Estimating bounds on the highest and lowest eigenvalues of any matrix

A method for estimating the bounds for the highest and lowest eigenvalues of a finite-dimensional matrix is presented. The method is tested for the Hamiltonian matrix for several particle-in-a-box-like systems. We also provide results for the well-studied benchmark case of the ro-vibrational states of H₃ ⁺, and consider bounds obtained for a completely random non symmetric matrix. Finally, we discuss how the error in a Chebychev expansion solution of quantum scattering depends on the error made in estimating the highest eigenvalue of the Hamiltonian matrix. Received: 7 May 1999 / Accepted: 30 July 1999 / Published online: 15 December 1999     

Bounds for the scattering length of spin-polarized helium from high-accuracy electronic structure calculations

We developed a series of correlation-consistent, polarized multiple zeta basis sets optimized specifically for the energy of the 2 3S state of helium atom. These basis sets were subsequently augmented with diffuse functions optimized for the van der Waals constants C6 through C14 which determine the asymptotic behavior of the second-order dispersion interaction between 2 3S helium atoms at large interatomic separation R. The resulting bases were applied to compute the Born-Oppenheimer (BO) potential for the lowest 5Sigmag+ state of the helium dimer. The coupled cluster and the full configuration-interaction techniques were employed to account for the electron correlation effects. The cardinal number extrapolation technique was used to obtain the complete-basis-set limit V(R) for the interaction potential and to find its lower VL(R) and upper VU(R) bounds. The resulting potentials were fitted to an analytical function containing accurate van der Waals constants C6 through C12 (including C11). We found that the complete-basis-set BO potential has a well depth De=1048.24+/-0.36 cm-1. The highest rotationless vibrational level is bound by D14=90.2+/-4.7 MHz, much stronger than the previous most accurate estimation of 15.2 MHz. The error bounds for De and D14 were obtained using the VL(R) and VU(R) potentials. The S-wave scattering length computed using the VL(R), V(R), and VU(R) potentials (assuming atomic masses) is aL=7.41 nm, a=7.54 nm, and aU=7.69 nm, respectively. We also computed the adiabatic, relativistic, and quantum electrodynamics (QED) corrections to the BO potential. When these corrections are taken into account the values of D14 and of a (both computed assuming nuclear masses) are 87.4+/-6.7 MHz and 7.64+/-0.20 nm; the error bounds reflect now also the uncertainty of the included adiabatic, relativistic, and QED corrections. The value of the scattering length resulting from our investigation lies outside the error bounds of all experimental determinations based on the properties of Bose-Einstein condensate of spin-polarized helium atoms.     

Spurious complex energies for confining potentials in the complex-coordinate method

Complex-valued energies are obtained analytically by application of the complex coordinate method to confining potentials of the form X^m (m > O). The results are used to analyse some properties of the multiple spectra of complex eigen-values recently obtained by Atabek and Lefebvre for the exponential well potential. Since molecular potential surfaces are commonly described by power-series expansions in the coordinates, our results represent possible pitfalls in the calculation of predissociation resonances by the complex-coordinate method.     

Theory of raman scattering from molecules. Summation over vibrational states

Upper and lower bounds of the vibrational sums involved in the polanzability tensor are derived. The lower bound leads to a simplified formula applicable to resonance Raman scattering in both weak and strong coupling cases. A new expansion of the vibrational sum is introduced and useful expressions for excitation profiles are obtained.     

Construction of the model radial distribution curves with regard to the features of X-ray diffraction experiment

A method to construct model radial distribution functions (RDFs) from the already known structural data is described. The method includes the procedures to calculate termination ripples that always appear on the experimental RDF because of the bounds of the integration limits in the Fourier transformation of the X-ray scattering curve. The introduction of this procedure increases the accuracy of the comparative RDF method used to elucidate the phase composition of nanodispersed materials and to determine the features of the local structure of the phases as compared to their well crystallized analogues. Cerium dioxide samples with different dispersion exemplify the applicability of this method to determine the features of the local structure.     

