anwendung der Methode letztere Information gewinnen kann, ohne Kenntnisse über das Spektrum zu besitzen, wird durchgeführt.

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Résumé Méthode variationnelle pour la détermination d'une fonction d'onde d'un état excité présentant le recouvrement maximum avec la fonction d'onde exacte. La méthode implique le calcul de E=¦H¦ et =¦(H-E)²¦ mais ne comporte pas la contrainte d'orthogonalité de la fonction d'essai à tous les états inférieurs de même symétrie. On doit cependant connaître une valeur approchée de la valeur propre exacte. Discussion sur la manière dont on peut obtenir cette valeur approchée sans connaître le spectre, par application répétée de la méthode.

     

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 Schrödinger 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 –2.9037243865₅E ₀–2.9037243769₆ i.e. E ₀ is determined with an absolute error smaller than 0.0022 cm^–1.     

On Kryachko's formula for the leaky aquifer function

Kryachko recently published in this journal a formula in which the incomplete Bessel function known as the leaky aquifer function is expanded in Legendre functions of the second kind. While Kryachko's analysis is ingenious, creative, and sound, it unfortunately contains errors that make its key result incorrect. The present contribution corrects and to some degree generalizes Kryachko's formula. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 332–334, 2001     

Upper and lower bounds for a serial dilution test

A Poisson-binomial model estimates the concentration of a target microbe from a serial dilution test. The maximum likelihood procedure gives an equation whose solution equals the estimate of the concentration. This paper gives bounds for the solution to this equation that require only minimal calculations.     

Upper and lower bounds to eigenvalues

The variational method based on the Temple-Kato formulae is applied to the determination of bounds for some ^1,3 P(1s np) states of He.

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Zusammenfassung Die Variationsmethode, die auf der Temple-Kato Formel beruht, wird angewandt zur Bestimmung der Grenzen für einige ^1,3 P(1s np)-ZustÄnde des He.

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Résumé La méthode variationnelle basée sur la formulation de Temple-Kato est appliquée à la détermination de limites pour certains états ^1,3 P(1s np) de He.

     

On the variational solution of morphed molecular potential in a diatomic molecule

The general one-dimensional potential energy function, including centrifugal distortion, for a diatomic molecule is morphed with a series of Morse-like functions for each of the rotational quantum numbers $J$ . For each of the morphed potential, explicit formulae for the matrix elements of the complete energy matrix, on the basis of the solutions of the one-dimensional harmonic oscillator, are given and these may be used in connection with the variational procedure to solve the corresponding vibrational Schrödinger equation. From the set of vibrational levels $\{\text{ E}_\mathrm{vJ}\}, J = 0, 1, 2,{\ldots }$ the ro-vibrational transitions can be deduced.     

Applications of some bounds to density functionals for atoms

In this article, some applications of lower bounds to density-dependent functionals in terms of radial expectation values are performed. These diverse applications include accurate upper bounds to the exact kinetic energy and to conjecture a wide set of relationships among radial and momentum expectation values. Some open questions for the improvement of these results are remarked upon. © 1995 John Wiley & Sons, Inc.     

On the ligand-field potential for f electrons in the angular overlap model

The angular overlap model is applied to f electrons. General relationships are evaluated to express the ligand-field potential in angular overlap parameters for arbitrary ligand-fields. Comparison with the electrostatic model is presented for a specific example.     

On higher order variational analysis in one and three dimensions for soft boundaries

For some problems in liquid crystal physics we need to use the Euler equation and the corresponding boundary equation in the three-dimensional case with soft boundaries. As a further complication the free energy expression, which should be minimized, might contain some second-order and third-order derivatives. These higher-order derivatives will cause the spatial derivatives of the boundary normal to appear in the boundary equation. Explicit formulae are given for the Euler equation and the corresponding surface equations for a general case. As an example, the theory is applied to nematic liquid crystals, where the general Euler equations and surface molecular fields are derived, including the effects of an imposed electric field.     

On the Coulson integral formula for total π-electron energy

The Coulson integral formula for total π-electron energy (E_π) is analysed, and an elementary derivation of it is presented. Three novel Coulson-type formulae are obtained and one is used for deriving the McClelland (approximate) expression for E_π. An approximation of a particular formula is proposed which results in a linear relationship between E_π and the coefficients of the characteristic polynomial of the molecular graph.     

Self-consistent-field variational approach to the interaction between a polymer and a small molecule

A variational SCF treatment based on a perturbational concept is developed and applied to the interaction between trans-polyacetylene and a small molecule. The validity of the present method is examined by comparing the results with those from the conventional tight-binding SCF crystal orbital method. The interaction energies and charge distributions obtained are in good agreement between the two methods. This result suggests that the present variational approach is promising for application to complicated interactions between a polymer and impurities.     

New translation method for STOs and its application to calculation of overlap integrals

A new translation method for Slater‐type orbitals (STOs) is proposed involving exact translation of the regular solid harmonic part of the orbital followed by the series expansion of the residual spherical part in powers of the radial variable. The method is positively tested in the case of the overlap integral, showing good rate of convergence and great numerical stability under wide changes in the relevant molecular parameters. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 333–340, 1999