Extrapolation and interpolation of asymptotic series by self-similar approximants

The problem of extrapolation and interpolation of asymptotic series is considered. Several new variants of improving the accuracy of the self-similar approximants are suggested. The methods are illustrated by examples typical of chemical physics, when one is interested in finding the equation of state for a strongly interacting system. A special attention is payed to the study of the basic properties of fluctuating fluid membranes. It is shown that these properties can be well described by means of the method of self-similar approximants. For this purpose, the method has been generalized in order to give accurate predictions at infinity for a function, whose behavior is known only at the region of its variable close to zero. The obtained results for fluctuating fluid membranes are in good agreement with the known numerical data.     

Additive self-similar approximants

A novel type of approximants is introduced, being based on the ideas of self-similar approximation theory, allowing for accurate extrapolation and interpolation of asymptotic series. The method is illustrated by the examples possessing the structure typical of many nonlinear problems in mathematical chemistry. Good numerical convergence is demonstrated for the cases that can be compared with exact solutions, when these are available. The method is shown to be not less and as a rule essentially more accurate than that of Padé approximants. Comparison with other approximation methods is also given.     

On the use of asymptotic expansions

Divergent asymptotic expansions in quantum chemistry often must be evaluated on Stokes lines, where the form of the expansion changes discontinuously and might appear to be ambiguous. Towards clarifying the use of asymptotic expansions on Stokes lines we discuss by numerical example the Airy function Bi(x) for real, positive x. Two physical problems to which this example is relevant, among others, are the Rayleigh-Schrödinger perturbation theory for the LoSurdo-Stark effect in hydrogen and the JWKB connection-formula problem, for which real series are associated with complex sums. The various roles of partial summation, Padé approximants, and Borel summation are compared. In addition, a derivation is given for an integral that occurs in a simple proof of the Borel summability of asymptotic expansions for the confluent hypergeometric function, which function is fundamental to certain quantum chemistry problems, and which integral is given incorrectly in several standard references.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

Resolvent technique and Padé approximants in configuration interaction calculations

A perturbation approach based on resolvent technique and Padé approximants is proposed. The eigenvalue of interest is part of a solution of two nonlinear algebraic equations. The nonlinear equations are arrived at by considering two different expression of the expectation value of the resolvent of an outer projection of the Hamiltonian. The first expression is based on the spectral resolution of the resolvent, and the second one is obtained by a power series expansion analogous to that applied in the derivation of the energy expression in the Brillouin–Wigner perturbation theory. The truncated power series is extrapolated by Padé approximants of type II. The method is tested on a CI calculation of the energy of the lowest ¹Σ state of the B₂ molecule.     

On Padé approximants to virial series

Padé approximants have long been used to predict virial series coefficients and to provide equations of state for low and high density materials. However, some justified criticism has appeared about this procedure. Although we agree to impose several restrictions on the use of Padé approximants, we indicate that the Padé approximant is still an excellent way to predict the first unknown virial series coefficients. As an example, we report a calculation of the B11=128.6 and B12=155 virial coefficients of the three dimensional hard sphere model that are in excellent agreement with the two most recent estimates. We also consider that the commonly used method to choose among Padé approximants is not completely reliable for this specific application and suggest an alternative new method.     

Robust interpolation between weak- and strong-correlation regimes of quantum systems

A robust interpolation between the weak- and strong-correlation regimes of quantum systems is presented. It affords approximants to the function E(ω) describing the dependence of the total energy (or other observable) on the coupling parameter ω that measures the correlation strength. The approximants conform to truncations of the asymptotic expansions of E(ω) at the ω → 0 and ω → ∞ limits with arbitrary (but given) numbers of terms. In addition, depending on the number of fitted parameters, they either reproduce or optimally (in the least-square or maximum-error sense) approximate the exact E(ω) at any given number of values of the coupling strength. Numerical tests demonstrate the high accuracy of even the low-order approximate expression for E(ω). The approximants, which do not suffer from spurious poles, possess a wide range of applicability that stems from their capability of accurately reproducing not only E(ω) but also its derivatives with respect to ω. They are equally useful for interpolation between the low- and high-temperature limits of energy and other quantities associated with various models of statistical thermodynamics. The new interpolation scheme is not applicable to the cases where the weak- and strong-correlation asymptotics involve non-analytic functions of ω or expressions dependent on logarithm of the coupling strength. Excluded are also the cases where the weak- and strong-correlation asymptotics pertain to de facto different states, e.g., the ground state of a homogeneous electron gas in three dimensions.     