Poisson ratios and upper bounds of intrinsic birefringence from brillouin scattering of oriented polymers

To overcome the difficulties in measuring high-frequency shear constants of polymeric materials by ultrasonic or Brillouin scattering technques, we extrapolate results from oriented materials to zero birefringence. Shear constants C₄₄ in the high-frequency limit (GHz) are determined for polyethylene, poly(ethylene terephthalate), poly(vinyl chloride), poly(methyl methacrylate), and polycarbonate using Brillouin scattering. Accurate values of Poisson's ratio are derived. The extrapolation to full orientation using an amended Moseley relation gives upper bounds for the “intrinsic birefringence.” Changes in the character of the orientation process are easily detected by monitoring the mode numbers, which are defined by analogy to Poisson's ratio. Extrapolation of these ratios to their upper bound 0.5 gives an independent check of the maximum intrinsic birefringence. © 1993 John Wiley & Sons, Inc.     

The role of model systems in the few-body reduction of the N-fermion problem

The problem of upper and lower ground state energy bounds for many-fermion systems is considered from the viewpoint of reduced density matrices. Model density matrices are used for upper bounds to, first uncoupled, then coupled fermions. Model Hamiltonians are developed for lower bounds in corresponding fashion. Both mathematical and physical models are constructed for setting up universally valid inequalities on density matrices. These are joined by both inequalities and equalities in which the explicit form of the system at hand is used. A few illustrative examples are presented.     

An alternative formalism for the method of intermediate Hamiltonians

The methods of intermediate Hamiltonians and of inner projections for the determination of lower bounds of eigenvalues of a Hermitian operator H are analysed and reformulated as linear matrix problems. Submatrices T which are only defined implicitly through the product TT^d? are best represented in triangular form. If the subspace complementary to the subspace determining the inner projection can be subdivided with different lower bounds for H, the bounds for the eigenvalues can be further improved. The new formalism is applied to obtain crude lower bounds for the ground state of the helium atom, using only 2 × 2 and 3 × 3 matrices.     

Determinations of bond energies by time-of-flight single-collision chemiluminescence

A pulsed beam of metastable atoms traverses a scattering chamber filled with oxidant gas at low pressures (beam + gas arrangement); the resulting chemiluminescence is spectroscopically resolved as a function of time to yield a time-of-flight (TOF) spectrum for different internal states. From this data, the initial relative translational energy distribution is derived for the reactants that populate the excited internal state observed. Lower bounds are placed on the barium halide (BaX) dissociation energies, using the reactions Ba(³D) + X₂ → BaX* + X, where X = Br, I. Arguments are presented to show that these lower bounds represent measurements of the true bond energies: it is concluded that D₀⁰(BaBr) = 85.8 ± 2 kcal/mole and D₀⁰(BaI) = 72.9 ± 2 kcal/mole. The present work corrects previous determinations of bond energies from single-collision chemiluminescent studies which were in error because of unrecognized metastable contamination in the high-temperature atomic beam.     

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.     

Use of augmented Lagrangians in the calculation of molecular conformations by distance geometry

Distance geometry is a technique widely used to find atomic coordinates that agree with given upper and lower bounds on the interatomic distances. It is successful because it chooses at random some relatively good "trial coordinates" that take into account the whole molecule and all constraints at once. Customarily, these trial coordinates must be refined by minimizing a penalty function until the structure agrees with the original bounds. Here we present an alternative to minimizing the penalty function, which has the advantage of more precisely satisfying the bounds, showing more clearly when the bounds are mutually contradictory, and simultaneously optimizing an objective function subject to precise satisfaction of the bounds.     

Improved lower bounds for the convergence radius of RS-perturbation theory

F. Rellich and T. Kato have given lower bounds for the convergence radius of Rayleigh–Schrödinger perturbation theory, if the perturbation operator is bounded relative to the unperturbed Hamiltonian. It is shown here that these lower bounds can usually be improved.     

Micellar kinetics of a fluorosurfactant through stopped-flow NMR

19F NMR chemical shifts and transverse relaxation times T2 were measured as a function of time after quick stopped-flow dilution of aqueous solutions of sodium perfluorooctanoate (NaPFO) with water. Different initial concentrations of micellar solution and different proportions of mixing were tested. Previous stopped-flow studies by time-resolved small-angle X-ray scattering (TR-SAXS) detection indicated a slow (approximately 10 s) micellar relaxation kinetics in NaPFO solutions. In contrast, no evidence of any comparable slow (>100 ms) relaxation process was found in our NMR studies. Possible artifacts of stopped-flow experiments are discussed as well as differences between NMR and SAXS detection methods. Upper bounds on the relative weight of a slow relaxation process are given within existing kinetic theories of micellar dissolution.     

