Alternatives to Bazley’s special choice for eigenvalue lower bounds

Bazley’s special choice operator is a lesser operator to a positive perturbation of a self-adjoint semi-bounded operator that possesses an exactly soluble base eigenvalue problem. It allows the construction of an exactly soluble intermediate problem that gives eigenvalues not less than the base problem and not greater than the perturbed problem so that lower bounds to the eigenvalues of the perturbed operator are produced. This paper considers alternate derivations of Bazley’s special choice which lead to two alternate methods to determine eigenvalue lower bounds. One is simpler, but gives poorer bounds; the other is more difficult, but sometimes yields superior bounds. Lower bounds to the particle in a box model with a linear perturbation and lower bounds to the helium atom are calculated using the two methods introduced and are compared to those given with Bazley’s special choice.      

Bounds to electronic expectation values for atomic and molecular systems

A generalization of a method to calculate lower bounds to expectation values of non‐negative observables is presented. We consider bounds to three electronic expectation values 〈r²〉, 〈r〉, and 〈r^?1〉 in the helium atom as an example. For both 〈r²〉 and 〈r〉, we are able to calculate improved lower bounds. The lower bound to 〈r^?1〉 does not improve, but we are able to calculate an upper bound which is much closer to the expectation value than the lower bound. Although our generalization allows for improved bounds and/or upper bounds, these bounds to general observables are much less precise than energy bounds and even the expectation values calculated from variational wave functions. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

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.     

Related upper and lower bounds to atomic binding energies

The method proposed by Singh for the calculations of lower bounds to atomic binding energies has been generalized to encompass upper bounds as well. The result is a pair of related matrix eigenvalue problems, constructed from similar sets of basic matrix elements, with the solution of one yielding the tower bound, and of the other, the upper bound. The upper bounds are identical to those calculated by the Rayleigh–Ritz method, which can be useful when the inversion of the normalization matrix is ill-conditioned. The lower bounds are comparable with the best available in the literature. © 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.     

Computing upper and lower bounds using Monte Carlo methods

We 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     

The mechanical moduli of lamellar semicrystalline polymers

Bounds on the elastic constants are derived for semicrystalline polymers whose local morphology is lamellar. Local response matrices (stiffness and compliance) are formulated in three dimensions that simultaneously incorporate uniform in-plane strain and additive forces from layer to layer of crystalline and amorphous phases and uniform stress and additive displacements normal to the lamellar surfaces. Spatial averaging of the stiffness and compliance matrices under the assumption of axially symmetric orientation gives the upper and lower bounds on the longitudinal and transverse tensile moduli and the axial and transverse shear moduli as functions of the separate phase elastic constants, the volume percent crystallinity, and the moments of the orientation 〈cos²θ〉 and 〈cos⁴θ〉. The bounds are much tighter than the Voight upper and Reuss lower bounds that do not recognize phase geometry. Using the known crystal elastic constants of polyethylene, sample calculations on isotropic unoriented materials show that the divergence of bounds at high crystallinity necessitated by the extreme crystal anisotropy shows up only at very high crystallinity. At low temperature the bounds are tight enough to specify G₁, the amorphous modulus, from the measured G and the known crystal elastic constants. At higher temperatures and lower G, the bounds are not tight enough for this purpose but the shear modulus versus crystallinity and temperature data are well fitted by the lamellar lower bound using a temperature-dependent, crystallinity-independent G₁.     

Variational bounds on transition probabilities

Complementary variational functionals are derived which impose upper and lower bounds on transition probabilities. These functionals are used to yield bounds in terms of sets of sum rules, and illustrative calculations are presented for hydrogen, helium and krypton atoms.     

DOUBLE-EDGED LIMIT OF TOTAL FERTILITY RATES

In this paper we present the accurate estimation of total fertility rates, prove the theorem of upper and lower bounds of double-edged limit and provide the formulas of upper and lower bounds of total fertility rates. Based on these estimations and statistical data, we calculate the upper and lower bounds of the double-edged limit for 46 countries including China. Furthermore, we describe the positive growth rate, negative growth rate and zero growth rate of population development. These results are of great importance for us to control population not only in theory but also in practice.     

Double-edged limit of total fertility rates.

In this paper we present the accurate estimation of total fertility rates, prove the theorem of upper and lower bounds of double-edged limit and provide the formulas of upper and lower bounds of total fertility rates. Based on these estimations and statistical data, we calculate the upper and lower bounds of the double-edged limit for 46 countries including China. Furthermore, we describe the positive growth rate, negative growth rate and zero growth rate of population development. These results are of great importance for us to control population not only in theory but also in practice.     

Bounds for Hartree-Fock perturbation theory

Upper and lower bounds for the second-order energy in both coupled and uncoupled Hartree-Fock perturbation theories are derived. Using these bounds inequalities are derived for the error in the geometric approximation.     

Upper and lower bounds and the generalized variation-perturbation approach of many-electron theory

The theory of upper and lower bounds is used in conjunction with the “varied-portions approach” (VPA ) of the “many-electron theory of atoms and molecules” (MET ). Upper and lower bounds equations are derived. It is recognized that if a very natural portion of the full variational expression is made stationary, according to (VPA ), a for of Löwdin's lower bound wave equation results. By using the bracketing theorem it is also possible to give lower bounds to the quantities arising from the (VPA ) of (MET ). This will be useful to study in the future the (MET ) pair correlations and to supplement the computational methods currently available with lower bound estimates as well.     

Finding more perfect matchings in leapfrog fullerenes

We use some recent results on the existence of long cycles in leapfrog fullerenes to establish new exponential lower bounds on the number of perfect matchings in such graphs. The new bounds are expressed in terms of Fibonacci numbers.     

Multipoint padé theory and semi-empirical method for the van der waals interactions

In the context of the multipoint Padé theory, the van der Waals coefficients obtained from the semi-empirical method are seen to be lower bounds. With a slightly different constraint, the method also gives upper bounds.     

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.     

Brackets to the eigenvalues of the Schrödinger equation,part 1. Tridiagonal matrices

The problem of upper and lower bounds to the first few eigenvalues of a very large or infinite tridiagonal matrix H is studied. Those eigenvalues of a comparison-matrix M _n which are lower than a characteristic limit, together with the corresponding eigenvalues of the variational matrix H _n are shown to bracket exact eigenvalues of H . M _n differs from H _n only in the last off-diagonal element and is easily obtained from H . Sufficient conditions for lower bounds are based on a low estimate of the characteristic limit. For increasing dimensions n, the lower bounds approach the exact eigenvalues from below. As a numerical illustration, brackets to the known eigenvalues of the harmonic oscillator with a linear perturbation are calculated.     

Lower bound problems and bounds to atomic ionization energies

A simple extension of a method by Calogero and Marchioro for constructing lower bound problems for ground states of systems of indistinguishable particles is applied to atomic systems. Their method is extended to yield an improved lower bound problem, which raises the ground‐state estimate and yields nontrivial lower bounds to excited states that were previously inaccessible by their method. Rigorous upper bounds to atomic ionization energies are also derived. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 779–784, 2000     

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.     

Upper and lower bounds to critical values of the Hartree operator

Upper and lower bounds for the minimal critical energy E of the Hartree operator for helium are calculated. We show that a Ritz analogous procedure for the calculation of the upper bounds converges to the exact value. The lower bound to E yields that the ground state E_H of the helium atom is strictly lower than 2E.