Padé approximants to physical properties via inner projections

The [N, M] Padé approximants to functions formally associated to power series expanssions are expressed in terms of expectation values of inverse matrices. These formulae, which can be derived by the inner-projection technique, lead to a simple analysis of the properties of serveral approximation methods and their inter-relationships, in particular Gaussian integration, continued factorization and Padé approximations, which are of current interest in the calculation of physical properties. A relation with Fredholm integral equations and expansions of the resolvent is also discussed. The use of operator inequalities in a systematic fashion is particularly convenient when both the function being approximated and the coefficients of the power series have physically meaningful expressions as moments of operators.     

Generalized Euler transformation in extracting useful information from divergent (asymptotic) perturbation series and the construction of Padé approximants

Euler transformation for accelerating convergence of a series is considered in the context of handling divergent (asymptotically convergent) perturbation series. A generalized (parametrized) version of this transformation is developed, based on the conjecture of Dalgarno and Stewart, which works better. Viewed from this standpoint, the Padé approximants follow as a special case of the parametrized Euler transformation (PET ), as is the case with the μ transformation procedure of Feenberg in a perturbative context. The PET is shown to serve as a more general method of handling a divergent series and is able to appreciate the construction and convergence behavior of specific sequences of Padé approximants. The role of parametrization in the context of the Z^?1 perturbation theory of atoms is also noted and the workability of the adopted strategy is demonstrated by choosing some specific test cases.     

Electrostatic energy of polygonal charge distributions

An asymptotic series for the electrostatic energy E₁(N){\mathcal{E}_1(N)} of an N-gonal charge distribution, i.e., a set of unit charges occupying vertices of a regular N-gon with a unit circumradius, is derived. Application of Padé approximants to truncations of this expansion produces compact approximate formulae capable of estimating E₁(N){\mathcal{E}_1(N)} with great accuracy. A closed-form expression for the energy of electrostatic interaction of two polygonal charge distributions is obtained from the respective Fourier series. The availability of this expression allows for a rapid calculation of the relevant energy with computational effort independent of the numbers of particles involved.     

Embedding scaling relations in Padé approximants: Detours to tame divergent perturbation series

A divergent perturbation series is known to yield very unreliable results for observables even at moderate coupling strengths. One of the most popular techniques in handling such series is to express them as rational functions, but it is often faithful only for small coupling. We outline here how one can gain considerable advantages in the large‐coupling regime by properly embedding known asymptotic scaling relations for selected observables during construction of the aforesaid Padé approximants. Three new bypass routes are explored in this context. The first approach involves a weighted geometric mean of two neighboring PA. The second idea is to consider series for specific ratios of observables. The third strategy is to express observables as functionals of the total energy in the form of series expansions. Symanzik's scaling relation, and the virial and Hellmann–Feynman theorems, are used at appropriate places to aid each of the strategies. Pilot calculations on the ground‐state perturbation series of certain observables for the quartic anharmonic oscillator problem reveal readily the benefit and novelty. © 2012 Wiley Periodicals, Inc.     

Accurate estimates of asymptotic indices via fractional calculus

We devise a three-parameter random search strategy to obtain accurate estimates of the large-coupling amplitude and exponent of an observable from its divergent Taylor expansion, known to some desired order. The endeavor exploits the power of fractional calculus, aided by an auxiliary series and subsequent construction of Padé approximants. Pilot calculations on the ground-state energy perturbation series of the octic anharmonic oscillator reveal the spectacular performance.     

A re-examination of the justification of neglect of differential overlap approximations in terms of a power series expansion inS

Neglect of differential overlap methods are treated as approximations to calculations in a symmetrically orthogonalized basis. The accuracy of this approximation is investigated in terms of a power series expansion of the overlap matrix. TheS-matrix can be transformed into a matrix which will give a convergent series, and this series is used in the examination. The only approximation having any justification from this point of view is the NDDO method and even this neglects certain important three-electron integrals. Corrected expressions for the repulsion integral scaling factors introduced by Chandrasekharet al. are also derived. On leave from The Chemistry School, University of Western Australia.     