Bounds for the Kirchhoff index of regular graphs via the spectra of their random walks

Using probabilistic tools, we give tight upper and lower bounds for the Kirchhoff index of any d‐regular N‐vertex graph in terms of d, N, and the spectral gap of the transition probability matrix associated to the random walk on the graph. We then use bounds of the spectral gap of more specialized graphs, available in the literature, in order to obtain upper bounds for the Kirchhoff index of these specialized graphs. As a byproduct, we obtain a closed‐form formula for the Kirchhoff index of the d‐dimensional cube in terms of the first inverse moment of a positive binomial variable. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

快速多模型回归分析方法的研究

逐步回归[1]是变量选择的常用方法．由逐步回归法得到的数学表达式不一定是最优数学模型，所以有其局限性．假如有m个变量x；，x。，……，x。，我们希望得到分别包含1个变量、2个变量、……、直至m个变量的最优回归方程，可以采用组合算法C：，对每个子集的所有组合进行回归分析，然后从C：个回归方程中选出最优的数学模型．由于随着变量个数m的增加，总的回归次数以2”的形式倍增，运算量大，时间长．此时可采用快速多模型（LeaPsandbounds）回归分析［‘j．此法为一种快速多模型回归，据此可以得到一组含有不同变量个数的最优方程．该…     

Pointwise bounds for the electron density and estimates of an expectation value for one-electron systems

Pointwise bounds for wavefunctions of a class of Schrödinger operators for one-electron systems are used to obtain pointwise bounds for the electron density, as well as estimates for the expectation value (r^?1).     

Variable neighborhood search for extremal graphs. 6. Analyzing bounds for the connectivity index

Recently, Araujo and De la Pe?a gave bounds for the connectivity index of chemical trees as a function of this index for general trees and the ramification index of trees. They also gave bounds for the connectivity index of chemical graphs as a function of this index for maximal subgraphs which are trees and the cyclomatic number of the graphs. The ramification index of a tree is first shown to be equal to the number of pending vertices minus 2. Then, in view of extremal graphs obtained with the system AutoGraphiX, all bounds of Araujo and De la Pe?a are improved, yielding tight bounds, and in one case corrected. Moreover, chemical trees of a given order and a number of pending vertices with minimum and with maximum connectivity index are characterized.     

Improved upper bounds for the atomic ionization potential

Two sets of rigorous upper bounds on the atomic ionization potential are derived from some known inequalities of the classical analysis. The first set of bounds are expressed in terms of radial expectation values of the electron density; they improve previously found bounds of the same kind and converge to the exact ionization potential. The other bounds depend on various atomic density functionals which describe global physical quantities such as the Thomas–Fermi and exchange energies and the Boltzmann–Shannon information entropy. The accuracy of some of the bounds is numerically analyzed within a Hartree–Fock framework. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 185–189, 1999     

Upper and lower bounds on the control field and the quality of achieved optimally controlled quantum molecular motion

A large class of problems in optimally controlled quantum or classical molecular dynamics has multiple solutions for the control field amplitude. A denumerably infinite number of solutions may exist depending on the structure of the design cost functional. This fact has been recently proved with the aid of perturbation theory by considering the electric field as the perturbating agent. In carrying out this analysis, an eigenvalue (i.e., a spectral parameter) appears which gives the degree of deviation of the control objective from its desired value. In this work, we develop a scheme to construct upper and lower bounds for the field amplitude and spectral parameter for each member of the denumerably infinite set of control solutions. The bounds can be tightened if desired. The analysis here is primarily restricted to the weak field regime, although the bounds for the strong field nonlinear case are also presented.     

New bounds for the atomic charge and momentum densities at the origin

The “Stieltjes moment problem” technique together with the positivity and monotonic decreasing properties of the electronic density of an atom is used to find new and more accurate lower bounds for the charge density at the nucleus and the momentum density at the origin, in terms of radial and momentum expectation values, respectively. Bounds depending on two and three expectation values are given explicitly and a Hartree-Fock study of their quality is carried out. Also, the behavior of the new bounds at largeZ's is discussed. The Stieltjes technique allows to find lower bounds of better accuracy by including expectation values of higher order.