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.     

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.     

Modeling flexible pharmacophores with distance geometry, scoring, and bound stretching

The study of pharmacophores, i.e., of common features between different ligands, is important for the quantitative identification of "compatible" enzymes and binding species. A pharmacophore-based technique is developed that combines multiple conformations with a distance geometry method to create flexible pharmacophore representations. It uses a set of low-energy conformations combined with a new process we call bound stretching to create sets of distance bounds, which contain all or most of the low-energy conformations. The bounds can be obtained using the exact distances between pairs of atoms from the different low-energy conformations. To avoid missing conformations, we can take advantage of the triangle distance inequality between sets of three points to logically expand a set of upper and lower distance bounds (bound stretching). The flexible pharmacophore can be found using a 3-D maximal common subgraph method, which uses the overlap of distance bounds to determine the overlapping structure. A scoring routine is implemented to select the substructures with the largest overlap because there will typically be many overlaps with the maximum number of overlapping bounds. A case study is presented in which 3-D flexible pharmacophores are generated and used to eliminate potential binding species identified by a 2-D pharmacophore method. A second case study creates flexible pharmacophores from a set of thrombin ligands. These are used to compare the new method with existing pharmacophore identification software.     

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     

Orthogonalization of overlapping orbitals by optimal polynomials. I. Polynomials constructed from spectral radii

Inverses and inverse square roots of overlap matrices are approximated by polynomials constructed on the basis of an integral transform of the metric resolvent and Kublanovskaya's conformal mapping technique. Given reasonable upper and lower bounds for the support of the overlap matrix spectral distribution, the described expansions have significantly better convergence properties than the well known power series methods, yet retain the simplicity of these schemes.     

Energy and energy gradient matrix elements with N-particle explicitly correlated complex Gaussian basis functions with L=1

In this work we consider explicitly correlated complex Gaussian basis functions for expanding the wave function of an N-particle system with the L=1 total orbital angular momentum. We derive analytical expressions for various matrix elements with these basis functions including the overlap, kinetic energy, and potential energy (Coulomb interaction) matrix elements, as well as matrix elements of other quantities. The derivatives of the overlap, kinetic, and potential energy integrals with respect to the Gaussian exponential parameters are also derived and used to calculate the energy gradient. All the derivations are performed using the formalism of the matrix differential calculus that facilitates a way of expressing the integrals in an elegant matrix form, which is convenient for the theoretical analysis and the computer implementation. The new method is tested in calculations of two systems: the lowest P state of the beryllium atom and the bound P state of the positronium molecule (with the negative parity). Both calculations yielded new, lowest-to-date, variational upper bounds, while the number of basis functions used was significantly smaller than in previous studies. It was possible to accomplish this due to the use of the analytic energy gradient in the minimization of the variational energy.     

Reduced first order density matrix for the Be ground state

A reduced first order density matrix for the Be ground state is computed from an extensive configuration interaction (CI ) wave function. A sequence of increasingly accurate CI wave functions Φ_q converging towards the exact Ψ is used to assess the quality of the results which include approximate bounds for the overlaps 〈Φ_q|Ψ〉, electron–nuclear coalescence cusp data, Weinhold's overlap between density matrices, virial ratios, occupation number spectra, and some expectation values. The nuclear magnetic shielding constant and the molar diamagnetic susceptibility are determined with 2.0 and 1.5% of uncertainty, respectively.     

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.     

Geometric properties of chiral bodies

The new developments concerning the possible metrization of structural chirality have drawn much attention recently. The main approach of such quantification is based on the maximal volume overlap between two enantiomorphs of a given chiral set. This approach raises an interesting problem concerning the shape of such a domain of overlap, namely, whether it is chiral or not. It is shown presently that for a two or three dimensional set if the maximal volume overlap is unique then it must beachiral. It is also shown that if a two-dimensional body is convex then by the Brünn-Minkowski theorem the maximal volume overlap of the body with its enantiomorph isachiral. In addition, universal upper bounds for chiral coefficients _n of convex sets in any dimensionn are given, being ₂ 0.3954 and ₃ 0.6977 for dimensions two and three, respectively.     

The dielectric constant of lamellar semicrystalline polymers

The problem of representing the dielectric constant of semicrystalline polymers in terms of the dielectric constants and volume fractions of the constitutent crystalline and amorphous phases is considered. For locally lamellar morphology, bounds based on uniform electric and displacement fields are derived. The equations also include the degree of crystal orientation as a parameter. For unoriented polymers the bounds are considerably tighter than the Hashin–Shtrikman bounds, the latter being the best possible without knowledge of phase geometries. The bounds presented here are sufficiently tight to represent the dielectric constant with practical accuracy for a number of examples of semicrystalline polymers. A treatment is also given of the dielectric constant where the lamellar morphology is further specified as being organized into spherulite-like structures. These bounds are somewhat tighter than the lamellar bounds.     

Perturbation theory with an approximate perturbation: The effect of charge overlap on interatomic interactions

The evaluation of interatomic interactions at large separations (R) typically involves neglecting electron exchange, treating the Coulomb interaction between atoms as a perturbation, neglecting third- and higher-order energy contributions, and approximating the Coulomb interaction by a short expansion in spherical harmonics and, usually, powers of R^?1. This last approximation, using an approximate perturbing Hamiltonian to evaluate a second-order perturbed energy, is examined here; error bounds and a simple correction are introduced. Three illustrative applications to the H? H⁺ interaction are given: the error incurred by truncating the spherical-harmonic expansion is bounded, the R^?1 expansion is corrected for the overlap of the “atomic” charge distributions, and the R^?1 expansion is analyzed to see why it works as well as it does.     

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.     

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     

Comparison of overlap-based models for approximating the exchange-repulsion energy

Different ways of approximating the exchange-repulsion energy with a classical potential function have been investigated by fitting various expressions to the exact exchange-repulsion energy for a large set of molecular dimers. The expressions involve either the orbital overlap or the electron-density overlap. For comparison, the parameter-free exchange-repulsion model of the effective fragment potential (EFP) is also evaluated. The results show that exchange-repulsion energy is nearly proportional to both the orbital overlap and the density overlap. For accurate results, a distance-dependent correction is needed in both cases. If few parameters are desired, orbital overlap is superior to density overlap, but the fit to density overlap can be significantly improved by introducing more parameters. The EFP performs well, except for delocalized pi systems. However, an overlap expression with a few parameters seems to be slightly more accurate and considerably easier to approximate.     

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.     

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.      

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.     

DISTORTIONS IN LASER FLASH PHOTOLYSIS ABSORPTION MEASUREMENTS. THE OVERLAP PROBLEM

Abstract Lack of overlap between the laser beam and the analyzed volume in laser flash photolysis experiments may lead to significant error in the analysis of transient absorbances. A simple theoretical treatment is given to calculate the error for a given overlap. The consequences of bad overlap is discussed for a number of standard experiments and finally a method to determine the quality of the overlap in an experimental setup is given.     

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.