A Critical Evaluation and Modification of the Padé–Laplace Method for Deconvolution of Viscoelastic Spectra

This paper shows that using the Padé–Laplace (PL) method for deconvolution of multi-exponential functions (stress relaxation of polymers) can produce ill-conditioned systems of equations. Analysis of different sets of generated data points from known multi-exponential functions indicates that by increasing the level of Padé approximants, the condition number of a matrix whose entries are coefficients of a Taylor series in the Laplace space grows rapidly. When higher levels of Padé approximants need to be computed to achieve stable modes for separation of exponentials, the problem of generating matrices with large condition numbers becomes more pronounced. The analysis in this paper discusses the origin of ill-posedness of the PL method and it was shown that ill-posedness may be regularized by reconstructing the system of equations and using singular value decomposition (SVD) for computation of the Padé table. Moreover, it is shown that after regularization, the PL method can deconvolute the exponential decays even when the input parameter of the method is out of its optimal range.     

Chromatography with Supercritical Fluids

Chromatography with supercritical fluids unites the features of both gas chromatography and liquid chromatography, yet retains special characteristics of its own. The diffusion coefficient and particularly the viscosity of fluid phases may approach values for low-pressure gases, while the solvent power may be similar to that of liquids. However, with supercritical fluids it is possible to control chromatographic separations very effectively by pressure programming, since the solubility increases with increasing density. Temperature programming, on the other hand, can have the opposite effect to that in gas- or liquid-chromatography since the density decreases with increasing temperature at a given pressure. Supercritical fluid chromatography is primarily of interest for the separation of higher molecular weight compounds. The efficiency of this method of separation is demonstrated on several homologous series. Thus, a styrene oligomer with nominal M_w=2200 can be resolved by a pressure and temperature program into 40 species.     

Estimation of eigenvalues using padé approximants. Stark effect for a model system

The perturbation series for a one-dimensional model system in a static electric field is obtained and shown to be divergent. Padé approximants to the series are calculated and found to converge, and to provide a good estimation of the Stark eigenvalue.     

Critical point,singularities and extrapolations in the helium iso-electronic sequence

The Z-expansion of two-electron systems is analyzed with the Padé technique with emphasis on establishing analytical properties of the function E(Z) formally associated with the power series expansion. The concept of critical point in this connection is stressed. For this sequence it occurs at Z_c = 0.911246 with E(Z_c) = ?0.415184. The structure of E(Z) for Z < Z_c is investigated. The use of Padé approximants to extrapolate values of electron affinities is emphasized.     

Method for computing protein binding affinity

A Monte Carlo method is given to compute the binding affinity of a ligand to a protein. The method involves extending configuration space by a discrete variable indicating whether the ligand is bound to the protein and a special Monte Carlo move, which allows transitions between the unbound and bound states. Provided that an accurate protein structure is given, that the protein-ligand binding site is known, and that an accurate chemical force field together with a continuum solvation model is used, this method provides a quantitative estimate of the free energy of binding.     

Improving the accuracy of ab initio methods with summation approximants and singularity analysis

The accuracy of Møller–Plesset (MP) perturbation theory and coupled‐cluster (CC) theory can be significantly improved, at essentially no increase in computational cost, by using summation approximants that model the way in which these theories converge to the full configuration interaction limit. Approximants for MP4 and CCSD(T) are presented, their size scaling is analyzed, and the functional analysis of the MP energy, on which the MP4 approximant is based, is discussed. The MP approximants are shown to have a form that is appropriate for describing resonance energies. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Thermogenesis: Identification by means of pade approximants

The paper describes how to obtain an analytic approximation to the transfer function of a conduction calorimeter, namely, a procedure to identify the calorimetric system. In this case Pade approximants are used on the Laplace transform of the thermogram. The feasibility of the method is tested on two models which span the frequency range usually attained by actual calorimeters. The influence of random noise and baseline drift have also been analyzed. The results show that three or four time constants are correctly obtained.     

Scaled particle theory for hard sphere pairs. I. Mathematical structure

We develop an extension of the original Reiss-Frisch-Lebowitz scaled particle theory that can serve as a predictive method for the hard sphere pair correlation function g(r). The reversible cavity creation work is analyzed both for a single spherical cavity of arbitrary size, as well as for a pair of identical such spherical cavities with variable center-to-center separation. These quantities lead directly to a prediction of g(r). Smooth connection conditions have been identified between the small-cavity situation where the work can be exactly and completely expressed in terms of g(r), and the large-cavity regime where macroscopic properties become relevant. Closure conditions emerge which produce a nonlinear integral equation that must be satisfied by the pair correlation function. This integral equation has a structure which straightforwardly generates a solution that is a power series in density. The results of this series replicate the exact second and third virial coefficients for the hard sphere system via the contact value of the pair correlation function. The predicted fourth virial coefficient is approximately 0.6% lower than the known exact value. Detailed numerical analysis of the nonlinear integral equation has been deferred to the subsequent paper.